Download Delocalized and Correlated Wave Functions for Excited States in

Delocalized and Correlated Wave Functions
for Excited States in Extended Systems
Alexandrina Stoyanova
Front cover: The ab initio energy bands associated with the two lowest electron states
in doped CaMnO3 (chapter 5) presented in a perspective.
The work described in this thesis was performed in the Theoretical Chemistry Group
of the Materials Science Centre at the University of Groningen, Nijenborgh 4, 9747
AG Groningen, The Netherlands.
Grants for computing time on super-computers awarded by the Stichting Nationale
Computer Faciliteiten (National Foundation for Computing Facilities (NCF)) contributed to the results, presented in this thesis. The author acknowledges the financial
aid of the Material Science Centre (MSC) plus.
Alexandrina Stoyanova,
Delocalized and Correlated Wave Functions for Excited States in Extended Systems,
Proefschrift Rijksuniversiteit Groningen.
c A. Stoyanova, 2006
RIJKSUNIVERSITEIT GRONINGEN
Delocalized and Correlated Wave Functions
for Excited States in Extended Systems
Proefschrift
ter verkrijging van het doctoraat in de
Wiskunde en Natuurwetenschappen
aan de Rijksuniversiteit Groningen
op gezag van de
Rector Magnificus, dr. F. Zwarts,
in het openbaar te verdedigen op
maandag 30 oktober 2006
om 14.45 uur
door
Alexandrina Stoyanova
geboren op 23 november 1977
te Gabrovo, Bulgarije
Promotor:
Prof. dr. R. Broer
Beoordelingscommissie: Prof. dr. M. Filatov
Prof. dr. C. de Graaf
Prof. dr. T. T. M. Palstra
ISBN
90-367-2789-8
vi
Contents
1 General Introduction
1.1 Prelude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Electron Correlations and Localized States in Solids . . . . . . . . . .
1.3 Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
7
12
2 Theoretical Framework
2.1 The embedded cluster approach . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Long- and short-range electrostatic interactions . . . . . . . . .
2.2 Wave function based theoretical approaches . . . . . . . . . . . . . . .
2.2.1 Multiconfigurational Self-Consistent Field Approach and Configuration Interaction . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Complete Active Space Second-Order Perturbation Theory . .
2.3 State Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
19
19
21
29
3 Delocalized and Correlated Wavefunctions for Excited States in Extended Systems
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Many-body bands . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Many-body excitation bands, many-body hole bands and manybody electron bands . . . . . . . . . . . . . . . . . . . . . . . .
IJ
IJ
3.2.3 Computation of Hab
and Sab
between localized states in extended systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The overlapping fragments approach . . . . . . . . . . . . . . . . . . .
3.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Overlapping fragments approach . . . . . . . . . . . . . . . . .
3.4.2 Non-orthogonal tight-binding approach . . . . . . . . . . . . .
4 Double Exchange Parameters in Lightly Doped Manganites
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
34
36
47
47
56
56
62
63
66
69
69
71
81
82
viii
CONTENTS
4.1.1
4.2
4.3
4.4
4.5
4.6
Local parameters in extended systems and localized orbitals
approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1.2 Double exchange in lightly hole and electron doped manganites 83
Definition and computation of the double exchange parameters . . . . 87
The Overlapping Fragment Approach ’at work’ . . . . . . . . . . . . . 89
Material model and computational information . . . . . . . . . . . . . 91
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.5.1 LaMnO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.5.2 CaMnO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.5.3 La0.75 Ca0.25 MnO3 . . . . . . . . . . . . . . . . . . . . . . . . . 111
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Analysis of Different Overlapping Fragment and Embedding Schemes
and Many-Electron Bands in Manganites.
129
5.1 Overlapping Fragment and Embedding Schemes . . . . . . . . . . . . . 129
5.1.1 Introduction and Computational Information . . . . . . . . . . 129
5.1.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.1.3 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2 Many-Body Bands in Manganites . . . . . . . . . . . . . . . . . . . . . 140
5.2.1 Introduction and Computational Information . . . . . . . . . . 140
5.2.2 Many-Body Bands in Manganites: Results and Discussion . . . 142
5.2.3 Discussion of Many-Body Hole Bands for LaMnO3 . . . . . . . 150
5.2.4 Discussion of many-body electron bands for CaMnO3 . . . . . 153
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6 d-d
6.1
6.2
6.3
Excitations and Charge Transfer States in LaM nO3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Crystal Structure, Material Model and Computational Information
Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Mn d-d excitations . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 CT excitations . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
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165
165
170
174
174
184
196
7 Delocalization of excited, hole and added-electron states in NiO
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Material model and Computational information . . . . . . . . . . . .
7.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Delocalization of d-d excitations . . . . . . . . . . . . . . . .
7.4.2 Delocalized hole and added-electron states . . . . . . . . . . .
7.4.3 Localized valence hole states . . . . . . . . . . . . . . . . . .
7.4.4 Added-electron states . . . . . . . . . . . . . . . . . . . . . .
7.4.5 Effective hopping matrix elements for hole states . . . . . . .
7.4.6 Many-body hole and electron bands . . . . . . . . . . . . . .
7.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
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203
203
209
210
212
212
215
216
221
222
229
234
CONTENTS
ix
A Matrix elements in the R- space for LaMnO3
241
List Of Abbreviations
245
List of Abbreviations
245
Samenvatting
248
x
CONTENTS
1
General Introduction
’The elegant equation, devised by Erwin Schrödinger in 1926 to describe the unfolding of quantum events offered not certainties, as Newtonian mechanics did, but only
an undulating wave of possibilities. Werner Heisenberg’s uncertainty principle then
showed that our knowledge of nature is fundamentally limited - as soon as we grasp
one part, another part slips through our fingers. The founders of quantum physics
wrestled with these issues. Albert Einstein, who in 1905 showed how Planck’s electromagnetic quanta, now called photons, could explain the photoelectric effect (in which
light striking metal induces an electric current), insisted later that a more detailed,
wholly deterministic theory must underlie the vagaries of quantum mechanics. Arguing that ”God does not play dice,” he designed imaginary, ”thought” experiments to
demonstrate the theory’s ”unreasonableness” ’.
John Horgan
1.1
Prelude
(or How the Quantum theory solved (almost) all of our problems)
The origins of the Quantum Theory may be ascribed to three main discoveries in
the stream of new physics phenomena revealed in the beginning of the 20th century:
the theoretical explanation of the blackbody radiation by Max Planck in 1900, followed
by the explanation of the photoelectric effect, formulated by Albert Einstein in 1905
and finally the novel atomic model by Niels Bohr in 1912. These phenomena could
not be explained using the laws of the classical physics. The quantum idea turned
out to be crucial for understanding the high conducting properties of the metalic
state. Few years after the introduction of the quantum model of the atom, Bloch and
Sommerfeld and Wilson independently developed the quantum theory of electrons in
a periodic lattice potential. The challenging properties of the metalic phase provoked
the development of this novel theory but its simplicity and adequateness allowed for
its further development into the band theory of the crystalline solids.
2
Chapter 1, General Introduction.
Basic Concepts of the Band Theory of Crystalline Solids
The crystalline materials are representative of a special type of solids often referred
to as periodic systems. They are built from an arrangement of atoms in a particular
stereometric form, so-called unit cell, which is repeated in all three dimensions of the
3D space through specific displacements, known as lattice constants ai , i=1, 2, 3.
The directions of periodicity are marked by the primitive lattice vectors ai a . In solid
state physics one defines a crystal lattice as a set of points associated with the unit
cells which are related by lattice vectors. The lattice vectors themselves are defined
as,
R = µ1 a1 + µ2 a2 + µ3 a3
where µi are integers. The primitive lattice vectors ai define the Bravais lattice of the
crystal. If the (crystalline) solids are built from an infinite number of particles (ions,
atoms) at lattice positions, the concept of the electronic structure of an individual
particle is no longer applicable. The formation of a band in this case may be viewed,
in the light of the molecular orbital theory, as the formation of an infinite number
of molecular orbitals, for an infinite system of atoms, being so close in energy, that
this manifold of energetic levels may be considered continuous rather than discrete.
Therefore the electronic structure of a solid is described by a large set of energy bands,
separated by band gaps. This description is based on the band theory, which may
be viewed as derived formally from the molecular orbital theory. In the band theory
one introduces the independent-electron approximation which is also invoked in the
orbital theories of atoms and molecules. Within this approximation each electron
moves independently in a periodic potential produced by all atomic nuclei and an
effective average field generated by all remaining electrons. Since the Hamiltonian
Ĥ in the Schrödinger equation which describes the motion of a single particle in the
external potential, and the potential itself are periodic, they possess simultaneous
eigenvectors with the translation operators T̂R for any arbitrary lattice translation
R,
Ĥψ(r) = ψ(r)
T̂R ψ(r) = c(R)ψ(r)
Simple considerations of the commutation relations between different T̂R (T̂R T̂R0 =
T̂R+R0 ) lead to the explicit form of the eigenvalues of the translation operator T̂R :
Y
c(R) = eik.R =
(e2πiηj )µj
j
where
k = η1 b1 + η2 b2 + η3 b3
(1.1)
a Detailed textbooks which discuss the theory of solids are those by C. Kittel, Introduction to
Solid State Physics, seventh edition, John Wiley and Sons, Inc., New York, Chichester 1996, and W.
Jones and N. H. March, Theoretical Solid State Physics, v. 1, Dover Publications, Inc., New York,
1985
Band Theory of Crystalline Solids
3
and bj are the reciprocal lattice vectors which satisfy the following, bj .ai = 2πδji and
ηj (j=1, 2, 3) are real numbers.
One needs to apply boundary conditions to the wave functions ψ(r), which restrict
the allowed values of k. To facilitate the theoretical description of the periodic infinite
system one may model the periodic system by a large number Nj of unit cells along
the three primitive lattice vectors aj , which form a super-cell, containing N1 N2 N3
unit cells and with a volume V=N1 N2 N3 Vunitcell . Then one can apply the Born-von
Kármán boundary conditions to the one- electron wave functions ψ(r):
ψ(r) = ψ(r)(r + Nj aj ) = eiNj k.aj ψ(r) j = 1, 2, 3
then
iNj k.aj
e
2πiNj ηj
=e
=1
Therefore the values of ηj are required to be real and equal to
ηj =
νj
Nj
where νj (j=1, 2, 3) are integers and 06 νj 6 Nj . This condition provides the general
permitted form for the Bloch wave-vector k (see Eq. 1.1):
k=
3
X
νj
bj
Nj
j=1
In other words the Born-von Kármán boundary conditions ensure that the number of
the distinct wave vectors k becomes discrete. The eigenvectors ψ(r) of the effective
one-electron crystal Hamiltonian are chosen in such a way that associated with each
ψ(r) is a Bloch vector k such that the symmetry condition,
ψ(r + R) = eik.R ψ(r)
is fulfilled for every Bravais vector R. This is Bloch’s theorem. According to Bloch’s
theorem the wave function, ψ(r) can be expressed as,
ψnk (r) = eik.r unk (r)
where unk (r) is a periodic function with the periodicity of the Bravais lattice, unk (r) =
unk (r + R). ψnk (r) are simultaneous eigenvectors of Ĥ and all T̂R . k is the Bloch
wave vector in the momentum space of the crystal. It can be confined to the first
Brillouin zone because any Bloch vector k0 , which does not lie in the first Brillouin
zone can be expressed as k0 =k+K and ψnk0 = ψnk . Because of the imposed periodic
boundary conditions the problem for an infinite number of electrons and an infinite
number of k points is reduced to a problem for a finite number of electrons (the
electrons in the single unit cell) and an infinite number of k points. In practice, the
consideration of an infinite number of k points is not necessary and one considers
only a finite set of k points, chosen in the first Brillouin zone. The corresponding
4
Chapter 1, General Introduction.
eigenvalue problem is defined for a finite volume of the crystal and the resulting
different solutions ψnk are associated with discrete eigenvalues, nk , labeled by the
band index n. The energy levels nk vary continuously with k and for a given n,
they are periodic functions of k, nk+K =nk . Thus the levels of an electron in a
periodic potential can be described by a set of continuous functions, nk , each obeys
the periodicity of the reciprocal lattice. An electron band is defined as the set of
electronic levels with energies nk for a specific n. The set of such bands is called
one-electron band structure of the solid.
Because unk (r) can be expressed as a Fourier series, ψnk (r) may be written as a
linear combination of plane-waves:
ψnk (r) = eik.r unk (r) =
X
λnk (G)ei(k+G).r
G
P
where the reciprocal lattice vector G = j bj mj ; mj are integers.
According to their one-electron band structure the crystalline solids can be classified in three main groups:
• Insulating materials have completely empty allowed bands above the Fermi
level, i.e. conduction bands, and completely filled allowed bands b below the
Fermi level, i.e. valence bandsc . The Fermi level is positioned in the middle of
the band gap between the lowest conduction and highest valence bands. Since
the bands below the Fermi level are completely full of electrons and the vacant
states are high in energy, these materials do not conduct electricity or have a
low conductivity.
• Semiconductors may be viewed as insulators for which the conduction and valence bands are positioned close enough with respect to each other, so that the
band gap in a semiconductor is smaller than that in an insulator. This gives
rise to temperature-dependent conducting properties.
• Metals possess partially filled bands regardless the temperature which determines their high conductivity.
Finally, we briefly comment on a somewhat different approach to the description
of the band structure of crystals, namely the tight-binding model. In this model the
one-electron wave functions are approximated by linear combinations of Bloch sums
of atomic orbitals (local basis functions),
1 XX
cin (Ri )eik.Ri φin (r − Ri )
ψnk (r) = √
N i Ri
where N is the number of unit cells and n denotes the band produced by the atomic
orbitals φin (r − Ri ). The atomic orbitals and Bloch sums are generally not orthogonal to each other. Due to the localized character of the atomic wave functions the
b at
absolute zero temperature, when the Fermi-Dirac distribution is not smoothed
occupation of the available energy levels at some temperature obeys the Fermi-Dirac statistic
c The
HF and DFT-LDA
5
Hamiltonian matrix elements and overlap integrals between these basis functions become exponentially small for large |Ri − Rj | distances. It therefore makes sense to
ignore all integrals outside some |Ri − Rj |>Rmax which would lead to only negligible corrections to the band structure. The simplest approximation is to neglect all
overlap integrals and to consider only the Hamiltonian matrix elements between the
atomic basis functions, localized at nearest neighbouring atoms. In the semi-empirical
tight-binding model those matrix elements are parametrized.
The method, developed in this research makes use of a non-orthogonal ab initio
tight-binding model for obtaining the translation symmetry-adapted wave functions
of excited, ionized and added-electron states in extended systems. The method is
applicable not only to extended systems with translational symmetry, such as crystalline solids but also to disordered extended systems. The objects of interest in this
thesis are crystalline structures and thus, the formalism is presented using a crystal
as an objective. We introduce in Chapter 3 the concept of many-body bands. The
many-body bands are considered as the energy bands associated with the dependence
of the ionization, electron-addition or excitation energies on the wave vector Kd . To
emphasize on the differences between these bands and the conventional one-electron
bands, we discussed above some basic aspects of band theory.
Approaches based on the independent electron approximation: HartreeFock and Density Functional Theory-Local Density Approximation
The concept of the independent electron approximation has been introduced by
Hartree [2], Fock [3] and Slater [4] by defining an effective interaction field which an
electron experiences in the presence of the other electrons. Within the Hartree-Fock
approximation, each electron moves in a mean field, generated by all other electrons
and the external potential (e.g. the nuclear potential). Consequently the potential
in the Hamiltonian of an N -electron system can be approximated by an effective
one-electron potential and hence, the wave function of the N -electron system can
be described by a Slater determinant, i.e an antisymmetrized product of one-electron
wave functions (orbitals). To obtain the best approximation to this wave function one
can apply the variational principle to minimize the expectation value of the Hamiltonian with respect to the Slater determinant. This results in a set of Hartree-Fock
(HF) equations. From the Hartree-Fock (HF) equations one can obtain the effective
one-electron potential, the Hartree-Fock potential. Because the Hartree-Fock potential depends on the one-electron orbitals, one needs to solve the set of Hartree-Fock
(HF) equations self-consistently. If one employs a finite set of basis functions one
arrives at the Hartree-Fock-Roothaan method (see for a detailed discussion, for example, [10]). The size of the finite basis set for a particular study may be defined in
the following manner: since one can not employ in practice a complete basis set, one
aims at using a basis set which allows for obtaining the closest possible approximation
to the Hartree-Fock calculation in a complete basis set.
d As opposed to k, which is the momentum of a single (quasi) particle, K is the total momentum
of the crystal , i.e., the sum of the individual momenta of (quasi) particles
6
Chapter 1, General Introduction.
The Hartree-Fock method does not account for electron correlation effects. The
electron correlation energy is defined as the energy difference between the exact nonrelativistic energy and the Hartree-Fock energy. To address the electron correlation
ab initio methods beyond Hartree-Fock are needed. The electron correlation can
be introduced straightforwardly within the multi-configurational wave function based
approaches. These approaches allow for description of the electronic structure of a
system beyond the one-electron model in a systematic and controlled manner. While
for molecular systems the ab initio correlated electronic wave function can be routinely constructed in many cases, for crystalline materials this is not straightforward
(see Chapter 3 ). The most straightforward manner of treating the electronic structure of crystalline materials is the band theory. Effective one-electron approaches
such as density functional theory (DFT) by Hohenberg, Kohn and Sham [7, 8] and
Hartree-Fock method are commonly used in periodic calculations of the ground state
properties of crystals. DFT is a theory for the ground state properties of finite and
infinite systems. A recent study of the validity of the Hohenberg-Kohn theorem [7]
for excited states has shown that there is a different energy-density relation for the
excited states compared to that for the ground state, and this represents problems
for the generalization of DFT to excited states [16]. The main problem arises from
the fact that there is no one-to-one mapping between the external potential and the
electron density associated with an excited state [16]. The excitation energies in
solids and molecules have been successfully computed within the Time-dependent
DFT formalism using the linear response of the systems to an external perturbing
electromagnetic field (see for example [11, 12]).
In DFT the many-body effects are introduced by the exchange-correlation functional. Various approximations to the exchange-correlation functional have been suggested which reproduce different properties of a system with a different accuracy but
there is no systematic way to improve a particular functional. Among the most commonly used DFT approaches are those which rest on the local-density approximation
(LDA) to DFT. The LD functional is derived by considering a homogeneous electron
gas. LDA approximates the true inhomogeneous system locally by a homogeneous
electron gas and hence, it is expected to be a reasonable approximation only for systems with a slowly varying density. Surprisingly, the LD approximation has been
successful in describing the properties of inhomogeneous systems such as atoms and
molecules.
A failure of LDA is the systematic underestimation of the band gaps of semiconductors and insulators (see for example [15]). This failure is due to the fact that
the LDA exchange-correlation energy functional changes continuously when an electron is removed or added to the system. This behavior is known to be caused by
the absence of derivative discontinuities in the LDA [18]. This failure of LDA has
been discussed by Birkenheuer, Fulde and Stoll [52]. They have pointed out that the
single self-consistent potential in DFT-LDA, which depends on the density only, can
not describe the different physical state of the electrons in the charge-neutral ground
state of a system and in a system where an electron is set into the conduction band.
Despite the drawbacks, the LDA is an attractive approach to band calculations
1.2 Electron Correlations and Localized States in Solids
7
because it incorporates the electron correlation effects preserving the effective oneelectron model. A general advantage of the DFT formalism is that one avoids constructing the many-electron wave function and obtains every physical observable as a
functional of the ground state electron density. This advantage however can not be
exploited when one is concerned with the complex, mostly multiplet, electron configurations of materials such as the transition metal (TM) oxides. The atomic and
molecular multiplets in general are not accessible using the LD functional. In TM
oxides with an open shell the LD functional does not produce the correct density
and consequently, the Hamiltonian of the open shell system no longer belongs to the
totally symmetrical representation of the (point) symmetry group.
Note also that the exchange energy is not treated exactly in LDA, and in general with any approximate density functionals, but it is approximated using exchange
energy functionals which are not self-interaction free. This leads to an incomplete cancellation of the self-interaction term in the Coulomb energy. As shown by Janssen [19]
in her study on the ionization spectrum and exciton states of CuX, this cancellation
error introduces significant problems in the description of transition metal 3d -states.
Different self-interaction corrections have been introduced to cure the self-interaction
error in the LD functional [20]. For further detailed information on the DFT formalism and the LDA approximation the reader is referred to the review papers [17, 18].
A significant advantage of the Hartree-Fock (HF) method is that it does not suffer
from the self-interaction error deficiency because within this method, the exchange
interaction is treated exactly.
In summary, the widely used DFT-LDA method is not applicable to band gap
transitions in insulating systems, and in general to the excited (ionized or addedelectron) states. As we will see in the next sections this approach can not describe
systems with strong electron correlation effects. As, in this thesis, we are interested in
treating systems for which the electron correlation effects are of particular importance
for the proper description of the many-electron TM 3d -states, the LD approach will
be out of consideration. In the next section, some theoretical approaches are discussed
which have been employed in studies of the electron correlations in solids.
1.2
Electron Correlations and Localized States in
Solids
The electron correlation in solids is a source of a rich variety of phenomena and
states related to the interplay between magnetism and electrical conductance. The
fascinating properties of the crystalline matter, in general are associated with the
electronic structure which has been a long-term research topic in condensed matter
physics as well as in theoretical physics and chemistry. The correct description of the
electronic structure of crystalline materials which exhibit strong electron correlation
effects is, in particular, an on-going challenge for both research fields. Materials for
which the electron-electron interactions can not be well described by effective oneelectron approaches, such as DFT-LDA and HF are referred to as strongly correlated
8
Chapter 1, General Introduction.
materials. An example for such a material is NiO for which those effective one-electron
approaches predict conducting properties. A most common definition of a strongly
correlated system in the literature refers to the electron-electron interactions in TM or
in rare-earth metal compounds. These materials are regarded as strongly correlated
because the on-site Coulomb repulsion between d (or f ) electrons of transition metal
or rare-earth metal ions, parameterized by the Coulomb parameter U, dominates the
kinetic energy of the electrons. The kinetic energy is considered to be of the order
of the width W of the corresponding one-electron energy bands. Namely, the strong
Coulomb repulsion between d -electrons in NiO makes the compound a wide band
gap insulator instead of metal.
As discussed above, the conventional one-electron band theory has been successful
in describing the properties of elementary metals. However the independent electron
approximation turns out to be inadequate for the strongly correlated materials. In
those materials the band-like effects and the strong intra-atomic-like electron-electron
interactions are either of a comparable magnitude or the latter prevail. Among those
strongly-correlated crystalline systems are, for example, the 3d- TM compounds.
DFT-LDA+U and GW approximations
The most widely used in the band calculations DFT-LDA method is not suited to
study in general systems with strong electron correlation effects because the LD functional does not incorporate the strong on-site Coulomb interaction. Within the LD
approach, there has been introduced an additional correction term to the LD functional in order to account for the strong on-site Coulomb interaction, U arising in TM
compounds, which is considered to be a source of the strong electron correlation. This
corrected LD functional gave rise to a new method denoted as LDA+U [21]. Within
LDA+U the electron system is separated into two subsystems, localized d (or f ) electrons for which the Coulomb d-d (f-f ) interactions are accounted for by the term
U, and delocalized s, p electrons, described by the orbital-independent one-electron
potential in LDA. Although it has been considered as a method with a complete absence of adjustable parameters, the values for U often vary significantly in different
studies on the same compound (see for example [22]). As the U term expresses the
strength of the Coulomb interactions, which determines the extent of localization of
the electrons, employing U with different magnitudes for the same compound results
in classifying the compound either or not as a system with strong electron correlation
effects.
Within the Green function theory, the many-electron effects are introduced through
a non-local and energy-dependent self-energy operator. Thus, calculating the selfenergy would provide the necessary account for the electron correlation effects. Since
the self-energy is hard to calculate, various approximations are introduced and among
the simplest ones is the GW approximation [23]. This approximation to the selfenergy operator is derived from the many-body perturbation theory. The self-energy
operator in GW is given as a product of the Green function and the dynamically
screened Coulomb potential [23]. The GW approximation may be viewed also as a
generalization of the HF approximation in which a potential term is introduced, that
Hubbard model and DMFT
9
incorporates the dynamical screening of the Coulomb potential. This potential term
however is not equivalent to the screened Coulomb potential in the Green function
theory [21]. It has been shown by Anisimov et al [21] that for localized states, such
as the TM 3d -states, the LDA+U theory may be viewed as an approximation to the
GW approximation [23]. These authors addressed also to what extent both LDA+U
and GW theories offer the same or a similar performance. They pointed out that this
issue depends crucially on a number of factors, such as the importance of the energy
dependence of the screened Coulomb potential, neglected in the LDA+U theory, as
well as the similarity in the values of U in LDA+U and the static screened potential
in GW.
Although the GW approximation offers a more sophisticated and a better account
for the electron correlation effects, note that for its practical realization, one makes use
also of the LDA method. To calculate the screened Coulomb potential term which
enters the expression for the self-energy operator, one needs to obtain the inverse
dielectric function. The inverse dielectric function is given itself in terms of the full
response function. To compute this response function, one considers most often energy
bands and wave functions derived from the LDA calculations. The performance of
these two theories, LDA+U and GW, for the cases of the strongly correlated LaMnO3
and CaMnO3 is discussed in more detail in Chapter 5, where the top of the valence
band of LaMnO3 , obtained in other studies using either LDA+U or GW, is compared
to the many-body bands derived from the new theoretical method, presented in this
thesis.
Hubbard model and Dynamical Mean-Field Theory
Hubbard model Another field describing the physics of the strongly correlated
electrons in solids rests on the model Hamiltonian methods. In the model Hamiltonian techniques the many-electron effects are included using various parameters.
Among the most widely used and simplest models are the Hubbard Hamiltonian [24]
and the Anderson model of a magnetic impurity coupled to a conduction band [26].
The Hubbard Hamiltonian method allows for a qualitative study of the competition
between the electron delocalization and localization effects in solids. The Hubbard
model considers only electrons in a single band. The second-quantized form of the
Hubbard Hamiltonian [24] is:
Ĥ =
XXX
σ
i
j
tij â†iσ âjσ + U
X
niα niβ
i
where â†iσ and âjσ are the creation and annihilation operators of the single band
electron at lattice sites i and j with spin σ (α or β) and niσ is the number operator.
The indices i and j sum over all lattice sites, and tij is the hopping integral which
quantifies the delocalization effects. The parameter U on the other side is viewed as
the on-site Coulomb repulsion term. The ratio between tij and U is often used to
determine whether a state is delocalized or localized. If t U, the states are regarded
as being delocalized while if t U, they are considered to be localized.
10
Chapter 1, General Introduction.
Janssen [19] has noted based on a work by Sawatzky [25], that the U and t
parameters are decreased by polarization effects occurring upon a hole or electron
creation, so that U is smaller than the atomic repulsion integral for a free atom and
tij becomes [19]:
tpol
ij = hφi |H|φj ihΨipol |Ψjpol i
where Ψipol and Ψjpol are the wave functions of the polarized extended system with
a hole or an electron either at i or j. In Chapters 4 and 7, we show that indeed
the polarization effects influence significantly the magnitude of the hopping integral
although a reduced value of tpol
ij is not always the final effect. This is so because there
are two different relaxation contributions to the values of hφi |H|φj i and hΨipol |Ψjpol i,
respectively, involved, which have an opposite sign. Depending on their magnitude,
tpol
ij may increase or decrease. Since the Hubbard model treats only one band, it is in
principle valid only when each atom has only one orbital. Imada et al [27] have pointed
out that when applying this model to the d -electron systems, one assumes that any
possible orbital degeneracy is lifted so that the relevant low-energy excitations can
be associated with a single band near the Fermi level. In addition, it is assumed by
Imada et al that the ligand p band in the transition metal compound is either well
separated from the relevant d band or a strong hybridization occurs between them,
which results into an effective single band. Furthermore, the inter-site interactions
in the short-range part of the Coulomb potential are also neglected which prevents
in some cases from obtaining, for example, the correct charge-ordering effects. It
is interesting to note that in spite of the great simplifications, the Hubbard model
reproduces the Mott insulating phase.
The model Hamiltonian approaches are most often employed to provide a qualitative picture of the mechanisms which drive a particular physical phenomena. The
use of those models for a quantitative description is restricted by the complexity of
the solutions, for example, the Hubbard model can be solved exactly only for the
1D case. The 2D and 3D models of correlated electrons require the use of various
numerical techniques.
Dynamical Mean-Field Theory The LDA +U method, discussed in the previous
subsection has been viewed by Anisimov et al [21] as a many-body extension of LDA.
However, the Coulomb interactions within LDA+U are treated essentially within the
Hartree-Fock approximation and thus, the approach would be better classified as an
effective one-electron approach.
Another method for treating local many-electron effects in solids rests on the assumption of a local or k-independent electron self-energy but preserving its frequency
dependence (dynamical mean field). This fundamental approximation leads to a mapping of the many-electron problem onto an effective single-site problem, which must
be solved self-consistently in a combination with the k-integrated Dyson equation,
where the latter connects the self-energy and the Green function for a certain frequency [34]. The formalism relates through a set of equations the Green function of
the correlated single-site impurity to the Green function of the crystal calculated at
The embedded cluster approach and beyond
11
the impurity site. The quantum impurity, deduced from the lattice, is embedded selfconsistently in an effective electron ”bath” representing the crystal. This approach
has been developed during the last decade and named Dynamical Mean Field Theory
(DMFT) (review paper: [29])
Originally the applications of DMFT were mainly directed to models such as
the Hubbard Hamiltonian. The DMFT approach has been combined with the LDA
approach where the LDA approach is exploited for the computation of the crystal
Green function [30, 34]. The single-site impurity in DMFT is replaced by a more
complicated cluster of sites in the cellular DMFT [31] and the dynamical cluster
approximation [32] and in addition, the short-range correlations are introduced.
The embedded cluster approach and beyond
The impurity cluster model considered in (cellular) DMFT is rather similar as a concept to the embedded cluster model. The embedded cluster approach will be discussed
in somewhat more detail in Chapter 2 because of its relevance to the present research.
Here, we outline the main characteristics of this method for ionic solids within the
framework of the Theory of the Electron Separability [35, 36]. In this theory an assumption is made that some of the physical properties of a compound can be ascribed
mainly to a particular electron group, and the effect of the remaining electron groups
on that electron group is treated approximately, supposing that the presence of the
remaining groups is of a little relevance for the property under investigation. That
particular electron group forms the ”quantum” cluster which is embedded in effective model potentials [37, 38] that account for the crystal Madelung potential and
short-range Pauli and exchange interactions with the nearest neighbour embedding
ions of the cluster. In this manner, the N- electron crystal Hamiltonian (N → ∞) is
mapped onto an effective M -electron cluster Hamiltonian (M is finite). To address
properly the many-electron effects, the cluster model is most often combined with an
accurate ab initio wave function based method. For example, configuration interaction techniques are widely used to calculate photo-emission and X-ray photoelectron
spectra [39–41] or to perform studies on ionized and added-electron states [42] and d-d
excitations and core hole states in different transition metal oxides [43–46]. Furthermore, second-order perturbation theory based on a multi-configurational reference
wave function [47] is applied to study d-d excitations and ligand to metal charge
transfer states [43, 48] as well as Heisenberg magnetic couplings. For a recent survey on the performance of the Multiconfigurational Perturbation Theory to predict
magnetic coupling parameters in different ionic insulators, see [49]. The embedded
cluster method has been used extensively to study excited, ionized, added-electron
states with predominantly localized character. The application of the wave function
based methods has been originally directed to the study of molecular and atomic
structures within quantum chemistry. The ab initio calculations which are based on
controlled approximations, that are subject of systematic improvement, are able to
address properly and with a high accuracy the many-electron correlation effects. On
the other side stand the effective one-electron approaches such as the ordinary DFT,
which in spite of their simplicity and a lack of a well formulated way for a system-
12
Chapter 1, General Introduction.
atic improvement, have been incorporated in the band theory studies and for some
compounds and properties provide a good description. They conserve the periodic
symmetry of the wave functions of the crystalline material. It seems logical to attempt to merge the two different theoretical fields in order to extract the advantages
from each of them. For a long time however, the inclusion of the electron correlation
effects in the band theory studies of the crystalline solids, using wave function based
techniques, has not been considered possible.
Fortunately, new algorithms were recently developed which incorporate electron
correlation effects explicitly in the studies of the extended periodic systems [50,51,53].
Some of these algorithms use the so-called effective Hamiltonian approach. This
approach allows one to construct the infinite 3D correlated Hamiltonian matrix for a
crystalline system, using the Hamiltonian matrix elements derived from calculations
on finite model clusters. In this manner correlation effects are easily incorporated
[50, 51, 53]. These and other algorithms which make use of a wave function based
formalism to treat the correlation effects are discussed in detail in Chapter 3. The
theoretical method which will be presented in this thesis belongs to those recently
developed algorithms.
1.3
Outline of This Thesis
The strong electron correlation effects lead to a localization of the electrons and consequently the wave functions describing the electrons state tend to localize spatially.
Then the zero-order wave function of a N -electron (N → ∞) system can be written in principle as an infinite sum of spatially localized N -electron wave functions,
provided that each localized wave function has an essentially negligible overlap with
all other localized wave functions [6]. We can facilitate the theoretical description
of a crystalline system by imposing periodic boundary conditions on the zero-order
wave function and by introducing Bloch sums of the localized many-electron wave
functions. Provided that the localized wave functions have almost negligible overlap
we may consider exploiting the formalism of a non-orthogonal many-electron tightbinding method in order to obtain the delocalized wave functions of the periodic
system. Since we are concerned with the description of excited, ionized or addedelectron states with predominantly localized character the energy bands associated
with these states are expected to be rather narrow.
In molecules containing, for example, transition metal ions the strong electron
correlation effects can be incorporated by constructing a multi-configurational (MC)
self-consistent field (SCF) wave function which is composed from configurations relevant for the description of a particular molecular state. A straightforward application
of the MCSCF approach to an infinite system would imply the inclusion of an infinite
number of different configurations in the MCSCF wave function expansion. This is
not feasible.
In this thesis a new theoretical method is developed for the generation of delocalized and correlated many-electron wave functions for excited, ionized or addedelectron states in extended systems with strong electron correlation effects. For the
Outline of this thesis
13
construction of the local correlated many-electron basis functions, in which the delocalized many-electron wave functions of the extended system are expressed, use is
made of the generalized antisymmetrized product wave function approximation [10].
This approximation leads naturally to the embedded cluster approach for the computation of relevant Hamiltonian matrix elements and overlap integrals between the
local correlated many-electron basis functions. Those Hamiltonian matrix elements
and overlap integrals are approximated by the matrix elements of a large embedded
cluster. In this context a new concept is introduced in which the large embedded
cluster, called super-cluster, is viewed as consisting of overlapping smaller embedded clusters, denoted as fragments. This overlapping fragment approach allows for
a balanced description of the electron distribution around the excitation sites in a
super-cluster and it circumvents the problems with SCF calculations for very large
clusters. The wave functions of the super-clusters, which describe excited states localized around different lattice sites, are expressed in terms of localized orbital basis
derived from fragment’s multiconfiguration SCF calculations. The use of localized
orbital sets allows for a rigorous treatment of local electron correlation and electronic
relaxation effects. The method is discussed in detail in Chapter 3.
The embedded cluster approach has been invoked throughout this thesis to model
the systems under investigation. The theoretical foundations of this well established
method are outlined in Chapter 2. Different techniques for modeling the surrounding
of the cluster, treated quantum mechanically, are described as well in Chapter 2.
Furthermore Chapter 2 discusses different methods for the approximation of the N electron cluster wave function and for the incorporation of the relevant non-dynamical
and dynamical electron correlation effects in the cluster calculations. Thereafter, we
discuss briefly the method which allows us to compute relevant Hamiltonian matrix
elements and overlap integrals between cluster wave functions of localized states, each
expressed in its own orbital set (the non-orthogonality problem).
In Chapter 4 the concept of the overlapping fragment approach is exploited for
obtaining localized multi-configurational wave functions for the hole or electron states
of super-clusters representing lightly doped manganites. The Hamiltonian matrix elements and overlap integrals between these localized wave functions are computed and
employed for the determination of effective hopping matrix elements. Those effective
hopping integrals, or double exchange parameters are important for characterizing
the mobility of the Mn 3d eg -like electrons in lightly doped LaMnO3 and CaMnO3
as well as in La0.75 Ca0.25 MnO3 .
Thereafter, in Chapter 5, we use the Hamiltonian matrix elements and overlap
integrals, derived from super-clusters in Chapter 4, to approximate the Hamiltonian
matrix elements and overlap integrals between the local many-electron basis functions,
each of them describing a hole or electron state of the doped manganite system, localized around a particular lattice site. We perform non-orthogonal tight-binding
calculations, using those ab inito -derived Hamiltonian matrix elements and overlap integrals, to obtain the bands associated with the delocalization of the holes or
electrons. These bands differ significantly from the one-electron bands obtained in
a conventional band calculation. They incorporate explicitly the electron correlation
14
Chapter 1, General Introduction.
effects. They are referred to as ”many-body” bands throughout this thesis. Since
we work with lightly doped compounds, the bands can, in principle, be compared to
those at the top of the valence band and at the bottom of the conduction band of
pure LaMnO3 and CaMnO3 , respectively.
In Chapter 6 we study low-lying excitations in orthorhombic and cubic LaMnO3 by
means of the embedded cluster model, combined with a multi-configurational wave
function based method, followed by second-order perturbation theory. The use of
this approach allows for an accurate account for the non-dynamical and dynamical
correlation effects on the localized excited states. The study aims at the computation
of parameters such as the Jahn-Teller, the Mn 3d exchange splitting and the crystal
field parameters. In addition, we investigate the cluster size effects on the energies
of the d-d excitations and charge transfer (CT) states which allows us to get insight
into the delocalization of those excited states.
In Chapter 7 our new theoretical approach is applied to the study of the lowest
hole and electron states in NiO. Those states are directly related to the states at the
top of the valence band and at the bottom of the conduction band in the compound.
While the bottom of the conduction band has clearly Ni 3d9 character, there has been
a debate concerning the character of the top of the valence band, either having Ni
or O character. This controversy classifies the compound either as a Mott-Hubbard
type insulator or as a CT type insulator respectively. To obtain the relative energies
of the lowest hole states localized either at Ni or O ions, we designed a super-cluster
which treats in a balanced manner the hole states, localized at either Ni or O ions.
Furthermore, the delocalization effects for both types of hole states are considered in
the light of the many-body bands.
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18
BIBLIOGRAPHY
2
Theoretical Framework
”If quantum mechanics hasn’t profoundly shocked you, you haven’t understood it
yet”
Niels Henrik David Bohr
2.1
2.1.1
The embedded cluster approach
Introductory remarks
As an alternative to the conventional band structure description of crystalline materials a different approach can be considered which is based on modeling the solid by
a small collection of atoms at lattice positions of the system, a cluster, embedded in
a potential which accounts for the part of the crystal that is not explicitly included
in the cluster. This approach is well known as the embedded cluster approach.
In this thesis the embedded cluster approach is employed as a starting point in
the studies of excited, ionized and added-electron states of insulating crystalline transition metal materials. In studying excited or ionized and added-electron states with
mainly localized character the local cluster approach is a more common starting point
than the conventional band theory. It has been shown by Nieuwpoort and Broer [1]
that in the case of closed-shell ionic systems, the ground state wave functions derived
from large enough embedded clusters and the corresponding crystal Wannier functions obtained both at the Hartree-Fock level are in fact equivalent. This is not the
case for ionized or added-electron systems where the large electronic relaxation effects
which accompany the ionization and electron addition process are not incorporated
in the delocalized description. The electronic relaxation effects can be well defined for
hole states and added-electron states. We consider an N -electron system, described
by an N -electron wave function. The latter is considered to be multi-configurational
SCF wave function. When an electron is removed from the N -electron wave function
without any further changes in the wave function, the resulting N-1 -electron wave
20
Chapter 2, Theoretical framework.
function is denoted as unrelaxed, frozen wave function. This wave function can be
optimized within the variational principle in order to obtain the best wave function
for the N-1 -electron system. The changes in the N-1 -electron wave function (orbitals and configuration interaction coefficients) that occur when the wave function
becomes the optimal wave function for the N-1 -electron system are called electronic
relaxation. When a hole or an electron is created in a very localized, almost atom-like
state, the relaxation energy gain is substantial. In the band description, for instance,
the hole is created in a Bloch state at the top of a band with a particular predominant
character. Because the hole-orbital is delocalized over all translation symmetry invariant lattice sites of a certain type, the change in the electron density at each site is
vanishingly small and hence, no orbital relaxation takes place (Koopmans’ Theorem
is valid) [1]. In a cluster description the hole is created in the highest occupied orbital
with the same predominant character, however it is localized, thus allowing for the
electronic relaxation. Therefore for those processes, the local approach provides a
better description. In addition the embedded cluster model facilitates the inclusion
of electron correlation effects because it allows for an ab initio many-body description of the electronic wave functions using state-of-the-art quantum mechanical wave
function based techniques.
As shown in numerous studies on local excitations in transition metal materials
(see [2–6] and the references therein), core level excitations (e.g. [7,8]), magnetic couplings and hopping integrals between localized states [9–12, 14, 15], the cluster model
combined with an accurate quantum-mechanical wave-function based treatment of the
cluster electrons allows for an accurate account for the local correlation and relaxation
effects. X-ray absorption, photo-emission, photo-electron, core X-ray photo-emission
etc. spectra of various TM materials [3, 19–22] are described accurately using the
local cluster approach.
In the cluster approach, the effective cluster Hamiltonian must be modified to
incorporate both the electrostatic interactions with the charges outside the cluster
region and the fact that the electrons described by the cluster wave function are
not confined to the cluster in the real crystal, i.e. the cluster wave functions may
extend beyond the cluster region [23]. It is clear that while for ionic compounds the
electrostatic interactions have more significance than the extension of the cluster wave
functions beyond the boundaries of the cluster, for covalent structures the importance
of these effects is vice versa.
In covalent materials and metals, the electrostatic interactions decrease fast with
the distance from the cluster region due to the effective screening processes (particularly metals). In ionic solids, however, the electrostatic fields have a long range.
Most often the interactions between the cluster and surrounding crystal are classified
as long-range and short-range electrostatic interactions. In addition long-range polarization effects are of a great importance, for example, for charged defects in ionic
crystals.
Different embedding techniques have been developed in the last decades to facilitate the incorporation of the electrostatic interactions between the cluster region and
the surrounding crystal in the electronic structure calculations. Some of them, devel-
Long- and short-range electrostatic interactions
21
oped mainly for ionic materials, the Ab Initio Model Potential (AIMP) method, by
Barandiarán and Seijo [24] and others, based on the so-called effective core potential
technique by Hay and Wadt [25], have been employed in this thesis and thus, will be
summarized briefly in the next sections.
2.1.2
Long- and short-range electrostatic interactions
As stated above, the electrostatic interactions between the cluster and surrounding
lattice are formally divided into long-range and short-range electrostatic interactions.
The long-range interactions are of importance mostly for the ionic materials. They
are usually included in the local model by adding the Madelung potential generated
by all ions outside the cluster region to the cluster effective Hamiltonian. This is done
by first calculating the Madelung potential using infinite lattice sums which are easily
handled by the Ewald summation method [26]. The Ewald summation is performed
using point charges at experimental lattice positions. Next the cluster contribution to
the total potential is subtracted in order to obtain the external Madelung potential.
In practical calculations, the external Madelung potential may be obtained by performing a summation over the contributions of a large enough set of point charges at
lattice positions aiming at the exact value of the external Madelung potential to some
predefined accuracy. Because of the slow convergence of the Madelung potential with
the distance, this approach is not practical since it requires a large set of formal point
charges. The approach used in this thesis is based on a fitting procedure using the
least squares numerical method. A finite (usually rather small) set of point charges
at lattice positions within a certain range around the cluster are optimized within the
fitting procedure in order to obtain the closest to the exact value of the Madelung
potential on a fine grid within the cluster region. This approach has, for example,
been exploited extensively by Janssen [27] and de Graaf [28].
The short-range interactions can not be treated using classical models. They consist of Pauli repulsions supplemented by Coulomb and exchange interactions between
the cluster charge distributions and the charge distributions of its nearest neighbouring ions. The modeling of these interactions has been a subject of long-term studies
and many various approaches have been suggested and developed [24,30,34–41,46,53].
The embedded cluster approach, subject of the current discussion, can be derived
starting from one of those approaches, the Theory of the electron separability (TES ))
in combination with the Group-Function Theory (GFT ) [24, 30, 46]. These two theories are also the starting point for the derivation of the embedding model potentials
that are employed in this thesis. In this study we consider excitations, ionizations
and electron additions in insulating transition metal compounds using crystal lattices
of the compounds with ions for which formal ionic charges are adopted at first approximation. When designing the cluster, which in the present study contains one or
few transition metal ions and the coordinating shells of O2− ions, one must take into
consideration that the nearest embedding shells of ions around the cluster are best
represented by effective model potentials. This requirement holds especially for the
positive embedding ions nearby the cluster counterions, because a point charge repre-
22
Chapter 2, Theoretical framework.
sentation of those cations may result in an artificial polarization of the cluster orbitals
towards the positive point charges (see also below). Because we use extensively in this
thesis the model potential techniques which are derived from the consideration of the
TES and GFT, we adopt the same line of presentation of the formalism of the embedded cluster approach, although the embedded cluster approach can be considered
without referring to the TES and GFT.
The simplest way to account for the rest of the crystal surrounding the cluster is to
account only for the long-range electrostatic potential by introducing corresponding
point charges at lattice positions. For very ionic materials such cluster models turn
out to be rather accurate because the electronic states have predominantly localized
character (e.g. Ref. [2, 7]). Furthermore the dependence of the energies of localized
excited, ionized or added-electron states on the cluster size can be accounted for by
enlarging the cluster (for example [64]). The size of the cluster required to study
states with predominantly localized character in materials with covalency, is usually
small enough to be handled computationally. However, representing the short-range
Coulomb interactions between the cluster and embedding ions by point charges as it
has been done in earlier studies (see for example [44]) may introduce problems, such
as delocalization of the cluster wave function towards the point charges. This problem
is particularly emphasized in the case of an ionic material for which the model cluster
has embedding ions (for example La3+ , Y3+ , Ge4+ ) with high formal ionic charges
(see, for example, [45]). In order to avoid the artificial charge flow the Pauli repulsion
between cluster and nearest embedding ions must be taken into account. A natural
way to do this is to introduce repulsive potentials set at those nearby ions.
In case of a material with a greater degree of covalency, the studies of localized
excited, ionized states or impurities can be carried out for sufficiently large clusters
terminated using appropriate termination schemes. For instance, Aissing has shown
in his study of the band width, band gap, ionization energy and electron affinity of
pure silicon, that these quantities can be extrapolated from clusters with an increasing
size in order to account for the cluster artifacts [31, 32, 89]. Some of the clusters in
that study are Si5 H12 , Si10 H16 , Si17 H36 , clusters. The H atoms are used to saturate
the dangling bonds of the surface Si atoms. In studies of, for example, transition
metal impurities in Si [31, 89], this is a simplistic but an efficient way to handle the
so called surface states, which are due to the dangling bonds of the surface Si atoms.
Theory of the electron separability and Group-Function Theory
Since the short-range electrostatic interactions are object of quantum mechanical
treatment, different quantum chemical approaches are developed to incorporate them
in the local model. One of those methods is based on the so-called theory of electron
separability (TES ) [30, 53]. The TES formalism uses conventional wave function
based methods for finite systems to describe the properties of extended structures.
The approach is denoted as Group-Function Theory (GFT ) and it is developed by
McWeeny and Kleiner [46] and Huzinaga and Cantu [30, 53].
The Theory of the Electron Separability assumes that it is possible to partition
the extended system into various electronic groups I, S, ..., each containing a number
TES and GFT
23
of electrons NI , NS , ... . The groups electronic states a, b, ... are described by the
antisymmetric group wave functions ΦIa , ΦSb , .... [24, 46]. The wave functions Φκ for
the extended system are then constructed as generalized antisymmetrized products
of the wave functions ΦX
x (r1 , r2 , ...., rNX ) [24, 46]:
Φκ (r1 , ..., rN )
= ÂMκ [ΦIa (r1 , ..., rNI )ΦSb (rNI +1 , ..., rNI +NS )....]
(2.1)
where N=(NI +NS + ...) → ∞ and Mκ is a normalization factor and  is the intergroup antisymmetrizer. Φκ corresponds to a particular configuration of states for the
different electron groups, namely, κ= (IaSb...). This wave function representation allows one to include in a straightforward manner the electron correlation locally within
each electron group I, since ΦIa can be a MC wave function expansion. This flexibility of the GFT is exploited by the embedded cluster approach, where the electronic
structure of one particular electronic group, the cluster, is described by a complicated
MC wave function expansion while the surrounding lattice is treated with the frozen
electron distribution.
Electron correlations between different electron groups I and S are not considered
and therefore, they should not be relevant to the properties under investigation. In
the case of weakly interacting electron groups, the approximate form (2.1) to the
wave function of the extended system is well suited. In case the different electronic
groups must be represented by more than one electronic state a, the wave function of
the extended system may be written as a linear combination of different generalized
product wave functions of the form (2.1):
X
Ψm =
cκ Φκ
(2.2)
κ
One of the main approximations in GFT are the so-called strong orthogonality conditions which are imposed between the different electronic groups. These so-called
strong orthogonality conditions are expressed as [24, 46],
Z
S
Φ∗I
(a 6= b) or both. (2.3)
a (r1 , ..., rNI )Φb,ex (r1 , rNI +2 , ..,)dr1 = 0 (I 6= S),
where ΦSb,ex is the wave function obtained from ΦSb by exchanging one (or more) of
its electron (s) with electron (s) of ΦIa . The integration is over the coordinates of that
electron.
The zero overlap in Eq. (2.3) can be achieved by constructing the different electronic group wave functions from different and mutually orthogonal spin-orbital sets.
McWeeny has shown that if a strong orthogonality condition is imposed, one can
apply the generalized Slater method to the generalized product wave functions in
Eq. (2.1), that allows one to handle the matrix elements of the crystal Hamiltonian,
Ĥ, between different product wave functions (2.1) as the matrix elements between
antisymmetrized orbital products [46]. The mathematical formulation of GFT is
presented in reference [46].
Very often the physical properties of the extended system can be ascribed to
a particular electron group I, the cluster, and the effect of the remaining electron
24
Chapter 2, Theoretical framework.
groups S is regarded to be somewhat smaller. Then the detailed electronic structure
of the groups S is not considered and the computational effort is redirected towards
an accurate MC expansion of the wave function ΦIa of cluster I, whereas adopting
the frozen electron distribution for all S groups, the latter are in their ground state
’0’, ΦS0 . The complicated set of SCF equations for the N-electron (N→ ∞) system
is reduced to a set of SCF equations for the cluster I in the effective field of all S6=
I which represent the embedding of the cluster. The effect of the effective field is
incorporated through the effective one-electron cluster Hamiltonian,
ĥIef f (i)
=
X
i∈I
JˆS (i)
X
X
Zµ
1
+
[JˆS (i) − K̂ S (i)]}
{− ∇2i −
2
|ri − Rµ |
µ
S(6=I)
X 1
= hΦSb |
|ΦS i
rij b
j∈S
K̂ S (i)
= hΦSb |
X P̂ (i, j)
j∈S
rij
|ΦSb i
(2.4)
where the summation over µ is extended to all the nuclei of the system. Z is the
nuclear charge. The effective one-electron Coulomb and exchange operators JˆS (i) and
K̂ S (i) in Eq. (2.4), however, are not used in practice in this form due to computational
deficiencies [24]. The latter can be circumvented by replacing the embedding operators
by model potentials which approximate the effect of the embedding on the cluster
electron distributions [24, 25].
As mentioned in the introduction, various embedding techniques have been suggested and developed [24, 33, 34, 40–43]. Within the framework of this thesis we
made use of two model potential techniques, namely the Ab Initio Model Potential
method [24] and the Total Ion Potential Technique based on the effective core potential technique by Hay and Wadt [25]. In the next subsection, we outline the basics of
the two techniques.
The GFT and TES formalisms have been considered here in connection with
the embedded cluster approach, but they may be also applied to other systems for
which it is possible to distinguish well -separated electron groups. For instance,
the formalism is well suited to study inter-molecular fields, since they concern the
interactions between two well-separated electron groups, which belong to two or more
molecules. The applicability of those theories may be extended to describe also the
solvent-solute dynamics [55, 56].
Furthermore, when considering the properties of a molecule which can be mainly
ascribed to its valence electrons, the valence molecular wave function ΦIa may be
constructed assuming frozen core-electron closed-shell atomic wave functions. Huzinaga and co-workers [34, 47] developed atomic model potentials to approximate the
core-electron density distribution in valence-only calculations.
Embedding techniques
25
Embedding techniques
The straightforward employment of the embedding effective one-electron operators,
that represent the crystal environment requires the computation of the same number
of two-electron integrals as that needed when the system is treated as a whole [24].
Therefore, various techniques were implemented that aim at the representation of
the local and non-local effective one-electron operators by model potentials [24]. One
possible representation of the embedding operators is within the AIMP approximation
of Seijo and Barandiarán [24].
The Ab Initio Model Potential approximation The AIMP method was initially developed by Huzinaga et al. to represent the core-electrons distribution in
valence-only calculations [34, 47]. In order to replace the embedding operators associated with the electron groups S by appropriate embedding potentials, one may
consider the long- and short- range Coulomb interactions and the exchange interaction of the cluster with the embedding groups, separately. If group I is chosen to be
the cluster, with state a, described by any MC wave function expansion, the effective
total cluster energy can be written as [24]:
I
I
(HI (aa))ef f = hΦIa |Ĥef
f |Φa i
X X Zµ Zν X
XX
Zµ
|ΦSb i +
+
hΦSb | −
|rj − Rµ |
|Rµ − Rν |
j∈S µ∈I
S6=I
(2.5)
µ∈I ν∈S
where
I
Ĥef
f
=
X
i∈I
X
S6=I
X
i>k∈I
X
1
Zµ
− ∇2i +
2
|ri − Rµ |
+
µ∈I
−
XX
i∈I ν∈S
X
Zν
+
[JˆS (i) − K̂ S (i) + P̂ S (i)] +
|ri − Rν |
i∈I
X
Zµ Zν
1
+
|ri − rk |
|Rµ − Rν |
(2.6)
µ>ν∈I
In the expressions above the Greek letters are reserved for nuclei, Z is the nuclear
charge, and JˆS and K̂ S are the effective one-electron Coulomb and exchange operators.
The interaction between the cluster electrons and the other electron groups is
contained in the first term of Eq. (2.5). Those interactions are basically introduced
as additional terms to the one-electron cluster hamiltonian ĥI ,
X Zν − Nν X
Nν
V̂ S (i) = −
−
+ JˆS (i) −K̂ S (i)
(2.7)
| {z }
|ri − Rν |
|ri − Rν |
ν∈S
ν∈S
S
|
{z
}|
{z
} V̂exch
(i)
S
V̂lr−Coul
(i)
S
V̂sr−Coul
(i)
The V̂ S (i) operator is a potential energy operator which contains the interaction of
a cluster electron i with the frozen electron group S. This operator can be decomposed
26
Chapter 2, Theoretical framework.
into different contributions of the embedding group S, namely the Coulomb potential
is separated into long- and short-range components and the non-local exchange operator K̂ S (i) constitutes the exchange contribution to V̂ S (i) (see Eq. (2.7)). Nν is a
number which is introduced to ensure the separation of the electrostatic Coulomb potential into long- and short-range components [24,53]. It is usually chosen to be equal
to the number of electrons associated with the nuclei ν that belong to the embedding
electron group S. The electrostatic Coulomb potentials include the attractions between the cluster electrons and the nuclei ν where the latter are arbitrary assigned to
a particular embedding group S. The assignment of a number of nuclei ν to a specific
embedding electron group S is arbitrary [24]. Next, the embedding operator, V̂ S (i),
S
in Eq. (2.7) is replaced by a model potential V̂ S,M P (i). The V̂lr−Coul
(i) operator
is replaced by the point charge potential corresponding to the long-range Coulomb
potential. The short-range Coulomb potential, which includes the deviations from
the point charge potential, is substituted by an analytical function [24],
S,M P
Vsr−Coul
(i)
=
X B S e−αSl ri 2
l
l
ri
(2.8)
in case the S electron group is atomic. The set of parameters (BlS , αlS ) is obtained
S
using a least-square fitting to V̂sr−Coul
(i), where the latter is computed with the
S
wave function Φb . For polyatomic electron groups S, one adopts the same model
S
potential representation of V̂sr−Coul
(i) as that of the non-local exchange operator,
S
V̂exch (i), namely, their corresponding spectral representations in the primitive basis
{|pS i} used for the wave function of S,
S,M P
V̂sr−Coul
(i)
= Ω̂S V̂sr−Coul (i)Ω̂S
S,M P
V̂exch
(i)
= Ω̂S V̂exch (i)Ω̂S
(2.9)
where the projection operator Ω̂S is given by
XX
S
Ω̂S =
|pS i(S S )−1
pq hq |
p
S
Spq
S
q
= hp |q S i
(2.10)
The expressions in Eq. (2.8) and Eq. (2.9) constitute the AIMP representation of
the embedding operator V̂ S . The AIMP of V̂ S is obtained by optimizing in a SCF
procedure the group wave function of S (most often an atom/ion) in the effective
field of the remaining ions in the crystal. To obtain the embedding wave functions
ΦSb and the cluster wave function ΦIa one needs to apply the restricted variational
principle of Huzinaga [30, 53]. As mentioned above, the embedding electron groups
S are usually taken in their ground state. The ground state wave function ΦS0 of
the embedding groups S can be most often described by a single configurational
state function (CSF). One employes the restricted variational principle of Huzinaga
which implies the introduction of a shift operator P̂ S in the expression for the AIMP
representation of V̂ S in order to prevent occupation of orbitals of the embedding ΦS0
Embedding techniques
27
in the configurational expansion of the wave function ΦIa of the cluster. The shift
operator is given as [24],
P̂ S (i)
=
occ
X
|ψgS i − xSg Sg hψgS |
(2.11)
g∈S
where Sg and ψgS are the orbital energies and the occupied orbitals in which the wave
function ΦS0 is expanded. xSg is a projection factor which has been shown to have a
value of 2 for the case of closed-shell electron groups at the Hartree-Fock level [48].
Various values of the factor can be used [49] to maintain the orthogonality conditions
between the cluster wave function and the wave functions of the embedding groups in
case the latter are represented by bare AIMPs without orthogonalizing basis functions.
The number of embedding groups SP
in the crystal is infinite and this requires an inS,M P
S,M P
S,M P
finite number of embedding potentials S(6=I) [Vlr−Coul
(i) + V̂sr−Coul
(i) + V̂exch
(i) +
P̂ S (i)] to be added to the one-electron cluster Hamiltonian in order to obtain a correct
expression for the effective one-electron cluster Hamiltonian. In practice however, the
embedded cluster calculations are carried out by including the long- and short-range
AIMP embedding operators for the nearest and next-nearest embedding ions of the
cluster while preserving only the long -range Coulomb potential associated with the
distant ions. The AIMP form of the effective cluster Hamiltonian, after replacing the
effective embedding operators by their model potential representations, is:
(
X
X
Zµ
1
S,M P
S
C
+ [Vlr−Coul
(i) + V̂sr−Coul
(i) +
ĤAIM P (1, ..., NI ) =
− ∇2i −
2
|ri − Rµ |
i∈I
µ∈I
)
X
S,M P
S
V̂exch (i) + P̂ (i)] +
ĝ(i, j)
(2.12)
i>j∈I
Note, that compared to the effective cluster operator before replacing the embedding
one-electron operators by their AIMP forms, its AIMP representation in Eq. (2.12)
includes explicitly only nuclear attractions associated with the cluster nuclei. The
ones associated with the ’embedding’ nuclei are collected in the long-range Coulomb
model potential.
Finally the second and the third terms in Eq. (2.5) express the interactions of the
cluster nuclei with the electrons and nuclei of the embedding groups. These terms
remain constant provided that the nuclei of the embedding groups are clamped and
the electron groups are frozen. More details of the analysis on the effective cluster
and cluster-embedding matrix elements are provided in reference [46].
The studies of local excitations and core and valence ionizations [16,17,28] in transition metal compounds have demonstrated that the AIMPs are good representation
of the effect of the embedding crystal. In some cases however, for example ionized
states in transition metal oxides [16], the bare potentials must be augmented with
so called orthogonalizing basis functions in order to maintain the strong orthogonality conditions between cluster wave function and embedding wave functions (see also
Chapters 6 and 7 ).
28
Chapter 2, Theoretical framework.
The AIMP embedded cluster approach performs equally well for clusters with
anion or cation peripherial ions when studying the energy potential surfaces of the
ground and excited states. An important advantage of this local approach is that the
cluster properties converge faster with the cluster size.
The Total Ion Potential Technique The Total Ion Potentials (TIPs) are basically effective core potentials without basis functions. They are sometimes employed to represent the cations external to the cluster. The interactions of the cluster
electrons with the external ions are incorporated through the effective one-electron
Hamiltonian of the cluster. Analogous to the AIMP approach the expression for the
cluster effective energy contains matrix elements of the embedding operators which
are approximated in the present study using the large effective core potentials by Hay
and Wadt [25].
To generate effective core potentials Hay and Wadt obtain first numerical HartreeFock (relativistic or non-relativistic) atomic valence orbitals from which they derive
so called smooth nodeless pseudo-orbitals, so that the latter are close enough in the
valence region of an atom to the initial numerical Hartree-Fock orbitals [25]. Then
Hay and Wadt generate numerically the effective core potentials Ul for each angular
O
momentum quantum number l using the valence pseudo-orbitals φP
l . Next the
PO
authors impose the condition that a φl is a solution of the atomic Hartree-Fock
equations in the field of Ul plus the core Coulomb and exchange operators, Jˆc and
K̂ c , and the valence Coulomb and exchange operators Jˆv and K̂ v , with the same
orbital energy as the initial numerical Hartree-Fock orbital φl .
Winter, Pitzer and Temple have employed first the effective core potentials as TIPs
in cluster calculations when studying the properties of Cu+ impurities in NaF [52].
Although it seems rather crude at first sight to approximate the charge distribution of
an ion (atom), external to the cluster, by an effective potential designed for the core
electrons of that external ion, the TIP technique often gives a satisfactory description
of the short-range interaction between the cluster and nearest surrounding ions. A
drawback of the embedding is that it is restricted only to external cations.
Long-Range Lattice Polarization The embedding model potentials, discussed
above, represent the effect of the embedding on the cluster under the condition of a
frozen electron distribution for the embedding groups. This means that the embedding potentials are derived using frozen wave functions for the embedding groups,
where the latter are obtained as they occur in a perfect cluster lattice. The lattice
response to the charge distribution within the cluster region due to the presence of,
for example, a defect or an ionization process, can not be accounted for in the frozen
electron distribution approximation. The presence of charged defects or impurities
in the crystal is usually accompanied by long-range dipole polarization and large local distortions, respectively. The local distortions that occur in the presence of an
impurity may extend in some cases beyond the cluster region [24]. The polarization
effects in the rest of the crystal may have a rather long range in the case, for example,
of ionization processes which are object of study in the core-level spectroscopy. The
2.2 Wave function based theoretical approaches
29
lattice polarization, on the other hand, affects back the electron distribution within
the cluster region and hence, the local properties under investigation. It seems natural that a self-consistent treatment should be applied in which the external potential
which the cluster experiences should be updated in order to incorporate the response
of the surrounding polarized crystal. One alternative in the case of an ionic solid is
to consider the embedding crystal as built from discrete polarizabilities and embed
the cluster in a set of point polarizabilities. Then one may invoke the so-called Direct
Reaction Field (DRF) method [55–59] in order to solve self-consistently the problem
of a cluster embedded in a set of polarizable entities. This method has been applied
to the solute-solvent interactions [58, 59] and to incorporate the long-range polarization effects on the computed ionization energies and electron affinities in NiO [44].
Janssen and Nieuwpoort have studied the band gap of NiO, i.e. the related ionization
energies and electron affinities, by employing a [NiO6 ] embedded cluster [44]. Their
study has shown that the 3d ionization energy at the Ni2+ site is strongly dependent
of the bulk polarization. We come back to this issue in Chapter 7, where we study
the lowest ionization energies at the Ni2+ and O2− sites in NiO.
The cluster model is particularly well suited for the studies of the local electronic
structure of defects, local excitations and impurities in crystals [31, 33, 64, 87–89]
as well as localized electronic states in transition metal materials (see e.g. [2–4, 44]).
Employing the periodic approach for isolated defects and impurities in infinite lattices
requires the introduction of a large unit cell to minimize the possible interaction of the
defects localized at two neighbouring unit cells. Neutral defects in bulk LiF have been
studied by Nada et al using an ab initio Hartree-Fock perturbed-cluster embedding
scheme [51]. This study has shown that even in the case of neutral defects, important
short-range polarization effects occur.
2.2
Wave function based theoretical approaches
Solving the time-independent Schrödinger equation for a N - electron system in the
non-relativistic approximation is a starting point in most quantum mechanical approaches. This equation reads,
ĤΨ(x1 , x2 , ..., xN ) = EΨ(x1 , x2 , ..., xN )
(2.13)
where Ψ(x1 , x2 , ..., xN ) is the exact many-electron wave function, E is the associated
total electronic energy and Ĥ is the Hamiltonian operator. Ψ(x1 , x2 , ..., xN ) is a
function of the space and spin electron coordinates, unified in the x-variables. Solving
this equation, is in practice, possible only for few simple one-electron systems, such
as the H atom and the H+
2 molecule- ion. This is so because the wave function
Ψ(x1 , x2 , ..., xN ) describes the complicated correlated motion of all N electrons. Thus,
approximations to the solutions of the Schrödinger equation are needed.
In the following sections we will use the second quantization formalism.
The Schrödinger equation in (2.13) in the Fock space is given as,
Ĥ|Ψi = E|Ψi
(2.14)
30
Chapter 2, Theoretical framework.
The most common approach to an approximate solution of the Schrödinger equation
is to expand the approximate N- electron wave function |Φi as a linear combination
of Slater determinants (SD), |ii, where the latter are built as N-electron normalized
antisymmetrized products within a set of orthonormal spin-orbitals φpσ ,
|Φi =
X
Ci |ii
(2.15)
i
Provided that we work in the limit of a complete set of spin-orbitals we may approach
the exact solution of the Schrödinger equation close enough, at the cost of an infinite
determinantal expansion.
In the Born-Oppenheimer approximation and in the absence of external fields, the
spin-free non-relativistic Hamiltonian is expressed as a sum of one- and two-electron
operators, ĥ and ĝ and a nuclear-nuclear repulsion term hnuc . The operators are
expressed in terms of the set of the orthonormal spin-orbitals φpσ [65],
Ĥ =
X
hpq Êpq +
pq
|
{z
1X
gpqrs (Êpq Êrs − δqr Êps ) +hnuc
2 pqrs
|
{z
}
(2.16)
epqrs
}
|
ĥ
{z
ĝ
}
where
epqrs
=
X
=
X
â†pσ â†rτ âsτ âqσ
στ
Êpq
â†pσ âqσ
(2.17)
σ
are the two-electron excitation operator and one-electron excitation operator, respectively. The one- and two-electron integrals are given here as,
Z
X Zµ
1
hpq =
φp (r)(− ∇2 −
)φq (r)dr
2
rµ
µ
Z Z ∗
φp (r1 )φ∗r (r2 )φq (r1 )φs (r2 )
gpqrs =
dr1 dr2
(2.18)
|r1 − r2 |
The integration is only over the spatial coordinates because the orthogonality of the
spin functions in the spin-orbitals leads to a cancelation of most of the terms in
the initial expression for the matrix elements of the spin-free two-electron operator
gpσ,qτ,rµ,sν In addition, the summations over the spin variables in Eq. (2.17) remove
the spin-index dependences of ĥ and ĝ.
In practice it is not possible to work with a complete set of (one-electron wave
functions) spin-orbitals and to construct a wave function expansion in a complete
(infinite) set of SDs. Therefore different methods have been developed to provide an
approximate solution to the Schrödinger equation within a finite one-electron basis
set and a truncated SD expansion.
CSFs
31
Within the Hartree-Fock (HF) approximation [61], the N-electron wave function
is approximated by a single SD (unrestricted HF) or a single spin- and symmetryadapted configuration state function (CSF). The HF wave function and its energy are
obtained by employing the variational theory in order to minimize the expectation
value of the Hamiltonian with respect to the spin-orbitals. The one-electron wave
functions φpσ are usually expanded into a discrete set of basis functions which leads
to so-called Hartree-Fock-Roothaan equations [62]. For further discussion on the
(restricted and unrestricted) Hartree-Fock approach the reader is referred to [46, 65,
66].
Configuration State Functions
The exact many-electron wave function is considered to be an eigenfunction of the
non-relativistic and spin-free electronic Hamiltonian within the Born-Oppenheimer
approximation. Ĥ commutes with the spin operators Ŝ 2 and Ŝz and hence, the
many-electron wave function must be also a spin eigenfunction with quantum numbers
S and M. When searching for an approximation to this exact wave function, most
often one carries out the optimization in a restricted space, which consists of spin
eigenfunctions with quantum numbers S and M. The Slater determinants are not
in general eigenfunctions of Ŝ 2 though, it is possible to generate eigenfunctions by
determining linear combinations of Slater determinants. Likewise, one sometimes
needs to determine linear combinations of SD in order to let them transform according
to one of the irreducible representations of the symmetry group. The CSF is a spinand symmetry- adapted linear combination of Slater determinants, i.e. eigenfunction
of the operators Ŝ 2 and Ŝz and transforming according to one of the irreducible
representations of the symmetry group of the system. In constructing the CSFs,
determinants which belong to the same orbital configuration are combined. An orbital
configuration contains all SDs with the same occupation number vectors of the orbital
basis they are built of.
2.2.1
Multiconfigurational Self-Consistent Field Approach and
Configuration Interaction
The presence of several important electronic configurations in the wave function expansion is a common case for the ground and excited states of transition metal oxides. The ground state of the ozone molecule and the dissociation of the nitrogen
molecule are other representative examples. The single-configuration Hartree-Fock
(HF) method [60–62] is by construction unable to describe multi-configurational electron systems. Perturbation methods as second- or higher-order Møller-Plesset theory
are designed to improve the description based on HF theory. This approach allows for
obtaining size-extensive correlation energies when the system is represented mainly
by a single electronic configuration. The orbitals generated self-consistently within
HF theory may be inappropriate for the MC system. The MCSCF method [63] then
provides a natural solution to the problems.
32
Chapter 2, Theoretical framework.
Within the framework of the MCSCF method the wave function is written as a
linear combination of Slater determinants (SD) |ii or spin- and symmetry-adapted
linear combinations thereof, denoted configuration state functions (CSFs) |ji,
X
X
|κ, Ci = e−κ̂
Ci |ii = e−κ̂
Cj |ji
(2.19)
i
j
In order to construct the wave function, one optimizes variationally the expectation
value of the energy
E = minκ,C
hκ, C|Ĥ|κ, Ci
hκ, C|κ, Ci
(2.20)
with respect to the parameters of the orbital-rotation operator e−κ̂ and the CI coefficients Ci or Cj . In practice, the parametrization of the MCSCF model Eq. (2.19)
is the same as that of the CI model, except for the presence of the orbital-rotation
operator. The CI wave function is constructed as a linear combination of SD or CSFs
and the coefficients in the CI expansion are determined by a variational optimization
of the expectation value of the energy but only with respect to Ci or Cj . The presence
of e−κ̂ in the MCSCF wave function allows for obtaining orbitals adapted to the CI
expansion. MCSCF may be also used as a generator of a reference wave function
for more elaborate electronic structure calculations [65]. If the CI expansions must
contain more than a single CSF for the proper description of the wave function, the
non-dynamical (static) electron correlation effects are considered to be significant.
In the case of the spin-free non-relativistic Hamiltonian which commutes with the
total and projected spin operators Ŝ 2 and Ŝz , one usually requires wave functions
with well defined spin quantum numbers.
The optimization of the expansion coefficients of the CSFs |ji in the MCSCF wave
function can be performed by using aPconfiguration vector which is orthogonal to the
(0)
MCSCF initial reference state |0i = j Cj |ji [65],
|0i + P̂ |ci
|Ci = e−κ̂ q
1 + hc|P̂ |ci
(2.21)
P
with a projection operator P̂ = 1 − |0ih0| and the state |ci = i cj |ji contains the
free parameters cj . |0i is the starting normalized approximation to the wave function
of an electronic state. Using this parametrization of the MCSCF wave function one
carries out the variations in the spin-orbital space and in the configuration space in
a SCF procedure to obtain the best approximation to the wave function and energy
of the electronic state. Efficient methods have been developed for the optimization of
the MCSCF wave function. One of the main schemes employed in the modern quantum mechanical packages is based on modifications of the full second-order NewtonRaphson method [65,66]. The second optimization approach is the so-called super-CI
approach [66, 72]. The Newton- Raphson method usually shows a fast convergence
of the optimization procedure provided that the initial wave function is a reasonable
CASSCF and RASSCF
33
start, however precautions must be taken to prevent from a convergence to a saddle
point or maxima. In the super-CI approach, each step in the optimization procedure
is faster, however more steps are needed to achieve a convergence. Owing to the
extensive description of both optimization schemes in textbooks as [66] we do not
discuss them here.
CASSCF and RASSCF
In this section we discuss a method for incorporation of the static electron correlation
or so called near-degeneracy correlation which is related to the presence of more than
one important CSFs in the wave function expansion of a particular state. This occurs because of the near-degenerate energies of two or more electronic configurations.
Different kinds of MCSCF wave functions may be constructed but the most used
ones are based on the Complete Active Space (CAS) SCF method [70]. Within this
method the (spin-) orbital space is partitioned into three subspaces: inactive, active
and secondary, or virtual. The orbitals which constitute the inactive space remain
doubly occupied in all configurations of the MC wave function expansion. The orbitals in the secondary or virtual subspace remain unoccupied in all configurations.
The active space contains a fixed number of electrons which are distributed over the
active orbitals in all possible ways, restricted by the spin and spatial symmetry of the
CASSCF wave function.
The Restricted Active Space (RAS) SCF method [71, 72] has been viewed as an
extension of the CASSCF method. The (spin-) orbital space contains again inactive
and secondary spaces, defined as within CASSCF, but in addition, the active space
is divided further into three subspaces RAS1, RAS2 and RAS3. In constructing the
different RAS subspaces, one imposes restrictions on the maximum number of holes
or electrons, allowed in the RAS1 and RAS3 sub-spaces, respectively. In the RAS2
space all possible configurations, arising from the distribution of the active electrons,
that are not in RAS1 and RAS3, over the RAS2 orbitals are included.
In the CASSCF approach one performs full CI within the active space, and thus,
the redundant active-active orbital rotations need not to be considered. The additional subdivision of the active space within RASSCF introduces active-active orbital
rotations between the three RAS spaces which are not redundant and thus, have to
be considered. An advantage of the RASSCF approach is the possibility to use more
active orbitals. The multi-reference (MR) singles and doubles (SD) wave function is
a RASSCF wave function, which has the CAS space as a reference space and at most
two holes and two electrons in RAS1 and RAS3, respectively. The RASSCF wave
functions are suited to incorporate not only the non-dynamical correlation effects but
also dynamical correlation effects.
Efficient methods have been developed for performing the RASSCF, RASCI and
CASSCF and CASCI calculations, based on SD-based CI algorithms [71].
34
2.2.2
Chapter 2, Theoretical framework.
Complete Active Space Second-Order Perturbation Theory
The dynamical correlation effects which are due to the two-electron interactions in the
cusp region (r12 ≈ 0) are often recovered in a post-MCSCF (or post- HF) calculation.
A common method used to treat those effects in the case of a MC reference wave
function is multi-reference CI (MRCI). The MRCI wave function is a CI expansion of
all excited configurations of a certain order (singly-, doubly-, triply- and so on excited
configurations) generated from a set of MC reference wave functions. An alternative
to MRCI is the many-body perturbation theory, (MBPT), which main advantage is
the smaller computational expense.
The most often used MBPT for the calculation of the dynamical correlation energy
is the second-order MBPT. We use in the current research a second-order perturbation
approach to the dynamical electron correlation based on a reference CASSCF [72]
wave function as it has been developed by Andersson, Malmqvist and Roos [81].
This method aims at the calculation of the second-order estimate of the dynamical
correlation energy and the first-order estimate of the wave function. The starting
point in this theory is to define an appropriate zero-order Hamiltonian operator and
a configuration space to expand the first-order wave function, so-called first-order
interaction space.
If one extends the one-electron Fock operator fˆ which is given in the orbital
basis within the Møller -Plesset second-order perturbation theory to the CASSCF
reference wave function, |0i, one obtains the following form of fˆ [65,81], the CASSCF
Fock operator,
fˆ =
X
pq
fpq Êpq =
1 XX
h0|[â†qσ , [âpσ , Ĥ]]+ |0iÊpq
2 pq σ
(2.22)
In case the reference CASSCF wave function is reduced to the closed-shell HF wave
function, the CASSCF Fock operator reduces to the Fock operator in the Møller Plesset theory. The CASSCF wave function is not an eigenfunction of the CASSCF
Fock operator, but one can still construct a zero-order operator Ĥ0 , which has the
reference, zero-order, CASSCF wave function as an eigenfunction. This can be done
by using a projection operator onto the reference wave function |0i, P̂0 =|0ih0|, and
orthogonal projection operators P̂K , P̂SD , P̂X , onto the rest of the configuration
space. Ĥ0 has the form [65, 81],
Ĥ0 = P̂0 fˆP̂0 + P̂K fˆP̂K + P̂SD fˆP̂SD + P̂X fˆP̂X
(2.23)
The first-order wave function |01 i is constructed using a so-called internally contracted scheme as a linear combination of all CSFs connected to all single and double
excitations from the reference CASSCF wave function |0i [65],
|01 i =
X
pq
1
Cpq
Êpq |0i +
X
pq>rs
1
Cpqrs
(Êpq Êrs − δqr Êps )|0i
(2.24)
CASPT2
35
where the Êpq are one-electron excitation operators. The use of the internally contracted scheme in the construction of |01 i ensures a rather limited number of terms
in the perturbation series.
On the other hand the internally contracted scheme leads to a rather complicated
formalism which involves the construction of three-particle density matrices over the
active space, needed in the evaluation of the second-order CASPT energy and the
coefficients of the first-order CASPT wave function correction [65]. An alternative to
the internally contracted scheme is to express the first-order wave function in terms
of Slater determinants. However, this leads to computationally demanding very long
perturbation series.
For further details on the CASPT2 formalism as well as on the use of different
forms of the zero-order Hamiltonian in CASPT2 the reader is referred to [65, 83].
Finally, we comment on the breakdown of the CASPT2 approach in case of the
presence of configurations in C1 with expectation values of Ĥ 0 that are very close
to or even lower than the expectation value of the reference wave function. If we
write the first-order wave function and second-order correction to the energy in the
following forms, disregarding first the correction µ,
Ẽ 2 = −
X |hj|Ĥ (1) |0i|2
j − E (0) + (µ)
j
C˜j1 = −
Ĥ = Ĥ 0 + λĤ (1)
X
j
j
hj|Ĥ (1) |0i
− E (0) + (µ)
the energy denominators of E 2 and Cj1 become very small when the expectation
values of a certain |ji and |0i, j and E (0) become close. This causes a breakdown
of the perturbation theory. As shown by de Graaf et al. in extensive studies of
the low-energy electronic excitations, magnetic couplings and core-hole excitations
in insulating 3d TM oxides [2, 10, 11, 28] the TM oxides are very susceptible to this
breakdown. In the case of non Fock-type zero-order Hamiltonians, Ĥ 0 , so called
intruder states may be contained in the configuration space of the first-order wave
functions. When the interaction matrix elements between the intruder state and
the reference wave function, in the expression for the first-order wave function, are
large, the best solution is to extend the reference wave function so to include the
intruder state. In the cases of small interaction matrix elements but conserved (near -)
degeneracy between the intruder and reference state, one may apply the so-called levelshift technique, developed by Roos and Andersson [84]. The energy of the intruder
state and all other configurations in the first-order interaction space is shifted up
artificially by adding a value µ to the expectation value of Ĥ 0 of this intruder state
(see Eq. (2.25)),
E 2 − Ẽ 2 = µ(1 −
1
)
w̃
(2.25)
where w̃ is the weight of the CASSCF wave function in the first-order wave function
after applying the level shift µ (see [84]). The first-order wave function looses its
36
Chapter 2, Theoretical framework.
significance after applying the correction shift, but the second-order energy can be
corrected back for the applied shift (see [84]), More details on the use of the technique
can be found, for example, in references [84]. In Chapter 6 we come back to this issue
in the context of the study of the lowest valence excitations in cubic and orthorhombic
LaMnO3 where we discuss as well a new level shift technique denoted as imaginary
level shift technique [85].
2.3
State Interaction
The CASSI and RASSI [73] methods enable an efficient calculation of matrix elements
of one- and two-electron operators and one- and two-electron transition density matrices, respectively for two CASSCF or RASSCF wave functions, each expressed in its
own optimized orbitals. The ground and excited electronic states of various molecular and crystal structures have been described with a good accuracy with CASSCF
and RASSCF wave functions (see e.g. [3, 5, 6, 16–18, 80]). Transition dipole moments
and oscillator strengths, on the other hand, involve two different electronic states.
Evaluating these quantities requires the computation of the matrix elements of oneelectron-operators, such as the dipole transition moments, between wave functions
expressed in different mutually non-orthogonal orbital basis. The computation of,
for example, oscillator strengths accurately requires high quality wave functions, but
those can be obtained in most cases best if the different wave functions, involved in the
transition, are expressed in their own orbital basis. If such wave functions are obtained
in separate CASSCF or RASSCF calculations, they are in general non-orthogonal and
interacting. Better wave functions are then obtained by solving the secular equations
within the space spanned by the non-orthogonal and interacting wave functions [73].
In order to do so, one must construct and evaluate the Hamiltonian matrix elements
and overlap integrals between those non-orthogonal wave functions.
To compute the matrix elements of the one-electron operators, relevant to the
transition strengths, as well as the Hamiltonian matrix elements and overlap integrals between the interacting non-orthogonal wave functions, Malmqvist developed
an efficient scheme based on non-unitary orbital transformations [75]. The development of the scheme was directed to the computation of the one- and two- electron
transition density matrices for two MC wave functions but it became an efficient and
powerful tool for the calculation of arbitrary one- and two-electron matrix elements
between wave functions expressed in different orbital basis, X and Y. Let us first
assume that the two sets are orthonormal but mutually non-orthogonal. The original
non-orthogonal MC wave functions are then given by the following CI expansions,
X
|CX i =
CiX |iX i
i
Y
|C i =
X
CiY |iY i
(2.26)
i
where |iX i and |iY i are the two different CSF bases, constructed from two different
orbital sets ϕX and ϕY . Both sets of CSF are constructed by the same spin-coupling
State Interaction
37
scheme which is otherwise arbitrary. The matrix elements of the one- (F̂ ) and twoelectron (Ĝ) operators, formed between |CX i and |CY i are [65, 66, 73–75]a :
hCX |F̂ |CY i =
X
CiX CjY
X
X ˆ Y
Aij
pq hϕp |f |ϕq i
pq
ij
1 X X Y X ij
A
hCX |Ĝ|CY i =
C C
hϕX ϕX |ĝ|ϕYq ϕYs i
2 ij i j pqrs pqrs p r
(2.27)
where
F̂ =
nelectrons
X
fˆ(k) =
X
pq
k
fpq â†p âq
|{z}
ˆ Y
fpq = hϕX
p |f |ϕq i
Êpq
Ĝ =
1X
2
k6=l
1X
ĝ(k, l) =
gpqrs
2 pqrs
â†p â†r âs âq
| {z }
gpqrs = hϕp ϕr |ĝ|ϕq ϕs i
Êpq Êrs −δqr Êps
Analogously, the overlap matrix is also obtained [73, 75],
hCX |CY i =
X
XY X Y
Sij
Ci Cj
(2.28)
ij
The coefficients A and S will be specified below. It has been shown by Malmqvist
[73, 75] for RASCI expansions that if the orbital overlap matrix is non-singular, other
orbital sets ϕA and ϕB can be obtained from the original orthonormal but mutually
nonorthogonal orbital sets ϕX and ϕY using non-unitary orbital transformations.
Provided that the non-singularity condition is imposed on the overlap matrix, two restricted orbital transformation matrices exist, ΥXA and ΥY B such that a biorthonormality condition between the transformed sets ϕA and ϕB holds,
B
hϕA
p |ϕq i = δpq
The transformation of the original orthonormal (individually) orbital sets ϕX and ϕY
to some new biorthonormal basis, ϕA and ϕB , is at the cost of loosing the individual
normalization of ϕA and ϕB . The transformation of the original orbital sets should
be accompanied by a corresponding transformation of the CI expansion coefficients
such that the new wave functions |CA i and |CB i are constructed from CSFs which
span the same space as those CSFs, expressed in the original orbital sets, ϕX and ϕY .
Provided that the biorthonormality condition holds for orbital sets ϕA and ϕB , the
ij
overlap matrix S AB becomes the unity matrix and the Aij
pq and Apqrs are the sparse
sets of one- and two-electron coupling coefficients, encountered in the conventional
case of wave functions expanded in a common set of CSFs, where the latter are
built from a common set of orthonormal orbitals (see [73] and [75]). The coupling
a In the following expressions we work with spin-free operators, provided that the trivial summation over the spin variables have been carried out properly
38
Chapter 2, Theoretical framework.
coefficients in the biorthonormal CSF basis have the form [75],
A
B
Aij
pq = hi |Êpq |j i
A
B B
B B
Aij
pqrs = hi |Êpq Êrs − δqr Êps |j i
Aij
pq are invariant under orbital transformations and take values 0 or ±1 (CAS and
RAS wave functions) as they do in the case of one orthonormal orbital basis for both
|CA i and |CB i. The one- and two-electron transition density matrices in biorthonormal orbital bases A and B have been expressed as the transition values of the spin-free
B
, and two-electron excitation operator êprqs , defined
singlet excitation operator Êpq
only in the basis B, respectively [75],
X
A
B
B
ΓAB
CiA CjB Aij
pq = hC |Êpq |C i =
pq
ij
ΓAB
pqrs
A
= hC
B B
|Êpr
Êqs
−
B
δrq Êps
|CB i
=
X
CiA CjB Aij
pqrs
ij
The biorthonormality of ϕA and ϕB allows one to use the same formulae as those
for the case of a single orthonormal orbital set for both wave functions. Thus the
calculation of the Hamiltonian matrix elements and overlap integrals, as well as the
matrix elements of various one-electron operators between non-orthogonal and interacting wave functions will be reduced to the calculation of the transition density
matrices within the biorthonormal basis sets ϕA and ϕB . This is naturally accompanied by a transformation of the very original orbital sets ϕX and ϕY into ϕA and
ϕB and the corresponding CI expansions CiX and CjY into CiA and CjB . The latter
transformation has been performed using the orbital transformation matrices ΥXA
and ΥY B [75]. The reader is referred to reference [75] for the detailed derivation of
these matrices. In order to obtain proper orbital transformation matrices that preserve the CSF spaces of the CI expansions, one may perform an LU -factorization of
S XY [73, 75]. In addition one needs to restrict the wave functions to RAS (CAS)
expansions.
Once obtained, ΥXA and ΥY B are LU -factorized and the CI coefficients CiX and
Y
Cj are transformed into the new CI coefficients CiA and CjB in a sequence of singleorbital transformations using the L and R−1 matrices of the LU -factorization and
the one-electron coupling coefficients (see above). Next the transition density matrix
ij
elements are calculated using CiA and CjB and Aij
pq and Apqrs . Finally the matrix
X ˆ Y
X
Y Y
elements of one- and two-electron integrals (hϕp |f |ϕq i and hϕX
p ϕr |ĝ|ϕq ϕs i) are
A A
B B
ˆ B
also transformed into the new basis set (hϕA
p |f |ϕq i and hϕp ϕr |ĝ|ϕq ϕs i) and the
corresponding matrix elements of the one- and two-electron operators between the
transformed wave functions, |CA i and |CB i, are computed. This algorithm has been
implemented in the quantum chemistry package MOLCAS [86] in a State Interaction code. The Hamiltonian matrix elements and overlap integrals are built between
the original individually optimized RASSCF or CASSCF wave functions and the
resulting secular equations are solved to yield a better set of non-interacting and orthonormal eigenfunctions expressed in terms of the original interacting and mutually
State Interaction
39
non-orthogonal wave functions,
|Ci =
X
X
ΛX
m |Cm i
(2.29)
m
Since the states are described by MCSCF wave functions, |CX
m i are many-electron
basis functions consisting of more than one CSF, hence the approach is denoted as
a state interaction (SI). In this context we mention another analogous approach,
known as non-orthogonal CI (NOCI) [76, 77]. NOCI is based on expressing the final
wave function as an optimized linear combination of non-orthogonal interacting wave
functions, which are single CSFs. The expansion coefficients, Λm , of the NOCI wave
function are obtained from solving the secular equations with the non-orthogonal
single CSFs as a many-electron basis. The Hamiltonian matrix elements and overlap
integrals between the non-orthogonal NOCI single CSF are evaluated by making use
of an efficient scheme, named the Factorized Cofactor Method [76, 78]. In this thesis,
we make an extensive use of the SI approach in order to obtain the Hamiltonian
matrix elements and overlap integrals between localized individually optimized excited
(ionized, added-electron) CASCI wave functions of large embedded clusters (superclusters). These matrix elements are needed, for example, for the computation of
double exchange parameters in manganites (see Chapter 4 ). Furthermore the matrix
elements of the super-clusters are used to approximate the matrix elements of an
extended system and then, employing those approximate matrix elements, to solve a
non-orthogonal many-body tight-binding problem in order to obtain the eigenvalues
and eigenvecotrs for the extended system (see Chapters 3, 5 and 7 ).
40
Chapter 2, Theoretical framework.
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Dijkman, H. Merenga, and B. T. Thole, J. Comp. Chem. 16, 37 (1995)
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[65] T. Helgaker, P. Jørgensen and J. Olsen Molecular Electronic-Structure Theory
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[69] C. Kittel, Introduction to Solid State Physics, edition seventh,1996;
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46
BIBLIOGRAPHY
3
Delocalized and Correlated
Wavefunctions for Excited States in
Extended Systems
ABSTRACT — In
this Chapter, an efficient method for the generation of
delocalized many-electron wave functions for excited states in strongly correlated
condensed matter is introduced. These wave functions are obtained within a
nonorthogonal tight-binding scheme. The trial functions in the calculation are
constructed as linear combinations of antisymmetrized products of localized and
correlated multiconfigurational cluster wave functions and wave functions for
the rest of the extended system. The multiconfigurational wave functions are
expressed in orbital sets that are obtained from calculations on small fragments.
Each fragment is built around a particular ion (or bond) and embedded in effective model potentials, representing the rest of the crystal. To compute the
Hamiltonian and overlap matrix elements between localized excited states at two
different ions (or bonds), large embedded clusters are employed. This approach
is applicable to structures where the many-electron effects are non-negligible and
conventional one electron band structure calculations lead to unphysical results.
3.1
Introduction
The electronic structure and properties of solids have been subject of intense theoretical and experimental investigations for a long time. This thesis is concerned with
the first principles description of the electronic structure of compounds that exhibit
strong electron correlation effects. The transition metal compounds are representative for this class of compounds. Different theoretical approaches emerged within the
domain of the quantum chemistry as well as within solid state physics. The methods
stemmed from the first domain are molecular quantum mechanical methods, that aim
48
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
to describe the properties of the system by deriving its many-electron wave functions.
The second domain, based on the Density Functional Theory (DFT) [1], emerged
later and offered an elegant alternative way to probe the ground and recently also
the excited state properties of solids. The first line of research is directly related and
relevant to the line of research in this thesis. In the following paragraphs, we discuss
in somewhat more detail some contributions, relevant to the studies on the transition
metal compounds. Next, some new tendencies in the development of wave functionbased methods for band structure calculations in solids are acknowledged. Finally, we
consider shortly also the domain based on the DFT formalism and its time-dependent
extension.
The most straightforward description of the electronic structure of solids rests
on the effective one-electron band structure theory [2]. The concept of considering
the state of an electron in a periodic effective potential, generated from the nuclei
and the rest of the electrons in the crystal is attractive and despite the conceptual
simplicity of this approximation, it has been successful in describing the properties
of many covalent and ionic compounds. The band structure method, combined with
the DFT formalism or its time-dependent extension is a conventional approach to
the description of the complex phenomena in the solid state physics. However, the
competition between the delocalization, band-like, effects and the strong (atomic-like)
electron-electron interactions in, for example, 3d-metal compounds leads to a failure
of the independent electron approximation. The strong electron-electron interactions
in these compounds usually lead to electron localization effects that consequently give
rise, for example, to local magnetic moments and Mott-Hubbard type metal-insulator
transitions.
Contrary to the effective one-electron approaches, the correlated many-electron
wave function based methods offer the ability to address explicitly and in a systematic manner the electron-electron interactions. Obtaining the many-electron wave
functions in this field is viewed as an efficient approach to probe the electron correlation effects in the systems. These methods are as well particularly well suited to
study the character of the excited states in a system, whereas the time-dependent
DFT (TDDFT), for example, accesses only the excitation energies and transition
dipole moments. Within the TDDFT formalism, one can obtain the correct excitation energy for a transition that involves a ground and excited state by using the
linear response of the system to an external perturbing electromagnetic field [46, 47].
The corresponding Kohn-Sham (KS) Slater determinants, however, are auxiliary determinants and do not describe the true wave functions of the ground and excited
states. Constructing their proper spin eigenfunctions is also beyond the limits of the
KS scheme. Within a wave function based method, the spin eigenfunctions for any
spin can be rigorously defined. The complicated character of the excited states can
be described reasonably by means of a MC computational algorithm. The wave functions are obtained by approximately solving the Schrödinger equation within a given
one-electron and many-electron basis set. They can be systematically improved by
enlarging the wave function expansion or the one-electron basis set. However, for most
of the systems, the number of terms in the MC expansions cannot grow up till the
Introduction
49
full configurational interaction limit. For some systems it is even impossible to reach
the length, that permits one to describe their properties with a satisfactory accuracy.
In case of solids, the wave function based methods face two main problems, one is
related to the difficulty to implement periodic symmetry in the MC expansion and
the other is that the ab initio calculation of the wave function scales unfavorable with
the size of the system. Therefore, treating the electron correlation effects explicitly in
extended systems with periodic symmetry by means of a wave function based method
was computationally not feasible for a long time. Recently, some new algorithms to
address the electron correlation explicitly in solids were introduced [42, 51–53]. We
discuss them below.
In Chapter 2, we introduced the embedded cluster method as a local approach to
describe processes with a predominantly localized character that are relevant for the
solid state physics. Combined with an accurate ab initio wave function based method
such as the MCSCF method [3, 4, 7], followed by CASPT2 [5], the local description
provides an insight into the large electron correlation and electronic relaxation effects
accompanying localized excitations, ionization and electron addition processes [8–11].
The proper treatment of the many-electron problem for a relevant portion of the extended system by means, for example, of CASSCF approach followed by CASPT2 allows for an accurate account for the non-dynamical and dynamical electron correlation
effects, that govern the physics of many insulating transition metal (TM) materials.
These techniques have permitted for the precise description of core level ionization
processes in transition metal oxides [12,15,16]. Using the CASSCF/CASPT2 method,
de Graaf studied in our laboratory successfully the different contributions to the electron correlation effects on the d-d excitations in NiO [14]. This technique was applied
recently by Hozoi [19] to study the Mn d → d and Mn 1s excited states in LaMnO3 ,
relevant for the interpretation of pre-edge features in the x-ray Mn K -edge absorption
and the anomalous Mn K -edge scattering in this compound. The potentials of the
CASSCF/CASPT2 approach to describe satisfactory magnetic couplings in different
TM materials have been explored in a number of studies, that aimed the accurate
computation of the exchange magnetic coupling constants in some TM oxides and
cuprates [25, 26, 29]. The CASSCF/CASPT2 approach has been successfully applied
by Roos and co-workers to obtain the spectral properties, molecular structures and
binding energies of TM complexes, such as Ni(CO)4 , Fe(CO)5 and Cr(CO)6 [13].
In spite of its local nature, the embedded cluster approach, combined with an ab
initio wave function based method has been also applied to compute hopping integrals
between localized states [21, 22]. A common approach applied to the physics of the
magnetic couplings is based on the Heisenberg model [30]. The effective parameters
in this model, exchange constants J and hopping integrals t have been successfully
derived within the local cluster approach [21, 22, 25, 26, 29]. The embedded cluster
approach, combined with a CI [33] treatment of the non-dynamical electron correlation
effects has been employed also by Janssen and Nieuwpoort in their study of the band
gap in NiO [31].
The question whether or not the treatment of the electronic properties of solids
can be based on a localized model (for example the embedded cluster model) or on a
50
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
delocalized (band) model depends on the system and property to be studied. The local
approach is an essential starting point in the treatment of excited and ionized states
when the strong on-site electron-electron interactions dominate in the system and give
rise to large local many-electron effects, such as electronic relaxation and correlation.
If these effects are larger than the inter-site interactions (delocalization), then the
best description is obtained if one considers first those effects and then restores the
translational symmetry of the system. This issue has been addressed by Nieuwpoort
and Broer in their review paper on the application of the cluster approaches to solid
state problems [34]. It has been shown that the electronic ground state of ionic
solids, built from closed-shell ions, such as NaF and CuCl are described identically
by the cluster model and periodic approaches both at the Hartree-Fock and DFT
level [34]. This is explained by considering the orbitals derived from a sufficiently large
embedded cluster calculation as being equivalent to Wannier functions. A unitary
transformation brings these Wannier functions into delocalized Bloch functions. This
consideration connects the embedded cluster approach and the band theory [34]. The
authors in reference [34] however also demonstrate, using as an example the Cu-3d
ionization in CuCl, that in the delocalized band description of the ionization process,
the large electronic relaxation effects, accompanying the process, can not be accounted
for. Such effects are crucial for the properties connected with the 3d states in TM
materials.
Whether or not the localized excitations for a system with few geometrically equivalent sites lead to a better description, compared to that obtained employing symmetry adapted one-electron approach has been extensively studied by Broer [35]. The
author used the concept of symmetry breaking that leads to symmetry unrestricted
localized states. The ratio between a polarization or relaxation parameter β and a
delocalization parameter b, introduced in this approach, determines whether the localized solutions or the delocalized symmetry adapted ones are energetically favorable in
a one-electron model [35]. Conventional approaches, based on the band theory or approaches that yield symmetry adapted wave functions face restrictions with respect
to their applicability to systems with predominant relaxation effects. Proper delocalized wave functions can be constructed afterwards from those symmetry broken
wave functions by applying group theoretical projection operators [36]. Obtaining
the energies and other electronic properties for the projected wave functions requires
computation of Hamiltonian matrix elements between determinants constructed from
mutually non-orthogonal orbital sets, that is not a trivial task. The matrix elements
between the non-orthogonal Slater determinants have been efficiently calculated by
using the factorized cofactor method, introduced by Van Montfort [37].
Recently, Shukla et al. [42] have suggested a wave function-based approach to correlated ab initio calculations on crystalline insulators of infinite extent. The method is
developed within the localized-orbital-based approaches and makes use of the so-called
incremental scheme [56]. Compared to some previous correlation calculations [43]
where the infinite crystal is modeled by a finite cluster, this new approach accounts
for the infinite nature of the system. The incremental scheme allows one to expand
the total correlation energy per unit cell of a solid in terms of interactions among
Introduction
51
the electrons assigned to localized orbitals. The correlation calculations are done employing a wave function based-approach without truncating the infinite system into
a finite cluster. Instead, their starting point is to represent the RHF ground state
of the solid in terms of Wannier functions and next, to choose a region of space in
which they calculate the correlation effects. They represent the infinite region outside the correlation region by frozen wave functions that allows them to sum up their
contribution and construct an effective one-electron potential. Their correlation calculation involves excitations from the RHF occupied space (Wannier functions) of
the correlation region to the virtual space in the presence of the frozen environment,
represented by this one-electron potential. At this point, the system Hamiltonian is
reduced to an effective Hamiltonian for the electrons located in the correlation region.
The correlation contribution to the total energy per unit cell is calculated within the
incremental expansion [42, 56],
4E =
X
i
i +
1X
1 X
4ij +
4ijk + ...,
2
3!
i6=j
(3.1)
i6=j6=k
where the summation i runs over the Wannier functions in the reference cell and j
and k run over all Wannier functions of the crystal. The authors define different
increments in this summation of increasing complexity. For example, the one-body
increment i is obtained if one considers the excitations only from the ith Wannier
function and the rest of the solid is described at HF level. Furthermore, the two-body
increment 4ij = ij − i − i includes the correlation energy obtained when one
correlates two distinct i and j Wannier functions. Employing this approach, Shukla
et al [42] obtained lattice constant, cohesive energy and bulk modulus of LiH, that
compared well with the experiment.
In a detailed review article on wave function methods in the electronic structure
theory of solids, Fulde [57] demonstrates within the method of increments that the
correlation energy can be expressed in terms of cumulants of the Hamiltonian operator
and the cumulant scattering operator which is essentially the same conclusion as that
of Shukla et al. This line of development resulted most recently into a study on
the accuracy of the incremental expansions for the correlation energy as well as the
convergence of different incremental expansions for solids with ionic, Van der Waals
or covalent bonding types [51].
Furthermore, the local incremental scheme has been successfully employed in studies of the valence band structure of covalent semiconductors [58,59]. Gräfenstein, Stoll
and Fulde [59] designed a method to determine the correlated valence band structure
of covalent semiconductors in which the band energies are expressed in terms of Hamiltonian matrix elements between many-electron states representing excited hole states
in an extended system. Within this method localized hole states are considered which
are described by localized many-electron wave functions. As a consequence, the band
energies are expressed in terms of the Hamiltonian matrix elements between those
localized states. In that manner, the calculation of the band energies is reduced to
the calculation of a smaller number of local Hamiltonian matrix elements. In order
to introduce the electron correlation effects, the authors determine the local matrix
52
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
elements from ab initio wave function based calculations, for example a multireference
CI calculation, on finite clusters [58] or a set of comparably small molecules [59]. The
formalism is in principle also applicable for obtaining the correlated conduction band
and thus the band gap of the system.
This formalism was extended by Albrecht and co- workers [60] to yield not only the
relative energies of valence band states but also the absolute energies. Extension of
the formalism to conduction band states has not been feasible because the localized
Hamiltonian matrix elements between states with an extra electron could not be
extracted from finite cluster calculations. For diamond and silicon, those calculations
did not lead to stable states for the extra electron [60]. Nevertheless, the authors
obtained information on the correlation-induced shift of the absolute position of the
upper valence band edge that allowed them to estimate the correlation effects on the
band gaps of diamond and silicon.
Recently, a new local Hamiltonian method by Bezugly and Birkenheuer [65] emerged
that stemmed from the approach to the valence bands, discussed above [58–60]. The
new method overcomes the difficulties to treat the conduction bands and allows one
to predict band gaps of non-conducting systems. The computational scheme makes
use of a multireference CI method with single and double excitations with a special size-consistency correction [66]. The determination of the delocalized electronic
Bloch states in a crystal is reformulated into the determination of localized orbitals in
the real space (Wannier representation). Within the local Hamiltonian approach, the
authors establish one-to-one mapping between the excited (hole or electron) state correlated wave functions and so-called localized one-particle configurations [65]. These
one-particle configurations are introduced by using Wannier orbitals. This results
in a local representation of an effective Hamiltonian whose matrix elements can be
evaluated in finite clusters of the periodic system [65]. For the hole states for covalent
semiconductors like diamond and silicon [60] (valence band states), simple hydrogen
or pseudo-potential terminated clusters are adequate to model the hole hopping. However, the anion clusters are electronically unstable. This electronic instability stems
from the fact that due to the negative extra charge, the added electron is weakly
bound to such a cluster and may ’leave’ it unless special techniques are employed to
stabilize the anionic cluster [65]. Thus, to treat the conduction bands, Willnauer and
co-workers [67] introduce special embedding techniques, denoted as cluster-into-solid
embedding schemes which cure the lack of electronic stability of anionic clusters. Using those techniques, Willnauer and co-workers perform calculations of the correlated
conduction bands of diamond in which the clusters are surrounded by an infinite host
crystal [67].
Another problem that arises when treating conduction bands within this approach
is obtaining properly localized virtual orbitals. The virtual orbitals are hard to localize
with the standard orbital localization schemes, such as those based on an extension of
the Foster-Boys localization criterion [68] for periodic systems [69]. Recently, Birkenheuer et al. [70] introduced a new localization scheme which is an extension of the
Wannier-Boys localization scheme [69].
Most recently Birkenheuer, Fulde and Stoll [71] presented an alternative scheme
Introduction
53
for calculating energy bands which is based on a quasiparticle approach. The authors
evaluate the relaxation and polarization effects around an extra electron residing in a
conduction band Wannier orbital, by allowing the valence orbitals to relax in a SCF
calculation due to the presence of the extra electron. In their studies of diamond and
silicon conduction bands, they find that the relaxation and polarization contributions
to the diagonal matrix elements of the local Hamiltonian lead to a shift of the center of
gravity of the conduction band while the off-diagonal matrix elements are responsible
for the reduction of the conduction band width [71].
Among the recently developed approaches to the N+1 and N-1 states of the crystal,
one needs to acknowledge as well those which employ a Green’s function formalism
[61, 62]. Here one introduces the correlation effects in the band calculations through
the self-energy that enters the Dyson equation [61,62]. The correlated band structure
is given by the poles of the space-time Fourier transformed Green’s function which
obeys in the reciprocal and frequency space the Dyson equation. Using local Wanniertype orbitals as a starting point, Igarashi [63, 64] developed a local Green’s function
approach using a model Hamiltonian within a space of so called three-particle states.
This local three-body correlation theory includes intra-atomic interactions. It has
been successfully applied to the 3d states of ferromagnetic nickel [63] and in studies
of the excitation spectra of the transition-metal oxides MnO, FeO, CoO and NiO [64].
Albrecht et al. [62] extended the procedure to the ab initio case. They introduced
a local-orbital based ab initio approach for the calculation of correlated valence and
conduction bands in insulating solids.
Most recently, Buth and co-workers [76] suggested a new general local orbital ab
initio Green’s function method for crystals. The band structure is given by the poles
of the one-particle Green’s function in terms of Bloch orbitals, which is evaluated
using the Dyson’s equation. A key advantage of the method is the fact that the
self-energy itself is evaluated in terms of Wannier orbitals and then transformed to
a crystal momentum representation. In this manner, the authors exploit the fact
that the electron correlations are predominantly local. The method is considered to
be applicable not only to the outer valence and conduction bands but also to inner
valence bands [76].
Pisani and co-workers [73] developed a second-order local- Møller-Plesset (MP2)
electron correlation method for non-conducting crystals. This post-Hartree-Fock approach makes use of the well developed Hartree-Fock method for periodic systems [74].
The authors exploit local correlation linear scaling techniques to account for the electron correlation effects. The translational symmetry facilitates the reduction of the
crystalline problem to a problem considering a cluster around a reference unit cell [73].
The method is implemented in a periodic local correlation code CRYSCOR as the
post-Hatree-Fock option of the CRYSTAL program [75].
Recently, Fink [72] developed a new wave function based approach to excited
states in periodic systems. The starting point of this method is a periodic HartreeFock calculation to obtain a reference wave function, for the following valence CI
calculation. The reference wave function is a Slater determinant expressed in terms
of localized orbitals. The author defines a space of active valence orbitals and a
54
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
number of active electrons for a reference unit cell and performs a full CI within this
space. The periodic CI method allows for including configurations which describe
simultaneous excitations in different unit cells as well as excitations between unit
cells.
This overview of newly developed wave function based approaches to systems with
strong electron correlation effects is by no means exhaustive. Apart from the methods
outlined above, which employ local Hamiltonian matrix elements in the real space
one can distinguish approaches which work directly in k space using, for example, an
atomic-orbital formulation of the MP2 theory to incorporate the electron correlation
effects in the calculation of quasiparticle band gaps in periodic systems [53]. Within
the framework of this MP2 theory, Ayala et al. [53] considered the MP2 quasiparticle
band gaps of different polyacetylenes.
Among the wave function-based approaches to excitations in infinite systems with
periodic symmetry, a valuable contribution is the dressed-cluster method, developed
by Malrieu and co-workers [55]. Within this method, the infinite periodic system is
approximated by a finite cluster in a self-consistent embedding. The authors perform
a CI for a finite cluster of atoms embedded in a periodic strongly localized zero-order
wave function. They exploit the resulting cluster wave function to account for the
effect of the elementary excitations occurring outside the cluster region through a
dressing of the cluster CI matrix [55].
Malrieu and co-workers [54] recently developed a method, based on the Direct
Space Renormalization Group (DSRG) approach. In this approach local parameters
are extracted from calculations on increasingly large clusters.
The development of different theoretical approaches based on DFT offered an
alternative way to probe the properties of solids. The DFT formalism combined with
the local density approximation (LDA) is widely applied in solid state physics. The
electron correlation effects are incorporated while preserving the effective one-electron
model. The exchange- correlation LDA functional is derived from considerations
on a homogenious electron gas. In spite of the local character of the functional,
the approach can handle rather inhomogenious electron systems. Although formally
restricted to the ground state properties of a system, it has been also employed in
studies of excited states. The TDDFT was developed by Runge and Gross [38] and
based upon the Hohenberg-Kohn theorem [39, 40] for the ground state energy of a
many-particle system. TDDFT extended the application of the theory to obtain the
exact dynamical response of the system to an external perturbing time-dependent
scalar potential [38, 41]. Further development of the theorem of Runge and Gross
[38] for a many-particle system in a general time-dependent electromagnetic field led
to the time-dependent-current density functional approach [44, 45]. This approach
is considered to be well suited to study the response of an extended system to an
external perturbation [46, 47]. The method was employed using the adiabatic LDA
for the exchange-correlation field [48, 50] and yielded optical dielectric functions of
various semiconductors and insulators [46, 47] that compared well with experiment.
However, treating excitonic effects in these systems requires, for example, the use
of orbital-dependent functionals [45]. DFT -based approaches are appealing, because
Introduction
55
they incorporate the many-particle effects (correlation) while maintaining the effective
one-electron description. Despite the fact that within the DFT-based formalism, one
avoids constructing explicitly the many-particle wave function of the system, one
needs a suitable approximation for the functionals in order to describe satisfactory
excitons in strongly correlated systems. Although the different approximations to
the form of the exact functional may yield accurate results for some properties of a
system, their applicability is still property dependent and in case of some strongly
correlated structure the known functionals experience significant failure.
Although recent developments in the TDDFT field have offered an attractive way
to probe the physical properties of the ground and excited states of the systems,
still a detailed insight in the electron correlations is gained by studying their manyelectron wave functions. It is well known, that the Kohn-Sham determinant is not
an approximation to the many-electron wave function but only a tool for reproducing
the correct density. Although the many-electron wave functions for the ground and
excited states as well as the excited-state energies are functionals of the ground-state
density, it is not possible to extract the many-electron wave functions from DFT.
In the following chapters, we aim to study excited, ionized or added-electron states
in extended systems with strong electron correlation effects. In a localized description,
one may observe a large change in the local electronic density accompanying the
excitation that is not present in an exact description of a periodic system. This effect
is a drawback of the local approach. On the other hand within a localized description
one can account for energy effects such as relaxation and correlation in a rigorous
manner. The wave functions describing excited states in a periodic system have to be
adapted to the translational symmetry of the crystal. A delocalized description yields
such properly translational-symmetry adapted wave functions but does not facilitate
the incorporation of the local relaxation and correlation effects. The delocalization
effects determine the width of the band associated to the excited state. If the band is
wide, the delocalization effects are large, in case of a narrow band the excitations may
be considered rather localized. The subject of this research are processes and systems
for which the local relaxation and correlation energies are larger than or comparable
to the delocalization energies. Thus, the best description is obtained if one accounts
firstly for the local effects and secondly for the translational symmetry of the system.
We designed a method that follows this order of considering the localization and
delocalization effects. The method goes beyond the local description and yields delocalized many-electron wave functions for excited (ionized, added-electron) states,
that are properly adapted to the crystal symmetry. Taking advantage of the rapid
decrease with distance of the interaction between the localized states, we employ a
non-orthogonal many-electron tight-binding scheme to obtain the delocalized wave
functions. These wave functions are expressed in terms of linear combinations of antisymmetrized wave function products of localized and correlated N-electron cluster’s
wave functions and a frozen ’embedding’ wave function, describing the other electrons
in the extended system. The crystal’s Hamiltonian and overlap matrix elements are
approximated by the Hamiltonian and overlap matrix elements of a large cluster. A
new method is presented, denoted as the Overlapping Fragments Approach (OFA),
56
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
for the computation of the Hamiltonian and overlap matrix elements of the large
cluster. In the following section, we introduce the approach, using as an objective an
electronically excited state.
3.2
3.2.1
Theory
Many-body bands
We treat the electronic excitations in extended systems within a new approach based
on expanding the wave functions of the extended system in terms of linear combinations of antisymmetrized products of correlated MC self-consistent field (SCF) cluster
wave functions and frozen ’embedding’ wave functions. These antisymmetrized wave
function products comprise the many-electron (ME) basis functions, which each represent an approximate, localized excited state in the extended system. We discuss in
detail below the case of an extended system with periodic symmetry. For comparison,
the many-electron crystal wave functions are described by single Slater determinants
within the Hartree-Fock band approach, where one makes use of the invariance of
the one-electron operators and the many-electron Hamiltonian under the symmetry
operators. The determinants are constructed from Bloch orbitals or in some cases,
when convenient, from their localized Wannier representation [2, 79]. The determinantal many-electron states can be labeled with a Bloch-vector which is a sum of all
individual momentum vectors of the Bloch orbitals, ki with i=1, M, where M→ ∞.
M is the number of electrons. The electron correlation effects can not be accounted
for by the single determinantal wave function. In the new approach, local electron
correlation effects are included by means of MC wave function expansions.
In the following paragraphs, we refer to the terms ion, center or site as to terms
with an equivalent meaning. The type of the site (ion, atom or bond) is not relevant
to the line of the derivations. It introduces later on some complications in employing
the model potential techniques. This does not change the validity of the theoretical
model, but requires adequate embedding representations. For simplicity, we consider
explicitly the case of a localized excited state centered on an ion.
Energies and Wavefunctions for crystalline systems
An essential part of the present work is to determine many-electron wave functions for
the ground state and electronically excited states of crystalline systems with periodic
symmetry. The wave functions for these states are expressed as linear combinations
of many-electron (ME) basis functions:
Ψn (r1 , r2 , ...., rM ) =
X
Cnλ Φλ (r1 , r2 , ...., rM )
(3.2)
λ
The basis contains apart from a ME basis function that represents a first approximation for the crystal ground state, Φ0 , a set of ME basis functions which each represent
an approximate, localized excited state, with the excitation centered around an atom
Many-body bands
57
or a bond in one particular unit cell of the crystal plus combinations of such excitations centered around different sites. In principle, the ME basis set is infinitely
large, but in practice it has manageable length, because only a limited number of
local excitations of each unit cell are included. Because this basis contains functions
representing locally excited states, we denote it a ”local” ME basis, in analogy with
the local one-electron basis sets that are sometimes used in one-electron band studies.
We note that, although the local excitations are centered at a particular unit cell,
the changes in electron density (with respect to that of the ground state) that are
associated with it are not confined to the unit cell, in fact, they are usually extended
far outside this unit cell. The ”ground state” basis function may usually be assumed
to be invariant under the symmetry operations of the crystal.
The best many-electron crystal wave functions or, in practice, the best expansion
coefficients Cnλ , are found by applying the variational principle: the variation of
the expectation value of the crystal Hamiltonian with respect to variations in these
coefficients is made zero. This leads to a set of secular equations to be solved and hence
to demanding the determinant of the secular matrix H-ES to be zero. The allowed
energy values E0 , E1 , ... are upper limits to the exact ground state energy, the energy
of the first excited state, ... and the corresponding wave functions Ψ0 (r1 , r2 , ...., rM ),
Ψ1 (r1 , r2 , ...., rM ), ... are approximations to the exact M-electron wave functions of
the crystal.
Local many-electron basis functions versus translation symmetry adapted
basis functions
The crystal can be regarded as a giant molecule. The Hamiltonian matrix H and
overlap matrix S in the secular determinant can then be expressed directly in terms
of the local ME basis mentioned above or, if the giant molecule has point group
symmetry, in point group symmetry adapted combinations thereof. This brute-force
giant-molecule approach gives rise to huge H and S matrices. It is feasible provided
that enough computer power is available and if that is the case it is easy to use.
On the other hand, if we are dealing with crystals it is more elegant to take
advantage of their translational symmetry. An obvious way to do this is to transform
the local ME excited state basis functions into translation-symmetry-adapted ME
basis functions, by forming appropriate linear combinations of the local ME basis
functions. How to derive a crystal-symmetry adapted ME basis from the original
ME basis in terms of localized excitations is shown below, after we have derived a
many-electron extension of Bloch’s theorem.
The procedure to be followed once the ME electron basis has been determined is
in principle straightforward. However, the computation of the matrix elements of H
and S in the secular matrix is somewhat complicated because we deal with a manyelectron basis. Furthermore, we have to keep in mind that the ME basis functions
are mutually non-orthogonal (the S matrix is not diagonal). This non-orthogonality
is -partly- maintained in the symmetry-adapted ME basis.
58
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
Extension of Bloch’s theorem to many-electron wave functions
In almost all text books and scientific papers the discussion of using translational
symmetry in order to obtain crystal wave functions is limited to one-electron functions
or ”orbitals”. A notable exception is the work by Calais [78] and co-workers. Yet,
the derivation of a many-electron extension of Bloch’s theorem is, albeit non-trivial,
not difficult. The reason for its absence in the scientific literature is probably the fact
that electronic structure calculations of systems with periodic symmetry are almost
always based on the use of effective one-electron models.
Below we show that the many-electron eigenfunctions of a many-electron BornOppenheimer crystal Hamiltonian can be chosen to be also eigenfunctions of the translation symmetry operators of the crystal. This leads to an extra symmetry label, K,
for each many-electron eigenfunction. It is important to note here, that this symmetry is present only for the many-electron wave functions. There is no question of
translation symmetry of orbitals, contrary to the usual one-electron band models. In
those models one works with an approximate one-electron crystal Hamiltonian, which
has the same symmetry properties as the many-electron Hamiltonian. Consequently,
the eigenfunctions of that approximate one-electron Hamiltonian, the orbitals, must
be eigenfunctions of the translation symmetry operators for the crystal. In our approach, there is no demand that orbitals in which the many-electron wave functions
are expressed transform according to the symmetry of the crystal, simply because we
do not make use of any approximate one-electron crystal Hamiltonian. This is an
important point, because it means that the orbitals can be localized, and this in turn
allows us to introduce the effects of electron correlation into the local many-electron
basis functions in which the crystal wave functions are expressed.
To keep the discussion simple we do not consider any point group symmetry here.
The line of reasoning followed here for ME wave functions is rather similar to that in
the derivation of Bloch’s theorem for crystal orbitals. See, for example [79].
A many-electron crystal Hamiltonian operator in the Born-Oppenheimer approximation, Ĥ, has the translational symmetry which is present in the crystal structure.
To be more precise, Ĥ commutes with all translation operators associated with the
crystal lattice. This means that we can in principle take advantage of the translation
symmetry at the many-electron level.
We define a translation operator T̂R associated with a lattice translation vector
R = µ1 a1 + µ2 a1 + µ3 a3 , where {a1 , a2 , a3 } are the primitive basis vectors of the
Bravais lattice and µi ∈ {1, 2, ...}. The translation operator T̂R has the following
effect on a many-electron function, F with arguments (r1 , r2 , ...., rM ):
T̂R F (r1 , r2 , ..., rM ) = F (r1 − R, r2 − R, ...., rM − R)
Since all translation operators T̂R , associated with the lattice translation vectors R,
commute with one another, and in addition the many-electron Hamiltonian operator
Ĥ commutes with all translation operators T̂R , we may choose the eigenfunctions
Ψ(r1 , r2 , ...., rM ) of Ĥ to be also eigenfunctions of all translation operators T̂R :
T̂R Ψ(r1 , r2 , ...., rM ) = γ(R)Ψ(r1 , r2 , ...., rM )
Many-body bands
59
Taking into account that in addition we have,
T̂R T̂R0 = T̂R+R0
it is straightforward to obtain that γ(R) must be an exponential function of R. Let
us write it as e−iK.R with K, real but not yet specified. The periodic boundary
conditions demand that for a translation over all Ni unit cells in one of the directions
i ∈ {1, 2, 3} the wave function Ψ(r1 , r2 , ...., rM ) remains unchanged,
T̂Ni ai Ψ(r1 , r2 , ...., rM ) = Ψ(r1 , r2 , ...., rM ),
for i ∈ {1, 2, 3}
We must therefore demand that e−iK.Ni ai =1 which results in the following choice for
the vector K,
K=
b2
b3
b1
ν1 +
ν2 +
ν3
N1
N2
N3
where {b1 , b2 , b1 } are the reciprocal lattice vectors which satisfy ai .bj = 2πδij and
νi ∈ {1, 2, ...}.
This means that the eigenfunctions Ψ(r1 , r2 , ...., rM ) of a many-electron crystal
Hamiltonian Ĥ in the Born-Oppenheimer approximation can be chosen in such a way
that associated with each Ψ(r1 , r2 , ...., rM ) is a wave vector K such that the symmetry
condition,
T̂R ΨK (r1 , r2 , ...., rM )
=
ΨK (r1 − R, r2 − R, ...., rM − R)
−iK.R
= e
(3.3)
ΨK (r1 , r2 , ...., rM )
is fulfilled for each lattice translation vector R.
The above statement is a many-electron generalization of Bloch’s theorem for
crystal orbitals. Since it is common practice to call all orbitals that fulfill the oneelectron version of this symmetry condition ”Bloch-type orbitals”, we will denote the
ME basis functions that fulfill the condition ”Bloch-type ME basis functions” and
analogously vector K is a ”Bloch-vector”,
Construction of Bloch-type many-electron basis functions
A simple and straightforward way to construct Bloch-type ME basis functions is
to project them from ME basis functions Φa (r1 , r2 , ...., rM )a representing localized
excitation a, in the unit cell that contains ion I,
ΘaK (r1 , r2 , ...., rM ) = √
1 X iK.R
e
Φa (r1 − R, r2 − R, ...., rM − R)
NR R
(3.4)
The summation runs over all lattice vectors R and NR = N1 N2 N3 .
a Φ (r , r , ...., r ) should satisfy the following condition Φ (r + T, r + T, ...., r
a 1 2
a 1
2
M
M + T) =
Φa (r1 , r2 , ...., rM ) for any lattice vector T of the form T=n1 N1 a1 + n2 N2 a2 + n3 N3 a3 (ni ∈ Z) in
order to satisfy the Born-von Kármán periodicity
60
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
To demonstrate that ΘaK (r1 , r2 , ...., rM ) transforms as a Bloch-type many-electron
function we apply a translation operation T̂R0 on it:
1 X iK.R
e
T̂R0 ΘaK (r1 , ..., rM ) = √
Φa (r1 − R0 − R, ..., rM − R0 − R)
NR R
0 X
0
1
e−iK.R
eiK.(R+R ) ×
= √
NR
R
Φa (r1 − (R + R0 ), ..., rM − (R + R0 ))
0
1 X iK.R
= e−iK.R √
e
Φa (r1 − R, ..., rM − R)
NR R
= γK (R0 )ΘaK (r1 , ..., rM ),
(3.5)
where γK (R0 ) is the eigenvalue of the translation operator. In the third step of Eq.
(3.5), we used the Born-von Kármán periodicity of Φa (r1 , r2 , ...., rM ).
An important advantage of employing Bloch-type ME basis functions instead of
localized ME basis functions is that the Bloch-type basis functions are already adapted
to the crystal symmetry. This implies that in the variational procedure to find the
crystal wave functions ΨnK (r1 , r2 , ...., rM ), we only need to consider Bloch-type ME
basis functions which have the same K-value,
X
ΨnK (r1 , r2 , ...., rM ) =
κna (K)ΘaK (r1 , ..., rM )
(3.6)
a
This gives a huge reduction in the dimension of the secular matrices that need to be
diagonalized. For example, if we take N1 , N2 and N3 unit cells in the {a1 , a2 , a3 }
directions, respectively, the secular matrix is blocked into N1 x N2 x N3 sub-matrices.
It is easy to demonstrate that the Hamiltonian and overlap matrix elements between Bloch-type ME basis functions which belong to different irreducible representations K and K0 of the translation group are zero,
hΘaK (r1 , ...., rM )|H|ΘbK0 (r1 , ...., rM )i =
1 X i(K0 .R0 −K.R)
e
×
NR
0
R,R
hΦa (r1 − R, ..., rM − R)|H|Φb (r1 − R0 , ..., rM − R0 )i (3.7)
By taking into account the periodicity of the Hamiltonian, the Hamiltonian matrix
elements between the two Bloch-type ME basis functions ΘaK and ΘbK0 become:
hΘaK (r1 , ...., rM )|H|ΘbK0 (r1 , ...., rM )i =
1 X iK0 .(R0 −R) i(K0 −K).R
e
e
×
NR
0
R,R
hΦa (r1 , ..., rM )|H|Φb (r1 − (R0 − R), ..., rM − (R0 − R))i
X
00
= δKK0
eiK.R hΦa (r1 , ..., rM )|H|Φb (r1 − R00 , ..., rM − R00 )i (3.8)
R00
where we changed in the last step the variable R0 − R to R00 and used the Born-von
Kármán boundary conditions in the reordering of the summations.
Many-body bands
61
The Hamiltonian matrix elements of H(K) between two Bloch-type ME basis
functions ΘaK and ΘbK are hence given by,
Hab (K)
= hΘaK (r1 , r2 , ...., rM )|H|ΘbK (r1 , r2 , ...., rM )i
X
=
eiK.R hΦa (r1 , ..., rM )|H|Φb (r1 − R, ..., rM − R)i
(3.9)
R
A similar expression can be obtained for the matrix elements of S(K).
Since the H(K) matrix elements in Eq. (3.9) are expressed in terms of the matrix
elements between the localized ME basis functions Φa and Φb , the summations above
can be restricted to only few (shells of) (next-) nearest neighbours. If the bands
we consider are derived from these localized ME basis functions, we need to solve a
generalized eigenvalue problem within the non-orthogonal tight-binding approach,
X
X
Hab (K)κbn (K) = nK
Sab (K)κbn (K)
(3.10)
b
b
The solutions of the system of secular equations in Eq. (3.10) have the following form,
X X
ΨnK (r1 , r2 , ...., rM ) =
{
eiK.R Φa (r1 , ..., rM )}κan (K),
(3.11)
a
R
with eigenvalues nK . The solutions in Eq. (3.11) are adapted to the crystal symmetry
K and form bands. Contrary to the one-electron bands in the conventional band
theory, these bands are many-electron bands.
Note, that despite introducing local ME basis functions as representing the locally
excited states, we have only specified their localized character without imposing any
constraints to their particular form. Thus, the theory is applicable to any ME basis
function, as long as its localized character is preserved.
Due to the periodicity of the Hamiltonian, the Hamiltonian matrix elements between localized ME basis functions describing excitations ’a’ and ’b’, centered around
sites I and J, respectively, are invariant under the operations T̂R :
IJ
Hab
= hΦIa (r1 , ..., rM )|H|ΦJb (r1 , ..., rM )i
J(−R)
= hΦI(−R)
(r1 − R, ..., rM − R)|H|Φb
a
(r1 − R, ..., rM − Ri (3.12)
An analogous expression exists for the overlap matrix S. Furthermore, we need to
compute only the matrix elements for distances |I − J| less than a certain limit.
The magnitudes of the matrix elements between (next-)(next-) nearest neighbours
determine that limit. It is shown in Chapter 4, that the matrix elements between excitations, centered around next-nearest neighbour Mn ions of manganites are already
an order of magnitude smaller compared to those between nearest neighbours. Hence,
IJ
IJ
the number of unique non-negligible matrix elements Hab
and Sab
is quite limited.
The next paragraphs demonstrate how to approximate the matrix elements of the
crystal Hamiltonian in Eq. (3.12) to the corresponding Hamiltonian matrix elements
of a large cluster. As explained above, employing the antisymmetrized wave function
Ansatz as an approximate form for the localized ME basis functions ΦIa and ΦJb allows
one to make use of the well-developed formalism of the embedded cluster approach
in the computation of the localized interactions and overlaps.
62
3.2.2
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
Many-body excitation bands, many-body hole bands and
many-body electron bands
In the following we will not only consider M electron states (neutral states) of the
extended systems, but also M-1 electron states (hole states) and M+1 electron states
(added electron states). The energy bands associtated with excited states of the
M electron system will be denoted many-body excitation bands. The energy bands
associated with hole states will be called many-body hole bands and the energy bands
associated with added-electron states will be called many-body electron bands.
In periodic Hartree-Fock theory the occupied bands of the bandstructure diagram
can be associated with (negative) ionization energies, whereas the conduction bands
can be associated with electron addition energies. In periodic Density Functional
theory there is no formal correspondence between the one-electron bands and ionization or addition energies. Nevertheless the DFT bands are often interpreted in
these terms. The many-body hole bands have a correspondence to the occupied part
of the Hartree-Fock band structure, whereas the many-body electron bands have a
correspondence to the conduction bands in the Hartree-Fock band theory.
It is important to remark that there is no one-to-one correspondence between the
many-body excitation bands treated in this thesis on the one hand and conventional
one-electron band theory on the other hand. Excited states are sometimes considered
in one-electron band approaches, but this is then done in a broken-symmetry supercell approximation. By employing so-called super-cells, which each consists of a large
number of unit cells, and allowing for one localized excitation in each super-cell, one is
sometimes able to consider localized excitations. The hope in the super-cell approach
is that one is able to make the super-cell large enough to prevent interactions between
the localized excitations in each super-cell. In that case there may be a correpondence
to the energies of our localized ME excited state basis functions and the excitation
energy per super-cell found in the super-cell approach. In Eq. (3.2) the ME wave
function for the neutral extended system was defined as,
X
Ψn (r1 , r2 , ...., rM ) =
Cnλ Φλ (r1 , r2 , ...., rM )
(3.13)
λ=0
where Φ0 represents a ME basis function for the ground state, and Φλ (λ = 1, 2, ...)
represent localized ME excited states. We now introduce two approximations. The
first approximation is
hΦ0 |H|Φλ i = 0
and hΦ0 |Φλ i = 0 for λ 6= 0
This implies that
Ψ0 = Φ0
and that for n6= 0 we have
Ψn (r1 , r2 , ...., rM ) =
X
λ=1
Cnλ Φλ (r1 , r2 , ...., rM )
(3.14)
IJ
IJ
Computation of Hab
and Sab
63
The second approximation consists of limiting the localized ME excited state basis to
excitations that are each centered around one single excitation site. This means that
we can write,
XX
Ψn (r1 , r2 , ...., rM ) =
CnIaI ΦIaI (r1 , r2 , ...., rM )
(3.15)
I
aI
Note, that for a size-consistent treatment of the excitations one also needs to introduce
two-site (three-site, ...) excitations, with an excitation around I as well as around J
(around I, as well around J and around K). For describing many-electron hole (or
added-electron) states the expression in Eq. (3.2) can also be used, be it that now
the localized ME states ΦIaI are M-1 (or M+1) states, with the hole (or electron)
localized around site I.
3.2.3
IJ
IJ
between localized states in
and Sab
Computation of Hab
extended systems
We choose the local ME basis function to be an antisymmetrized wave function product of a multiconfigurational (MC) wave function ΩIa for a N-electron cluster, representing an excited state ’a’ centered around I, and a wave function ΩIJ
E , describing a
frozen electron distribution for the remaining M-N electrons.
ΦIa (r1 , ..., rM ) = Â[ΩIa (r1 , ..., rN )ΩIJ
E (rN+1 , ..., rM−N )]
(3.16)
For the computation of the matrix elements, it turns out convenient to construct
the wave function ΦIa (r1 , ..., rM ) for an arbitrary excitation ’a’, centered in a large
cluster that includes both excitation centers I and J. Analogously, ΦJb (r1 , ..., rM ) is
constructed as an antisymmetrized wave function product of a MC wave function ΩJb ,
representing an excited state ’b’ centered around J, and the wave function ΩIJ
E . Using
this form of the local ME basis function, the Hamiltonian matrix elements become:
IJ
Hab
≈
J
IJ
hÂ[ΩIa (r1 , ..., rN )ΩIJ
E (rN+1 , ..., rM−N )]|H|Â[Ωb (r1 , ..., rN )ΩE (rN+1 , ..., rM−N )]i
(3.17)
Employing the wave function Ansatz in Eq. (3.16) for the localized ME basis functions allows one to invoke the embedded cluster approach in the computation of the
Hamiltonian matrix elements in Eq. (3.17).
We now approximate the matrix elements in Eq. (3.17) by the matrix elements
of the large cluster. The large cluster is embedded appropriately by using model
potential techniques [81], as described in Chapter 2. The sensitivity of their values
to the quality of the model potential employed in the computation must of course
be investigated. This problem is addressed once again in Chapter 4, where we treat
matrix elements between localized hole or electron states in hole-doped or electrondoped manganites.
64
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
The electronic polarization of the embedding crystal, caused by the localized excitation in the cluster, is not accounted for at this level of approximation. The shortrange effects are introduced by choosing a cluster that is sufficiently large, whereas the
long-range effects can be accounted for in an approximate manner. The effect of the
lattice polarization on the Hamiltonian and overlap matrix elements will be addressed
in Chapter 7. In this section, we invoked two main approximations. First, the crystal
wave function that describes an excitation ’a’ localized in a cluster, is constructed as
an antisymmetrized product of the cluster MC wave function and frozen embedding
wave function. This product wave function provides a localized description of the
excited state. Using these embedding techniques, the crystal matrix elements of such
product wave functions are conveniently approximated with the matrix elements of a
large cluster effective Hamiltonian over the cluster MC wave functions.
An important issue is the choice of the large embedded cluster. Within the framework of our overlapping-fragments approach, which will be described in detail below,
the large embedded cluster is viewed as built from two or more mutually overlapping small embedded clusters, denoted as fragments. A requirement for the size and
configuration of the fragment is that the localized excitation is adequately described.
This means that the density change, related to the excitation, does not extend to
the outer parts of the fragment. In analogy with the super-molecule approach, we
denote the large cluster as a super-cluster. The super-cluster contains at least two
excitation centers I and J that are involved in the interaction. Furthermore, it should
be sufficiently large to account for all relevant relaxation and correlation effects. The
size and configuration of the super-cluster should be chosen in such a manner, that
if the interacting excitation centers I and J are equivalent in the crystal, they should
remain equivalent in the super-cluster. In other words, the cluster environment of I
and J should not introduce untrue difference between the relative energies of localized
excited states ’a’ at I and ’b’ at J.
In Figure 3.1 we have visualized overlapping fragments in a super-cluster. We have
used for this purpose the crystal structure of LaMnO3 , the latter will be an object
of investigation in the next chapter. The fragments are [MnO6 ] clusters which are
enclosed in circles in Figure 3.1. Each [MnO6 ] cluster contains a Mn ion surrounded
by its nearest O counterions. Every two fragments which overlap in Figure 3.1 form a
super-cluster, in this case the super-clusters are [Mn2 O11 ] super-clusters, containing
two Mn ions and the nearest 11 O ions. If four fragments overlap they form a [Mn4 O20 ]
super-cluster.
We re-direct our attention towards construction of the wave function ΩIa for a
super-cluster that contains ions I and J as well as their nearest environment. This
wave function describes an excited state ’a’ centered around ion I. Analogously, we
can construct for this super-cluster wave function ΩJb for an excited state ’b’ centered
around ion J. The Hamiltonian matrix elements between the two wave functions ΩIa
and ΩJb can be computed using this super-cluster model. It is quite common to
calculate inter-site parameters, such as magnetic coupling parameters and hopping
parameters between (next-) neighbouring sites using a large super-cluster that conIJ
IJ
tains all non-unique interactions. The matrix elements Hab
and Sab
are small if I and
IJ
IJ
Computation of Hab
and Sab
65
Figure 3.1: Visualization of overlapping [MnO6 ] fragments in [Mn2 O11 ] and [Mn4 O20 ]
super-clusters (see text). Each [MnO6 ] fragment is enclosed within a circle.
J are at significant distance from each other. The point, at which we can neglect them
is defined by the ab initio values of the matrix elements, and depend on the structure
of the extended system and the nature of the excited state. In Chapter 4, we calculate
the effective Hamiltonian and overlap matrix elements associated with the hopping of
a localized hole or electron in hole-doped or electron-doped manganites. In this study,
we employ two- and four- center super-clusters to obtain the nearest neighbour (nn)
and next- nearest neighbour (n-nn) hopping parameters. We find that the decrease
in the absolute value of the effective Hamiltonian matrix elements from nn to n-nn
is an order of magnitude. The fast decrease in the values of the matrix elements with
distance is caused by the local character of the excitations at I and J. This decrease
justifies the tight -binding approach to obtain the solutions in Eq. (3.11).
The matrix elements can in principle be computed within the conventional approach, where the orbitals are optimized for the super-cluster, for example, the wave
functions describing the localized excitations are CASSCF wave functions for the
super-cluster. Such wave functions may provide, when the excitation centers (for
example transition metal ions) are not in the centre of the cluster, an unbalanced
description of the electron densities around these ions. The different densities, which
correspond to excitations of the same type but localized at two different centers I and
J of the super-cluster, may lead to two localized excitations which are not related
by a translation operation. Another problem that arises is the orbital optimization
that may become rather cumbersome for a large super-cluster, especially if the point
group symmetry is low. In addition, the optimization often leads to partially occupied
orbitals that are delocalized over all sites. Conventional methods often fail to provide
a wave function that preserves the excitation localized at site I. Although this delo-
66
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
calization problem does not prevent in all cases to extract localized parameters [21],
it is advantageous to use a localized orbital basis. It also allows for including the local
electron correlation and electronic relaxation effects in a rigorous manner.
In the next section, we present an efficient scheme for constructing wave functions
ΩIa of super-clusters using wave functions of smaller clusters, that we denote fragments.
The fragments are usually centered around one crystal site, for example I. We refer to
this scheme as the overlapping fragments approach (OFA). We describe in detail the
construction of local wave functions for a super-cluster I{K}n Jb , which is designed
to compute the interaction (Hamiltonian matrix elements and overlaps) between an
excitation centered around site I, wave function ΩIa , and an excitation around site J,
wave function ΩJb . {K}n indicates that the super-cluster contains in general also other
fragments, in addition to the fragments around the sites I and J that are involved in
the Hamiltonian matrix elements.
3.3
The overlapping fragments approach
A procedure was developed to construct super-cluster wave functions ΩIa (1, ..., Ns ),
starting from fragments wave functions ΓIa (1, ..., NI ), ΓJ0 (1, ..., NJ ) and n ΓK
0 (1, ..., NK ).
Note, that the number of electrons in the super-cluster Ns is in general smaller than
the sum of the number of electrons in the fragments, NI + nNK + NJ , because the
fragments share ions and hence electrons. First, the MCSCF wave function for an
excitation ’a’, for a fragment built around center I is constructed with an embedded
cluster calculation. Next, one obtains the MC SCF wave functions for the ground
state of fragments centered around sites K and site J. All fragment electrons except
those in the active space are in doubly occupied, or inactive, orbitals. The active
space of the fragment, at which the excitation is centered, is chosen large enough to
give an adequate description of the excited state. For the other fragments, which are
in their ground state, we commonly choose the minimum active space that is needed
to treat the open shell electrons of the fragment. For example, a NiO6 fragment representing the ground state of NiO will have two electrons in two Ni 3d (eg )- like active
orbitals. All other electrons are in doubly occupied orbitals, which includes the Ni 3d
(t2g )- like orbitals.
The occupied orbitals for a super-cluster I{K}n J are determined by using the
occupied MC SCF orbitals of the fragments. An I{K}n J super-cluster can be seen
as consisting of n+2 overlapping fragments where, depending on the geometry of the
super-cluster, one fragment can share one or more ions with one or more other fragments. The electrons associated with these shared ions are doubly counted. Therefore,
we need a procedure to eliminate the doubly counted electrons. A corresponding orbitals analysis [20] is performed in order to determine which fragment orbitals should
be disregarded in the super-cluster.
We start with the occupied orbital sets of two fragments, each expanded in the
symmetry adapted basis functions of the super-cluster. A corresponding orbitals
b In the following discussion, we denote a fragment built around a lattice site I by just using the
site notation I. What we really mean is the site I together with its nearest environment;
The overlapping fragments approach
67
transformation is performed for the inactive orbitals (IO) of two fragments that share
ion (s). The two fragments can be, for example, the fragments Kk and J. Fragments
Kk and J have m1 - and m2 -IO sets, respectively that are not mutually orthogonal.
Fragment Kk :
ϕKk 1 , ϕKk 2 , ........, ϕKk m1
Fragment J:
ϕJ1 , ϕJ2 , ........, ϕJm2
The overlap matrix between the two IO sets is,


S11
S12 ... S1m2

.........
Sm1 1 Sm1 2 ... Sm1 m2
Amos and Hall [88] and Löwdin [89] proved that any matrix S of dimension m1 × m2
can be diagonalized by two unitary matrices U (m1 × m1 ) and V (m2 × m2 ),
S 0 = U † SV
0
0
Sij
= Sjj
δij
(3.18)
where S 0 is a matrix of dimension m1 × m2 . We choose m1 larger or equal to m2
0
and all non-zero overlaps Sjj
to be positive. This unitary transformation is always
possible even in case of a singular S [90] In fact, we perform a standard singular
value decomposition of S to find the matrices S 0 , U and V . The matrices U and V
transform the m1 -IO and m2 -IO sets into two new m1 -IO and m2 -IO sets, which
are the corresponding orbitals of Amos and Hall [88].
ϕ0Kk i
ϕ0Jj
=
=
m1
X
ϕKk m Umi
m=1
m2
X
ϕJl Vlj
(3.19)
l=1
where
0
hϕ0Kk i |ϕ0Jj i = Sjj
δij
The new orbital sets are called biorthogonal. Each of the orbitals in one of the new
sets has a non-zero overlap with only one orbital (its ’corresponding’ orbital) in the
other new set. Some of these non-zero overlap matrix elements are close to unity,
the others are rather small. The orbitals with large mutual overlap are the inactive
orbitals localized at the shared ion (s). If their number is 2l (l at fragment Kk and l
at fragment J), l linear combinations are disregarded for the two-center {K}k J part
of the super-cluster I{K}n J, for example,
ϕ0Kk a − ϕ0Ja , a = 1, l
68
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
whereas the linear combinations
ϕ0Kk a + ϕ0Ja , a = 1, l
are used as inactive orbitals for {K}k J. Note, that using the ’+’ linear combinations,
above, as inactive orbitals for the super-cluster is one particular choice (for a discussion
see Chapter 5 ). The remaining inactive orbitals of the two fragments are also used
as inactive orbitals for {K}k J.
In the most common case that the shared ions do not contribute to the active
space of the fragments, the active orbital set of the super-cluster is taken as the
sum of the active sets of the fragments. Otherwise, a corresponding orbitals analysis
is performed also for the active orbitals of the fragments. The procedure becomes
slightly more complicated if an orbital at the shared ion is an active orbital in the
calculation for one of the fragments and inactive for the other one. Most often, we are
interested in this case in keeping the active orbital and disregarding its corresponding
inactive orbital. The biorthogonalization involves the inactive and active orbitals of
both orbital sets, separately for each block,
InKk InJ InKk AcJ
AcKk InJ AcKk AcJ ,
where InJ and AcJ are the inactive and active orbital sets of fragment J and analogously for InKk and AcKk . The block InKk InJ that represents the overlap matrix
between the IO sets of fragments J and Kk is handled within the procedure described
above. Next, to determine which of transformed orbitals at the first biorthogonalization ϕ0Jj should be eliminated due to their presence in the active space of Kk , the
biorthogonalization procedure is performed between these inactive ϕ0Jj and the active
0
00
A
ϕA
Kk i orbitals. This leads to new unitary transformed inactive ϕJj and active ϕKk i
00
00
00
00
A
A
orbital sets. We finally obtain ϕKk i , ϕJj , ϕJj and ϕKk i orbital sets. They are used
to construct the inactive and active spaces of {K}k J.
Once the orbital set of {K}k J is formed, the next fragment {K}l is added to
{K}k J, using the same algorithm for construction of the orbital set of {K}l {K}k J.
Combining the inactive orbital sets of all fragments after the unitary transformations
and removal of the double-counted orbitals, we construct the doubly occupied orbitals
of the super-cluster I{K}n J. Next, its active orbital set is formed from the active sets
of the fragments and Gramm-Schmidt orthogonalized to the doubly occupied orbitals.
The localized wave function ΩIa (1, ..., Ns ) of the super-cluster and its energy Ea =
CII
Haa
are determined by a RASCI calculation. Since we do not perform an orbital
optimization for the super-cluster, but rather use localized orbital sets derived from
fragments, we avoid the delocalization of the orbitals of each fragment over the entire
super-cluster. We prevent also to a high extent an unbalanced description of the
electron distribution around the excitation centers.
In principle, the RASCI expansion may allow for electron transfer between different fragments, for example, from the ”excited” fragment to the surrounding ”ground
state” fragments. In practice, the choice of the RAS spaces and, in particular, the
3.4 Implementation
69
minimal active spaces at the surrounding fragments prevent such charge transfer. We
demonstrate this in Chapter 4, for double exchange interactions in manganites.
It should be noted that depending on the super-cluster’s size and geometry, some
equivalent crystal sites may experience different cluster environment. If this is the
C
CII
case for sites I and I’, then Haa
and HaaI 0 I 0 may differ and we speak about a cluster
artifact. Therefore, the super-cluster should be chosen in a wise manner to minimize
this artifact.
The localized wave function ΩJb (1, ..., Ns ), that describes an excited state ’b’ localized around J is determined in the manner outlined above.
CIJ
The computation of the off-diagonal elements Hab
is not a trivial task, since
ΩIa (1, ..., Ns ) and ΩJb (1, ..., Ns ) are expressed in mutually non-orthogonal orbital sets.
We make use of the RASSI approach [6], implemented in the program package MOLCAS [93].
3.4
Implementation
3.4.1
Overlapping fragments approach
In this section, we discuss briefly a few aspects of the implementation of the OFA
in MOLCAS. They are related mainly to relevant transformations of the occupied
fragment orbitals. It turns out convenient for the implementation to express in the
first step each fragment occupied orbital in terms of the fragment’s atom-centered
basis functions χk ∈ {χ} instead of the symmetry-adapted combinations φµi ∈ {φ}µ
thereof. This is done for each irreducible representation µ of the fragment’s point
group G of the fragments separately, using a matrix dµ . The explicit form of dµ can
be deduced from the following considerations. Each of the fragment orbitals ϕµJj is
initially expanded in terms of the symmetry-adapted basis functions φµi ,
X µ µ
ϕµJj =
φi cij
i
where
φµi
are expressed in terms of the atom-centered basis functions χk :
X
µ
φµi =
χk αki
k
A particular column of matrix αµ that defines a particular φµi has non-zero elements
only for those χk that belong to centers related by symmetry operations. We consider
only abelian groups, and the matrix elements of αµ are +(-)1, +(-) 21 and 0 (the
symmetry-adapted combinations are not normalized to one). Using the expressions
above, we expand ϕµJj in terms of χk ,
ϕµJj
=
X
=
X
k
dµkj
i
χk
X
i
µ µ
αki
cij
µ µ
αki
cij
70
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
If the atom-centered basis set of the super-cluster is {ξ}, we can relate {χ} and {ξ}
through a transformation matrix T ,
χk =
X
ξn Tnk
n
and ϕµJj can now be expressed in terms of ξn ,
ϕµJj
=
X
=
X
n
µ
fnj
ξn
X
Tnk dµkj
k
Tnk dµkj
(3.20)
k
T may contain translation and rotation of χk . While the translation operation does
not change the matrix dµ , the centers of a particular fragment may be rotated with
respect to their corresponding centers in the super-cluster and thus dµ is transformed
into f µ . This implies that each component of a fragment atomic basis function with
an azimuthal quantum number l 6= 0 will turn into a linear combination of the components of l in the super-cluster. For example, a pz basis function becomes a linear
combination of px , py and pz basis functions in the super-cluster. The rotation properties of our basis functions, which are real spherical Gaussians, are in fact the same
as those of the real spherical harmonics. Therefore, we are interested here in the real
orthogonal rotation matrices Rl for the transformation of the real spherical harmonics [91]. The non-zero matrix elements of T are in fact the matrix elements of the
different Rl matrices. The rotation matrices for the different quantum numbers l are
obtained by recurrence relations, which allow one to express the rotation matrix Rl+1
in terms of the rotation matrices Rl and R1 [91]. The reader is referred for more
details to reference [91], where these recurrence relations are derived explicitly for
the different values of quantum number ml . The implementation of those recurrence
relations by those authors was adapted and implemented in our code to obtain the
elements of Rl .
At this step, we obtained fragment orbitals ϕµJj expressed in terms of the atomic
basis functions ξn of the super-cluster. The fragment orbitals transform according
to a point group F in the super-cluster. Next, we express them in F -symmetry
adapted basis functions of the super-cluster. All fragment orbitals ϕµJj which belong
to irreducible representation µ of G belong to a particular irreducible representation
ν of F,
X
ν
ϕνJj =
ζpν gpj
p
where ζpν are the symmetry-adapted basis functions for that irreducible representation
ν. They are given by
X
ν
ζpν =
ξn γnp
(3.21)
n
Non-orthogonal tight-binding approach
71
The columns of γ ν are generated by applying standard projection operators to the
atom-centered basis functions ξn [92]. Finally, we obtain the explicit form of matrix
g ν by using the expressions in (Eq. (3.20)) and (Eq. (3.21)):
X X
ν −1 µ
ϕνJj =
ζpν
(γpn
) fnj
p
ν
gpj
=
X
n
ν −1 µ
(γpn
) fnj
(3.22)
n
The orbitals ϕνJj of J and the ones of {K}k are involved afterwards in the biorthogonalization procedure, described above. The reformulation of the fragment
orbitals in terms of the super-cluster basis set and the bi-orthogonalization scheme
were implemented in the program package MOLCAS [93].
3.4.2
Non-orthogonal tight-binding approach
It is practical to choose a reference unit cell 0 with ME basis functions Φa (r1 , ..., rM ),
Φb (r1 , ..., rM ), ..etc. Symmetry equivalent ME basis functions in other unit cells are
given by,
Φa (r1 − R, ..., rM − R)
=
T̂R Φa (r1 , ..., rM ),
Φb (r1 − R, ..., rM − R)
=
T̂R Φb (r1 , ..., rM ),
...
Having evaluated the Hamiltonian and overlap matrix elements between localized
ME basis functions, we transform these to matrix elements between Bloch-type i.e.,
translational symmetry-adapted ME basis functions. Their concise form is (see Eq.
(3.9)):
X
R
Hab (K) =
eiK.R Hab
R
Sab (K)
=
X
R
eiK.R Sab
(3.23)
R
where
R
Hab
R
Sab
= hΦa (r1 , ..., rM )|H|Φb (r1 − R, ..., rM − R)i
= hΦa (r1 , ..., rM )|Φb (r1 − R, ..., rM − R)i
(3.24)
and the sum is over all lattice vectors. These matrix elements can be re-written in
the following form, which is convenient for the implementation,
X
X
R
R
Hab (K) = Hab (0) +
cos(K.R)Hab
+i
sin(K.R)Hab
R6=0
Sab (K)
= Sab (0) +
X
R6=0
R
cos(K.R)Sab
R6=0
+i
X
R6=0
R
sin(K.R)Sab
(3.25)
72
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
In practice, the summations are performed only over M nearby unit cells. The trunIJ
cation criterion for M is estimated and checked a priori when we computed Hab
and
IJ
Sab . We recall the generalized eigenvalue equation we need to solve
X
X
κbn (K)Hab (K) = n (K)
κbn (K)Sab (K)
(3.26)
b
b
or its equivalent form,
H(K).κn (K) = S(K).κn (K)n (K)
(3.27)
where H(K) and S(K) are Hemitian matrices, which are represented by the sum of
the real and imaginary parts,
H(K)
= <H(K) + i=H(K)
S(K)
= <S(K) + i=S(K)
(3.28)
As is well known the complex n×n eigenvalue problem can be converted to the 2n×2n
real problem by separating the Hermitian matrices in Eq. (3.27) into their real and
imaginary parts in Eq. (3.28). One proves easily that Eq. (3.29) is equivalent to Eq.
(3.27) by substituting Eq. (3.28) in Eq. (3.27),
<H(K) −=H(K)
<S(K) −=S(K)
<κn (K)
<κn (K)
= n (K)
(3.29)
=H(K) <H(K)
=S(K) <S(K)
=κn (K)
=κn (K)
where <κn (K) and =κn (K) denote the real and imaginary parts of the vectors κn (K),
respectively and if the vector
<κn (K)
=κn (K)
is a solution of the eigenvalue problem in Eq. (3.29), then the vector
−=κn (K)
<κn (K)
is also a solution and as a result the eigenvalue n (K) appears twice. The eigenvectors in each degenerate pair are the same up to an inessential phase. Thus,
one can solve the augmented 2n×2n eigenvalue problem, obtain the 2n eigenvalues
1 (K), 1 (K), 2 (K), 2 (K), ..., n (K), n (K) and corresponding pairs of eigenvectors
and then choose one eigenvalue and one eigenvector from each pair to be the eigenvalues and eigenvectors of the original complex eigenvalue problem (Eq. (3.27)).
Analogous to the conventional one-electron band structure theory, we refer to the
distribution of the n (K) values for a given n as a many-body energy band.
In a Bravais lattice with an infinite number of unit cells the eigenvalues n (K)
and their gradients with respect to K, ∇K n (K), are continuous functions of K. In
practice, the energies n (K) along a symmetry line are computed for some discrete set
Non-orthogonal tight-binding approach
73
of K values within the first Brillouin zone and then, one makes use of an interpolation
scheme with an accurate analytical function to produce the band structure. The
interpolation scheme, used here, is developed by Kertész and Hughbanks and based
on symmetry-adapted Fourier functions as fitting functions [94].
The theory shown above, allows one to obtain the eigenvectors, defined in Eq.
(3.11). The studies in Chapters 5 and 7 are concerned with probing the K-dispersion
of a particular excited (ionized) state that determines the width of the band related
to this excited (ionized) state. The band width is proportional to the hopping matrix
elements between many-electron basis functions that represent a particular exited
(ionized) state localized around different lattice sites I or J.
74
Chapter 3, Delocalized and Correlated Wavefunctions in Extended
Systems.
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[85] S. Huzinaga, L. Seijo, Z. Barandiaran, and M. Klobukowski, J. Chem. Phys. 86,
2132 (1987); Z. Barandiaran and L. Seijo, J. Chem. Phys. 89, 5739 (1988)
[86] N. W. Winter, R. M. Pitzer, and D. K. Temple, J. Chem. Phys. 86, 3549 (1987)
[87] P. J. Hay and W. R. Wadt, J. Chem. Phys. 82, 270 (1985)
[88] A. T. Amos and G. G. Hall, Proc. R. Soc. 263, 483 (1961)
[89] P.O. Löwdin J. Appl. Phys. (Suppl.) 33, 251 (1962)
[90] I. Mayer, Adv. Quantum Chem. 12, 189 (1980)
[91] J. Ivanic and K. Rüdenberg, J. Phys. Chem. 100, 6342 (1996)
[92] P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics, third edition
[93] G. Karlström, R. Lindh, P.. Malmqvist, B. O. Roos, U. Ryde, V. Veryazov, P.O.
Widmark, M. Cossi, B. Schimmelpfennig, P. Neogrady, L. Seijo, Computational
Material Science, 28, 222 (2003).
[94] M. Kertész and T. Hughbanks, Phys. Rev. B 24, 6870 (1981)
80
BIBLIOGRAPHY
4
Double Exchange Parameters in
Lightly Doped Manganites
ABSTRACT — In Chapter 3,
an approach was introduced for the computation of interactions and overlaps between localized excited states in extended
systems. In this chapter, the method is applied to study the double exchange
parameters in lightly hole and electron doped manganites. The hopping matrix elements between localized states are obtained as the matrix elements of the
Hamiltonian of a large embedded cluster between mutually non-orthogonal multiconfigurational wave functions. Each wave function describes a localized ionized
or added-electron state. These wave functions are expressed in terms of localized
orbitals. Expressing the electronic states in terms of localized orbitals allows one
to account in a rigorous manner for the local electron correlation and relaxation
effects. We compare the hopping matrix elements with those obtained within the
conventional embedded cluster approach.
In the lightly hole doped LaMnO3 and lightly electron doped CaMnO3 , we find
different effective hopping matrix elements or ”double exchange” parameters in
the ab planes (in the Pbnm space group) which are larger than along the c axis.
This anisotropy is less pronounced in La0.75 Ca0.25 MnO3 , where we find the double exchange parameters within the ab planes and along the c axes to be of the
same magnitude. In LaMnO3 and CaMnO3 , a simple model for the interactions
reveals predominant σπ -type interactions. In La0.75 Ca0.25 MnO3 the interactions
are found to be of σσ- type.
In all cases, we find nearly perfect agreement for the spin-dependence of the
double exchange parameters with the model of Anderson-Hasegawa. The nextnearest neighbours parameters are more than an order of magnitude smaller than
the nearest neighbours ones.
82
4.1
4.1.1
Chapter 4, Double exchange in manganites
Introduction
Local parameters in extended systems and localized orbitals approaches
A considerable amount of investigations on different materials has shown, that the local properties of transition metal compounds can be probed, using embedded clusters
containing one or two transition metal ions and a shell of anions embedded in effective model potentials [2–5,8–11]. For example, the energies of crystal field excitations
in TM oxides are accurately described within the embedded cluster model, making
use of multiconfigurational representation of the wave functions [66, 67] of the ground
and excited states that incorporates the dominant part of the non-dynamical correlation effects. The dynamical correlation effects are often estimated by a second-order
perturbation theory with reference multiconfigurational wave functions [72].
Furthermore, the cluster approach has been validated by the authors of reference
[11] who considered the possible presence of collective effects in magnetic coupling in
ionic solids by calculating the Heisenberg constant J in different cluster models having
more than two magnetic centers. They found, that indeed in some nickel fluorides, TM
oxides and cuprates the interactions between two magnetic centers are local and the
two-center clusters are sufficient to describe them. The magnetic exchange parameter
J emerges from the phenomenological Heisenberg Hamiltonian [7]. Another simple
model Hamiltonian is parameterized by only two parameters, the nearest neighbour
exchange parameter J and a hopping integral t. More elaborate models, for example
the Hubbard-Heisenberg models, make use of a larger set of local effective parameters,
that includes also interactions between next-nearest neighbours. The local effective
parameters are t, t’ and J. Values for these effective parameters have been successfully
determined over the last decade within the conventional embedded cluster approach
[8–15].
Recently, a study of J of some cuprate semiconductors carried out at the same
theoretical level by means of both embedded cluster model and periodic calculations has validated the cluster model [18] for this problem. The hopping integrals t
and other localized parameters can also be well described within the embedded cluster approach [10]. The most recent validation of this statement are studies of the
hopping parameters t in spin ladder compounds SrCu2 O3 , Sr2 Cu3 O5 and CaCu2 O3
performed by Bordas and co-workers [8, 9]. Although this conventional embedded
cluster approach has proven to be successful in the applications outlined here, it has
some drawbacks, that complicate or prohibit in some cases the extraction of localized
parameters. One drawback is related to the cluster model itself whereas the other is
relevant to obtaining the correct wave function with localized electron states on either
one of the cluster centers. Since the site symmetry of the TM ion in the cluster is
lower than its actual symmetry in the crystal, it may lead to some cluster artifacts.
The orbital optimization may become rather cumbersome in larger clusters. It may
also lead to delocalization of the orbitals over the entire cluster. Although this orbital
delocalization has not restrained in all cases the extraction of localized parameters,
the use of a localized orbital basis as a starting point in the computation of localized
Introduction
83
parameters is more attractive.
The concept of employing localized orbital sets in describing localized properties
in molecules and solids has evolved considerably the last decades. Among the most
used ones are the localization schemes of Boys [73], Edmiston-Ruedenberg [74] and
Pipek [75]. All these localization methods belong to so called a posteriori localization
schemes that make use of relocalizing unitary transformations. The a posteriori localization schemes however do not resolve all problems outlined above. The localized
orbital set should be generated from the start.
Along with the a posteriori methods, a priori methods emerged that allow one
to start from the beginning with a localized orbital set [16, 17]. Among these are
the one proposed by Adams [43] and recently Maynau and co-workers developed a
priori method that uses the one-particle density matrix to generate directly localized
orbitals [77].
We developed an alternative method to compute inter-site interactions that makes
use of super-cluster wave functions, expanded in a localized orbital basis, generated
within a priori localization scheme. The localized orbitals are obtained from CASSCF
calculations on smaller embedded clusters that we denote as ”fragments”. Within the
new scheme, the super-clusters are viewed as consisting of two or more overlapping
”fragments”, each centered around a TM ion. The symmetry point group of the
fragment is the same as the site symmetry at the TM ion in the extended system.
The localized states of the super-clusters are represented by CASCI wave functions
expressed in terms of the localized orbital set. This Overlapping Fragment method
was described in Chapter 3. The inter-site parameters are defined as the effective
matrix elements between different localized states of the super-cluster.
4.1.2
Double exchange in lightly hole and electron doped manganites
To demonstrate the method we choose an application, that is relevant to the conductivity properties of the perovskite manganites La1−x Cax MnO3 . These materials have
attracted considerable interest due to the interplay between magnetism and transport
properties [36]. In the present study, the effective hopping matrix elements are considered for localized hole (for compounds with x≈0) or electron (for x≈1) states in
lightly hole or electron doped manganites. In addition the compound with a hole doping corresponding to x=0.25 is also studied. The new approach allows one to study
the hopping of a hole or electron, introduced in the material by doping, with their
electronic polarization clouds. The effective hopping matrix elements are obtained
in terms of the Hamiltonian matrix elements between states with a localized hole or
electron.
The first studies of the perovskite type manganites La1−x Ax MnO3 (A=Ca, Sr,
Ba) have been carried out by Jonker and van Santen [30,31]. The ground state of the
perovskite manganites La1−x Cax MnO3 in the concentration region 0.2≤x≤0.5 and
a temperature T below the Curie temperature Tc for a given x, is a ferromagnetic
(FM) metallic phase. At concentrations x=0 and x=1 the materials are antiferro-
84
Chapter 4, Double exchange in manganites
magnetic insulators. Shortly after the discovery of these compounds, Zener [19, 20]
has introduced the concept of the ”double exchange”, DE, that correlates the conductivity and ferromagnetism that had been found by Jonker and van Santen. Zener’s
mechanism, despite its simplicity, seems to describe qualitatively the relation between
ferromagnetism and conduction in the FM phase below Tc .
The concept of the DE model has been invoked in many studies as the basic
microscopic mechanism that explains the coexistence of a long-range FM order and
a drastic increase in the electrical conductivity below the Curie temperature Tc for
0.2≤x≤0.5 [19, 20]. This effect is the so called colossal negative magnetoresistance
(CMR) [54]. Although the DE model provides a qualitatively correct description of
these materials, recent theoretical and experimental studies [39–41] have addressed
the two-phase metal-insulator competition that occurs at Tc at concentration x for
which the FM phase represents the ground state at the low temperatures (T lower than
Tc ). Contrary to the prediction of the DE model, the phase at x∼0.2 is insulating
and FM [44, 45]. Above Tc , so called charge ordered (CO) structures have been
detected in neutron scattering studies [42,43]. A recent informal paper by Dagotto [37]
addresses the existence of a competition between the CO states and the FM metalic
state. Hotta and co-workers [39] discussed also the possibility of a contribution of
the FM insulating phase near Tc to the CMR. In fact, Millis and co-workers [46]
pointed out that the CMR effect, predicted only on the basis of the DE model is
expected to be much smaller than observed. They also provided an explanation for
this discrepancy, including in the model a large lattice distortion above Tc [47]. This
distortion is expected to be partially removed below Tc . Although these studies
indicate that the DE model is alone insufficient to explain the CMR effect, the double
exchange mechanism is a necessary ingredient for its understanding. Some of the DE
parameters, obtained in the current manuscript, are concerned with the FM ground
state of the doped LaMnO3 at a doping concentration 0.25 for a crystal structure
refined at T=100 K below Tc ∼ 220 K.
The compounds with x=0 (LaMnO3 ) and x=1 (CaMnO3 ) are the end members
of the series La1−x Cax MnO3 . The Mn ions formally have either four or three 3d
electrons. Depending on the stchiometry and T, the intermediate compounds display
either a CO phase or contain 3d electrons shared between two or more Mn sites [48].
The charge state of the O ions in these compounds has also been questioned. The
study of Jonker and van Santen [30, 31] assumes that the O ions can be regarded as
O2− independent of the values of x and the nature of A. Ab initio embedded cluster
studies [48] on the half doped La0.5 Ca0.5 MnO3 have shown that indeed the ionic
ansatz with O2− ions is a reasonable first approximation to the electron distributions
of the ions. Periodic unrestricted Hartree-Fock studies [49, 50] of Zheng and Ferrari
questioned the ionic ansatz with O2− ions in the same half doped form. In the
following discussions, we will adopt this ionic ansatz, however, we note that in fact
the O ions are involved more in the physics of the compounds than being just present
in the structure. Locally, the Mn ions have high spin coupled electron configurations,
that arise from the weak crystal field. In the undoped compounds, LaMnO3 and
CaMnO3 , the exchange couplings between the high-spin Mn ions were found to be
Introduction
85
weak [51, 52]. The nature of the magnetic interactions is often described in terms of
effective Heisenberg Hamiltonian models [7] with only nearest-neighbour interactions
hi, ji,
X
Ĥ = −
Jij Ŝi .Ŝj
(4.1)
hi,ji
The nearest-neighbour exchange couplings within this model Hamiltonian in the ab
planes (in the Pbnm space group) of LaMnO3 and CaMnO3 are ≈ 0.8 meV [51, 53]
and ≈ -2.0 meV [52], respectively. The Mn spins are ordered ferromagnetically in
the ab planes of LaMnO3 and antiferromagnetically in CaMnO3 . Both compounds
have antiferromagnetic coupling perpendicular to the ab planes, ≈ of -0.6 meV. The
MnO6 fragments are almost octahedral in CaMnO3 , whereas static distortions with
so called orbital ordering are present in LaMnO3 [36]. The ratio of the magnitudes
of the DE parameter and the magnetic exchange parameter J will have an impact on
the formation of small self-trapped ferromagnetic polarons [55–57]. We will address
this question in section 4.5.
Zener’s DE formalism assumes that both electric conduction and ferromagnetic
coupling arise from an electron transfer from a Mn3+ ion through the closed shell
of an O2− ion to an adjacent Mn4+ ion. Although this picture is oversimplified, it
captures the tendency to ferromagnetism [37]. In section 4.2.1, following Zener, we
introduce two normalized wave functions ΨIa and ΨJb that reflect two electron configurations of a system of two neighbouring Mn ions, MnI and MnJ , before and after an
IJ
electron transfer from MnI to MnJ . Consequently, we obtain the Hamiltonian Hab
IJ
and overlap Sab matrix elements between these localized wave functions and define
IJ
the DE parameters in terms of these matrix elements. According to Zener [19,20] Hab
are non-vanishing only if the spins of the two 3d-shells at MnI and MnJ are coupled
in parallel. Zener deduces the conclusion that the conduction electrons are itinerant
only in an environment of parallel spins and therefore, conduction only occurs for
ferromagnetic coupling. Furthermore, the DE mechanism that leads to lowering the
energy of the system only occurs if conduction electrons are present. Zener concludes
that ferromagnetism does not occur in the absence of conduction electrons or other
indirect couplings. This concept of strongly coupled ferromagnetism and conduction
IJ
is re-considered by Anderson and Hasegawa [23]. Although Hab
is indeed expected
to be largest in case of high spin coupling between the MnI and MnJ sites, the asIJ
sumption that for all other spin couplings Hab
is zero is not confirmed. Anderson and
Hasegawa [23] demonstrated that in first approximation, the DE parameter t between
two sites, one having spin S0 and the other spin S0 + 12 is proportional to the spin
multiplicity 2S+1 of the combined system,
t(S) =
1
(2S + 1)t 12
2
(4.2)
with S=|2S0 + 21 |... ... 12 .
In the present investigation, we determine t in lightly hole doped LaMnO3 , in
La0.75 Ca0.25 MnO3 and in lightly electron doped CaMnO3 . As noted above, we study
86
Chapter 4, Double exchange in manganites
the dependence of t on the total spin S. This allows us to verify the approximate
relation in Eq. (4.2). Recently, Guihéry and co-workers [58, 59] have confirmed the
validity of the Anderson-Hasegawa model in their analysis of the low-energy spectrum
of DE systems.
The manganites are materials with strong electron correlation effects. Independent
electron approaches such as Hartree-Fock or Density Functional Theory are unable
to express the complex multiconfigurational nature of the wave functions of the systems. Quantum chemical methods, that account explicitly for electron correlation
and relaxation effects are more appropriate in describing the properties of these systems. The cluster models enable the explicit inclusion of local electron correlation
effects. They also ensure that the electronic relaxation effects at the hole (or electron)
site and its nearest environment are taken into account, because the hopping particle
moves together with its electronic polarization cloud. This local approach provides
an elegant approximation of the matrix elements of the Hamiltonian of an extended
system by those of the effective Hamiltonian of an embedded cluster. Extracting
two-center parameters requires the use of super-clusters which contain at least two
and sometimes more TM centers. For such large clusters, the orbital optimization
may become cumbersome especially for low point group symmetries. Employing the
overlapping fragment approach circumvents this problem.
In CaMnO3 , the Mn spins have weak antiferromagnetic coupling in all directions
of the crystal. We analyze the possibility of a formation of a local ferromagnetic spin
region in the AFM lattice of CaMnO3 , induced by the DE coupling. In LaMnO3 ,
the spins are coupled ferromagnetically in the ab planes and antiferromagnetically
along the c axes. The relevant 3d (eg -like) Mn orbitals are oriented in the ab planes
and hence also the holes are mainly oriented in these planes. Therefore, we expect
larger t in these planes than along c. The manuscript addresses also the question
whether or not t is strongly reduced when a Jahn-Teller distortion is present in the
system. We calculate t in LaMnO3 not only for the experimental structure with JahnTeller distortions, but also for an idealized JT undistorted but yet tilted geometry.
We expect similar DE parameters in all three directions in CaMnO3 , because of the
absence of Jahn-Teller distortions in this compound and similar magnetic couplings
in all three directions. In La0.75 Ca0.25 MnO3 , the spins are coupled ferromagnetically
in all three directions. For this compound, we consider the hopping of a single eg -like
electron between two Mn sites which are in different valence states, Mn3+ and Mn4+
(see section 4.4). We anticipate the magnitude of t, associated with this hopping
to be similar in all three directions. In X-ray diffraction experiments [28, 29] the
precise position of the Ca ion in the unit cell is general and hence the positions of
the nearest-neighbouring Ca ions around our model Mn2 O11 clusters are defined with
a certain randomness (see section 4.4). In the Mn2 O11 cluster models, studied in
this manuscript, this randomness is constrained to some chosen La/Ca configurations
around Mn2 O11 . These configurations lead to a different orientation of the relevant
3d (eg -like) Mn orbitals at two neighbouring Mn sites and therefore, different types
of hopping matrix elements are anticipated. In CaMnO3 , the presence of two nearly
degenerate low lying electron states, leads also to various hopping matrix elements.
4.2 Definition and computation of the double exchange parameters
87
We consider the anisotropy of the DE parameters for the three compounds.
The rest of the chapter is organized as follows. First, we give a brief overview
of the theoretical approach used to calculate the hopping matrix elements between
localized hole or electron states in extended systems with strong electron correlation
and relaxation effects: in section 4.2, we provide a definition of the so called double
exchange (DE) parameters in terms of the Hamiltonian matrix elements between the
localized hole and electron states. Section 4.3 shows how to use the OF method for
the construction of the super-clusters CASCI wave functions for the localized hole or
electron states. Section 4.4 gives computational information. Next, we report our
results and analyses for the double exchange parameters for three members of the
series La1−x Cax MnO3 : LaMnO3 , La0.75 Ca0.25 MnO3 and CaMnO3 . Furthermore,
the spin dependence of the DE parameters is compared to the Anderson-Hasegawa
model [23]. A comparison with the conventional embedded super-cluster approach
using CASSCF wave functions is provided as well.
4.2
Definition and computation of the double exchange parameters
We consider the hopping of a hole or an electron, introduced via doping in an extended
system with strong electron correlation effects. The normalized many-electron wave
functions ΨIa and ΨJb describe localized states, a and b, before and after an electron
transfer between the neighbouring ions MnI and MnJ ,
2−
Mn4+
ΨIa → ...Mn3+
I O
J ...
4+
J
2−
Ψb → ...MnI O Mn3+
J ...
One constructs linear combinations of ΨIa and ΨJb wave functions for the hole
(electron) states. Their form is given by,
I
J
Ψ+
n = Ψa + Ψb
I
J
Ψ−
n = Ψa − Ψb
(4.3)
Assuming orthogonality between the localized states ΨIa and ΨJb , the energy gain is
−
equal to one-half the energy splitting 4E between Ψ+
n and Ψn , i.e., to the HamilIJ
I
J
tonian matrix element Hab =hΨa |H|Ψb i. Ĥ is the Hamiltonian of the extended system. We define the quantity t= −4E
as DE parameter. When the overlap integral
2
IJ
Sab
=hΨIa |ΨJb i is non-zero, the energy difference 4E is no longer a single integral but
has the form,
4E = 2
(H II +H JJ )
IJ
IJ
Hab
− H av Sab
,
IJ
2
1 − (Sab )
(4.4)
where H av = aa 2 bb . We can still define an effective hopping matrix element
II
JJ
I
t= −4E
2 . If, in addition the energies Haa and Hbb of the localized states Ψa and
J
Ψb are slightly different, we get the following expression,
o 12
1n
42
t = − 4E 2 −
,
(4.5)
IJ )2
2
1 − (Sab
88
Chapter 4, Double exchange in manganites
JJ
II
where 4=Hbb
-Haa
The Hamiltonian matrix elements and overlap integrals in Eq. (4.4) and Eq. (4.5)
are the necessary ingredients for the computation of the DE parameters. They are
calculated within the framework of the new computational method introduced in
Chapter 3. The method makes use of the embedded cluster approach, but uses the OF
approach for constructing the super-cluster wave functions. The many-electron wave
functions ΨIa and ΨJb are considered to be antisymmetrized product wave functions [24]
of a MC wave function ΦIa (r1 , r2 , ...., rN ) or ΦJb (r1 , r2 , ...., rN ) for a N -electron supercluster and a wave function ΦE
0 (rN+1 , rN+2 , ...., rM−N ), describing frozen electron
density distribution for the remaining M-N electrons of the extended system,
ΨIa (r1 , r2 , ...., rM ) ≈ Â[ΦIa (r1 , r2 , ...., rN )ΦE
0 (rN+1 , rN+2 , ...., rM−N )],
(4.6)
ΦIa and ΦJb a represent hole or electron states of the super-cluster. The N -electron
super-cluster is built for a relevant portion of the extended system where the extra
hole or extra electron reside. In this case, the super-cluster contains at least MnI
and MnJ and their neighbouring O ions. Choosing the approximation Eq. (4.6) for
the wave functions ΨIa and ΨJb allows one to use the embedded cluster approach for
IJ
IJ
IJ
the computation of Hab
and Sab
. Within this formalism, the matrix elements Hab
IJ
and Sab between antisymmetrized product wave functions are approximated by the
CIJ
CIJ
CIJ
matrix elements Hab
and Sab
for the super-cluster. Hab
is the matrix element
C
of the N-electron Hamiltonian Ĥ between MC super-cluster wave functions ΦIa and
ΦJb ,
CIJ
IJ
J E
I
C
J
Hab
≈ hÂ[ΦIa ΦE
0 ]|H|Â[Φb Φ0 ]i ≈ hΦa |H |Φb i = Hab
(4.7)
Model potential techniques are used to incorporate in an approximate manner the
effects of the electrons and nuclei that are not included in the super-cluster [24, 62].
CIJ
The overlap matrix elements Sab
are obtained similarly. The MC expansions of ΦIa
J
and Φb ensure the explicit inclusion of the local electron correlation and relaxation
effects accompanying the ionization (hole states) or electron addition (electron states).
Within the conventional embedded cluster approach, the super-cluster MC wave
functions are constructed using orbital basis sets obtained in an orbital optimization
for the super-cluster. This well established formalism has been routinely employed
in the calculation of various localized magnetic and electronic parameters in solids,
including hopping matrix elements [8–15]. As mentioned in section 4.1.2, in some cases
the orbital optimization for large clusters with low point group symmetry becomes
cumbersome and it may lead even to delocalization of the orbitals involved in the intersite interaction over all sites of the cluster. We re-direct our attention in the following
sub-section towards the construction of the super-clusters MC wave functions in an
alternative manner, starting from localized orbital sets derived from CASSCF [66,67]
calculations on fragments. Each fragment is centered around a different TM site. By
expressing the electronic states of the super-cluster in those localized orbital sets, we
also avoid an unbalanced description of the electron density around the TM sites.
a The
wave function arguments ri are omitted for simplicity
4.3 The Overlapping Fragment Approach ’at work’
89
The final super-clusters wave functions are constructed as CI wave functions, using
the OF scheme.
4.3
The Overlapping Fragment Approach ’at work’
The Hamiltonian matrix elements that we aim to calculate, are associated with the
hopping of a localized hole or electron in hole or electron doped manganites. The effective matrix elements are calculated using embedded super-clusters containing either
two or four Mn ions plus the first neighbouring shell of O ions. This leads to [Mn2 O11 ]
and [Mn4 O20 ] super-clusters, respectively. The four-center clusters, containing MnI ,
MnJ , MnK and MnL , are illustrated in Figure 4.1 and Figure 4.2. The Mn ions in
these clusters are in a square-like arrangement. The two-center clusters [Mn2 O11 ] are
employed to obtain the nearest- neighbour DE parameters. The four-center clusters
allow us to probe the next-nearest neighbour hopping parameters and to investigate
the dependence of the nearest-neighbour parameters on the presence of the electron
distribution clouds of other ions in the extended system. The super-clusters are viewed
as consisting of smaller overlapping fragments. In the present study, the fragments
are embedded [MnO6 ] clusters. In LaMnO3 , La0.75 Ca0.25 MnO3 and CaMnO3 , the
cluster charges are chosen according to the ionic model. We describe in somewhat
more detail the algorithm used to construct the inactive and active orbital sets of the
[Mn2 O11 ] clusters. The inactive and active orbital sets of the [Mn4 O20 ] clusters are
built analogously, applying the algorithm multiple times.
We start with performing CASSCF calculations for the [MnO6 ] fragments. The
occupied orbitals of the fragments are classified in inactive orbitals, which are doubly
occupied in all Slater determinants (SD) in the CAS wave function expansion and
active orbitals, which are not doubly occupied in all SD. In our case, the set of active
orbitals for the fragments calculations consists of the Mn-3d -like orbitals. Considering
the [Mn2 O11 ] and [Mn4 O20 ] super-clusters consisting of two and four overlapping
[MnO6 ] fragments respectively, we denote the fragments sets of doubly occupied or
inactive orbitals as follows,
Fragment I, ϕI1 , ϕI2 , ..., ϕIn
Fragment J, ϕJ1 , ϕJ2 , ..., ϕJn0
Fragment K, ϕK1 , ϕK2 , ..., ϕKn00
Fragment L, ϕL1 , ϕL2 , ..., ϕLn000
Here, the [MnO6 ] fragments, denoted by I, J, K, L, have equal number of doubly occupied orbitals (n=n’=n”=n”’). Each pair of overlapping fragments in either
[Mn2 O11 ] or [Mn4 O20 ] shares one O2− ion. If the inactive orbital sets of two fragments
are combined to form the inactive orbital set of the super-cluster, this will result into
double counting of the electrons of the shared O2− ion. In our case for each pair of
[MnO6 ] fragments, ten electrons, occupying five orbitals, localized at the shared O2−
ion, are double counted. In order to determine which five doubly occupied orbitals
should be eliminated, we perform a corresponding orbital transformation [68] of the
doubly occupied orbital sets of the two [MnO6 ] fragments, for example fragments I
and J. This involves a separate unitary transformation for each of the two original
90
Chapter 4, Double exchange in manganites
sets ϕIn and ϕJn , resulting into two new orbital sets which are bi-orthogonal (for the
explicit form of the unitary transformation matrices, see Chapter 3 ):
0
hϕ0Ii |ϕ0Jj i = SIjJj
δij
These unitary transformations leave the fragment wave functions unchanged. Each of
the orbitals in one of the new sets has a non-zero overlap with only one orbital from
the other new set, its ”corresponding” orbital. By choice, all non-zero overlaps are
positive. For [Mn2 O11 ] clusters, built for example from fragments I and J, five of the
transformed orbitals of one fragment have an overlap close to unity with an orbital of
the other fragment. These five orbitals are associated with the O 1s, O 2s and O 2p
shells. Each of the remaining doubly occupied orbitals of one fragment has a small
overlap integral with its corresponding orbital of the other fragment. The orbitals
with large mutual overlap are the doubly occupied orbitals localized at the shared
O2− ion (s). We form ten normalized linear combinations of these ten transformed
orbitals, five at each fragment: ϕ0I1 , ϕ0I2 , ..., ϕ0I5 and ϕ0J1 , ϕ0J2 , ..., ϕ0J5 . Five of these
linear combinations are disregarded, namely,
1
(ϕ0Ia − ϕ0Ja ), a = 1, .., 5
0
)
2(1 − SIaJa
p
whereas the linear combinations
1
(ϕ0Ia + ϕ0Ja ), a = 1, ..., 5
0
2(1 + SIaJa
)
p
are used as doubly occupied orbitals for [Mn2 O11 ]. All other transformed fragment
”inactive” orbitals are also used as doubly occupied orbitals for the super-cluster.
For a discussion of this particular choice, see Chapter 5. The active orbital sets of
the fragments have very little contribution from the shared oxygen ion. The active
orbital sets of the super-cluster are constructed by super-posing the active orbitals
of the fragments, and Gramm-Schmidt orthogonalizing them to the doubly occupied
orbitals and to each other. Since the CASCI wave functions for the super-clusters
are expressed in terms of only the occupied (inactive and active) orbitals, there is no
need to generate virtual orbitals for the super-clusters from fragment calculations.
Next, the localized wave functions ΦX
x of the super-clusters and their energies Ex
CIJ(KL)
(Hab(cd) ) are determined by CASCI calculations in terms of the combined transformed orbital basis. This calculation yields the best wave functions that can be
obtained by distributing the electrons that are not in the doubly occupied inactive
orbitals in all possible ways over the active orbitals. The CASCI wave functions obtained in this manner usually have the added hole or added electron localized either
around one or another Mn ion, i.e. the hole or electron does not delocalize.
Note also that each localized CASCI wave function of a super-cluster is expressed
in its own orbital basis. Let us illustrate this statement with an example for a
super-cluster [O5 MnI OMnJ O5 ]15− representing hole doped LaMnO3 . For a state
with the hole localized around MnJ , the CASCI wave function is expressed in terms
4.4 Material model and computational information
91
of [MnI O6 ]9− -derived orbitals combined with [MnJ O6 ]8− -derived orbitals, whereas a
state with the hole localized around MnI is described with [MnI O6 ]8− -derived orbitals
combined with [MnJ O6 ]9− orbitals. The resulting localized CASCI wave functions of
[O5 MnI OMnJ O5 ]15− are mutually non–orthogonal. Furthermore, if we consider the
CASCI wave function for [O5 MnI OMnJ O5 ]15− with a hole localized around MnJ ,
there are two choices for the embedding of the ’undoped’ [MnI O6 ]9− fragment. The
first is to neglect the presence of the hole at MnJ ; the second is to include in the
embedding for [MnI O6 ]9− the presence of the hole at MnJ by increasing the effective
charge at this embedding ion by one. In section 4.5, we study both choices and discuss
the differences.
Finally, we compute the Hamiltonian matrix elements and overlap integrals between the localized wave functions and use them to determine the DE parameters t.
CIJ
As mentioned in Chapter 3, the computation of the off-diagonal elements Hab
and
CIJ
overlap integrals Sab is not a trivial task because the localized CASCI wave functions
ΦIa and ΦJb are expressed in mutually non-orthogonal orbital sets. The calculation of
these matrix elements is performed with the RASSI approach discussed in Chapter 2
( [71, 81]).
4.4
Material model and computational information
The three manganites crystalize in a perovskite crystal structure [25–27] in which each
Mn ion is surrounded by six oxygen ions. The MnO6 octahedra are tilted with respect
to each other and the Mn-O-Mn angles vary in the range 155-160◦ . The crystal space
group is Pbnm and the unit cell contains two different types of oxygen ions O(1) in
the ac and bc planes and O(2) in the ab planes (see Figure 4.1).
In LaMnO3 , the Mn-O bond lengths within the ab planes and along the c axes differ
considerably and can be regarded as long (2.18 Å), short (1.91 Å) and intermediate
(1.97 Å), respectively. The ionic anzsatz leads to charges of +3 for Mn, -2 for O
and +3 for La. The electronic configurations are Mn3+ (....3d4 ), O2− (....2p6 ) and
La3+ (....5p6 ). The ground state of Mn3+ in nearly Oh site symmetry arises from the
weak field electron configuration t32g e1g and in terms of the irreducible representations
of the Oh point symmetry group is a high spin coupled 5 Eg state. The degeneracy
of the ground state favours a Jahn-Teller distortion of the MnO6 octahedra. The
distorted MnO6 octahedra have alternating short and long Mn-O bonds in the ab
plane. The occupied eg -like orbitals are along the long Mn-O bonds that alternate
between neighbouring Mn3+ ions [36, 37, 80].
The crystal structure of La0.75 Ca0.25 MnO3 has been refined at 100 K [27]. In
this temperature region, low below the Curie temperature (Tc ) of the compound
(∼ 220-250 K), the ground state is ferromagnetic and metalic [27–29, 36, 39]. The
system can be viewed as a doped antiferromagnet with a hole concentration of 25 %,
in which according to conventional studies [27, 28, 36, 61] the double exchange plays
an important role. The bond lengths are more homogenous compared to those in
LaMnO3 , but one can still distinguish alternating long (1.974 Å) and short (1.957 Å)
Mn-O bond lengths in the ab plane and an intermediate (1.968 Å) one along the c
92
Chapter 4, Double exchange in manganites
axes. An ionic model may lead to distinct Mn sites with charges +4 and +3, O with
-2 and La and Ca with +3 and +2, respectively. Often, La1−x Cax MnO3 systems have
been considered within an ionic model. However, the eg electrons are assumed to be
delocalized over the Mn sites in the metalic ferromagnetic phase. Studies based on
X-ray absorption near edge structure spectroscopy at the Mn K edge [33] have shown
that the description of the valence state of the Mn atoms in terms of intermediate
valence between 3+ and 4+ is more appropriate than the ionic model. Therefore,
the ionic model of the crystal can only be a starting point. Within our method, we
consider the hopping of a eg -like electron between Mn sites and the magnitude of the
effective hopping matrix elements, associated with this hopping, reflects indirectly the
delocalization of the hole. We computed these matrix elements in a simplified model
that considers the hopping of a single eg -like electron between two Mn sites. In terms
of the irreducible representations of Oh the three electrons of an M n4+ ion occupy
the three 3d (t2g )-like orbitals and are coupled to a 4 Ag state. The formal weak-field
electronic configuration of M n3+ is t32g e1g coupled to 5 Eg . The states with an electron
in either one of the Mn 3d (eg -like) orbitals are close in energy due to the absence of
large JT distortions. In Oh symmetry they are the two components of a 5 Eg state.
X-ray diffraction experiments do not detect any long-range order of La and Ca
cations at the A-site (A1−x A0x M nO3 ) in La0.75 Ca0.25 MnO3 [27, 28] and assume fully
randomly mixed La/Ca at A. A long range order in the La/Ca lattice would result
in a large unit cell and/or breaking of the Pbnm symmetry [28]. These observations
lead to a randomness in the positions of the nearest neighbouring Ca ions around the
model [M n2 O11 ] clusters.
Taking into consideration the presence of either one of the two different cations
La/Ca at each site A (A1−x A0x M nO3 ), we investigate for chosen distributions of the
nearest neighbouring cations La/Ca around the [M n2 O11 ] clusters the effect on the
magnitude of the effective hopping matrix elements. The latter is directly coupled
to the orientation of the occupied eg -like orbitals in the lowest in energy states at
next-neighbouring Mn sites.
Unlike in LaMnO3 and La0.75 Ca0.25 MnO3 the bond lengths in CaMnO3 are almost equal (1.900±0.005 Å). The ionic anzsatz leads to charges of +4 for Mn, -2
for O and +2 for Ca, with electronic configurations Mn4+ (....3d3 ), O2− (....2p6 ) and
Ca2+ (....3p6 ). The corresponding ground state of Mn4+ in nearly Oh site symmetry
is 4 Ag , resulting from the high spin coupling of the three electrons that occupy the
3d (t2g ) sub-shell of the open Mn (3d) shell.
As explained in section 4.1.2, we expect a spatial anisotropy of the DE parameters
in LaMnO3 . In CaMnO3 , the states with an extra electron in either one of the Mn
3d (eg -like) orbitals are quite close in energy, because of the almost octahedral Mn
site symmetry. Moreover, the magnetic coupling is antiferromagnetic in all directions.
Therefore, for CaMnO3 , the DE parameters are expected to be similar in all spatial
directions. In order to investigate the spatial anisotropy of the DE parameters, we
designed various super-clusters, both in the ab crystal plane and perpendicular to
it. To distinguish the two alternatives, the super-clusters are indexed by an ab or c
subscript. La0.75 Ca0.25 MnO3 is metalic [27, 29, 36, 39] and hence, t is anticipated to
4.5 Results and Discussion
93
have a larger magnitude.
The nearest La, Ca and Mn ions around the fragments and around the superclusters are represented by effective one-electron potentials, or Total Ion Potentials
(TIPs). In the present case, we use the effective core potentials by Hay and Wadt [60]
as TIPs. The long-range interactions are described by an array of point charges at
lattice positions, optimized to reproduce the Madelung potential of the rest of the
crystal within the fragment or the super-cluster region.
The orbitals are expanded in atomic natural orbitals (ANO) type basis functions
[34, 35]. A primitive set of [21s, 15p, 10d, 6f] Gaussians is contracted to (5d, 4p,
3d, 1f) for Mn and for O a [14s, 9p] set is contracted to (4s, 3p) Gaussian type
of basis functions. All calculations are performed with MOLCAS-5.4 [81] and our
CIJ
TRANSFORM program, which is integrated in MOLCAS-5.4. The Hamiltonian Hab
CIJ
and overlap Sab matrix elements are computed using the State Interaction module
in MOLCAS [79]. Mulliken population analysis (MPA) of the natural orbitals and
MPA are carried out in order to determine the character of the singly occupied Mn
3d (eg )-like orbital. In addition, some difference spin density plots are produced to
illustrate the character of the singly occupied Mn 3d (eg )-like orbital.
4.5
4.5.1
Results and Discussion
LaMnO3
We consider the hopping of a single hole in LaMnO3 . We start with constructing
the wave functions ΦX
x (Xx ∈ Ia, Jb) of the super-clusters [Mn2 O11 ] as CAS CI wave
functions (see section 4.2.2). As noted above, the ΦX
x for super-clusters [Mn4 O20 ] are
obtained in the same way.
CASSCF calculations are carried out for the ’undoped’ [MnO6 ]9− and ’hole doped’
[MnO6 ]8− fragments with an active space containing four electrons in five 3d-like
orbitals and three electrons in three 3d (t2g )-like orbitals, respectively. The orbitals
of the ’undoped’ [MnO6 ]9− fragment are optimized for the ground state of the pure
LaMnO3 . As explained in section 4.2.2, we perform the CASSCF calculation for this
ground state using two different embeddings: without or with an increased positive
charge at one of the neighbouring Mn sites. We discuss this issue in more detail below.
In orthorhombic LaMnO3 , the ground state of the structure is viewed as derived
from one of the components of the 5 Eg -like state. The other component is found at 1.2
eV higher energy (see Chapter 6 ). Considering the z axes directed along the long MnO bond, the occupied 3d (eg )-like orbital of the ground state of the [MnO6 ]9− fragment
has predominantly 3dz2 character and according to MPA consists of 94% Mn- 3d. The
occupied 3d (eg )-like orbital of the higher 5 Eg -like state has predominantly 3dx2 −y2
character. The orbitals of the ’hole doped’ [MnO6 ]8− fragment are optimized for
the 4 Ag -derived ground state of the fragment arising from the Mn 3d3 electron
configuration. For the [MnO6 ]8− fragment, we have only three active electrons in
three 3d (t2g ) -like orbitals. In all cases the singly occupied t2g -like orbitals consist
of 96-98% Mn- 3d.
94
Chapter 4, Double exchange in manganites
Table 4.1: Overlap integrals between transformed doubly occupied orbitals of two overlapping
fragments, representing LaMnO3 . Ground states wave functions. Values below 0.03 are not
listed.
A
B
C
1.0000
1.0000
1.0000
0.9990
0.9988
0.9991
0.9945
0.9923
0.9927
0.9940
0.9908
0.9918
0.9755
0.9524
0.9624
0.0431
0.0393
0.0409
0.0334
0.0305
0.0322
A. [Mn(1) O6 ]9− fragment overlapping with [Mn(2) O6 ]9−
B. [Mn(1) O6 ]8− fragment overlapping with [Mn(2) O6 ]9− (nearby hole not included).
C. [Mn(1) O6 ]8− fragment overlapping with [Mn(2) O6 ]9− (nearby hole accounted for).
The super-clusters [Mn2 O11 ] are built from overlapping [MnO6 ] fragments according to the method of the overlapping fragments, outlined in sections 4.2.2 and 3.3.
The occupied orbital basis of the super-cluster is formed from the occupied CASSCF
orbitals of the fragments after elimination of the double-counted doubly occupied orbitals associated with the common oxygen ion. The orbitals to be disregarded are
determined by a corresponding orbital transformation [68] of the inactive orbitals of
the two fragments. As an illustration for the case of LaMnO3 , the overlap integrals
between the transformed inactive orbitals of a pair of overlapping [MnO6 ]9− fragments
in their ground state are listed in Table 4.1.
We also show the overlap integrals between the transformed inactive orbitals of a
[MnO6 ]9− fragment overlapping with [MnO6 ]8− . Five orbitals at each fragment have
an overlap integral of 0.95 or larger with their ”corresponding” orbital at the other
fragment. These five orbitals are the double counted orbitals, localized at the common
oxygen ion. Five (normalized) plus linear combinations, one for each of the five pairs
are constructed and employed as doubly occupied orbitals for the super-cluster. The
remaining transformed inactive orbitals that have overlap integrals below 0.05 with
their ”corresponding” orbital at the other fragment are also used as inactive orbitals
for the super-cluster.
Within the ab planes and perpendicular to them, there are two different Mn-OMn units with slightly different geometries. We designed different [Mn2 O11 ]15− and
[Mn4 O20 ]27− clusters in order to compute the different parameters t. The overall
charges of the clusters correspond to the formal charge of a system with a hole. We
choose to describe in detail the results for two [Mn2 O11 ]15− clusters, one within the
ab plane and the other perpendicular to the ab plane. The results for these clusters
are representative for all [Mn2 O11 ]15−
and [Mn2 O11 ]15−
clusters. They are listed in
c
ab
Tables 4.2 and 4.4.
The clusters chosen to illustrate the analyses contain Mn ions 1 and 2, both in the
ab planes, (Figure 4.1 (a)) for [Mn2 O11 ]ab and Mn ions 2 and 4 for [Mn2 O11 ]c (Figure
4.2 (a)). Let us start with [Mn2 O11 ]ab . This cluster contains two different Mn-O
bond lengths of 2.18 Å and 1.91 Å. The Mn-O-Mn angle with the bridging oxygen
is 155.1◦ . The super-cluster energies (Table 4.2) and the corresponding CASCI wave
functions are obtained within an active space of eight Mn-3d-like orbitals: five derived
Results and Discussion: LaMnO3
95
Figure 4.1: Representation of the [Mn4 O20 ] super-clusters in the ab plane for a) LaMnO3
and b) CaMnO3 . Eight oxygen ions above and below the ab plane are not shown for clarity.
They are present in the clusters.
Figure 4.2: Representation of the [Mn4 O20 ] super-clusters in a perpendicular to the ab
plane for a) LaMnO3 and b) CaMnO3 . Eight oxygen ions above and below the ab plane are
not shown for clarity. They are present in the clusters.
96
Chapter 4, Double exchange in manganites
Table 4.2: DE parameters tab , overlap integrals S12 and Hamilton matrix elements H12
between two localized CASCI wave functions in a [Mn2 O11 ]15−
super-cluster representing
ab
hole doped LaMnO3 . 7 electrons in 5+3 active orbitals; Last row: tab obtained with 5+5
active orbitals.
a) The effect of the nearby hole on the orbitals of the ’undoped’ [MnO6 ]9− fragment is not
included.
2S+1
−tab (meV)
S12
-H12 (hartree)
H22 , H11 (meV)a
−tab (meV) 5d+5d
a
8
213
0.392.10−3
1.283911
0; 37
213
6
160
0.293.10−3
0.960215
-1; 37
160
4
106
0.190.10−3
0.621238
-3; 37
107
2
53
0.095.10−3
0.310576
-4; 38
53
These CASCI energies have been shifted upwards by 3292.321008 hartree
b) The hole at one embedding Mn ion of the [MnO6 ]9− fragment in a ground state configuration
is represented by increasing the effective charge at that ion by one.
2S+1
−tab (meV)
S12
-H12 (hartree)
H22 , H11 (meV)a
a
8
263
4.307.10−3
14.188303
0; 105
6
197
3.232.10−3
10.647287
-1; 104
4
132
2.154.10−3
7.095911
-2; 106
These CASCI energies have been shifted upwards by 3292.325211 hartree
2
66
1.077.10−3
3.547820
-3; 106
Results and Discussion: LaMnO3
97
from the [MnO6 ]9− fragment and three from the [MnO6 ]8− fragment. We denote this
active space as 5d+3d. The CASCI wave function for each localized state of [Mn2 O11 ]
is expressed in its own dedicated orbital basis. The 5d+3d active space prevents a
charge transfer from the 3d (eg ) -like orbitals around one Mn ions to those around
the other. We compare the results obtained within the 5d+3d active space with those
obtained within 5d+5d active space (last row of Table 4.2 (a)), i.e. an active space
of five 3d-like active orbitals around each Mn ion, where the charge transfer can take
place. The two choices for the embedding of the [MnO6 ]9− fragments, discussed in
section 4.2.2, are investigated and the results are compared in Tables 4.2 (a) and (b).
In the first case, we neglect the presence of the nearby hole (Table 4.2 (a)). Secondly,
we account for the presence of the hole at the Mn ion in [MnO6 ]8− when designing the
embedding for the [MnO6 ]9− fragment. We do so by increasing the effective charge of
that embedding Mn ion of [MnO6 ]9− , which corresponds to the Mn ion in [MnO6 ]8− ,
by one (Table 4.2 (b)).
We investigated all possible spin couplings of the seven active electrons of [Mn2 O11 ]15−
but found for the lowest state of each spin, as expected, high spin coupling at the
individual Mn ions. The different total spin values, S, arise from different couplings
between the Mn ions. The first column of Table 4.2 shows results for the maximum
coupling of the two local spins, S1 =2 and S2 = 23 , to S= 72 . The next columns are for
lower S values. In the first row of Table 4.2 (a) and (b), we list the results for tab as
a function of different S values. The values of the DE parameters are obtained from
Eq. (4.5). Since the two localized CASCI wave functions are each expressed in their
own orbital set, they are non-orthogonal. In the second row are listed the overlap
n2
n1
and ΦM
, localized at
integrals S12 between the two CASCI wave functions, ΦM
a
b
Mn1 and Mn2 , respectively. The third row shows their Hamiltonian matrix elements
H12 . We found indeed a proportionality of tab to the total spin multiplicity 2S+1, in
perfect agreement with the simplified model of Anderson and Hasegawa [23] (see Eq.
(4.2)). The overlap integrals S12 are small for all spin values but significant for the
computation of tab . Note, that H12 and S12 are also roughly proportional to 2S+1.
This proportionality holds for all [Mn2 O11 ]15−
ab clusters, considered. As we will see
later it also holds for all [Mn2 O11 ]15−
clusters.
In row 4 in Table 4.2 (a) and (b) are
c
shown the relative energies (H11 and H22 ) of the localized states with a hole around
either one of the ions, Mn1 and Mn2 . Their energy difference is caused by the slightly
different crystal (and cluster) environment of Mn1 and Mn2 . In larger clusters, where
the two Mn1 and Mn2 ions have equivalent environments, this energy difference is not
present. We discuss this issue in more detail in the first part of Chapter 5.
The CASCI energies of the localized states (i.e., H11 and H22 ) are also quite
similar with only small variations of few meV when the total spin is varied from
S= 27 to S= 12 . Although the signs of the exchange couplings are incorrect and their
magnitude is systematically underestimated at this level of approximation, we do not
anticipate the magnitude of the exchange interaction between neighbouring Mn3+
and Mn4+ ions in the ab planes to become comparable to that of the DE interactions
between them if the level of approximation is improved. We address this question
below. Finally, we list in the last row in Table 4.2 the values of t, obtained with 5
98
Chapter 4, Double exchange in manganites
active 3d-like orbitals not only for the [MnO6 ]9− fragment, but also for the [MnO6 ]8−
fragment. The results obtained within 5d+5d active space differ insignificantly from
those obtained in 5d+3d. Although in the 5d+5d case, charge transfer processes from
Mn 3d4 to Mn 3d3 are allowed by providing empty eg -like orbitals at the Mn4+ site,
the results remain the same.
Comparing the results in Table 4.2 (a) and (b) reveals that indeed the effect on
the values of S12 , H12 and tab , of accounting for the presence of a neighbouring hole in
the calculation for the [MnO6 ]9− fragments is substantial. One observes significantly
larger values of S12 , H12 and t when the fragment [MnO6 ]9− embedding contains
information regarding the presence of a hole at the neighbouring Mn4+ ion in the
[Mn2 O11 ]15−
super-cluster. This is because the eg -like orbital is more delocalized
ab
which enhances the DE interaction. The results in Table 4.2 (b) indicate that t
is underestimated with 19 % if the presence of the hole is not accounted for. As
stated in Chapter 1, the hopping integral t is influenced by relaxation effects. Let
us create holes, associated with particular orbitals, φ1 and φ2 , localized at sites Mn1
and Mn2 , respectively within the frozen orbital (FO) approximation. Then let us
compute within the FO approximation the Hamiltonian and overlap matrix element
between the corresponding localized many-body wave functions Φ1 and Φ2 of the
cluster system with a hole either in φ1 or φ2 . If all other cluster orbitals except those
involved in the interaction, φ1 and φ2 , are allowed to relax in response to the created
hole, the overlap integral hΦrelax
|Φrelax
i becomes smaller since the orbital polarization
1
2
relax
and Φrelax
. This also leads to a decrease in
effects have opposite direction in Φ1
2
relax
relax
hΦ1
|H|Φ2
i and t (see, for example, [89]). If all cluster orbitals are allowed
to relax including the orbitals with the holes, φ1 and φ2 , the latter are allowed to
delocalize. This compensates the reduction due to relaxation of the other orbitals and
may even lead to an increase in the magnitudes of hΦrelax
|Φrelax
i, hΦrelax
|H|Φrelax
i
1
2
1
2
and t [90].
The CASCI states of the super-cluster of Table 4.2 (b) have still the extra electron localized around the longest bond of either one of the two Mn ions but their
energy differs now by about 105 meV compared to 37 meV, when the presence of the
neighbouring hole is neglected. The larger energy difference arises from the different
orientation of the occupied Mn 3d (eg )-like orbital at either one of the two Mn ions
with respect to the neighbouring hole and hence, depending on its orientation different relaxation effects on the Mn 3d (eg )-like orbital occur. Taking into consideration
that the Mn ions have ferromagnetic spin coupling in the ab planes, the DE parameter, relevant for the physics of LaMnO3 , is the one for maximum spin coupling S,
t( 72 )ab =-263 meV. This is the bold number in Table 4.2 (b).
In Table 4.3, we show the same quantities as in Table 4.2, but now obtained with
the more traditional approach, i.e. from CASSCF wave functions of the [Mn2 O11 ]ab 15−
super-cluster. Starting with the CASCI orbitals within the 5d+3d active space we
were able to obtain converged CASSCF wave functions in which the occupied Mn
3d (eg )-like orbital is still localized around each of the two Mn ions. However, the
near-degeneracy of the two localized states is lost, because the super-cluster environment is quite different for the Mn 3d (eg ) electrons, localized either at Mn2 or Mn1 ,
Results and Discussion: LaMnO3
99
Table 4.3: DE parameters tab , overlap integrals S12 and Hamilton matrix elements H12 , obtained from the interaction between two localized CASSCF wave functions in a [Mn2 O11 ]15−
ab
super-cluster representing hole doped LaMnO3 . Single root CASSCF, 7 active electrons in
5+3 active orbitals.
2S+1
−tab (meV)
S12
-H12 (hartree)
H22 , H11 (meV)a
a
8
294
73.096.10−3
240.671373
0; 319
6
220
52.928.10−3
174.268121
4; 320
4
148
33.483.10−3
110.243453
6; 321
2
75
16.241.10−3
53.475248
7; 321
These CASSCF energies have been shifted upwards by 3292.394046 hartree
respectively. This cluster artifact is emphasized more in this case compared to the
CAS CI calculation, because the relevant Mn 3d (eg )-like orbitals are optimized for
the super-cluster [Mn2 O11 ]15−
ab and reflect the incorrect electron distributions around
the Mn ions due to their different environment in the super-cluster. The two localized
states differ by ≈ 0.3 eV. Furthermore, the relaxation process leads to an increase in
S12 and H12 and an overestimation of the DE interaction by 12 %.
By using the CASCI orbitals within the 5d+5d active space as starting orbitals,
we were unable to obtain converged CASSCF wave functions. The reason is that the
correlating Mn 3d’ t02g -like orbitals for the singly occupied Mn 3d (t2g )-like orbitals
at the Mn4+ ion replace in the active space the Mn 3d (eg ) -like orbitals at this ion.
Next, we discuss the magnitude of the magnetic exchange parameter J between
the Mn3+ and Mn4+ ions in connection with the magnitude of the DE parameter between these ions. Numerous studies of the magnetic exchange parameter J in various
crystal structures (cuprates, vanadates, ets.) [6, 8, 9, 13] have shown that a quantitative estimate of the parameters can be extracted within the difference dedicated
CI (DDCI) [83, 84] or the iterative DDCI [85] approaches as well as within second
order perturbation theory based on complete active space expansion of the zero-order
wave function (CASPT2) [72]. A recent study on the performance of the different approaches [63], has classified the DDCI and CASPT2 methods based on an extended
reference wave functions as the most appropriate computational methods to obtain
magnetic coupling constants in ionic TM insulators with average errors of less than
10 % [63]. Although the CASPT2 values of J with a reference wave function, that
includes only the magnetic orbitals, so called minimal active space, are about 20 %
smaller than the DDCI and experimental results [63], they still provide a rough estimate of J. We will extract the values of the nearest neighbour J by mapping the
CASPT2 energies onto the eigenvalues of the Heisenberg Hamiltonian (Eq. 4.8). The
values of J obtained from the energy differences between each of the states with lower
total spin S’ and the state with maximum total spin S,
E(S) − E(S 0 ) =
−J
(S(S + 1) − S 0 (S 0 + 1))
2
(4.8)
show a maximum deviation from the average value of J of ≈ 0.1 meV. Although the
100
Chapter 4, Double exchange in manganites
system does not behave as a strict Heisenberg system, the estimates of J are rather
similar. Taking into account, that the CASPT2 average value of J between the Mn3+
and Mn4+ ions, obtained within a minimal 5d+3d active space is only 1.2 meV, we
conclude that the DE parameter t in the ab planes is an order of magnitude larger
than the exchange coupling J.
The hole-delocalization energy is proportional, at least for small doping, to the
hole concentration. The hopping interaction in the ab planes is large enough to lead
to an appreciable ”band” stabilization energy of about 1.1 eV per hole (see part 2
of Chapter 5 ), due to delocalization in this plane. Since the energy scales of DE
parameters and JT distortions are comparable, there will be present a competition
between JT stabilization and hole-delocalization energy. This issue will be addressed
again in part 2 of Chapter 5.
We recalculated the quantities in Table 4.2 for an idealized structure within 5d+3d
active space, where the Jahn-Teller distortion has been ”undone” but the tilting of
the octahedra is still present. The analyses have shown that similar dependencies as
shown in Table 4.2 are present also for the undistorted LaMnO3 . The relevant DE
interaction in the ab plane is indeed much stronger, twice as large: t( 72 )ab (idealized)
=-0.57eV. This result points out that indeed the JT distortion is unfavorable for DE.
If we consider in detail the mutual relative orientation of the occupied Mn 3d-like
orbitals localized at either one of the Mn ions and relevant to the DE interaction, we
observe that they are ordered within the ab planes, however they are rotated with
respect to their orientation in the distorted structure. The rotation causes a larger
overlap and a twice as large value of t. The relative orientations of the occupied
Mn 3d-like orbitals localized at either one of the Mn ions in both JT distorted and
undistorted LaMnO3 is illustrated in Figure 4.3. Since the hopping interaction in the
ab planes is larger, the band stabilization energy, due to the hole delocalization in
this plane, is also larger: 2.3 eV per hole.
In Table 4.4, we list the same quantities as in Table 4.2, but now for the model
[Mn2 O11 ]c 15− cluster. We found very similar behavior of t, H12 and S12 as functions
of the total spin S. As expected, S12 and H12 increase when the [MnO6 ]9− fragment
orbitals are obtained, accounting for the presence of a hole in the neighbourhood
(Table 4.4 (b)). The exchange coupling between the Mn ions along the c axis is
antiferromagnetic and hence the relevant parameter is t( 12 )c . This is the bold number
in Table 4.5. The value of t( 12 )c is -59 meV and the delocalization of a hole along
c adds only 0.1 eV to the 1.1 eV band stabilization energy for a hole delocalization
in the ab planes. We notice, that the relative energies H11 and H22 of the localized
states with a hole around either one of the ions Mn2 and Mn4 (see Figure 4.1) are the
same, as expected. In this cluster, the Mn2 and Mn4 ions have the same environment
and we do not observe cluster artifacts. Since the occupied Mn 3d (eg )-like orbital
is oriented in the same manner at either one of the two Mn ions with respect to the
neighbouring hole, the CAS CI states of the cluster with the extra electron localized
around the longest bond of either one of the two Mn ions remain degenerate.
In order to investigate whether or not the DE interactions depend significantly on
the cluster size, we compared the t values obtained from the super-clusters [Mn2 O11 ]15−
Results and Discussion: LaMnO3
101
Figure 4.3: Mutual orientation of the occupied eg - like orbitals at sites Mn1 and Mn2 in the
[Mn2 O11 ] super-cluster, representing hole doped JT distorted and undistorted but yet tilted
LaMnO3 .
102
Chapter 4, Double exchange in manganites
Table 4.4: DE parameters tc , overlap integrals S12 and Hamilton matrix elements H12
between two localized CASCI wave functions in a [Mn2 O11 ]15−
super-cluster representing
c
hole doped LaMnO3 . 7 electrons in 5+3 active orbitals; Lowest row: tCASSCF
obtained in a
c
single root CASSCF calculation.
a) The effect of the nearby hole on the orbitals of the ’undoped’ [MnO6 ]9− fragment is not
included;
2S+1
−tc (meV)
S12
-H12 (hartree)
H22 , H11 (meV)a
(meV)
−tCASSCF
c
a
8
171
0.132.10−3
0.427311
0; 0
259
6
129
0.097.10−3
0.313216
-2; -2
195
4
86
0.064.10−3
0.206390
-3.5; -3.5
130
2
43
0.032.10−3
0.102528
-4.4; -4.4
68
These CASCI energies have been shifted upwards by 3292.910723 hartree
b) The hole at one embedding Mn ion of the ’undoped’ [MnO6 ]9− fragment is represented by
increasing the effective charge at that ion by one;
2S+1
−tc (meV)
S12
-H12 (hartree)
H22 , H11 (meV)a
a
8
237
2.862.10−3
9.433617
0; 0
6
178
2.150.10−3
7.085123
-1; -1
4
119
1.436.10−3
4.734186
-3; -3
These CASCI energies have been shifted upwards by 3292.913226 hartree
2
59
0.718.10−3
2.369066
-4; -4
Results and Discussion: LaMnO3
103
Table 4.5: DE parameters, for [Mn4 O20 ]27− clusters representing hole doped LaMnO3 in
the ab plane and perpendicular to the ab plane. In Figures 4.1 and 4.2 is illustrated the atom
labeling. Maximum spin multiplicity (2S+1=16) of 15 active electrons in 5+5+5+3 active
orbitals.
a) The effect of the nearby hole on the orbitals of the ’undoped’ [MnO6 ]9− fragment is not included.
1.
ion pairs
[Mn4 O20 ]ab 27−
-t 72 (meV)
1 2, 3’ 4’
1 3’, 2 4’
1 4’
2 3’
200
213
2
16
2.
ion pairs
[Mn4 O20 ]c 27−
-t 72 (meV)
1 2, 3 4
13
24
14
23
222
158
186
5
4
b) The hole at one embedding Mn ion of the ’undoped’ [MnO6 ]9− fragment is represented by
increasing the effective charge at that ion by one;
1.
ion pairs
[Mn4 O20 ]ab 27−
-t 72 (meV)
1 2, 3’ 4’
1 3’, 2 4’
1 4’
2 3’
245
258
5
18
2.
ion pairs
[Mn4 O20 ]c 27−
-t 72 (meV)
1 2, 3 4
13
24
14
23
267
214
252
2
2
with those obtained employing larger [Mn4 O20 ]27− super-clusters. Moreover, these
four-center clusters provide also the values of t for the other Mn-O-Mn units in the ab
plane and along the c axes, respectively. These two other [Mn2 O11 ]15− super-clusters
were not discussed, because as we noted already, we did not anticipate any significant
differences for the DE parameters, that would be calculated for these clusters.
In Table 4.5 (a) and (b), we show the results for a [Mn4 O20 ]27−
cluster, repab
resenting part of the ab plane (see Figure 4.1 (a)) and for a [Mn4 O20 ]c 27− cluster
perpendicular to this plane (see Figure 4.2 (a)). To simplify the computation, we first
used [MnO6 ]9− fragment orbitals constructed without accounting for the nearby hole,
just as in Table 4.2 (a). These results are reported in Table 4.5 (a).
The calculated magnitudes of the DE parameters illustrate the minor cluster dependence of the DE parameters: for example, t( 27 )ab between Mn ions 1 and 2 (see
Figure 4.1 (a)) and Figure 4.2 (a)) obtained from the different clusters varies between -200 meV to -222 meV. In analogy with the [Mn2 O11 ]15− super-clusters, we
studied the effect of accounting for the presence of the nearby hole in the [MnO6 ]9−
fragment calculations on the DE interaction. As expected, the magnitude of the DE
interactions is underestimated when the presence of the nearby hole is neglected, for
104
Chapter 4, Double exchange in manganites
example, the values of the DE parameters t( 72 )ab between Mn ions 1 and 2 (see Figure 4.1 (a) and Figure 4.2 (a)) are underestimated by about 17-18 % (Table 4.5 (a))
compared to the values obtained in the presence of the nearby hole: -245 meV and
-267 meV (bold numbers in Table 4.5 (b)). Comparing the results for the other DE
parameters in Tables 4.5 (a) and 4.5 (b) reveals similar or even larger underestimations of t. Furthermore, comparing the results in Table 4.5 (b) with the relevant
results for the [Mn2 O11 ]15− clusters in Tables 4.2 (b) and 4.4 (b) demonstrates again
the very little cluster dependence of t. The other nearest-neighbour DE interaction
in the ab plane (between Mn ions 1 and 3’) is indeed similar to that between Mn
ions 1 and 2 (see above): -258 meV in [Mn4 O20 ]27−
ab cluster. We recall, that the DE
interaction between Mn ions 1 and 2 in the [Mn2 O11 ]15−
cluster is -263 meV. The
ab
next nearest-neighbour DE interactions in the ab plane between ions 1 and 4’ and
ions 2 and 3’ are an order of magnitude weaker than the nearest-neighbour interactions. The two symmetry-unique nearest neighbour interactions along c, t( 72 )c in the
super-cluster between Mn ions 1 and 3 and ions 2 and 4 are -214 and
[Mn4 O20 ]27−
c
-252 meV, respectively (see Table 4.5 (b)). The exchange coupling between Mn ions
along the c axis is antiferromagnetic, so the relevant parameter is t( 21 )c . Owing to
computational limitations, we could only consider the states with high spin coupling
between the four Mn ions. However, assuming 2S+1 proportionality as found in the
smaller [Mn2 O11 ]15− super-clusters allows us to deduce the low spin parameters in
the [Mn4 O20 ]27− clusters. We estimate for t( 12 )c : -54 meV and -63 meV, respectively.
The result for t( 21 )c between ions 2 and 4 in the [Mn2 O11 ]c 15− cluster is -59 meV (see
Table 4.4 (b)). Finally, the next nearest neighbour interactions out of the ab plane
are negligible.
4.5.2
CaMnO3
The Mn-O bond lengths within the ab plane of CaMnO3 are similar although one
can still distinguish alternating bond lengths (1.900 Åand 1.903 Å). The shortest
bonds (1.895 Å) are perpendicular to this plane. Since the nearest-neighbour exchange interactions are antiferromagnetic in all directions [52], the most relevant DE
parameters are those for low spin coupling, t( 21 ) . Analogous to LaMnO3 , we first perform CASSCF calculations for the ’undoped’ [MnO6 ]8− fragment and for the ’electron
doped’ [MnO6 ]9− fragment within an active space containing three electrons in three
Mn- 3d-like (t2g -like) active orbitals and four electrons in five Mn- 3d-like active orbitals, respectively. This active space, analogous to that for the LaMnO3 , is referred
to as 5d+3d. The orbitals of the ’undoped’ fragment were optimized for the 4 Ag derived ground state, taking into account the nearby added electron via an decreased
effective nuclear of the embedding Mn ion at hand. The t2g -like orbitals consist of 96
% of Mn 3d. An ’electron doped’ [MnO6 ]9− fragment has two low-lying states, corresponding to the two components of 5 Eg in Oh symmetry. Contrary to LaMnO3 , these
two components are near-degenerate, the energy splitting between them is only 0.05
eV. The [MnO6 ]9− fragment orbitals are optimized for the lower of the two states. In
this state, the occupied eg -like orbital has predominantly Mn 3dz2 character, with the
Results and Discussion: CaMnO3
105
z axes along the longest bond 1.903 Å. It consists of 92 % Mn 3d. This orbital configuration is the same as in LaMnO3 . In the second 5 Eg -derived state, the occupied
eg -like orbital is predominantly Mn 3dx2 −y2 . In the ’undoped’ [MnO6 ]8− fragments,
the presence of the nearby extra electron is represented by decreasing the effective
nuclear charge at the relevant embedding Mn ion by one.
Again like in LaMnO3 , one can distinguish within the ab plane and perpendicular
to it two different Mn-O-Mn units with slightly different geometries. We choose to
analyze in detail only one of the model [Mn2 O11 ]15− super-clusters, the one within
the ab plane. As in the case of LaMnO3 , the results for this cluster are representative
clusters. Like in LaMnO3 two [Mn4 O20 ]25−
for all [Mn2 O11 ]15−
and [Mn2 O11 ]15−
c
ab
super-clusters, one within the ab plane, and the other perpendicular to this plane
were selected. The [Mn4 O20 ]25− super-clusters are illustrated in Figures 4.1 (b) and
4.2 (b).
The results for the [Mn2 O11 ]15−
ab cluster modeling the ab plane (Mn atoms 1 and
2 in Figure 4.1 (b)) are summarized in Table 4.6 (a) and (b). The Mn-O-Mn angle
is 157.2o . The two lowest CAS CI states of this super-cluster have the extra electron
localized around the longest bond of either one of the two Mn ions. Their energy differs by ≈ 50 meV (see Table 4.6 (a)) if one neglects the presence of the nearby extra
electron and increases up to about 80 meV otherwise (Table 4.6 (b)). This energy
difference arises from the slightly different environment of the two Mn ions. Because
the exchange interactions in all directions of the crystal are antiferromagnetic, the
relevant parameter is t( 12 )ab and the DE interaction between the two lowest localized
states is t( 21 )ab =-171 meV. As in LaMnO3 , we find almost perfect 2S+1 proportionality of t. The results obtained within 5d+5d active space differ again insignificantly
from those obtained within 5d+3d upon the condition that we provide empty eg -like
orbitals at the Mn4+ site.
Unlike in LaMnO3 , there are four low-lying localized states in [Mn2 O11 ]15− . The
lower two were already described above, and the higher two states have the extra
electron in one the other two available Mn 3d (eg )-like orbitals at each Mn site. The
two additional states are at only about 0.16 eV higher energy. We performed also
calculations with all four low-lying localized states, allowed to interact. In this case
the orbitals of the fragment with an electron were optimized for an average of the
states. The energy splittings are larger due to the interaction of the four localized
states of [Mn2 O11 ]15−
ab . This results into a larger effective value of the DE parameter
(see the one-but-last row in Table 4.6 (a)). The resulting CASCI wave functions for
the two lower states, Φ1a and Φ2b , having the extra electron localized around Mn1 and
Mn2 , respectively, show a mixing of the nearly degenerate higher localized states,
described by Φ1a0 and Φ2b0 , in order to maximize the DE interaction present in the
super-cluster. This mixing occurs only in clusters where not all nearest neighbour
DE interactions are present and therefore, it is considered as a cluster artifact and
the result is not relevant for the physics of the doped CaMnO3 . The orientation and
character of the occupied eg -like orbitals at Mn1 and Mn2 for the four localized lowlying states are illustrated in Figure 4.4. To clarify the role of the two higher states,
we performed a new set of calculations based on a simple analysis.
106
Chapter 4, Double exchange in manganites
Table 4.6: DE parameter tab , overlap integrals S12 and Hamilton matrix elements H12 in
15−
[Mn2 O11 ]ab
super-cluster representing electron doped CaMnO3 .
a) The effect of the nearby electron on the orbitals of the ’undoped’ [MnO6 ]8− fragment is not
included.
2S+1
−tab (meV)
S12
-H12 (hartree)
H11 , H22 (meV)a
−tab (meV) 5d/5d
4E/2 (meV) 5d/3d
−tCASSCF
ab
8
452
2.039.10−3
6.757602
0; 52
450
682
939
6
339
1.542.10−3
5.109918
-2; 51
339
513
707
4
227
1.038.10−3
3.438945
-6; 47
225
343
474
2
113
0.521.10−3
1.727428
-7; 46
113
173
238
CASCI: 7 electrons in 5d+3d active electrons; in line five: t obtained with 5d+5d active space; last
obtained from the interaction between two localized CASSCF wave function.
row: -tCASSCF
ab
Single root CASSCF, 7 active electrons in 5d+3d active orbitals. 4E/2: half the energy splitting
4E obtained from a configuration interaction between all four low-lying states (see text);
a
These CASCI energies have been shifted upwards by 3306.229857 hartree
b) The added electron at one embedding Mn ion of the ’undoped’ [MnO6 ]8− fragment is
represented by decreasing the effective charge at that ion by one;
2S+1
−tab (meV)
S12
-H12 (hartree)
H11 , H22 (meV)a
a
8
683
4.490.10−3
14.870406
0; 79
6
513
3.389.10−3
11.224388
-3; 78
4
342
2.274.10−3
7.532520
-7; 73
2
171
1.141.10−3
3.779066
-9; 71
These CASCI energies have been shifted upwards by 3306.239710 hartree
CASCI: 7 electrons in 5d+3d active electrons.
Figure 4.4: Illustration of the mutual orientation of the relevant occupied eg -like orbitals
at Mn1 and Mn2 for all four low-lying localized states of the [Mn2 O11 ]15−
super-cluster,
ab
described by the localized CASCI wave functions, Φ1a , Φ1a0 , Φ2b and Φ2b0 .
Results and Discussion: CaMnO3
107
Let us consider a simplified model of the ’electron doped’ [Mn2 O11 ]15− supercluster in the ab plane, for which the tilting of the [MnO6 ] octahedra is effectively
”undone”. The longer bond l at one of the Mn ions (Mn2 ) within the ab plane
is directed along the x axis whereas at the other Mn ion (Mn1 ) along the y axis.
The interaction between the two lower localized CASCI states, discussed above, is of
strength t =-0.17 eV and involves the states for which the orientation of the occupied
eg -like orbitals is the following: (Mn2 dx2 )-(Mn1 dy2 ). These two lower states are
described by the CASCI wave functions Φ2b and Φ1a . This interaction alone leads to
a stabilization energy t and an energy splitting 2t. If we account, in addition, for the
presence of the two higher localized states, that have the two complimentary eg -like
orbitals occupied, (Mn2 dy2 −z2 )-(Mn1 dx2 −z2 ), described by Φ2b0 and Φ1a0 , we find that
one of those higher states, namely the one that √has the (Mn1 dx2 −z2 ) -like orbital
occupied (i.e., Φ1a0 ) interacts with a strength of 3t with the lower state that has
the (Mn2 dx2 ) -like orbital occupied (i.e., Φ2b ). This higher CASCI state, Φ1a0 , has a
negligible interaction with the other higher CASCI state with the (Mn2 dy2 −z2 ) -like
orbital occupied (i.e., Φ2b0 ). If we allow for all four states to interact, the problem is
reduced to diagonalizing the following symmetric 4 x 4 matrix of effective hopping
matrix elements,


Φ1a0
Φ2b
Φ2b0 Φ1a √
 Φ2b
0
0
t
3t

 2
Φ 0
0
4E ≈ 0 ≈ 0 

 b
 Φ1
t
≈0
0
0 
a
√
Φ1a0
3t ≈ 0
0
4E
where we have assumed orthogonality between the two low-lying localized wave functions at each Mn site. Furthermore, the two lower localized states at both Mn1 and
Mn2 are considered at ’0’ relative energy and the two complementary higher-lying
localized states are at energy 4E=t. The non-zero effective hopping matrix elements
are for an Mn1 - O -Mn2 angle of 180 degrees. The interactions between the states
described by wave functions Φ2b0 and Φ1a and Φ2b0 and Φ2a are considered to be negligible and hence, set to zero at this level of approximation. Solving this problem, we
estimate that the stabilization energy due to the additional interactions introduced
with the presence of the higher CASCI states is 1.7t and the energy splitting is found
to be 4.16t. To justify our findings, based on this simple model, we designed a set of
calculations that allow for introducing the two higher states in the interaction. We
prevent the artificial rotation of the occupied Mn 3d (eg )-like orbitals at each Mn ion.
We performed a state average CASSCF calculation for the ’electron doped’ fragment
within an active space, containing the two Mn 3d (eg )-like orbitals. The orbitals for
the ’undoped’ fragment were obtained in a single root CASSCF calculation within
an active space that contains 0 Mn 3d (eg )-like orbitals. Using the natural orbitals
for the lower state of the fragment with the added electron, we construct two CASCI
wave functions localized either at Mn2 or Mn1 within an active space that contains all
Mn 3d (t2g )-like orbitals from both fragments and only one of the natural 3d (eg )-like
orbitals of the fragment with the added electron, that one that has an occupation
number different from 0.0. Analogously, we construct the two higher CASCI wave
108
Chapter 4, Double exchange in manganites
functions, using now the other natural 3d (eg )-like orbital of the ’electron doped’
[MnO6 ]9− fragment which is occupied in the higher Eg -like state of this fragment.
All four CASCI states are allowed to interact. We find indeed a significant interaction
of -0.18 eV between the two states, Φ2b (Mn2 dx2 ) and Φ1a0 (Mn1 dx2 −z2 ), localized at
Mn
√ 2 and Mn1 , respectively. This interaction is larger than t but it is smaller thano
3t. The reason for this is the fact that the Mn1 -Mn2 angle is smaller than 180
in the tilted form of the compound, hence the interaction is overestimated using the
simplified model above. The energy splitting is calculated to be about 0.5 eV and the
stabilization energy becomes 1.4t.
The CASSCF approach for the super-clusters does lead to localized occupied 3d
eg -like orbitals at either one of the Mn ions just as it did for LaMnO3 . However
the occupied 3d eg -like orbital at one of the two Mn ions is no longer directed along
the longer Mn-O bond. This is due to an artificial rotation of the initially occupied
3d eg -like orbitals derived from the ’electron doped’ [MnO6 ]9− fragment. The two
lower localized CASSCF wave functions for [Mn2 O11 ]15−
ab display that rotation of the
occupied Mn 3d eg -like orbitals. This artificial rotation leads to an increase in the
DE interaction (see the last row in Table 4.6 (a)).
For the [Mn2 O11 ]15− cluster along the c axes, (Mn atoms 2 and 4 in Figure 4.2 (b))
with an Mn- O(1)-Mn angle of 158.6 0 , t( 12 )c = -84 meV. This value of t is obtained by
considering only the two lower localized CASCI states. The DE interaction associated
with the higher CASCI states, which can be characterized by wave functions Φ2b0 and
Φ4b0 , is expected to be larger due to the σσ -type mutual orientation of the occupied
super-clusters.
eg - like orbitals localized at the two Mn sites in the [Mn2 O11 ]15−
c
We made an estimate of the nearest-neighbour exchange interactions based on
CASPT2 results obtained with CASSCF wave functions, where the latter are built
within a minimal 5d+3d active space. The estimates of J derived from the energy
differences E(Smax )-E(S’) between the state with Smax = 27 and these with S’= 52 , 32
and 21 are similar. They show a maximum deviation from the average value of J of
≈0.4 meV. The CASPT2 average value of J between the Mn3+ and Mn4+ ions is
3.2 meV and if we take into consideration that we underestimate J at this level of
approximation by about 20 % compared to the experimental results, then the increase
in the value is expected to be up to 4.0 meV. We find a ferromagnetic exchange
coupling between the Mn3+ and Mn4+ ions with a small magnitude. A neutron
diffraction study, performed already in the 50’s of the magnetic properties of the
series La1−x Cax MnO3 led to the conclusion that the exchange coupling in general is
ferromagnetic between Mn3+ and Mn4+ ions, antiferromagnetic between Mn4+ ions
and either ferromagnetic or antiferromagnetic between Mn3+ ions [32].
Like in the case of LaMnO3 , we compared the DE parameters obtained from the
[Mn2 O11 ]15− super-clusters with those obtained from the [Mn4 O20 ]25− super-clusters,
for high-spin coupling between the four Mn ions. The results are listed in Table 4.7 (a)
and (b). First the [MnO6 ]8− fragment orbitals are constructed without taking into
account the nearby added electron. As we already showed in the case of LaMnO3
disregarding the presence of the nearby effective nuclear charge leads to an underestimation of the values of t, but does not restrain one from deducing the cluster size
Results and Discussion: CaMnO3
109
Table 4.7: DE parameters, for [Mn4 O20 ]25− clusters representing hole doped CaMnO3 in
the ab plane and perpendicular to the ab plane. In Figures 4.1 and 4.2 is illustrated the atom
labeling. Maximum spin multiplicity (2S+1=14) of 13 active electrons in 5+3+3+3 active
orbitals.
a) The effect of the nearby extra electron on the orbitals of the ’undoped’ [MnO6 ]8− fragment is
not included.
1.
ion pairs
25−
[Mn4 O20 ]ab
-t 72 (meV)
1 2, 3’ 4’
1 3’, 2 4’
1 4’
2 3’
432
428
19
24
2.
ion pairs
[Mn4 O20 ]c 25−
-t 72 (meV)
1 2, 3 4
13
24
14
23
416
119
195
1
0
b) The added electron at one embedding Mn ion of the ’undoped’ [MnO6 ]8− fragment is
represented by decreasing the effective charge at that ion by one;
1.
ion pairs
[Mn4 O20 ]ab 25−
-t 72 (meV)
1 2, 3’ 4’
1 3’, 2 4’
1 4’
2 3’
620
613
17
23
2.
ion pairs
[Mn4 O20 ]c 25−
-t 72 (meV)
1 2, 3 4
13
24
14
23
593
153
303
1
1
dependencies of t due to the local and ’Mn ion’-selective effect of this nearby effective
charge on the electron relaxation processes at the different Mn4+ ions in the supercluster. Nevertheless, we carried out calculations for the [Mn4 O20 ]25− super-clusters
using localized orbital sets for the ’undoped’ [MnO6 ]8− fragments which ”know” for
the presence of the nearby extra electron. Comparing the various DE parameters
obtained without and with accounting for the effect of the nearby electron (Table 4.7
(a) and (b)) shows, as expected, that the first set of calculations underestimates the
DE couplings even more than it did in LaMnO3 (∼ 30 %). Clearly the electronic relaxation and polarization effects at the ions involved in the hopping and their closest
counterions enhances the DE interaction between these ions.
Although the low-spin coupling configurations are not accessible due to computational limitations, assuming 2S+1 proportionality as found in the smaller clusters, we
could deduce the relevant t( 12 ) parameters. Like in LaMnO3 , we find only a modest
cluster-size dependence of those low-spin coupling parameters: the value for t( 21 )ab
varies from -171 meV (Table 4.6 (b)) in [Mn2 O11 ]15−
ab to -148 meV and -155 meV in
25−
the [Mn4 O20 ]25−
and
[Mn
O
]
super-clusters.
The cluster-size dependence for
4 20 ab
c
110
Chapter 4, Double exchange in manganites
the nearest neighbour interactions along the c direction is also little: t( 21 )c between
ions 2 and 4 is -76 meV in [Mn4 O20 ]25−
(Table 4.7 (b)) and -84 meV in [Mn2 O11 ]15−
.
c
c
The other symmetry-unique DE interaction between Mn ions 1 and 3 is -38 meV.
The next nearest neighbour DE interactions along c are negligible and within the ab
plane they are an order of magnitude smaller than the nearest neighbour interactions:
t’( 27 )ab is -17 or -23 meV, depending on the ion pair.
At this point, we can summarize that we find the DE parameters associated with
nearest neighbour hopping of a single electron, introduced by doping in pure CaMnO3
to be t( 12 )ab ≈ -0.17 eV and t( 12 )c ≈ -0.06 eV.
Analogously to the calculation of the stabilization energy associated with the interaction of the four localized low-lying CASCI states of the [Mn2 O11 ]15−
ab super-cluster
(see above), we calculate the stabilization energy associated with the interactions between all eight low-lying localized CASCI states (two per Mn site) in the [Mn4 O20 ]25−
super-clusters. If we allow only for the four low-lying localized CASCI states in the
[Mn4 O20 ]25−
ab super-cluster to interact, the computed stabilization energy is ∼ 0.21 eV
and the energy splitting is about twice as large: ∼ 0.43 eV. The CASCI wave functions
describing the four low-lying states, localized at either Mn1 , Mn2 , Mn30 or Mn40 of
the [Mn4 O20 ]25−
(see Figure 4.1 (b)) can be denoted as in [Mn2 O11 ]15−
ab super-cluster
ab
2
30
40
1
(see above) Φa , Φb , Φb and Φa . The other four higher CASCI states with the second
3d (eg ) -component occupied are at about 0.2 eV above the lower-lying CASCI states.
0
0
Their wave functions are denoted Φ1a0 , Φ2b0 , Φ3b0 and Φ4a0 analogous to the CASCI wave
functions of the [Mn2 O11 ]15−
ab super-cluster. The stabilization energy associated with
the interactions between all eight low-lying localized CASCI states is then computed
allowing for all eight states to interact. The stabilization energy, obtained as the
difference between the energy of the lowest CASCI states, localized at Mn2 and Mn30 ,
and the lowest delocalized state, is ∼ 0.26 eV and the total energy splitting is ∼ 0.8
eV. This analysis clearly confirms that the role of the higher-lying localized states
in the delocalization of the added electron in CaMnO3 is significant. In Chapter
5, we explicitly derive the many-body electron bands associated with the two lowlying localized CASCI states and quantify the contribution of the higher states to the
formation of the many-body electron bands in lightly electron doped CaMnO3 .
It becomes clear, that the energies associated with the nearest neighbours DE in
the ab planes for both compounds are much larger compared to those of the corresponding nearest neighbours exchange couplings. The energy difference between the
parallel and anti-parallel spin pairing is about 12 meV for pure CaMnO3 and about
-13 meV for pure LaMnO3 . The calculations on the nearest neighbour exchange interactions show that the presence of an extra electron or extra hole does not lead to
a significant enhance of J.
We can make further the following analysis: In lightly doped CaMnO3 if one
considers electron delocalization over the complete system, the DE mechanism leads
to a band stabilization energy of about 0.8 eV per added electron. This estimate is
obtained by considering only the two lower-lying localized states which have the 3d
(eg ) -like orbital along the long Mn-O bond occupied and by using the simple relation
between the stabilization energy Edeloc and the hopping matrix elements t, Edeloc ≈
Results and Discussion: La0.75 Ca0.25 MnO3
111
4t( 12 )ab +2t( 21 )c . We note that the low-lying localized states with the second 3d (eg )
-component occupied contribute also to the delocalization of the extra electron, along
the c axes and within the ab planes. This issue will be considered in detail in Chapter
5. In case the electron is delocalized over only an octahedral cluster that contains the
central Mn ion and its six nearest Mn neighbours, this incomplete delocalization of
the extra electron leads to a stabilization energy of about ≈ 0.35 meV, i.e. ∼ 40-50
% of the band stabilization energy. If a single spin flip occurs at the central Mn ion it
will multiply the DE parameter with its six nearest Mn neighbours by a factor of four,
and it is therefore more favorable. This leads to a total stabilization energy of about
1.4 eV. Since the nearest neighbour exchange interactions do not increase significantly
due to the presence of the added electron, the energy loss associated with the nearest
neighbours exchange coupling is an order of magnitude smaller. Hence, for small
doping concentrations, the DE interaction in CaMnO3 is large enough to produce for
each added electron a so-called ”self-trapped magnetic polaron” [55–57], i.e. a local
ferromagnetic cluster in which the electron can lower its energy by delocalization.
4.5.3
La0.75 Ca0.25 MnO3
In this section we calculate the hopping matrix elements between two localized CASCI
wave functions associated with the hopping of a single eg -like electron between two
Mn sites in La0.75 Ca0.25 MnO3 . We discussed already in section 4.3 the arbitrariness
in the positions of the Ca and La ions in the unit cell of the compound. Since
it is equally probable to assign Ca or La to any of the four positions (the La/Ca
sites) in the unit cell, this introduces arbitrariness in the positions of the nearest
neighbouring Ca ions around the [Mn2 O11 ]15− super-clusters. We have restrained
this arbitrariness to some extent by distributing the nearest neighbouring Ca ions
as it is shown in Figure 4.5. Although one may find in the literature reports for
structural inhomogeneity in lightly doped manganites [86] such as La1−x Srx MnO3 ,
we have maintained in our model clusters the proportion of Ca:La to be 1: 3. Any
deviation from the random distribution of Ca at the La/Ca sites would lead to La-rich
and Ca-rich clusters in the crystal, that is accompanied by local structural disorder
on the cation sub-lattice in the vicinity of the La-rich and Ca-rich regions [86]. Since
the lattice parameters of Ca-rich and Ca-poor compounds are different, the stress
between the two regions may lead to cracking, bad mosaicity of the crystal or even
to polycrystalline samples [27, 28]. In case this inhomogeneity in the distribution of
the Ca ions exists at the doping concentration 0.25 and T=100 K, it would lead to
inhomogeneity in the hole distribution or even to phase co- existence [27, 28]. The
crystal structure refined at 100 K and at the doping concentration x=0.25, chosen in
this study, does not support this inhomogeneity. Assuming that at doping x=0.25, the
system is rather disordered, we designed our model clusters to maintain the ratio 1: 3
in the nearest Ca/La embedding around [Mn2 O11 ]15− . We assigned to these nearest
ions, represented by TIPs, formal charges of +3 for La and +2 for Ca, whereas we
assumed a complete charge averaging at 2 34 for the optimized point charges of the
embedding. The charges of all embedding Mn ions are averaged at 3 14 . There are
112
Chapter 4, Double exchange in manganites
Figure 4.5: Illustration of the crystal structure of La0.75 Ca0.25 MnO3 ; Pbnm space group;
Representation of the model [Mn2 O11 ] super-clusters in the ab plane and the distribution of
the nearest 12 embedding La/Ca ions around the super-clusters.
8 La/Ca ions around each [MnO6 ] fragment. For a particular [MnO6 ] fragment,
positioning an embedding Ca ion at a particular lattice La/Ca site determines the
position of the second Ca ion within the nearest La/Ca embedding array, namely, the
second Ca ion is positioned at a La/Ca site, connected to the first La/Ca site through
the cube body diagonal between them (see Figure 4.5). The same ”building” principle
is applied to the nearest La/Ca embedding of the [Mn2 O11 ]15− super-clusters.
Employing this ”building” principle, we construct four different distributions of
the 8 La/Ca ions around the [MnO6 ] fragments. They differ in the positions of the
two nearest Ca ions with respect to the fragment Mn ion. One distinguishes four
different Mn-Ca distances in our four models: 6.304 atomic units (a.u.), 6.380 a.u,
6.160 a.u. and 6.519. a.u.
We perform CASSCF calculations for all four [MnO6 ] fragments in which the Mn
ion has a formal valency 3+ as well as for all four [MnO6 ] fragments with formal
valency 4+. The orbitals of the [MnO6 ]9− fragments are optimized for the lower
component of the 5 Eg -like state in Oh symmetry. The other component is found at
about 0.1-0.3 eV higher energy. According to MPA the occupied (eg )- like orbital
consists of 92-94 % Mn -3d. The orbitals of the [MnO6 ]8− fragments are obtained for
the 4 Ag -derived ground state. In all cases, the t2g -like orbitals consist of 96-98 %
Mn 3d.
For La0.75 Ca0.25 MnO3 , relevant is the orientation of the [MnO6 ]9− fragments occupied Mn 3d (eg )- like orbital in the presence of the nearby hole at a neighbouring
Mn4+ ion, which is introduced in the present CASSCF fragment calculations by in-
Results and Discussion: La0.75 Ca0.25 MnO3
113
Table 4.8: DE parameter tab , overlap integrals S12 and Hamilton matrix elements H12
between CASCI wave functions for a [Mn2 O11 ]15− super-cluster in the ab plane representing
La0.75 Ca0.25 MnO3 . 7 electrons in 5d+3d active orbitals; Mn1 -Ca (6.380 a.u.); Mn2 -Ca(6.519
a.u.)
The effect of the nearby electron on the orbitals of the [MnO6 ]8− fragment is included, so is the
effect of the nearby hole on the orbitals of the [MnO6 ]9− fragment.
2S+1
−tab (meV)
S12
-H12 (hartree)
H11 , H22 (meV)a
a These
8
743
10.551.10−3
34.801994
0; 14
6
557
7.928.10−3
26.151784
-2; 12
4
371
5.293.10−3
17.458771
-3; 11
2
186
2.648.10−3
8.734525
-4; 10
energies have been shifted upwards by 3295.992952 hartree
creasing the effective nuclear charge of that neighbouring Mn ion by one. Analogously
the orbitals of the [MnO6 ]8− fragments are optimized accounting for the presence of
the nearby electron at a neighbouring Mn3+ ion by decreasing the effective nuclear
charge of that neighbouring Mn ion by one.
Like in LaMnO3 and CaMnO3 , two different Mn-O-Mn units with slightly different
geometry can be distinguished within the ab planes and likewise there are two different
Mn-O-Mn units perpendicular to the ab planes. For one of the [Mn2 O11 ]15− clusters,
we describe in detail our analysis in Tables 4.8 to 4.10. As already pointed out for
LaMnO3 and CaMnO3 , these results are representative for all [Mn2 O11 ]15− clusters.
The cluster analyzed in detail contains Mn ions 1 and 2, both in the ab plane (see
Figure 4.5). The distances between Mn ions 1 and 2 and the bridging oxygen ion are
1.974 Å, and 1.957 Å, respectively and the Mn-O-Mn angle is 160.3◦ . Because the
[Mn2 O11 ]15− super-clusters are viewed as built from [MnO6 ] fragments, we consider
also for them four different distributions of the nearest embedding 12 La/Ca ions, that
differ in the positions of the nearest embedding 3 Ca ions. These distributions of the
embedding La/Ca ions are derived in consistency with the four model distributions
of the embedding 8 La/Ca ions around the fragments. This leads to four different
models for the [Mn2 O11 ]15−
ab super-cluster and for each of them the Mn1 -Ca and Mn2 Ca distances have two different values.
In Table 4.8, we list the results for a [Mn2 O11 ]15−
ab super-cluster for which the Mn1 Ca and Mn2 -Ca distances are 6.38 a.u. and 6.52 a. u., respectively. The two lowest
CASCI states of this super-cluster have the eg - like electron localized around the long
bond of Mn1 with Ob and mostly around the short bond of Mn2 with Ob b . The active
orbital set for the CASCI wave functions of the super-cluster is designed as a superposition of the active orbital sets of the two [MnO6 ]9− and [MnO6 ]8− fragments. The
active spaces contain four electrons in five Mn 3d-like orbitals and three electrons
in three Mn 3d (t2g )-like orbitals, respectively. Analogous to the notations adopted
for LaMnO3 and CaMnO3 , we refer to this active space as to 5d+3d active space.
Taking the x -axis along the long Mn-O bond in the [MnO6 ]9− fragment with Mn-Ca
bO
b
denotes the shared O ion between Mn1 and Mn2
114
Chapter 4, Double exchange in manganites
Table 4.9: DE parameter tab , overlap integrals S12 and Hamilton matrix elements H12
obtained from the interaction between two localized CASSCF wave function in a [Mn2 O11 ]15−
super-cluster in the ab plane representing La0.75 Ca0.25 MnO3 . 7 electrons in 5d+3d active
orbitals; Mn1 -Ca (6.380 a.u.); Mn2 -Ca(6.519 a.u.)
2S+1
−tab (meV)
S12
-H12 (hartree)
H11 , H22 (meV)a
a
8
784
0.196737
648.485328
0; 15
6
585
0.144220
475.378761
7; 22
4
392
0.092318
304.300945
12; 27
2
196
0.044517
146.737535
14; 30
These energies have been shifted upwards by 3296.060606 hartree
distances of 6.38 a.u., the relevant occupied Mn 3d (eg )- like orbital has predominantly
3d2x2 −y2 −z2 character i.e., it is directed along the long Mn-O bond. For the other
[MnO6 ]9− fragment with Mn-Ca distances of 6.52 a.u., and with the y -axis along the
long Mn-O bond, the occupied Mn 3d (eg )- like orbital is mostly 3dx2 −z2 , oriented
mainly along the short Mn-O bond.
The orientations of the occupied Mn 3d (eg )- like orbital in the two localized
CASCI states of [Mn2 O11 ]15−
are those obtained from the fragments calculations.
ab
The energies of the two CASCI states differ by about 15 meV (see Table 4.8) due to
the slight difference in the environment of the two Mn ions. The mutual orientation
of the occupied Mn 3d (eg )- like orbitals at Mn1 and Mn2 in the two localized CASCI
states, respectively results into a σσ type interaction. The interaction between the
magnetic moments at the Mn ions within the ab planes is ferromagnetic and hence, the
relevant parameter t is that for high spin coupling S= 27 . The DE interaction between
the localized states gives t( 72 )ab =-743 meV. Analogous to LaMnO3 and CaMnO3 , we
obtain almost perfect 2S+1 proportionality of t. Furthermore, note that H12 and S12
are also roughly proportional to t just as in LaMnO3 and CaMnO3 . As we noted
above, the [MnO6 ]9− fragment orbitals incorporate the effect of the presence of a hole
at the relevant neighbouring Mn4+ ion. Analogously the orbitals of the [MnO6 ]8−
fragment are derived considering the nearby electron at the relevant neighbouring
Mn3+ ion.
Furthermore, in Table 4.9 we list, for comparison, the DE quantities obtained with
the CASSCF wave functions of the [Mn2 O11 ]15−
ab super-cluster. In Figure 4.6, we have
illustrated by means of difference spin density maps the character and orientation
of the relevant occupied Mn 3d (eg )- like orbitals for the two localized states in the
[Mn2 O11 ]15−
ab super-cluster. As expected, the CASSCF leads to an increase in H12 and
S12 but the overestimation of the DE interaction in this case is only modest. Note,
that the near degeneracy of the two localized CASCI states is preserved because the
super-cluster environment in the directions in which the relevant occupied Mn 3d
(eg )- like orbital at Mn1 or Mn2 is oriented is quite similar. We also notice that the
exchange interaction between neighbouring Mn3+ and Mn4+ ions in the ab plane is
more than an order of magnitude smaller than the DE interaction between them,
again just as in LaMnO3 and CaMnO3 .
Results and Discussion: La0.75 Ca0.25 MnO3
115
Figure 4.6: Character and orientation of the occupied Mn 3d (eg )- like orbitals localized
around Mn1 and Mn2 for the two CASCI states of the [Mn2 O11 ]15−
super-cluster with Mn1 ab
Ca and Mn2 -Ca distances of 6.380 a. u. and 6.519 a. u., respectively.
It is interesting to make an analysis of the interactions that arise between two
CASCI states localized at Mn1 and Mn2 in the other three model [Mn2 O11 ]15−
ab
super-clusters. Here, we present in detail this analysis for the model [Mn2 O11 ]15−
ab
super-cluster with Mn1 -Ca and Mn2 -Ca distances of 6.160 a. u. and 6.380 a. u.,
respectively. The results for this super-cluster are reported in Table 4.10 (a) and (b).
The importance of choosing an adequate embedding in the CASSCF calculations for
the fragments can be illustrated by the following considerations. If the embedding
of the [MnO6 ]9− fragment does not contain information about the nearby hole at a
relevant Mn4+ ion (all nearest embedding Mn ions are assigned an averaged charge
at 3 41 ) and if the embedding of the [MnO6 ]8− fragment is not ”informed” about the
presence of the nearby electron at a relevant Mn3+ ion, the use of the orbitals derived
for those fragments leads to an underestimation of the DE interaction between the
neighbouring ions Mn3+ and Mn4+ (see Table 4.10 (a)).
The DE interaction is obtained as the hopping matrix elements between the two
CASCI states of the [Mn2 O11 ]15−
ab super-cluster with the (eg )- like electron localized
around either Mn1 and Mn2 . It is about twice as small -335 meV, compared to the
σσ type interaction, obtained in the model [Mn2 O11 ]15−
ab cluster, above.
To classify this interaction, we referred to a simple consideration of the orientation
and character of the occupied Mn 3d (eg )- like orbital localized around Mn1 and Mn2
for the two CASCI states.
If the x -axis is taken along the long Mn-O bond in the [MnO6 ]9− fragment with
Mn-Ca distances of 6.16 a. u., the occupied Mn 3d (eg )- like orbital has predominantly
3d2x2 −y2 −z2 character, i.e., it is oriented along the long Mn-O bond. The occupied
Mn 3d (eg )- like orbital for the other [MnO6 ]9− fragment with Mn-Ca distances of
6.38 a. u. is directed also along the long Mn-O bond. It is mostly 3d2y2 −x2 −z2 if the y
-axis is taken along this long Mn-O bond. Since the active orbital set of [Mn2 O11 ]15−
ab
is constructed as a super-position of the active orbital sets of the two fragments, the
two lowest CASCI states of this super-cluster have the occupied Mn 3d (eg )- like
116
Chapter 4, Double exchange in manganites
Table 4.10: DE parameter tab , overlap integrals S12 and Hamilton matrix elements H12
between CASCI wave functions for a [Mn2 O11 ]15− super-cluster in the ab plane representing
La0.75 Ca0.25 MnO3 . 7 electrons in 5d+3d active orbitals; Mn1 -Ca (6.160 a.u.); Mn2 -Ca(6.380
a.u.)
a) The effect of the nearby electron on the orbitals of the [MnO6 ]8− fragment is not included,
neither is the effect of the nearby hole on the orbitals of the [MnO6 ]9− fragment.
2S+1
−tab (meV)
S12
-H12 (hartree)
H11 , H22 (meV)a
a These
8
335
0.37074.10−3
1.209642
0; 23
6
252
0.273.10−3
0.892058
0; 21
4
168
0.178.10−3
0.580424
-3; 19
2
84
0.087.10−3
0.283376
-4; 18
energies have been shifted upwards by 3295.993016 hartree
b) The effect of the nearby electron on the orbitals of the [MnO6 ]8− fragment is included, so is the
effect of the nearby hole on the orbitals of the [MnO6 ]9− fragment.
2S+1
−tab (meV)
S12
-H12 (hartree)
H11 , H22 (meV)a
a These
8
729
9.691.10−3
31.966760
0; 66
6
547
7.281.10−3
24.018799
-1; 64
4
365
4.863.10−3
16.041383
-3; 63
2
183
2.434.10−3
8.028608
-4; 62
energies have been shifted upwards by 3295.998786 hartree
orbital and consequently the eg - like electron localized around the long bond of either
one of the two Mn ions. This mutual orientation leads to a σπ -type interaction just
as in LaMnO3 and CaMnO3 . The strength of this interaction is expected and found
to be twice as small, compared to a σσ type interaction. In Figure 4.7, we illustrate
the orientation of the relevant eg -like orbitals for the two localized CASCI states.
Next, the orbital sets for the CASCI wave functions of the [Mn2 O11 ]15−
ab supercluster are derived from fragments calculations in which the [MnO6 ]9− fragments
orbitals carry information about the presence of the nearby hole at Mn4+ and the
[MnO6 ]8− fragments orbitals incorporate the effect of the nearby electron at Mn3+ .
The results for the DE parameters, S12 and H12 are reported in Table 4.10 (b).
Comparing the results of Tables 4.10 (a) and 4.10 (b) reveals, that significantly
larger values of S12 and H12 and t ab occur when the fragments orbitals are optimized
in the presence of the neighbouring effective charges, representing the hole or eg like electron at Mn4+ and Mn3+ , respectively. The character of the occupied eg like orbitals for the two lowest CASCI states shows a change for that CASCI state
which is localized around the Mn2 ion. For this CASCI state, the eg -like orbital
has predominantly 3dx2 −z2 character and its lobes are oriented along the short and
intermediate bonds. The interaction is no longer of σπ -type but rather σσ -type and
the DE parameter has in accordance a larger magnitude of -729 meV (the bold number
in Table 4.10 (b)). The DE parameter obtained with the CASSCF wave functions
of [Mn2 O11 ]15−
is overestimated, as expected, but only modestly. In Figure 4.8, we
ab
have illustrated the new mutual orientation of the relevant eg -like orbitals of the two
Results and Discussion: La0.75 Ca0.25 MnO3
117
Figure 4.7: Character and orientation of the occupied Mn 3d (eg )- like orbitals localized
around Mn1 and Mn2 for the two CASCI states of the [Mn2 O11 ]15−
super-cluster with Mn1 ab
Ca and Mn2 -Ca distances of 6.160 a. u. and 6.380 a. u., respectively.
Table 4.11: DE parameters tc ( 27 ), obtained from the interaction between two localized
CASCI wave functions for [Mn2 O11 ]15− super-clusters in a plane perpendicular to the ab
plane. In all cases the presence of the nearby hole or eg - like electron is included;
d(Mn-Ca) (a. u.) in [Mn2 O11 ]15−
c
−tc (meV)
6.519
750
6.304
724
6.380
728
6.160
739
localized CASCI states. Clearly, incorporating the effect of the presence of the hole or
eg - like electron in the optimization of the fragments orbitals is necessary for obtaining
meaningful results for the DE interaction. Introducing adequate embeddings for the
fragments calculations which carry information about the nearby hole or eg -like
electron, we obtained that for all model [Mn2 O11 ]15−
super-clusters the two lowest
ab
CASCI states have the eg -like orbital oriented along the bond (s) of either one of the
two Mn ions that leads to a σσ -type interaction and maximizes t.
The values of t ab for the other two model [Mn2 O11 ]15−
ab super-clusters with Mn1 Ca and Mn2 -Ca distances of 6.519 a. u. and 6.304 a. u. and 6.304 a. u. and 6.160 a.
u., respectively are -789 meV and -745 meV, depending on the eg -like orbitals at Mn1
and Mn2 involved in the interaction. For all super-clusters, the values of t obtained
with CASSCF wave functions for the super-clusters show a modest overestimation of
the DE interaction.
Analogous analyses were carried out for four model [Mn2 O11 ]15−
super-clusters
c
in which the Mn1 -Ca and Mn2 -Ca distances are equal. The values of t c ( 72 ) obtained
from the different super-clusters are summarized in Table 4.11. Since the nearest
neighbour spin coupling along c is ferromagnetic the relevant parameters are again
these for S= 72 . For all four model [Mn2 O11 ]15−
super-clusters, we find the nearest
c
neighbour DE parameters along c to be of the same magnitude as those within the
ab planes. The strong DE couplings within the ab planes and along the c axes are
118
Chapter 4, Double exchange in manganites
Figure 4.8: Character and orientation of the occupied Mn 3d (eg )- like orbitals localized
around Mn1 and Mn2 for the two CASCI states of the [Mn2 O11 ]15−
super-cluster with Mn1 ab
Ca and Mn2 -Ca distances of 6.160 a. u. and 6.380 a. u., respectively.
consistent with the high mobility of the eg -electrons, expected in this compound. The
homogeneity in the values of t in all directions leads to a band stabilization energy of
about 4 eV per one hole due to delocalization of the hole over the complete system.
Although we do not check explicitly the cluster size dependence of the parameters,
we anticipate very little cluster dependencies in analogy with our findings for LaMnO3
and CaMnO3 . The next-nearest neighbour DE parameters are not considered in the
current study. Based on our investigations, these parameters are expected to be at
least an order of magnitude smaller than the nearest neighbour parameters.
4.6
Summary and Conclusions
In the present chapter, we employed the overlapping fragments approach (OFA) for
the calculation of DE parameters, in doped perovskite manganites. We determine
the DE parameters t defined in terms of hopping matrix elements between localized
hole or electron states in lightly hole doped LaMnO3 and electron doped CaMnO3 .
Furthermore, the DE parameters defined as the hopping matrix elements between
localized hole states in La0.75 Ca0.25 MnO3 are also considered. The hopping matrix
elements are defined as effective Hamiltonian matrix elements between mutually nonorthogonal many-electron states, associated with the hopping of an electron or hole
between two lattice sites MnI and MnJ . These effective matrix elements are expressed
in terms of the diagonal and off-diagonal Hamiltonian matrix elements and overlap
integrals of localized mutually non-orthogonal states i and j, described by ΦIa and ΦJb
that have a hole or an extra electron localized around ions MnI and MnJ . The multiconfiguration wave functions ΦIa and ΦJb and their matrix elements are obtained by
Conclusions
119
performing CASCI calculations on sufficiently large embedded super-clusters, which
in the present study are [Mn2 O11 ] and [Mn4 O20 ]. Employing the OFA, we construct
an individual localized orbital basis set for each ΦIa and ΦJb . The orbitals in each set
are determined from CASSCF calculations for small embedded [MnO6 ] fragments,
centered around one particular Mn ion. Constructing the total super-cluster orbitals
from smaller overlapping fragments has several advantages, compared to a direct wave
function optimization for the super-cluster. One is that the site symmetry is usually
preserved in the smaller fragment whereas this is not the case in the super-cluster
and another is the fact that the molecular orbitals remain localized. In this way, the
orbital relaxation effects and the polarization effects are included in a natural way.
CIJ
CIJ
Let us summarize the steps involved in the computation of Hab
and Sab
,
1. Perform a series of CASSCF calculations for the [MnO6 ] fragments in d4 and
d3 electronic configurations, in order to determine a set of doubly occupied or
inactive orbitals as well as a set of active orbitals for the ground and lowest
ionized (added-electron) states of the fragments.
2. Construct the occupied orbital set of the super-cluster, by combining the occupied orbital sets of the fragments after their unitary transformation. It is
important to account for the nearby extra charge in the orbital optimization of
the fragments in the ground state configurations. The unitary transformation is
followed by a corresponding orbital analysis in order to eliminate the occupied
orbitals associated with double- counted electrons.
3. Determine localized wave functions ΦIa and ΦJb (ΦK
c ....) and their corresponding
CIJ
CIJ
energies, Haa
and Hbb
, by performing CASCI calculations on the supercluster. The CASCI wave functions are expressed in terms of the localized
orbital basis sets, generated in the previous step separately for each CASCI
wave function. They are mutually non-orthogonal.
CIJ
4. Finally compute the Hamiltonian matrix elements Hab
and the overlap inteCIJ
CIJ
grals Sab and determine t. t is an effective one-electron operator and Hab
and
CIJ
Sab are matrix elements between multi-configuration many-electron wave functions. Since ΦIa and ΦJb are expressed in mutually non-orthogonal orbital sets,
CIJ
CIJ
the computation of Hab
and Sab
is more complicated than the computation
of a one-electron integral.
CIJ
In all cases the overlap integrals Sab
between two localized CASCI wave functions
−2
are less than 1.1x10 , so in practice, we can compute t always according to t ≈
IJ
IJ
IJ
IJ (Haa +Hbb )
. Because of the slightly different cluster-environment of I and
Hab
− Sab
2
CIJ
CIJ
J in some super-clusters, Haa
and Hbb
, differ by a small energy, even if the sites
I and J are equivalent in the crystal. In all cases, this energy difference is small
and it does not influence the outcomes. The next nearest neighbour parameters in
LaMnO3 and CaMnO3 were found much smaller compared to the nearest neighbour
parameters. Therefore, they will be disregarded in the discussion below. Although
120
Chapter 4, Double exchange in manganites
not explicitly computed the next nearest neighbour parameters in La0.75 Ca0.25 MnO3
are also expected to be smaller than nearest neighbour parameters.
The approach ensures that the local electronic relaxation effects are taken into
account. Dynamical correlation effects can be included by choosing an appropriately large active space in the CASCI calculation. In the present study, we have not
investigated the dependence of the hopping matrix elements on dynamical electron
correlation. Recent work on comparable systems [8] has shown that the hopping matrix elements are only weakly dependent on the inclusion of dynamical correlation, so
we do not anticipate large changes.
In all cases incorporating the effect of the presence of an extra charge at a relevant
neighbouring Mn ion while optimizing the orbitals of the [MnO6 ]9− and [MnO6 ]8−
fragments is important for evaluating the magnitude of the DE interactions.
We obtained anisotropic values for the DE parameters in LaMnO3 within the ab
planes t ab ( 72 )=-0.26 eV and perpendicular to them t c ( 12 )≈ -0.05 eV. The hopping
interaction in the ab plane is large enough to yield a band stabilization energy of
about 1 eV per hole, due to delocalization in this plane. Delocalization along c adds
only 0.1 eV to this stabilization energy. If the JT distortion were not present in the
lattice, the DE parameter in ab would be about twice as large: -0.57 eV. Clearly, the
JT distortion is unfavorable for double exchange. Assuming comparable energy scales
of the JT distortions and the DE parameters, one may expect a competition between
JT stabilization and the hole-delocalization energy, where the latter is, at least for
small doping, proportional to the hole concentration.
For CaMnO3 , we found t ab ( 12 ) ≈-0.17 eV and t c ( 12 ) ≈-0.06 eV. The interactions
within the ab planes are weaker than in LaMnO3 , despite the absence of JT distortion.
This is due to the fact that the nearest-neighbour spin coupling in the ab planes is
antiferromagnetic. According to simple models such as the Zener model and models
found in textbooks [22] (also [21]), electron hopping in the ab planes of CaMnO3
should be prohibited due to the antiferromagnetic couplings. In contrast to these
simple models we found in all cases an almost perfect 2S+1 proportionality of the DE
parameters, where S is the combined spin of the two magnetic ions. This result is in
excellent agreement with the more refined model of Anderson and Hasegawa [23]. This
allows one to conclude that the DE mechanism is active not only in a ferromagnetic
phase but also for antiferromagnetic coupling, although in the latter case it is indeed
not as effective as in the former case. We did not find in the literature direct estimates
of values for DE deduced from experiments. However, it is widely accepted that the
hopping parameter is just a fraction of an eV [38]. We found the energies associated
with the nearest neighbour DE within the ab planes considerably larger than the
corresponding nearest neighbour Heisenberg exchange coupling energies. The energy
effects of different competing factors such as JT distortion, exchange and double
exchange interactions are accounted for by using complex effective Hamiltonians [88],
but solving these is beyond the scope of this study.
A simple analysis of the stabilization energy associated with the incomplete delocalization of an extra electron over an octahedral cluster, consisting of a central Mn
ions and its six nearest Mn neighbours, allows us to conclude that, for small dop-
Conclusions
121
ing concentrations, the DE interaction in CaMnO3 is strong enough to produce for
each electron, introduced in the material by doping, a so-called self-trapped magnetic
polaron [55–57]. We considered as well the role of the higher-lying localized CASCI
states with the other 3d eg -component occupied in the delocalization of the extra
electron. We found an important contribution of those states to the stabilization
energy obtained, if we allow also for those states to interact in between and with the
lower-lying CASCI states in the [Mn2 O11 ] and [Mn4 O12 ] super-clusters. The contribution of the higher-lying localized CASCI states to the band stabilization energy
will be considered in more detail in Chapter 5.
For La0.75 Ca0.25 MnO3 we found similar values of t within the ab planes and along
c. It turns out important not only for the magnitude of the DE parameter but also
for the type of the interaction to include the presence of the hole or the eg -electron
at a neighbouring embedding Mn ion while optimizing the orbitals of the [MnO6 ]9−
and [MnO6 ]8− fragments. We find rather large width of the eg -like band. The band
stabilization energy per hole is about 4 eV. The results obtained for La0.75 Ca0.25 MnO3
show that the simple Anderson model, that predicts an effective electron mobility in
an environment of parallel spins, offers an adequate qualitative description of the
relation between ferromagnetism and conductivity.
122
Chapter 4, Double exchange in manganites.
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5
Analysis of Different Overlapping
Fragment and Embedding Schemes
and Many-Electron Bands in
Manganites.
ABSTRACT — In this chapter, three different schemes for constructing localized orbital sets within the Overlapping Fragment approach are compared. The
chapter addresses as well the effect of different embedding schemes for the fragments and super-clusters, on the super-cluster matrix elements that approximate
the crystal matrix elements for the orthorhombic LaMnO3 and CaMnO3 .
The Hamiltonian and overlap matrix elements between localized hole states in
LaMnO3 and added-electron states in CaMnO3 are obtained with an OF scheme
which is optimal for these two systems. Within the many-electron tight-binding
approach, we obtain the delocalization energies as well as the dispersion of the
bands related to the hole and to the electron states.
5.1
5.1.1
Overlapping Fragment and Embedding Schemes
Introduction and Computational Information
The Overlapping Fragment (OF) approach was applied in Chapter 4 for constructing
the localized orbital sets for CASCI wavefunctions representing hole states in hole
doped LaMnO3 and electron states in electron doped CaMnO3 , respectively. In these
calculations, the localized doubly occupied or inactive orbitals of the super-clusters
which are associated with the ion that is shared between any two fragments, were
constructed as normalized linear combinations with equal coefficients of the corre-
130 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
sponding orbital sets generated from the doubly occupied orbitals of the fragments.
The remaining transformed doubly occupied orbitals of the two fragments were combined to complete the inactive space of the super-clusters. The active orbitals of the
two fragments were combined to form the active space of the super-clusters. This
OF scheme, that we denote here as OF1, must be viewed as a particular choice and
therefore, one needs to investigate whether that particular choice provides a balanced
description of all localized CASCI wavefunctions.
When considering localized excitations one can also choose to obtain the orbitals
associated with the shared ion from a MCSCF calculation for the excited fragment,
i.e. the fragment at which the excitation is localized. This is different from the OF1
procedure. For future reference, we denote this OF scheme OF2.
The active space of the excited fragment should be chosen large enough to ensure
an adequate description of the excited state whereas for the remaining ”embedding”
fragments which are in their ground state one can use the minimum active space
required to describe the open shell electrons of those fragments.
In case of hole states and added-electron states, two different choices for the orbitals of the shared ion can be considered. One is based on the overlapping scheme
OF1, i.e. the orbitals for the shared ion are obtained as a super-position of the orbitals derived for the ”doped” fragment where the hole or added-electron is localized
and those derived for the ”embedding” fragment. The other choice is to use only
the orbitals derived for the ”doped” fragment and disregard those obtained for the
”embedding” fragment, i.e. OF2.
For hole states, a third option is considered, in which the shared ion orbitals are
obtained from calculations for the fragment without hole. This scheme is denoted as
OF3.
The studies in Chapter 4 were performed by making use of the OF1 scheme. In
the following paragraphs, we discuss the other two OF schemes, OF2 and OF3 for the
construction of localized orbital sets. The three schemes are compared for obtaining
the effective hopping matrix elements, associated with the hopping of a single hole in
doped LaMnO3 and a single electron in doped CaMnO3 , respectively. These effective
matrix elements were employed in the previous chapter to deduce the double exchange
parameters t in lightly doped manganites [11–13].
We use for this comparison the model [Mn2 O11 ]ab super-clusters in the ab planes
of LaMnO3 and CaMnO3 (Pbnm space group) which we designed in Chapter 4 to
compute the different t parameters. We used the same computational details for
LaMnO3 and CaMnO3 as outlined in Chapter 4, section 4.3 unless stated otherwise.
Analogous to the studies performed in Chapter 4, the super-cluster wavefunctions and
energies are obtained from CASCI calculations with eight 3d -like active orbitals: five
obtained from the [MnO6 ]9− fragment, and three from the [MnO6 ]8− fragment. This
active space, 5d+3d, minimizes a charge transfer from the 3d (eg ) orbitals around
one Mn ion to those around the other. The analysis in Chapter 4 has shown that the
results obtained within 5d+3d space differ insignificantly from the results obtained
with an active space of five 3d -like orbitals around each Mn ion where this charge
transfer can take place. We also concluded that accounting for the presence of the
Method
131
nearby hole or electron in the CASSCF [14, 15] calculations for the [MnO6 ]9− and
[MnO6 ]8− fragments, respectively, is important for obtaining correct values for the
hopping integrals. All calculations in this chapter are performed using embeddings for
the fragments which incorporate the presence of the extra charge on a neighbouring
ion (for more details see Chapter 4 ).
5.1.2
Method
We consider a [Mn2 O11 ]ab super-cluster in the ab plane of the crystal (Pbnm space
group), representing either hole doped LaMnO3 or electron doped CaMnO3 (see Chapter 4, Figures 4.1 and 4.2). This super-cluster is viewed as consisting of two overlapping [MnO6 ] fragments built around TM sites Mn1 and Mn2 . The numbering of the
Mn ions is chosen according to Figures 4.1 and 4.2. Mn1 has a short bond with the
central oxygen ion whereas Mn2 has a long bond with this ion. As in Chapter 4, we
denote the CASCI wavefunctions that describe two localized hole states or electron
states of the super-cluster as Φ1 and Φ2 . Φ1 represents a state with a hole localized around Mn2 in LaMnO3 or an added-electron localized around Mn1 in CaMnO3 .
Analogously Φ2 represents a state with a hole localized around Mn1 in LaMnO3 or an
added-electron localized around Mn2 in CaMnO3 , respectively. To derive the localized
orbital sets for each two states in LaMnO3 and CaMnO3 , respectively we employed
in Chapter 4 the OF1 scheme. Here, we investigate the OF2 and OF3 schemes for
the same two localized states in each of the two structures.
Overlapping fragment schemes OF2 and OF3
In OF2, the localized orbitals for the super-cluster are derived analogously to the OF1
except for those doubly occupied orbitals which are localized at the shared ion (s).
For [Mn2 O11 ]ab , the two [MnO6 ] fragments share an O ion and thus at each [MnO6 ]
fragment there are five doubly occupied orbitals associated with the common O ion.
To determine which five doubly occupied orbitals should be eliminated, we perform a
corresponding orbital transformation [1] of the doubly occupied orbital sets of the two
[MnO6 ] fragments. Till this point there is no difference between the procedures OF1
and OF2. Next, in OF2 we choose to use as doubly occupied orbitals for the shared
ion those five orbitals which are derived from the ’doped’ fragment a and disregard
the five corresponding orbitals, derived for the ’undoped’ fragment. So, in the OF1
scheme for the CASCI wavefunction Φ1 the linear combinations,
1
(ϕ01a + ϕ02a ),
0
2(1 + S1a2a
)
p
a = 1, .., 5
(5.1)
are used (see Chapter 4 ). In the OF2 scheme in the case of LaMnO3 , these linear
combinations are replaced by the transformed doubly occupied orbitals obtained for
the hole doped fragment, i.e. ϕ02a , a = 1, .., 5 whereas the orbitals of the ’undoped’
fragment ϕ01a , a = 1, .., 5 are disregarded.
a For LaMnO the ’doped’ fragment is the [MnO ]8− fragment with the hole whereas for CaMnO
3
6
3
the ’doped’ fragment is the [MnO6 ]9− fragment with the added-electron.
132 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
For CaMnO3 in OF2, the linear combinations are replaced by the doubly occupied
orbitals derived for the electron doped fragment, i.e. ϕ01a , a = 1, .., 5 and ϕ02a , a =
1, .., 5 are disregarded (see Figure 5.1). Analogously to OF1 all other transformed
doubly occupied orbitals of the two fragments are also employed as doubly occupied
orbitals for the super-cluster. For LaMnO3 , we applied as well the OF3 scheme (see
Figure 5.1). For this third option, the linear combinations in Eq. (5.1) are substituted
by the transformed doubly occupied orbitals obtained for the fragment without a hole,
i.e. the ”embedding” fragment, ϕ01a , a = 1, .., 5 while the orbitals obtained for the hole
doped fragment are disregarded. We compare the performance of the three procedures
Figure 5.1: Choice of doubly occupied orbitals at the shared O ion of OF1, OF2 and OF3
for the CASCI wavefunctions (see the text for details)
for providing adequate localized orbital sets for the two CASCI wavefunctions Φ1 and
Φ2 in LaMnO3 and CaMnO3 respectively. For this, we computed the energies, H11
and H22 , of the two localized hole states in LaMnO3 and the two localized addedelectron states in CaMnO3 , respectively using for both systems the model supercluster [Mn2 O11 ]ab (see Chapter 4 ). For LaMnO3 , the high spin coupled states,
with total spin S= 72 , were considered being relevant for the physics of the magnetic
couplings in the ab plane of the system (see Chapter 4 ). These high spin states imply
a high spin or ferromagnetic coupling between the (high spin) Mn ions in the ab plane.
For CaMnO3 , the relevant states to be considered are the lowest spin coupled states
with total spin S= 21 . The lowest spin states are those with low spin coupling between
the different (high spin) Mn ions in the ab plane.
5.1.3
Results and analysis
LaMnO3
We first discuss the performance of the three procedures for the cases of orthorhombic
and idealized LaMnO3 . The CASCI energies H11 and H22 of the two localized hole
states are shown for OF1, OF2 and OF3 in Figure 5.2 and Table 5.1. CASSCF
Results and analysis
133
Figure 5.2: Super-cluster CASCI energies H11 and H22 obtained within OF1, OF2 or OF3
for the description of the localized hole states in a [Mn2 O11 ]15− super-cluster in the ab plane
representing LaMnO3 . H11 is the CASCI energy of the localized state depicted as Mn 1
(d4 ) -Mn 2 (d3 )) in the Figure, H22 is the CASCI energy of the Mn 1 (d3 ) - Mn 2 (d4 )
configuration. Super-cluster CASSCF energies are also shown.
energies are also shown. Figure 5.2 demonstrates that the three OF schemes provide
descriptions that differ in quality for the two hole states. The super-cluster CASCI
energies H11 and H22 differ by 1.21 eV within OF2 and 1.11 eV within OF3. In
the crystal, the Mn1 and Mn2 sites are not crystallographically equivalent, although
they have very similar environments. The external Madelung potential at the Mn1
and Mn2 sites is -2.36821 a.u. and -2.36763 a.u, respectively. This small potential
difference of about 0.02 eV determines the order of magnitude of the difference in
the energies of a state with a hole localized around Mn1 and another one having the
hole localized around Mn2 . In OF1 the energy difference is slightly larger: 0.11 eV.
Comparing the energy difference H11 -H22 obtained within the three procedures shows
clearly that OF2 and OF3 provide a non-balanced description of the two states. In
addition to the slightly different cluster environment of Mn1 and Mn2 (see Figure 4.1)
in the [Mn2 O11 ]ab super-cluster, it is especially the description of the shared O ion
within OF2 and OF3 that contributes to the large energy difference.
Figure 5.2 reveals that the OF2 scheme lowers H22 but raises H11 . This is mostly
due to the different cluster environment of Mn1 and Mn2 in [Mn2 O11 ]15−
ab . In the
localized CASCI state, described by wave function Φ2 , the relevant occupied 3d eg like orbital at Mn2 is directed towards the central O ion (see also Chapter 4 ), whereas
in the localized state, described by Φ1 , the relevant occupied 3d eg -like orbital at Mn1
is oriented along the long Mn1 -O bond, the latter is approximately perpendicular to
the bond between the Mn1 and central O ion. This opens the question whether the
134 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
Table 5.1: LaMnO3 . Comparison of the performance of OF1, OF2 and OF3 for the description of the localized hole states; CASCI energies, H22 , of the lower localized hole state and
energy difference H11 -H22 for a [Mn2 O11 ]15−
super-cluster; The last column lists CASSCF
ab
results (see for details Chapter 4)
Scheme
H22
H11 -H22 (meV)
OF1
-3292.325211
105
OF2
-3292.341577
1210
OF3
-3292.253835
-1110
CASSCF
-3292.394046
319
state Φ2 will be better described by using orbitals for the shared oxygen ion which
have been derived for the fragment, [Mn(2) O6 ]9− , that contains the occupied eg -like
orbital, i.e. employing OF3. This choice raises both H11 and H22 , but by very different
amounts (see Figure 5.2 and Table 5.1), i.e. while H11 differs only slightly in OF2
and OF3, H22 is largely raised in OF3.
Apparently a better description of the electron density around the shared oxygen
ion is obtained if the orbitals for the shared O ion are optimized in the presence of
the Mn ion, which has the shortest Mn-O bond with the shared O ion, i.e. Mn1 . In
the CASSCF calculation for the ’undoped’ [Mn(2) O6 ]9− fragment, the effect of the
hole around the nearby Mn1 ion was accounted for by simply increasing the effective
nuclear charge of Mn1 by one. This purely electrostatic interaction between the extra
charge at Mn1 and the [Mn(2) O6 ]9− fragment electrons is not sufficient to reproduce
the correct electron distribution around the shared O ion in the presence of a hole
around Mn1 . Although the shared O orbitals, derived for the [Mn(2) O6 ]9− fragment
are optimized and adapted to the 3d eg -like electron, they are not optimal for the
description of the localized state because they incorporate only partially the effect of
the nearby hole.
As pointed out above, the difference in the external Madelung potential at Mn1
and Mn2 is 0.02 eV and thus, we expect that the CASCI energy difference between
the two states localized around either Mn1 or Mn2 is of the same order of magnitude.
In both OF2 and OF3, this energy difference is above 1 eV. Indeed the OF2 favors
the CASCI energy of the localized state, represented by Φ2 . Note however, that we
aim at obtaining a balanced description of the two localized states rather than the
lowest CASCI energies. A balanced description of the states is achieved when their
energies are obtained at the same level of accuracy. Because OF2 leads to an increase
in the CASCI energy of the other state described by Φ1 , the balanced description
of the two states is lost within OF2. A non-balanced treatment of the states is also
observed within OF3 (see Figure 5.2 and Table 5.1). Within OF1 the CASCI energy
difference of 0.1 eV is not much larger than the difference in the external Madelung
potential at Mn1 and Mn2 .
Because of the unbalanced description of the two localized states, Φ1 and Φ2 , in
both OF2 and OF3, determining Hamiltonian matrix elements and overlap integrals
between Φ1 and Φ2 , and consequently calculating the hopping integrals t is not adequate. The unbalance observed for OF2 and OF3 is also present in the CASSCF
Results and analysis
135
Table 5.2: Idealized LaMnO3 . Comparison of the performance of OF1, OF2 and OF3 for
the description of the localized hole states. CASCI energies, H22 , of the lower localized hole
state and energy difference H11 -H22 for [Mn2 O11 ]15−
super-cluster; The last column lists
ab
CASSCF results (see for details Chapter 4)
Scheme
H22 (hartree)
H11 -H22 (meV)
OF1
-3292.161280
15
OF2
-3292.162710
62
OF3
-3292.112763
48
CASSCF
-3292.224902
32
wave functions, be it not as large. The CASSCF energy difference is 0.3 eV.
We consider as well the idealized LaMnO3 where the Jahn-Teller distortion has
been ”undone”. The CASCI energies of the two states with the hole localized at
Mn1 or Mn2 are summarized for all three OF schemes in Table 5.2. Comparing the
results from OF1 and OF2 reveals again a non-balanced description of the CASCI
wavefunctions Φ1 and Φ2 obtained within OF2. The difference in the CASCI energies
of the two localized states in OF2 and OF3 is larger than in OF1 and CASSCF.
Note that these energy differences are much smaller than those found for the real
structure because the bonds between the shared O ion and the Mn1 and Mn2 ions are
equivalent and thus, the orbitals for the shared O ion are almost equally described in
the presence of either Mn1 or Mn2 . Nevertheless even those smaller energy differences
make impossible the correct determination of the effective hopping matrix elements.
CaMnO3
Next, we revisit the analysis in Chapter 4 of the effective hopping matrix elements
associated with the hopping of a single electron in electron doped CaMnO3 . The
quantities listed in Table 4.6 (b) of Chapter 4 were calculated using OF1. We report
in Table 5.3 the results for some of those quantities, i.e. the CASCI and CASSCF
energies of the two localized electron states, but now obtained using the OF2, i.e.
with the orbitals ϕ01a or ϕ02a , a=1,..., 5 derived from the electron doped fragments
[Mn(1) O6 ]9− or [Mn(2) O6 ]9− instead of the normalized linear combinations used in
Chapter 4. The CASCI wavefunction Φ1 and Φ2 are constructed and the corresponding super-cluster energies are computed. Analyzing the results in Table 5.3 reveals
that the OF2 scheme performs worse than the OF1 also for the case of the electron
doped system. Analogous to idealized LaMnO3 , the difference in the energies of Φ1
and Φ2 , yielded by OF2, is rather small because the bonds of Mn1 and Mn2 with
the shared O ion are almost equal and hence, the orbitals for that central O ion are
described equivalently in the presence of one or the other Mn ions.
The energy order of the two states is however exchanged in the OF2. The absolute
CASCI energies of Φ1 and Φ2 obtained within OF2 are also much higher compared
to those obtained within OF1. The CASSCF energies are almost degenerate but this
degeneracy is accompanied by an artificial rotation of the occupied Mn 3d eg -like
orbital at the Mn1 site (see for details Chapter 4 ).
136 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
Table 5.3: CaMnO3 . Comparison of the performance of OF1 and OF2 for the description
of the localized electron states. CASCI energies, H22 , of the lower localized electron state and
energy difference H11 -H22 for [Mn2 O11 ]15−
super-cluster; The last column lists CASSCF
ab
results (see for details Chapter 4)
Scheme
H22 (hartree)
H11 -H22 (meV)
OF1
-3306.240036
80
OF2
-3306.160710
-13
CASSCF
-3306.303482
15
In summary, OF2 and OF3 have been considered as alternative schemes for deriving a localized orbital set for a super-cluster. In the cases of localized hole and
electron states, we find that these schemes provide an unbalanced description of the
corresponding localized wavefunctions. OF1 provides a balanced description for the
ionized and electron-added states considered in this study. In OF1 the localized
CASCI wavefunctions Φ1 and Φ2 , are obtained by expressing them in terms of the
normalized linear combinations, with equal weights, of the unitary transformed doubly occupied orbitals ϕ01a and ϕ02a (a=1,..., 5) of the ’doped’ and ’undoped’ fragments
associated with the common O ion (see Eq. 5.1). The remaining transformed orbitals
ϕ01i and ϕ02i (i6=a) of both fragments complete the set of doubly occupied orbitals for
the super-cluster. Although the OF1 scheme has been found to be optimal for the
hole and added-electron states, its performance for localized excitations has not yet
been discussed and it might not lead to the optimal description of localized excited
states. We explore the performance of different schemes in the case of localized excited
states in NiO in Chapter 7.
Embedding schemes
Having established that the performance of the OF scheme OF1 is best, we address the
effect of different embedding representations on the effective hopping matrix elements.
In the next calculations the nearest neighbouring embedding Mn and La/Ca ions of
the [Mn2 O11 ] super-clusters and [MnO6 ] fragments are represented by effective oneelectron potentials of different type: Total Ion Potentials, bare AIMPs [3] or AIMPs
with orthogonalization basis functions. The AIMPs [17] are designed to represent
the short-range Coulomb repulsions between the cluster ions and the embedding ions.
In the present case, the results obtained using the effective core potentials, designed
by Hay and Wadt [2] as TIPs are compared to those obtained using AIMPs derived
from self-consistent field embedded ion calculations [3]. In Tables 5.4. A and 5.5.
A we summarize the effective hopping parameters and effective Hamiltonian matrix
elements and overlap integrals for total spins S= 72 and S= 12 in LaMnO3 and CaMnO3 ,
respectively.
For LaMnO3 , we find a moderate, even minor change in the magnitude of the
effective hopping parameters if we change the embedding. The CASCI energies of
Results and analysis
137
the two localized states, however, differ more substantially in case bare AIMPs are
employed. This energy difference is more emphasized for LaMnO3 . The change in
the effective hopping parameter for CaMnO3 is more pronounced than for LaMnO3
but it is still smaller than 0.04 eV.
Previous studies [16] on ionization and excitation energies in CuCl and NiO have
shown however that the lowest ionization energies in both structures are underestimated in case bare AIMPs are employed to represent the short-range interactions
between the cluster ions and the nearest embedding ions. This happens because the
bare AIMPs overestimate the effect of the Pauli repulsion [16]. The same study has
also demonstrated that augmenting the bare AIMPs with few orthogonalization basis
functions (1s1p) enables the strong orthogonality condition between cluster electrons
and embedding electrons. Using AIMP-1s1p the authors have obtained ionization
energies that are close to those obtained with the representation of the cluster surroundings by a frozen charge distribution of the ions.
Taking into consideration this well established performance of the bare AIMPs
and the AIMPs augmented with orthogonalization basis functions, we designed a
set of calculations employing AIMPs+1s1p in order to check the dependence of the
effective hopping parameters and super-cluster CASCI energies of the augmented
AIMPs. Due to computational restrictions one of the embedding Mn ions for the
fragments, the one which represents either Mn1 or Mn2 in the [Mn2 O11 ] super-cluster,
could not be modeled by augmented AIMPs. This results into unbalanced treatment
of the closest Mn ions embedding of the [MnO6 ] fragments which leads to unphysical
CASCI energy difference between the two localized CASCI states. The localized state
for which the relevant occupied Mn 3d eg -like orbital in the fragment [MnO6 ]9− is
oriented towards that embedding Mn ion, represented by bare AIMPs, is particularly
affected by this bare AIMP embedding. However employing the orbital set derived
from the calculations for the fragments in a CASSCF calculation for the super-cluster
circumvents the computational shortcomings. In Chapter 4 we concluded that for
LaMnO3 the CASSCF approach overestimates the effective hopping parameter by
about 12 %. For CaMnO3 this overestimation is even larger, however in that case
the optimization of the orbitals for the super-cluster is accompanied by an artificial
rotation of the relevant occupied Mn 3d eg -like orbital localized around one of the Mn
ions (see also Chapter 4 ). Nevertheless performing the CASSCF calculations within
the three embedding models, TIPs, bare AIMPs and AIMPs+1s1p, will demonstrate
the effect of the embedding on the relevant t and CASCI energies H11 and H22 . The
results of the CASSCF calculations are listed in Tables 5.4. B and 5.5. B.
The results for the hopping parameters and CASSCF energies H11 and H22 demonstrate a comparable performance of the TIPs and the AIMPs augmented with 1s1p
basis functions. For LaMnO3 , the two embedding schemes produce the same H22 H11 difference of 0.3 eV and yield values for t which differ by 6.10−3 eV. The bare
AIMP representation leads to an increase in H22 -H11 by about 0.1 eV compared to
the TIPs but it introduces only a small deviation in the value of t compared to the
other two embeddings.
In the case of CaMnO3 , the results for the hopping parameters and CASSCF
138 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
Table 5.4: A) Different embeddings for LaMnO3 . Effective hopping parameter t 7 , energies,
2
H11 and H22 , and overlap integrals S12 and Hamiltonian matrix elements H12 between CASCI
wavefunctions for a [Mn2 O11 ]15− super-cluster in the ab plane. Active orbitals: 5d+3d; B)
t 7 , S12 and H12 between the CASSCF wavefunctions for the [Mn2 O11 ]15− super-cluster in
2
the ab plane; Active space: 5d+3d;
A)
2S+1
−t 27 (meV)
S12
-H12 (hartree)
H11 -H22 (meV)
H22 (hartree)
TIP
263
4.307.10−3
14.188303
105
-3292.325211
AIMP
271
2.992.10−3
9.858185
69
-3291.358816
B)
2S+1
−t 72 (meV)
S12
-H12 (hartree)
H11 -H22 (meV)
H22 (hartree)
TIP
294
73.096.10−3
240.671373
319
-3292.394046
AIMP
306
77.514.10−3
255.143206
389
-3291.448783
AIMP-1s1p
300
75.121.10−3
247.305703
321
-3291.967519
Results and analysis
139
Table 5.5: A) Different embeddings for CaMnO3 . Effective hopping parameter t 1 , energies,
2
H11 and H22 , and overlap integrals S12 and Hamiltonian matrix elements H12 between CASCI
15−
wavefunctions for a [Mn2 O11 ]
super-cluster in the ab plane. Active orbitals: 5d+3d; B)
t 1 , S12 and H12 between the CASSCF wavefunctions for the [Mn2 O11 ]15− super-cluster in
2
the ab plane; Active space: 5d+3d;
A)
2S+1
−t 21 (meV)
S12
-H12 (hartree)
H11 -H22 (meV)
H22 (hartree)
TIP
171
1.141.10−3
3.779066
80
-3306.240036
AIMP
133
0.105.10−3
0.352390
108
-3305.310348
B)
2S+1
−t 12 (meV)
S12
-H12 (hartree)
H11 -H22 (meV)
H22 (hartree)
TIP
238
51.466.10−3
170.171261
0; 15
-3306.303482
AIMP
233
50.533.10−3
167.041822
0; 14
-3305.415297
energies difference H22 -H11 obtained, using either the effective core potentials by Hay
and Wadt or the bare AIMPs, differ insignificantly. We do not expect that augmenting
the bare AIMPs with 1s1p basis functions will change the trend differently compared
to the results in LaMnO3 (Table 5.4 B). Note, that the CASSCF values for t 21 and
H22 -H11 obtained using TIPs or bare AIMPs differ less than the corresponding CASCI
values.
This is due to the fact that the CASSCF wave functions contain configurations
expressing a rotation of the relevant occupied Mn 3d eg -like orbital localized around
one of the Mn ions (see above and for more details Chapter 4 ).
140 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
5.2
5.2.1
Many-Body Bands in Manganites
Introduction and Computational Information
To demonstrate the method derived in section 3.2 of Chapter 3, the energy bands
(i.e. the K- dependent energies) associated with the lowest Mn 3d - type hole state
in LaMnO3 , and the lowest added-electron states in CaMnO3 are considered. Use
was made of the localized character of the interactions between the ionized or addedelectron states, centered around different lattice sites. Employing localized manyelectron (ME) basis functions, which represent localized ionized or added-electron
states, requires expressing the H(K) matrix elements in terms of those ME basis functions (see Chapter 3 ). The bands, considered here, are derived from such localized
ME basis functions. Therefore, to obtain those bands, i.e. the K- dependent energies,
we solve a generalized eigenvalue problem within the non-orthogonal tight-binding approach (see Eq. (3.11)) in Chapter 3 ). Furthermore, the Hamiltonian and overlap
matrix elements between the localized ME basis functions are approximated by the
corresponding matrix elements of a large embedded super-cluster. They are transformed from the direct R-space to the reciprocal K-space according to the expressions,
given in section 3.4.2 of Chapter 3. The band calculations were performed using a version of the BICON-CEDIT program [4] modified in our laboratory. The modifications
concern mainly the implementation in section 3.4.2 of Chapter 3 for the evaluation of
the H(K) and S(K) matrices using the H(R) and S(R) matrix elements. The latter
are approximated by the matrix elements of the super-cluster effective Hamiltonian
between super-cluster localized CASCI wavefunctions. The super-clusters [Mn4 O20 ]
used here to extract the relevant Hamiltonian matrix elements and overlap integrals
were designed in Chapter 4 to study the different hopping matrix elements t and t’
within and between the ab planes of the two compounds. Since only the high spin
coupled CASCI states of these super-clusters were accessible due to computational
limitations, the effective matrix elements for the corresponding low spin coupling were
deduced using the observed simple Andersson-Hasegawa spin dependence of t and the
Hamiltonian matrix elements and overlap integrals, respectively. We specify the twocenter Mn-Mn interactions corresponding to the low spin couplings between the Mn
ion for LaMnO3 and CaMnO3 in section 5.2.2
The unit cells of the compounds contain 4 units LaMnO3 or 4 units CaMnO3
respectively. The electronic configuration of the Mn ions in LaMnO3 or CaMnO3
respectively as well as the orbital occupation in the ground state of the starting
[MnO6 ]9− and [MnO6 ]8− fragments were described in detail in section 4.4 of Chapter
4. In the case of LaMnO3 we assigned a local many-electron basis function approximated by the [Mn4 O20 ] CASCI wave functions ΦX
x (X ∈ 1, 2, 3, 4) per lattice site
MnX within the reference unit cell. In the case of CaMnO3 two different situations
arise because unlike in LaMnO3 there are two low-lying localized states per Mn site
and the higher of them is at only about 0.2 eV higher energy as calculated for the fourcenter clusters. Like in LaMnO3 the four lower CASCI states of either the [Mn4 O20 ]ab
or [Mn4 O20 ]c super-clusters have the extra electron localized around the longer Mn-O
bond of each of the Mn ions within the ab -plane and along the c axes. The higher
Many-Electron Bands in Manganites
141
Figure 5.3: The Mn ions in the reference unit cell and in the nearest neighbour cell along
the three crystal directions are shown. The reference unit cell is arbitrarily chosen to contain
those Mn ion for which the localized ME basis functions are depicted as Φ1a , Φ2b , Φ3c , Φ4d . All
other ME basis functions, also depicted, are obtained by translation of the ME basis functions
in the reference unit cell.
four states have the extra electron in the other available Mn 3d eg -like orbital at
each of the Mn ions. In one band calculation we considered the bands associated
with the delocalization of the added-electron in the Mn 3d (eg -like) orbital along the
longer Mn-O bond, i.e. the lower CASCI states were used as approximations to the
corresponding local many-electron basis functions. In another separate calculation
only the higher localized states were allowed to interact. In a final calculation all
eight localized CASCI wave functions of the [Mn4 O20 ] super-clusters are employed as
approximations to the corresponding eight local many-electron basis functions. The
resulting bands are discussed in the section 5.2.2. The bands are calculated along high
symmetry directions (see section 5.2.2 ) on a fine discrete grid along each symmetry
direction (step of 0.01 between two consequtive K points) in the K space.
A motivation for the present study is the on-going investigation of the hole doped
manganese oxides with perovskite structure due to the rich physical phenomena observed in these systems. These phenomena originate from the interplay between
so-called spin, charge, orbital, and lattice degrees of freedom arising from the strong
electron correlation effects [7, 8, 10, 42]. It is well known, that while the parent compounds LaMnO3 and CaMnO3 are A-type and G -type antiferromagnetic insulators,
respectively, the hole doped and electron doped forms exhibit a ferromagnetic state
at low temperatures, below the Curie temperature (Tc ) [10] and colossal magnetoresistance (CMR) effect appears near the ferromagnetic transition temperature. Till
recently the transition to ferromagnetic metal around the Tc , induced by doping was
142 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
explained by means of the double exchange mechanism [11, 12] (see also Chapter 4 ).
After recent experimental discoveries which revealed that the double exchange alone
can not explain the complex nature of the CMR effect (see for example, [37,38]), new
mechanisms were invoked which imposed the possibility of a contribution of a FM insulating phase near Tc to the CMR (for a review on this subject see for example [10]).
An important factor to be discussed in relation with the rich phenomena observed in
those hole doped and electron doped systems is the property of holes and electrons
in Mn oxides.
In this study the delocalization of an extra hole in LaMnO3 and an extra electron
in CaMnO3 are considered. These properties are directly related to the mobility of
the electrons in Mn 3d eg -like bands. For example, the theoretical study of Ravidran
et al [19] on the ground and excited state properties of LaMnO3 carried out within
the gradient corrected full -potential linearized augmented plane-wave (FPLAPW)
method [21] suggests that indeed the transport properties of the hole doped LaMnO3
are dominated by Mn 3d eg -like electrons. In the next section, while discussing the
characteristics of the many-body hole and electron bands in LaMnO3 and CaMnO3 ,
respectively, we also compare with relevant HF, DFT and GW -based studies on the
mobility of the Mn 3d eg -like electrons in manganites.
5.2.2
Many-Body Bands in Manganites: Results and Discussion
Many-body hole bands in LaMnO3
We consider first the bands related to the delocalization of a single hole in hole doped
orthorhombic LaMnO3 . Whether the extra holes in LaMnO3 are mainly localized on
Mn ions rather than in oxygen bands is not yet well established. Since those hole
states are relevant for the character of the top of the valence band (VB) of LaMnO3 ,
it is interesting to consider the following simple analysis.
CASSCF calculations, carried out for establishing the relative energies of the lowest localized Mn 3d and O 2p hole states in a model [MnO6 ] cluster, representing
orthorhombic LaMnO3 , pointed out (see Chapter 6 ) that the lowest O 2p ionized
state is 1.0 eV above the lowest Mn 3d ionized state. Since the two lowest localized
Mn 3d and O 2p ionized states are close in energy, the delocalization effects for both
states are considered in order to determine with a higher certainty the character of
the top of the VB. Using the estimate, made in Chapter 4, of the band stabilization
energy associated with the Mn 3d hole states in LaMnO3 (1.1 eV) and the estimate,
from Chapter 6, of the band stabilization energy associated with O 2p hole states (1.5
eV), we can conclude that the top of the VB most likely contains contributions from
both ionized states. More details on these calculations are provided in Chapter 6.
The ultraviolet photoemission spectroscopy (UPS) and Bremsstrahlung isochromat spectroscopy (BIS) studies by Chainani et al [35] indicate as well that the top of
the valence band contains a strong mixing of O 2p and Mn 3d states. Furthermore,
the oscillator strength of a peak in the optical conductivity data of Jung and coworkers [34], associated with Mn 3d eg → eg transition appears to be larger than that
Many-Electron Bands in Manganites
143
expected for such a dipole forbidden transition. The authors explain the magnitude
of the oscillator strength as caused by a strong interaction between the Mn 3d and O
2p bands. The theoretical studies of Ravidran et al [19] within the FPLAPW method
have yielded as well partial DOS, indicating a considerable amount of O 2p and Mn
3d states present at the top of the VB. Another theoretical study of the electronic
structure of LaMnO3 carried out by Muñoz et al [36] by means of periodic calculations
within the framework of either hybrid DFT or UHF has shown that the character of
the occupied states near the Fermi level can be described satisfactory neither within
UHF nor within LDA. While the local DOS obtained by the UHF method [36] show
a large contribution of O 2p states and a low contribution of Mn 3d states near
the Fermi level, the L(S)DA calculations by Pickett and Singh [33] and Satpathy et
al [23, 24] suggest that the DOS near the Fermi level contain predominantly Mn 3d
states. Muñoz et al [36] demonstrate, using different hybrid functionals, amongst
which the B3LYP functional, that the hybrid DFT methodology yields a reasonable
estimate of the band gap of the compound and the qualitative description of the DOS
near the Fermi level is mostly not sensitive to the details of the mixing of non-local
Fock exchange and local Dirac-Slater exchange in the functionals (for example, the
B3LYP functional). They also show that the use of the B3LYP functional leads to a
large contribution of Mn 3d states near the Fermi level contrary to the UHF results.
Because this hybrid DFT approach provides as well a reasonable band gap value,
Muñoz and co-workers suggest that indeed there are substantial contributions in a
similar proportion of both O 2p and Mn 3d to the DOS near the Fermi energy [36].
In all studies, the Mn 3d hole states play a relevant role for the formation of the
top of the VB and for the mobility of the electrons in Mn 3d eg -like bands. Thus in
this study we focus attention on the Mn 3d hole states in orthorhombic LaMnO3 at
a low doping level.
The H(R) and S(R) matrix elements, employed in the non-orthogonal tightbinding calculation, were extracted in Chapter 4 from CASCI calculations on the
four-center [Mn4 O20 ] clusters. As pointed out in Chapter 4, due to computational
limitations only high spin coupling between the four Mn ions within the ab planes
and perpendicular to them were considered. The relevant interactions between the ab
planes, however, are antiferromagnetic and therefore we deduced the H(R) and S(R)
matrix elements for the low spin coupling using their Anderson-Hasegawa dependence
on the total spin S [13] which we established in Chapter 4.
In Table A.1 in Appendix A, we summarize the Hamiltonian matrix elements and
overlap integrals in the R space between localized CASCI wavefunctions within the
reference unit cell. The small values of the overlap integrals reflect the localized character of the CASCI states. Another confirmation of the localized character of the
interactions is provided by the studies of the effective hopping parameters in Chapter
4. These investigations have shown that the next-nearest neighbour interactions are
an order of magnitude smaller than the nearest neighbour interactions. This finding
justifies the use of a tight-binding approach in the derivation of the bands associated
with the hole states. In Table A.2 the two-center (Mn-Mn) effective hopping matrix
elements between the same CASCI wavefinctions are reported. These effective hop-
144 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
ping matrix elements are employed below in an orthogonal tight-binding calculation
in order to establish the validity of this approximation to the effective hopping matrix elements deduced otherwise in a super-cluster calculation where all four localized
CASCI states within the ab plane or perpendicular to it are allowed to interact (see
Figure 5.3). Taking into account only the nearest and next-nearest interactions in the
three crystal directions leads to the consideration of the interactions only between the
first neighbour cells and the reference unit cell. Figure 5.3 displays the reference unit
cell and the first -neighbour cells in the 3D space. The localized CASCI wavefunctions are depicted as well, Φ1a is the CASCI wavefunction representing a hole state,
localized at lattice s ite Mn1 in the reference unit cell. Analogously we define the
wavefunctions Φ2b , Φ3a and Φ4b . Note that in Table A.1 the wavefunctions Φ3a and Φ4b
are depicted as Φ3c and Φ4d because the energies of the states localized at Mn3 and Mn4
are slightly different from those at Mn1 and Mn2 , respectively. This energy difference
is less then 0.01 eV and does not affect the outcome of the tight-binding calculation.
Employing the Hamiltonian matrix elements and overlap integrals derived from the
accurate cluster calculations leads to a non-orthogonal many-electron tight-binding
model with ab initio derived parameters.
LaMnO3 has a primitive orthorhombic lattice, classified with the crystal space
group Pbnm [18]. The energy bands associated with the lowest hole state are calculated along high symmetry lines Σ, ∆ and Λ, H, B, G and P in the first irreducible
Brillouin zone of the Pbnm orthorhombic lattice (Figure 5.4). The energy bands are
displayed in Figure 5.5. The zero energy is taken to be the average energy of the
four localized hole states, one per Mn site, in the unit cell, which is 4.6 eV above the
ground state energy. The latter is approximated by the ground state energy of the
relevant [Mn4 O20 ] super-cluster. Unlike the bands obtained within an effective oneelectron approximation, the energy bands derived here are referred to as many-body
ab inito bands. Compared to the one-electron bands produced, for example, by DFT
band structure calculations within the local (spin) density approximation (LSDA) the
many-body ab initio energy bands incorporate electronic relaxation and electron correlation effects, the latter refer to so-called on-site strong Coulomb interactions. They
reflect the dispersion of the energy of the many-electron N-1 ionized states, in different symmetry directions whereas the one-electron bands in the common one-electron
band pictures reflect the dispersion of the Bloch orbital energies. The Bloch orbital
energies are associated also with the ionization energies but within the Koopman’s
theorem, i.e. no electronic relaxation effects are accounted for. Within this study we
are concerned only with the lowest Mn hole state in the limit of x≈0 (La1−x Cax MnO3 )
and thus we consider only the Mn 3d eg -like band below the Fermi level in lightly
hole doped manganites. Because of the very low doping level this band will closely
resemble the top of the valence band in pure A-antiferromagnetic (AFM) LaMnO3 .
The delocalization or band stabilization energy, Ed , obtained as the difference
between the energy of the lowest localized CASCI state (see Table A.1) and the
lowest band energy (K) is 1.13 eV. The latter is at the highest symmetry point Γ.
The width W of the Mn 3d eg -like band is the difference in the energy of the lowest
and highest bands at the Γ point and it has value of 2.42 eV. The band stabilization
Many-Electron Bands in Manganites
145
Figure 5.4: Brillouin zone of a simple orthorhombic Bravais-lattice representative of the
crystal symmetry Pbnm.
E, eV
1
0.5
0
-0.5
-1
Δ
Γ
Η
ϒ
Β
Τ
Λ
Ζ
Σ
Γ
G
Χ
Ρ
U
R
Γ
Figure 5.5: Many-body hole energy bands associated with the lowest Mn hole state obtained
for LaMnO3 .
146 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
energy concerned with the delocalization of a single hole in LaMnO3 was estimated
in Chapter 4 by making use of the simple dependence of W and Ed , respectively on
the effective hopping parameters t and t’. t and t’ parametrize the hopping of the
hole within the ab -plane and along the c -axes, respectively. The estimate was based
on the simple relation Ed ∼ 4t+2t’ and yielded 1.1 eV for Ed .
To demonstrate that indeed the mobility of the electrons in the Mn 3d eg -like
band in lightly doped manganites is captured satisfactory, if we parameterize it by
the effective hopping parameters t and t’, we performed an orthogonal tight-binding
calculation using the two-center effective matrix elements (see Table A.2) rather than
the H(R) and S(R). Comparing Figures 5.5 and 5.6 reveals that the bands derived
within the non-orthogonal and orthogonal tight-binding calculations are indeed very
similar. This demonstrates that the parameterization of the mobility of the single
hole in LaMnO3 , used throughout Chapter 4 provides a realistic description of the
delocalization of the hole. The Mn 3d eg -like band associated with this delocalization
E, eV
1
0.5
0
-0.5
-1
Η
Δ
Γ
ϒ
Β
Τ
Λ
Ζ
Γ
Σ
G
Χ
U
Ρ
R
Γ
Figure 5.6: Many-body hole energy bands associated with the lowest Mn hole state obtained
for LaMnO3 , using effective hopping matrix elements.
preserves in practice its width W, 2.37 eV, when the effective hopping matrix elements
are employed in the orthogonal tight-binding calculation, and Ed remains in practice
unchanged 1.12 eV. This similarity of the two band pictures is expected because of
the small values of the overlap integrals between the localized CASCI wavefunctions
(see Table A.1). The high symmetry of the bands also originates from these small
Many-Electron Bands in Manganites
147
overlap integrals.
Comparison of the many-body hole bands with the conventional oneelectron bands
The most straightforward comparison of the many-body Mn 3d eg -like bands is with
the bands derived within a restricted open-shell HF, with a particular canonization,
effective one-electron band study in which the Koopmans theorem is valid and the
one-electron orbital energies can be regarded as approximations to the ionization
energies. Within the Kohn-Sham DFT approach the physical meaning of the oneelectron orbital energies has been regarded as less clear although it has been discussed
by Perdew et al [45] that the highest-occupied Kohn-Sham orbital energy is equal to
the exact first ionization energy of the system. A recent review paper by Gritsenko
et al. [57] discusses the relation between the Kohn-Sham valence orbital energies and
the relaxed vertical ionization potentials. These authors obtain from KS calculations
for prototype second- and third-row closed-shell molecules valence KS orbital energies
corresponding to the experimental vertical ionization potentials with a small energy
deviation. Hence, they conclude that an analog of the Koopmans’ theorem can be also
applied to DFT. This finding justifies the comparison of the many-body bands derived
in this thesis with the one-electron bands derived using DFT based approaches.
In practice however Kohn-Sham potentials derived within the LDA or GGA introduce an artificial upshift of the one-electron orbital energies (see for example the
review article by Baerends and Gritsenko [46]). By constructing potentials with an
appropriate behavior this artificial shift can be cured [47]. Within the RHF and UHF
band calculations, the band gaps are usually largely overestimated (see for example [36, 48]) because the virtual orbitals, which the conduction band is formed from,
are derived in the Coulomb and exchange fields of all electrons in the unit cell [48] and
thus, the corresponding virtual orbital energies are very high. It has been also shown
by Moreira et al [48] for the electronic structure of NiO and by Muñoz et al [36] for
LaMnO3 that the nature of the highest occupied states and lowest unoccupied states
obtained within the UHF band studies, does not reflect the experimental findings.
Periodic HF calculations on the electronic structure of LaMnO3 have been carried
out by Su et al [50]. This study yields a distribution of the projected DOS in which
the Mn 3d bands are located well below the O 2p bands with only a small contribution
immediately below the Fermi level and throughout the valence band. A large peak
in the Mn 3d projected DOS is observed in the range 8.7-10.3 eV below the Fermi
level [50] which suggests a width of the Mn 3d band of 1.6 eV. It is not clear however
what is the character of the Mn 3d band. In this study, the top of the valence band is
constituted mainly from O 2p states. As pointed out by Su et al [50], their results for
the character of the valence bands are consistent with the photoemission and x-rayabsorption spectroscopy data of Saitoh et al [51], who determined the character of the
band gap of LaMnO3 to be O 2p to Mn 3d CT type. As discussed above, Muñoz et
al [36] have performed as well UHF band calculations on the orthorhombic LaMnO3 .
Their local DOS predict a similar distribution of the Mn 3d and O 2p states at the top
of the valence band as that obtained by Su et al [50]. The contribution of the Mn 3d
148 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
states to the local DOS immediately below the Fermi level is almost negligible as found
in the UHF band study by Su et al [50]. These authors however consider the band
calculations within the formalism of the hybrid DFT methodology as more reliable for
the description of the electronic structure of the strongly correlated compound. The
local DOS obtained within the B3LYP method show more significant contributions
of the Mn 3d states near the Fermi level. One can observe immediately below the
Fermi level DOS, assigned by the authors to Mn 3d, spread over an energy range of
1.5 eV. It is not completely clear however whether those Mn 3d states are mainly of
Mn 3d eg -like character, although this seems most probable.
The width of the Mn 3d eg -like bands obtained by Satpathy et al [23, 24] within
the LDA+U [25] approach using the linear muffin -tin orbitals with the atomic-sphere
approximation [26] is in the range 1.0-1.6 eV. Note also that within the L(S)DA
method the Mn 3d bands are the bands closest to the Fermi level while the L (S)DA+U
[25] approach yields a band picture in which the O -2p states contribute at most to
the top of the valence band.
We compared the Mn 3d eg -like bands obtained in this study with those obtained from different one-electron band calculations, based on the linearized augmented plane-wave method [19] and the GW method [20].
Figure 5.7: Calculated electronic energy eg - like bands immediately below the Fermi level,
for A-AFM LaMnO3 in orthorhombic GdFeO3 -type structure (see [19])
Figure 5.8: eg - like energy band of the A-AFM LaMnO3 by the LSDA (a), and by the
GWA (b) approximation. The energy zero-th is fixed at the top of the respective valence band
(see [20])
We discuss in more detail the comparison between the ab initio many-body bands
and the one-electron bands obtained by Ravidran and co-workers [19] using the rel-
Many-Electron Bands in Manganites
149
ativistic full -potential linearized augmented plane-wave method [21]. The authors
used the generalized-gradient corrected approximation [22] for the electron exchange
and correlation in order to introduce some inhomogeneity effects. Their bands immediately below the Fermi level are displayed in Figure 5.7. The bands are characterized
as mainly Mn 3d eg -like bands with a somewhat smaller contribution of O 2p [19].
When comparing the bands in Figure 5.5 and 5.7 one must take into consideration
the fact that the bands in Figure 5.7 are ”upside down” with respect to the many-body
bands in Figure 5.5. As mentioned above the plots in Figures 5.5 and 5.6 illustrate the
energy bands associated with the lowest Mn ionized state of LaMnO3 as a function of
K while the plots in Figures 5.7 and 5.8 display the energies of the Bloch states, (k),
i.e. the one-electron bands. Since the authors in reference [19] have employed a Pnma
space group whereas in the present study we adopted the alternative Pbnm space
group, the second line of symmetry points in Figure 5.7 provides the corresponding
symmetry points in the Pbnm reference system.
Comparing the Mn 3d eg -like bands in Figure 5.7 with the ab inito bands in Figure
5.5 reveals a difference in the band widths obtained in the two different approaches by
about a factor of 2. Note however that we extracted from the band structure displayed
in reference [19] the band immediately below the Fermi level which was assigned by
the authors to be mainly an Mn 3d eg -like band. The orbital-projected DOS obtained
within their approach displays an energy spread of about 2 eV for the relevant Mn
3d eg -like -states with two separated peaks (see Figure 6 in Reference [19]). Because
of the presence of significant DOS attributed to the t2g -like and the counter Mn 3d
eg -like electrons (3z2 -r2 ) in the same energy region (-2-0 eV) it is difficult to extract
from the band structure shown in [19] the Mn 3d eg -like bands within the -2 and -1
eV region.
The shape of our ab initio many-electron hole bands is also rather different although we observe some similar bands dispersion at the ”lower” part of the Mn 3d eg
-like one-electron bands and its corresponding higher part in the many-body band picture. Note however that the most relevant part of the many-body bands is the lower
part because the higher part of the bands may interact with other many-electronbands in the same energy region.
The degeneracy of the bands along the symmetry directions Υ → T , X → U and
U → R and close to the high symmetry points Υ, T and X which is observed in the
common one-electron band structure is lifted in the many-body bands. We observe
also degeneracy of the many-body bands below and above the zero energy along the
symmetry line T → Z.
It is well known that the Coulomb, Hubbard parameter U and the exchange parameter J in the L(S)DA+U method influence significantly the electronic structure
of the perovskites [27] (see for an extensive study on the influence of the parameters
U and J in the LSDA+U method reference [27]). This study has shown that the
positions of the occupied Mn 3d eg -like and t2g -like states move gradually to higher
binding energies with increasing the value of U and this behavior reflects the strong
electron correlation present in the manganites. The authors probe a rather broad
range of values for U and J and determine optimal values which reproduce results
150 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
for the energy band gaps close to experimental values. Although the optimal values
reproduce well the optical gaps deduced from an optical reflectivity spectrum measurements [28] there is no consensus in the literature with respect to the value of the
parameter U, it varies from 5.5 [30] to 9.1 [29] and even larger 10.1 eV [23, 24]. It has
been recently discussed also by Nohara et al [20] that the L(S)DA+U produces as an
artifact a strong bondig state between the O -2p and Mn 3d states in the deep valence
region. The emphasized parameter-dependence of the electronic structure makes it
difficult to compare the many-body ab initio bands for which the electronic correlation and relaxation effects are explicitly incorporated with the one-electron bands for
which the electron-electron interactions are parametrized, with a strong dependence
of the value of U (see for example [31]).
The larger band width of 2.4 eV deduced for the many-body bands arises from the
explicit inclusion of those electron relaxation and correlation effects in the ab initio
non-orthogonal tight-binding calculation.
The importance of inclusion of the electron correlation effects in the electronic
structure calculations has been pointed out by Solovyev [29]. A straightforward way
to incorporate the electron-electron correlation effects is provided by the GW approximation based on the many-body perturbation theory. Within this approximation the
exchange-correlation self-energy is estimated and the LSDA eigenenergies (k) are corrected by making use of the correlation and exchange contributions to the self-energy
as well as the LSDA exchange -correlation potential [20].
We compared the many-body bands of the A -AFM LaMnO3 with those obtained
from the GW calculation [20]. In Figure 5.8 the bands positioned immediately below
the Fermi level, derived within the LSDA and GW approximations, respectively are
depicted. Let us first comment on the difference in the bands obtained with (GWA)
and without (LSDA) accounting for the strong on-site electron correlation effects in
the calculation. The authors of reference [20] point out that the main difference in the
bands within this energy region (the top of the valence band) consists of a change in
the width of the band which is described as mainly Mn -3dx2 −r2 / Mn -3dy2 −r2 in the
Pbnm reference system. Accounting for the strong electron correlation effects leads to
a broader band width of 1.1 eV. The approach does not require any parametrization
[20]. Although the same trend for the change in the band widths is observed for both
the GW bands and the many-body bands when the strong electron-correlation effects
are taken into consideration there is still a rather large energy difference. The GW
calculations have been performed using the linear-muffin-tin orbital method with the
atomic sphere approximation which is a rather crude approximation that does not
include the non-spherical part of the potential.
5.2.3
Discussion of Many-Body Hole Bands for LaMnO3
The Mn 3d eg -like electrons are considered to be rather mobile and the transport
properties of the hole doped LaMnO3 are thought to be dominated by Mn 3d eg -like
electrons [19]. Indeed the broad width of the bands shows that the Mn 3d eg -like
electrons are more delocalized than for example the t2g -like electrons.
Many-Electron Bands in Manganites
151
Within the framework of the theoretical approach described in Chapter 3 we treat
the electronic relaxation and correlation in a straightforward manner without introducing any parametrization. By choice, there is no enforcement of translation
symmetry on the orbitals in contrast with all one-electron band models considered
above. Since the translation symmetry is introduced only for the many-electron crystal wave functions, the orbitals in which the local many-electron basis functions are
expressed are localized and thus the effects of electron correlation and relaxation are
included. These effects are consequently introduced into the eigenenergies (K) of the
many-electron crystal wave functions.
The bands associated with the delocalization of the hole are derived without accounting for the nuclear relaxation accompanying the introduction of the hole at a
particular Mn lattice site. This issue has been discussed in detail by Allen and Perebeinos [52], who introduced the concept of a formation of an anti-Jahn-Teller polaron
in lightly hole doped LaMnO3 . This concept has been recently revisited by Perebeinos and Allen [53] who have considered not only the lightly hole doped LaMnO3
but also the polaron formation in lightly electron doped CaMnO3 . These authors apply model Hamiltonian approaches, where the model Hamiltonians contain essentially
terms accounting for the hopping of the hole and the electron-phonon interactions.
The anti-Jahn-Teller polaron is formulated by those authors [52, 53] in terms of a
self-trapped hole in LaMnO3 which is a result from the strong electron-phonon interactions in the compound. Moreover, it lowers its energy by a lattice relaxation of the
local oxygen atoms, surrounding the Mn lattice site where the hole is localized, rather
than by a delocalization. Within the polaron model, Allen and Perebeinos [52] and
Chen, Perebeinos and Allen [53] define the minimum energy of the delocalized hole
as the difference between the energy of the localized hole in an unrelaxed lattice and
the magnitude of the nearest-neighbour hopping integral, the latter is associated with
the mobility of the hole. If then they allow for a local lattice relaxation around the
hole, the energy of the localized hole state is lowered to below the minimum energy
of the delocalized hole in the unrelaxed local environment.
The energy gain related to the local lattice distortions has been studied extensively
by Millis and co-workers (see, for example, [39] and the references therein). They also
suggest that the energy gain associated with the Jahn-Teller distortion is large enough
(& 0.1 eV [39]) to imply a strong electron-phonon coupling and thus polaronic models
of the transport in doped manganites. Van den Brink et al [40] have considered the
holes strongly coupled to the excitations of the Mn 3d eg orbitals which coupling
produces a quasiparticle type energy dispersion. Extensive studies on the free hole
dispersion in doped LaMnO3 as opposed to the dispersion of the hole in the presence
of orbital polarons have been performed by Bala et al [41] (see also [42]).
The discussion above illustrates the importance of accounting for the nuclear relaxation in the nearest neighbourhood of an excess hole. A relevant question is whether
the energy gain associated with the geometry relaxation of the nearest oxygen atoms
around the lattice Mn site, where the hole resides, is of the same magnitude as the
energy gain associated with the band formation due to the delocalization of the Mn
3d eg -like hole in the unrelaxed lattice, i.e. within the Born-Oppenheimer approx-
152 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
imation. To obtain an estimate of the energy scale of the geometry relaxations we
performed the following simple analysis, using a method which has been justified by
Moraza, Sejio and Barandiaran [56]. We considered the changes in the total CASSCF
energies of the ground states, 5 Eg and 4 Ag of Mn3+ and Mn4+ in the 3d4 and 3d3
electron configurations, respectively, as a function of the Mn-O bond length, the one
which forms initially the longest Mn-O bond in the ’undoped’ Jahn-Teller distorted
[MnO6 ] cluster. The wave functions of 5 Eg and 4 Ag are constructed within CASSCF
-d active spaces for the [MnO6 ]9− and [MnO6 ]8− clusters, respectively. The CASSCF
energies of 5 Eg and 4 Ag are computed for Mn-O bond lengths varying from the JahnTeller value of 2.18 Å to 1.88 Å. This Mn-O bond is chosen as a relevant distortion
coordinate, RM n−O , because it undergoes the most significant displacement inwards,
i.e. towards the Mn ion with the hole.
The oxygen ions which form the shorter Mn-O bonds in the undoped distorted form
are only slightly displaced further inwards, i.e. towards the Mn ion with the hole.
Thus, the geometry relaxation gain associated with those oxygen ions is expected
to contribute at least to the overall relaxation energy gain. The displacement of the
oxygen ions along the c axes of the crystal is not considered, because it is not expected
to change significantly the energy of the geometry relaxation.
The difference in the energy of the 4 Ag state, at RM n−O for which the CASSCF
curve of 4 Ag shows a minimum, and the energy of 4 Ag at another RM n−O for which the
CASSCF curve of 5 Eg has a minimum, is a measure for the nuclear relaxation. In this
study this CASSCF energy difference is 0.8 eV. This means that the energy scales of
the nuclear relaxation and the band stabilization are very similar although the energy
gain from the formation of the band is larger. These results clearly demonstrate that
there is a competition between the localization of the hole at a particular Mn lattice
site, favoured by an energy gain of the nuclear relaxation, and the delocalization of
the hole favoured by the stabilization energy, gained from the band formation.
Many-body electron bands for CaMnO3
Next we consider the bands related to the delocalization of a single electron within
the ab plane and along the c axis of orthorhombic G- type AFM CaMnO3 . The
non-orthogonal tight binding calculations were performed using the H(R) and S(R)
matrix elements obtained in the calculations for the [Mn4 O20 ] super-clusters (see
5.2.1 ). The relevant interactions within and between the ab planes are antiferromagnetic and therefore we deduced the Hamiltonian matrix elements and overlap integrals
corresponding to the low spin couplings between the Mn ions from those obtained for
the high spin couplings. To do this we made use of the Andersson-Hasegawa spin
dependence of the matrix elements (see Chapter 4 and the subsection on LaMnO3
in this chapter). Unlike in LaMnO3 two localized low-lying states per each Mn ion
are considered since their relatively small energy difference of 0.2 eV suggests that
the higher state may contribute to the formation of the band associated with the
delocalization of the extra electron (see also subsection 5.2.1 ). We first consider the
contribution of the lower and higher states separately.
Analogous to LaMnO3 in CaMnO3 the nearest-neighbour Mn-Mn interactions
Many-Electron Bands in Manganites
153
have a significant magnitude whereas the next-nearest neighbour interactions are
found to be an order of magnitude smaller (see also Chapter 4 ). Therefore in the
sums in Eq. (3.23) only terms corresponding to the nearest - and next-nearest interactions in all three crystal directions are considered, i.e. only the interactions between
the first neighbour cells and the reference unit cell in the 3D crystal are included.
Since CaMnO3 has the same crystal structure as LaMnO3 Figure 5.3 displays also for
this compound the reference unit cell and the first neighbour cells in the 3D space.
The localized CASCI wavefunctions Φ1a , Φ2b , Φ3a and Φ4b within the reference unit cell,
depicted in the Figure, correspond to the lower states. The higher states Φ1a0 , Φ2b0 ,
Φ3a0 and Φ4b0 are not shown but they are considered as well in the tight -binding cal0
00
culations. The wavefunctions Φ1a , Φ1a are CASCI wavefunctions of the same type as
Φ1a but localized at Mn ions in the nearest neighbour unit cells of the reference unit
cell along the lattice vectors in the ab crystal plane (see Figure 5.3).
Analogously to LaMnO3 , CaMnO3 has a primitive orthorhombic lattice with Pbnm
crystal space group (see also section 5.2.1 ). We calculated first the energy bands associated with the delocalization of the electron assuming that it occupies the same Mn
3d eg -like orbital at all Mn sites, the one directed along the long Mn-O bonds. This
situation corresponds to performing a non-orthogonal tight-binding calculation with
only four local many-electron basis functions, one per Mn site, within the reference
unit cell. The bands derived from these lower local many-electron wave functions are
displayed in Figure 5.9 along the same high symmetry lines Σ, ∆ and Λ, H, B, G
and P in the first irreducible Brillouin zone of the Pbnm crystal structure. The zero
energy is introduced as the average energy of either the lower four localized CASCI
states or the higher four localized CASCI states (in the next set of calculations). The
case of considering all eight localized CASCI states in the unit cell is discussed also
below and in that case the zero energy is the average energy of all eight localized
states.
5.2.4
Discussion of many-body electron bands for CaMnO3
As expected the shape of the bands is similar to that of the bands associated with the
lowest hole states in LaMnO3 . Indeed the lower localized CASCI states representing electron doped CaMnO3 and the localized CASCI states representing hole doped
LaMnO3 characterize the lowest states in both systems which are related to the mobility of the electrons in the Mn 3d eg -like band. The band stabilization energy Ed
is reduced to 0.71 eV compared to LaMnO3 due to the smaller values of the Mn-Mn
effective hopping matrix elements, tab = -0.16 eV and tc ∼ -0.06 eV (see results for the
four-center clusters in Chapter 4 ), determined by the antiferromagnetic couplings in
all three crystal directions. The lowest band energy (K) is at the highest symmetry
point Γ. The width W of the Mn 3d eg -like band is evaluated to be 1.47 eV and it is
the difference in the energy of the lowest and highest bands at the Γ point. The manybody bands in Figure 5.9 reflect the change in the first electron affinity of CaMnO3
along different symmetry directions, i.e. E (K). Since only lightly doped systems are
considered the bands associated with the lowest electron states are not expected to
154 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
E, eV 0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
Δ
Γ
Η
ϒ
Β
Τ
Λ
Ζ
Σ
Γ
G
Χ
Ρ
U
R
Γ
Figure 5.9: Energy bands associated with the lowest electron state obtained in the cluster
calculations for CaMnO3 .
Many-Electron Bands in Manganites
155
differ substantially from the eg -like conduction bands of the undoped CaMnO3 . The
single electron is introduced in the conduction eg -like band of CaMnO3 and occupies
the lowest eg -like orbital at each Mn site maintaining high-spin coupling with the
t2g -like electrons at the same site. The excess electron is delocalized over the entire
lattice which remains antiferromagnetic. The band stabilization energy associated
with this delocalization was estimated in Chapter 4 to be about 0.8 eV (the effective
hopping matrix elements tab = -0.17 eV and tc ∼ -0.06 were used as obtained for the
[Mn2 O11 ] cluster). This estimate agrees well with the outcome 0.71 eV of the tightbinding non-orthogonal calculation. However in Chapter 4 the higher-lying localized
states with the second Mn 3d (eg -like) component occupied were not considered. As
we see below they also play a role in the delocalization of the extra electron. Before
introducing the higher-lying localized states in the tight-binding calculation we consider the bands associated with the delocalization of the excess electron assuming that
it occupies always the higher eg -like orbital when residing at a particular Mn site.
The resulting bands are displayed in Figure 5.10. The shape of these bands differs
from that of the bands associated with the lower electron states in Figure 5.9. One
may note however that in fact the shapes of the bands in Figure 5.10 and Figure 5.9
are closely related and to some extent similar. The bands along the symmetry line
U → R are nearly degenerate in both band pictures. Furthermore, the interactions
E, eV
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
Δ
Γ
Η
ϒ
Β
Τ
Λ
Ζ
Σ
Γ
G
Χ
Ρ
U
R
Γ
Figure 5.10: Energy bands associated with the higher electron state obtained in the cluster
calculations for CaMnO3 .
between the higher-lying localized CASCI states along the c axis have the same mag-
156 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
0 0
nitude, tac a = -0.22 eVb as the interactions between the lower-lying CASCI states
within the ab planes, tab
ab = -0.16 eV. Analogously, the hopping matrix elements between the higher localized CASCI wavefunctions within the ab planes are of the same
magnitude ∼ 0.04 eV as those between the lower CASCI wavefunctions along the c
axis, ≈ 0.06 eV. These similar magnitudes of the effective hopping matrix elements
and consequently of the Hamiltonian matrix elements and overlap integrals manifest
in the resulting bands. The bands derived from the lower localized many-electron
basis functions along the symmetry directions Γ → Y and Γ → X (Figure 5.9) are
similar to those derived from the higher localized many-electron basis functions along
the symmetry direction Γ → Z (Figure 5.10). Moreover the bands along the symmetry direction Γ → Z in Figure 5.9 are similar to those along the symmetry directions
Γ → Y and Γ → X in Figure 5.10. The same considerations hold for the bands along
Z → T in Figure 5.9 and those along Y → T and X → U in Figure 5.10 as well as
along Z → T in Figure 5.10 and Y → T and X → U in Figure 5.9, respectively.
Ed associated with the higher-lying electron states is taken as the difference in
the energy of the lowest from all four higher-lying localized CASCI states, considered
within the reference unit cell, and the lowest band energy (K) which is at the Γ
point. Its value is 0.57 eV. W of the many-body Mn 3d eg -like bands, defined in
the same manner as above is 1.19 eV. Both calculated values of Ed and W agree well
with the values estimated using the simple relation Ed ∼ 4t+2t’.
In the next tight-binding calculation all eight localized CASCI wavefunctions,
four for the lower-lying states and four for the higher-lying states were considered
as localized many-electron basis functions within the reference unit cell. This manyelectron basis set allows one to account not only for the interactions between the lower
or higher-lying states ((Figure 5.9) and (Figure 5.10)), but also for the interaction
between the lower-lying state at one Mn site and the higher-lying state at another
Mn site. The resulting bands are plotted in Figure 5.11 A. We find indeed the role
of the higher-lying states in the delocalization of the extra electron to be significant
and to lead to an increase in the band stabilization energy Ed up to 1.05 eV. W of
the many-body Mn 3d eg -like bands also increases up to 2.30 eV and becomes of the
same magnitude as that of the many-body Mn 3d eg -like bands in LaMnO3 .
Let us consider the changes in the bands associated with the lower and higher
states when the interactions between them are incorporated. To facilitate the comparison, in Figure 5.11 B the bands of the lower (Figure 5.9) and higher (Figure 5.10)
states are combined in the same plot. Comparing Figure 5.11 A and Figure 5.11
B reveals changes in the shape of the bands arising with the introduced additional
interactions although the resulting bands retain some of the features of the bands
associated with the higher- and lower-lying states. The bands along the symmetry
direction U → R can be regarded as a super-position of the bands in Figures 5.9 and
5.10 along the same symmetry direction. The interactions between the higher and
lower states open a small gap between the bands along the symmetry line R → Γ.
Along the symmetry directions Γ → Y , Γ → X, Y → T and X → U , respectively the
b the
0 0
superscript in the notation tca a shows that the hopping parameter is concerned with the
higher-lying localized CASCI states a’ and a’ at the Mn ions 1 and 3
Many-Electron Bands in Manganites
157
Figure 5.11: A. Energy bands associated with the two lowest electron states obtained in the
cluster calculations for CaMnO3 . The interaction between the lowest state at one site and
the higher state at another site is considered. B. The interaction between the lowest state at
one site and the higher state at another site is not considered.
E, eV
1
0.5
0
-0.5
-1
Δ
Γ
Η
ϒ
Β
Τ
Λ
Ζ
Σ
Γ
G
Χ
Ρ
U
R
Γ
R
Γ
E, eV 0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
Δ
Γ
Η
ϒ
Β
Τ
Λ
Ζ
Σ
Γ
G
Χ
Ρ
U
158 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
bands can be viewed as ”combinations” of the bands of the lower- and higher- lying
states along these symmetry lines. Furthermore the bands in Figure 5.11 A along
T → Z and Z → Γ can also be viewed as a super-position of the bands in Figures
5.9 and 5.10 along the same symmetry lines. Due to the introduced additional interactions the bands in Figure 5.11 A have on the ∆ and Σ lines, and H and G lines,
respectively points of contact located at about the inter-section points between the
bands in Figure 5.11 B along those lines.
In Chapter 4, we considered the possibility for formation of a ”self-trapped magnetic polaron” region [32] within the AFM lattice favored by the double exchange
interaction between the nearest neighbour Mn ions. Meskine and co-workers have
shown that indeed such magnetic polaron states can be formed below the eg -like
bands in AFM CaMnO3 [32]. The formation of such ”magnetic polaron” region is
favorable for the delocalization of the excess electron over the nearest neighbour Mn
sites. We have shown (see Chapter 4 ) that the stabilization energy associated with
the delocalization of the excess electron over its six nearest neighbour Mn ions is ∼
40%-50% the band stabilization energy in the case of AFM lattice. Furthermore, this
stabilization energy increases by a factor of 4 upon the formation of the seven Mnsite ferromagnetic cluster. Taking into consideration also the higher-lying localized
CASCI states, this stabilization energy increases up to 2.1 eV when a ”self-trapped
magnetic polaron” is formed. This means that the excess electron in the lightly electron doped CaMnO3 will delocalize within the ferromagnetic region, which it forms,
rather than over the entire lattice, contrary to the predictions of Gennes [55].
We did not attempt to make a direct comparison between the many-body Mn 3d
eg -like bands derived for the AFM CaMnO3 and the one-electron Mn 3d eg -like
bands obtained in the more common one-electron band pictures [23, 24]. These oneelectron Mn 3d eg -like bands are located immediately above the Fermi level, i.e. at
the bottom of the conduction band, however it is not very clear how to extract them
from the band structures because of the presence of significant DOS associated with
the Mn 3d t2g -like electrons in the same energy region (see for example [23,24,33] and
also the band diagrams derived by Jung et al [34] from optical conductivity analysis).
However some discussion is in place on the band calculations within the periodic HF
approach, performed by Fava at al [54]. Fava and co- workers [54] have obtained a HF
band structure of AFM CaMnO3 in which they identified the highest occupied bands
as formed from mainly O -2p states while the lowest unoccupied states as having
mainly Mn 3d eg -like character. Their projected density of states show a significant
contribution of Mn 3d eg -like states at the bottom of the conduction band, spread
over an energy range of about 2.5-2.7 eV.
5.3
Conclusions
In this Chapter, two topics were discussed which are related to the application of
the theoretical approach derived in Chapter 3 to the transport properties of doped
manganites. In the first part of the Chapter, we presented a detailed analysis of the
performance of three different overlapping fragment schemes, OF1, OF2 and OF3 for
5.3 Conclusions
159
the construction of localized orbital basis for localized CASCI wavefunctions representing hole and electron states in doped manganites. The comparison between the
three overlapping schemes revealed that a balanced description of the two localized
CASCI states of two-center clusters, representing either hole doped LaMnO3 or electron doped CaMnO3 is obtained only within OF1. Within OF1 the localized doubly
occupied orbitals of the super-cluster, associated with the O ion shared between two
[MnO6 ] fragments, are constructed as normalized linear combinations of the fragments doubly occupied orbitals. The remaining unitary transformed doubly occupied
orbitals of the two fragments are combined to form the complete inactive orbital basis
set of the super-cluster. The active orbitals of the two fragments are also combined
to yield the active orbitals of the super-cluster. The localized CASCI wavefunctions
Φ1 and Φ2 of the super-cluster are expressed in terms of this newly generated set of
doubly and singly occupied orbitals. Furthermore, the effect of three different embeddings on the effective hopping matrix elements as well as CASCI energy differences
H22 -H11 was investigated and only a small change in those quantities was obtained
as a function of the embedding scheme.
The effective Hamiltonian matrix elements and overlap integrals between the localized CASCI wavefunctions Φ1a , Φ2b , Φ3a and Φ4b of the [Mn4 O20 ] super-clusters within
the ab plane and along the c axis are used to approximate the Hamiltonian matrix elements and overlap integrals, H(R) and S(R), between the local many-electron basis
functions. The latter represent hole or electron states localized at different lattice sites
(see Chapter 4 ). The final delocalized many-electron wave functions of the extended
system are expressed in terms of linear combinations of Bloch sums of the local manyelectron basis functions. These delocalized wave functions and their corresponding
eigenvalues (K) are obtained from a non-orthogonal tight-binding calculation.
Using this formalism, the many-body bands associated with the lowest hole state
in LaMnO3 and the lowest added-electron states in CaMnO3 are derived and compared, in case it is possible, to the one-electron bands obtained in conventional band
structure calculations. We find a larger width of 2.4 eV, of the many-body bands,
associated with the mobility of the Mn 3d eg -like electrons at the top of the valence band of LaMnO3 , compared to the Mn 3d eg -like one-electron bands obtained
by Ravidran et al. within the generalized gradient corrected full potential linearized
augmented plane-wave method [19]. Comparing the many-body bands with those
obtained within LDA+U band calculations [23, 24] as well as within the GW approximation revealed also a larger width of the many-body bands. Note however that
although these effective one-electron approaches incorporate to some extent the manyelectron effects, either by parametrizing them (LDA+U) or by introducing corrections
to the L(S)DA eigenenergies based on the correlation and exchange contribution to the
self-energy (GW ), they do not include the electronic relaxation effects accompanying
the ionization and electron addition.
The method, introduced in Chapter 3, allows for the explicit inclusion of the
strong electron correlation and relaxation effects in the final many-body bands. The
larger band width of 2.4 eV is a result of the explicit inclusion of those effects in the
tight-binding calculations.
160 Chapter 5, OFA ”at work” and Many-Electron Bands in Manganites.
We also confirmed the adequateness of the parametrization of the mobility of the
electrons in the Mn 3d eg -like band by the two-center effective hopping matrix elements introduced in Chapter 4. Using those effective hopping matrix elements rather
than the Hamiltonian matrix elements and overlap integrals between the localized
CASCI wavefunctions, we obtained the many-body bands within an orthogonal tightbinding calculation. The energy difference in the band widths obtained within both
tight-binding schemes is negligible, which is expected because of the small overlap
integrals between the localized CASCI wavefunctions.
The many-body bands in CaMnO3 are associated also with the mobility of the
Mn 3d eg -like electrons, however in this case the bands refer to the bottom of the
conduction band of CaMnO3 . We find a similarity in the shape of the bands associated
with the lowest hole state in LaMnO3 and the lowest electron state in CaMnO3 . This
similarity is anticipated because the lowest states in doped LaMnO3 and CaMnO3
are both related to the mobility of the electrons in the Mn 3d eg -like band. The
difference in the band widths is directly related to the difference in the Mn-Mn effective
hopping integrals. The smaller effective hopping integrals in CaMnO3 within the ab
plane determine the lower band stabilization energy of 0.71 eV compared to that in
LaMnO3 , 1.1 eV.
A significant difference between the two compounds arises from the presence of an
extra localized electron state in CaMnO3 with an energy close to that of the lowest
electron state. The two lowest electron states in CaMnO3 , with either one or the other
Mn 3d eg -like orbital occupied, are nearly degenerate in the fragments calculations
but the cluster artifacts occurring for the super-clusters introduce a somewhat larger
energy difference between them. The consideration of the lower and higher localized
CASCI states in the non-orthogonal tight-binding calculations shows that the higher
state contributes significantly to the Mn 3d eg -like band formation in CaMnO3 .
Accounting also for the presence of those higher-lying localized CASCI states at the
different Mn lattice sites leads to an increase in the band stabilization energy up to
1.05 eV and a correspondingly larger band width of 2.3 eV. Hence, the width of the
many-body Mn 3d eg -like electron bands in CaMnO3 becomes of the same magnitude
as that of the many-body Mn 3d eg -like hole bands in LaMnO3 in spite of the lowspin couplings between the different Mn ions and consequently, the lower values of
the effective hopping parameters.
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6
d-d Excitations and Charge Transfer
States in LaM nO3
ABSTRACT —
Ab initio embedded cluster calculations are presented on
d-d and charge transfer excitations in cubic and orthorhombic LaM nO3 . This
theoretical approach allows for an independent evaluation of controversial quantities related to the Mn d-d excitations, such as Jahn-Teller and crystal field
splitting parameters as well as Mn 3d exchange interaction. We investigate the
effect of the JT distortion and tilting of the MnO6 octahedra on the lowest d-d
transitions.
We find 1.3 eV for the JT splitting, ∼ 2.0 eV for the exchange splitting and
1.8 eV for the crystal field splitting in the orthorhombic compound. Cluster size
effects on the calculated d-d transitions are found to be modest. As expected,
the computed charge transfer excitations are influenced more significantly by
cluster size effects. We find that the changes in the relative energies of the
charge transfer states when the [MnO6 ] cluster is extended are mainly caused by
electronic relaxation effects and a change in the external potential. The highest
spin quintet state in the Mn 3d4 manifold and the lowest state (a spin septet)
in the charge transfer manifold are separated by ∼1.3 eV in the model [MnO6 ]
cluster. This separation decreases to ∼ 0.4 eV in the largest cluster.
6.1
Introduction
The intriguing physical properties of the Mn-based perovskite materials have been
a focus of interest in the physics of strongly correlated condensed matter in the last
years. Depending on the composition A1−x Dx M nO3 , where A is a rare-earth element,
e.g. La, Nd, Pr and D is a divalent ion such as Ca, Ba, Sr, these transition metal
(TM) compounds show a large variety of interrelated physical phenomena. These
include ferromagnetic (FM), antiferromagnetic (AFM) ordering, so-called orbital or-
166
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
dering and charge ordering [1], colossal magnetoresistance (CMR) effect [2] and diverse
phase transitions [3]. The La1−x Dx M nO3 series has particularly attracted much attention mainly due to the colossal negative magnetoresistance effect (CMR), observed
in this system [4]. This phenomenon has been described as an AFM insulator to FM
metal phase transition [4,6,7], induced by a magnetic field. Different physical mechanisms such as the double exchange model [8], the dynamical Jahn-Teller (JT) effect [9],
lattice and magnetic polarons [3, 18] etc. have been invoked to clarify the complex
nature of the charge, magnetic and lattice order in this material. Clearly, understanding the nature of the interrelated phenomena in these systems, requires information
about the electronic structure and properties of the end members of the series, i.e.,
LaM nO3 and CaM nO3 . Numerous experimental and theoretical efforts have been
dedicated to elucidate the electronic structures of the two compounds [6, 10–24]. In
particular the electronic structure of LaM nO3 has been intensively studied theoretically within the periodic UHF approach [5,10], the local (spin) density approximation
(L(S)DA (+U )) [14], generalized-gradient corrected relativistic full potential method
(GGCRFPM)a [15], etc. GGCRFPM is a DFT approach which employs the fullpotential linearized augmented plane wave method including spin-orbit couplings and
GGA corrections.
LaM nO3 has orthorhombic symmetry at RT [31, 32] with space group Pnma (the
equivalent Pbnm system is also used in some studies [30]). The formal electronic
configuration of the M n3+ ions is 3d4 . The 5 Eg ground state is in a (nearly) Oh
site symmetry high spin coupled, characterized by a weak-field electron configuration
t32g e1g . The two-fold orbital degeneracy of the ground state implies a JT type distortion, which splits the degenerate components of 5 Eg and produces large asymmetric
oxygen displacements of the M nO6 octahedra. In addition, the M nO6 octahedra are
titled and the Mn-O-Mn angles are less than 180◦ [30, 32].
The valence electronic structure of LaM nO3 contains the local Mn d-d excited
states and local charge transfer (CT) O 2p → Mn 3d states. Important model parameters associated with the Mn d-d excitations are the crystal field (CF) splitting
parameter, 10 Dq, the Mn 3d exchange splitting and the Jahn-Teller splitting parameter. In a simplistic orbital model the 10 Dq parameter measures the energy splitting
between the Mn 3d (eg ) and (t2g ) levels of the Mn3+ ion with four 3d electrons in
the weak, nearly octahedral crystal field of the oxygen ligands. The Mn 3d exchange
parameter determines the energy splitting between two (eg ) levels which differ by a
spin flip of the electron. The JT parameter is a measure of the energy splitting of
the Mn 3d (eg ) levels in the distorted structure b . These model parameters have
been deduced in several theoretical and experimental studies. In Table 6.1, we have
listed these parameters, extracted from some selected studies. We outline briefly some
relevant findings in those studies.
The optical conductivity studies performed by Jung et al. [21] suggested that the
lowest optical conductivity peak in the spectra should be associated with a transition
a We
adopt the GGCRFPM acronym for future reference
description of the crystal structure of LaM nO3 was presented in Chapter 4 in section 4.4.
The reader is referred to this section also for the description of the Mn3+ configuration.
bA
Introduction
167
Table 6.1: CF, JT splitting and Mn 3d exchange parameters, evaluated in some theoretical
and experimental studies; The values in the table are in eV;
Reference
Jung et al. 21
Picket and Singh 5
Ravindran et al.15
Elfimov, Anisimov et
al.20
Satpathy et al.14
Lawler et al.24
Model parameters
Approach
Optical conductivity spectra
LAPW
GGCRFPM/FPLAPWa
LSDA +U
JT
1.9
1.5
1.41
exchange CF
3.1
3.5
3.3
1.1a
-
LDA +(U)
Optical transmission spectra
'1.5
-
'3.0
-
'2.0
1.7
across the JT gap which gives rise to a JT splitting parameter of about 1.9 eV. The
theoretical study of Allen and Perebeinos [16, 17] within the framework of a model
Hamiltonian produces the same JT splitting parameter. To explain the character
of the lowest optical conductivity peak in the optical spectrum by Jung [21], Allen
and Perebeinos [16, 17] introduced the self-trapped exciton theory within which the
lowest electronic excitation is found to be a d-d transition. This excitation is selftrapped on a single Mn site because the energy gain accompanying a local lattice
undistortion (relaxation) is larger then the energy gain associated with an exciton
band formation [16, 18]. The lattice relaxation leads to a reduction of the energy of
the d-d excited state to about half the energy associated with the maximum intensity
of the optical conductivity peak [18].
An alternative interpretation of the optical peak at ∼ 2.0 eV is suggested by
Quijada and co-workers [59], who studied the temperature dependence of the optical
conductivity of LaM nO3 . These authors found a gradual increase in the oscillator
strength of the 2.0 eV feature within the 300-150 K interval [59]. They argued that
the dominant contribution to the optical spectral weight of the conductivity peak
at ∼ 2.0 eV arises from the optical charge-transfer hopping transition between the
JT-split eg levels at adjacent Mn ions rather than the on-site optical transition of a
self-trapped exciton [59].
The density functional band structure (DFBS) studies by Satpathy et al [14] and
Elfimov et al. [20] within LD(S)A (+U) yielded a value of '1.4-1.5 eV for the JT
parameter.
Clearly the magnitude of the JT splitting parameter is a question of on-going
controversy, so is the interpretation of the optical excitation.
Similar controversy exists with respect to the value of the CF splitting parameter
(see the last column in Table 6.1). For instance, Lawler et al. [24] deduced from
the optical transmission spectra of LaM nO3 a value for the CF splitting parameter
of ∼ 2.0 eV. The DFBS calculations carried out by Satpathy and co-workers [14]
yielded also a value for the CF splitting parameter of ∼ 2.0 eV. Ravindran and
co-workers [15] obtained within the full potential linearized augmented plane-wave
168
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
(FPLAPW) method a twice as small CF parameter.
Contrary to the JT and CF splittings, the estimates of the Mn 3d exchange splitting parameter deduced from experimental and theoretical studies are more consistent. The optical conductivity spectra studies by Jung et al. [21], the GGLSDA band
calculations by Ravindran and co-workers [15], the LAPW [19] band calculations by
Picket and Singh [6] as well as the LDA+U calculations by Satpathy [14] lead to
an estimate of the Mn 3d exchange parameter in the range of 3.0-3.5 eV (see Table
6.1). The discussion above indicates that in spite of the numerous studies dedicated
to LaMnO3 the CF spectrum and the magnitude of the CF and JT splittings are still
controversial. LaMnO3 is often cited as a representative of the strongly correlated
electron materials. A correct account for the electron correlation effects in these 3d
TM compounds is compulsory for obtaining quantitative results.
The model parameters, discussed above, characterize the spectra of the Mn 3d4
electronic configuration. The d-d excitations which constitute this spectra are excitonic states with a predominantly localized character. A relevant question to be
addressed in connection with the valence electronic structure of LaM nO3 is also the
relative position of the hole-electron coupled O 2p → Mn 3d CT states with respect
to the d-d excitations. These CT excitations are also considered to be local but not as
local as the d-d excitations. In the following discussion, we refer to those hole-electron
coupled states as local d-d and CT excitations.
The focus of the present study are some of those excitonic states, which are related
to the optical spectra and the optical band gap of the structure. This optical gap
is defined at the onset of an intensity increase in the measured optical conductivity,
i.e. it is related to a transition at the same k point of the Brillouin zone. Arima and
co-workers [11] carried out an optical conductivity study of 3d TM oxide compounds
based on reflectivity spectra measurements and their conductivity data for LaM nO3
suggested a CT optical gap of ∼1.1 eV.
The O 2p → Mn 3d CT excitations have been assigned in the optical conductivity
analyses of Jung and co-workers [21] to the high frequency peaks at 4.6 eV and
7.71 eV in the optical conductivity spectra. Quijada et al. [59]. show also that
the peak centered around 4.0-5.0 eV in their room temperature (RT) conductivity
measurements corresponds to CT transitions between the O 2p and Mn 3d (eg ).
The dielectric tensor spectra of the orthorhombic LaMnO3 obtained by Ravindran et
al. [15] within GGRFPM calculations shows also peaks at 4.7 and 8.8 eV associated
with O 2p → Mn 3d CT transitions.
In some earlier photoemission spectra analysis by Saitoh and et al. [12], LaM nO3
has been referred to as a CT insulator with a conductivity band gap of 1.7 eV. They
also suggested that the ground state has mixed d4 - d5 L character, i.e. strong covalency of the compound. Chainani, Mathew and Sarma [23] claimed a strong mixing of
the Mn 3d and O 2p derived states (from X-ray and ultraviolet photoelectron valence
band spectra) and suggested that according to their bremsstrahlung isochromat spectroscopy (BIS) data the conductivity band gap in LaM nO3 should be about 1.3 eV.
Contrary to Saitoh and et al. [12] the analysis of Chainani, Mathew and Sarma [23]
placed LaM nO3 in the Mott -Hubbard insulator regime rather than in the CT regime.
Introduction
169
Previous studies on related compounds [25–27, 33, 55, 56] have shown that an ab
initio quantum chemical study on embedded clusters may provide an accurate determination of the CF and CT excitation energies. Indeed the embedded cluster model
combined with a well chosen MC wave function (WF) ensures the proper incorporation of the predominant non-dynamical electron correlation effects and relaxation
effects. The dynamical correlation effects are often estimated using the CASPT2 [42]
approach. Furthermore the cluster approach allows for extracting information about
the validity of the material model, the covalency of the compound, and the degree of
localization of the excitation.
In the present study we employ the embedded cluster model combined with the
CASSCF / CASPT2 [35] method to investigate the lowest CF and CT excitations in
cubic and orthorhombic LaMnO3 . A recent similar embedded cluster study on the
d-d excitations in cubic and JT distorted LaMnO3 , performed by Hozoi et al [33] has
shown that the lower part of the valence excitation spectrum is constituted of CF
excitations. Hozoi et al [33] have however neglected the tilting of the MnO6 octahedtra. The cluster size effects on those d-d excitations have also not been addressed
explicitly. Hozoi et al [33] have also not found excitations with predominantly CT
character below 3.0 eV. The model cluster employed in their study is a [MnO6 ] embedded cluster. Taking into consideration that the CT excitations are anticipated to be
more delocalized than the CF excitations, the CT energies computed using the small
[MnO6 ] cluster might be overestimated. A recent study on the CF and CT excitations
in the related compound CaMnO3 , performed by Bordas [56], has demonstrated that
indeed the CT energies are affected significantly by the cluster model.
The rest of the chapter is organized as follows. Computational details and structural data for the calculations of the valence electronic structure of LaMnO3 are given
in section 6.2. We organize section 6.3 in two parts. In the initial discussion, we focus
on the Mn 3d4 CF excitations in the local Oh , D2h and Ci symmetries in order to
probe the role of the distortions on the CF spectra. We include subsequently the
local distortion and the titling of the MnO6 octahedra. Furthermore the present
study allows for an ab initio determination of the model parameters, JT, CF and
Mn 3d exchange splitting parameters by relating those parameters to the calculated
CF excitation energies. The cluster size effects on the calculated d-d excitations are
also discussed. The effect of the cluster expansion on the energy is expected to be
insignificant if the excitation is reasonably well localized, which is the case for d-d
excitations in many ionic systems.
The second part of section 6.3 is dedicated to the character and relative energies
of some lowest local CT excitations. An important question is how localized these
CT excitations are. The extent of delocalization is probed by investigating the cluster
size dependence of the CT energies. Furthermore the relative position of the lowest
CT states with respect to the Mn 3d4 states is addressed.
Finally, we provide an estimate of the error in the excitation energies due to
the ideal cubic geometry, used in some calculations, and due to the active space
restrictions. We compare our results for the lowest CF excitation energies with the
results obtained in other theoretical and experimental works to point out the new
170
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
insights in the valence electronic structure of the compound.
6.2
Crystal Structure, Material Model and Computational Information
The cooperative JT coupling between the Mn 3d eg electrons and the oxygen distortions at neighbour M nO6 sites leads to a simultaneous ordering of the octahedral distortions as well as so-called ordering of the occupied eg orbitals (d(3x2 −r2 )/d(3y 2 −r2 )
ordering in the ab plane in Pbnm reference system) [43]. To obtain a reliable picture
for the Mn 3d4 manifold of the system, the study is carried out for the crystal structure at room temperature. However, we estimate in two separate sets of calculations
the influence of the JT distortion and the tilting on the structure of the manifold.
First, we consider the ideal cubic geometry with equal M n − O distances of 1.95
Å and no tilting. At the next step we take into account the JT distortion, while
keeping the tilting angle at 0 degrees. In this JT distorted M nO6 octahedra each
M n − Oi bond length has a value, which is the average between the corresponding
experimental M n − Oi0 estimate in the tilted geometry and its projection on the
i axis, where i ∈ {x, y, z} [33]. Hence, the Mn-O bond lengths have the following
values: d( M n − Ox )=2.14 Å, d( M n − Oy )=1.90 Å, and d( M n − Oz )=1.95 Å, where
the z direction is along the c axis of the Pbnm reference system. The third step
involves accounting for the tilting. At this step we consider the experimental Mn-O
distances for the room-temperature tilted structure, which are d( M n − Ox0 )=2.18 Å,
d( M n − Oy0 )=1.91 Å, and d( M n − Oz0 )=1.97 Å [30]. Finally, the estimate of the
cluster size effect on the computed d - d spectra in the real geometry is discussed. For
practical reasons, we choose to investigate the character and relative energies of the
CT states and their cluster size dependence in the ideal cubic geometry. We suggest,
that this geometry already incorporates the main features of the CT manifold.
In the initial calculations on the Mn 3d4 and CT manifolds, the crystal is modeled
with an embedded cluster, centered around a transition metal (Mn) ion: [M nO6 ]9− .
We denote this cluster as [A]. In order to study the cluster size effects on both manifolds, we introduce three types of larger clusters: [B]-[M nO6 M n6 ], [C]-[M nO6 M n6 La8 ]
and [D]-[M nO6 M n6 La8 O24 ]. In the [B], [C] and [D] clusters the six outer M n3+ ions
are represented by Al3+ . The clusters for the cubic LaMnO3 are illustrated in Figure
6.1. Al3+ is a reasonable representation of M n3+ , because both ions have equal net
charges and similar ionic radii (r(Al3+ )=0.67 Å and r(M n3+ )=0.75 Å). Al3+ is a
better representation for M n3+ than a crude point charge representation. The reason for replacing M n3+ by Al3+ is, that the number of orbitals in the active space
increases significantly with the number of M n3+ ions in the cluster. We note that
with our clusters the delocalization of the open Mn-3d orbitals is not possible. The
short-range interactions between the cluster oxygen ions and the nearest Mn3+ and
La3+ ions outside the cluster are taken into account by modeling the latter with total
ion potentials (TIPs) [46]. There are no basis functions present on these neighbouring ions. The next shells of ions, surrounding the cluster, are represented by a finite
Crystal Structure, Material Model and Computational Information
171
Figure 6.1: Cluster models; The figures are in the ideal structure to facilitate the views.
The calculations are however performed in the real structure with local Ci geometry; [A]:
[MnO6 ], [B]: [MnO6 M n6 ], [C]: [MnO6 M n6 La8 ], [D]: [MnO6 M n6 La8 O24 ]
array of point charges (PCs) at lattice positions. The first few shells of PCs have the
formal charges of the fully ionic model of the material. The PCs at the outermost
shell are optimized to obtain the best representation of the Madelung potential due
to the crystal represented by formal ionic charges on a fine grid within the cluster
region [33].
The basis sets are atomic natural orbital (ANO) Gaussian-type basis sets. The
contraction scheme for each basis set is as follows: Mn (21s15p10d6f)/(6s5p4d2f),
O (14s9p4d)/(5s4p2d), Al (17s12p5d4f/3s2p) and O (14s9p4d)/(3s2p) [45] for the
oxygen ions O(24) in the largest cluster. In clusters [C] and [D], the core shells up to
(....4d 10 ) of the La3+ ions within the clusters are represented by relativistic ab initio
model potentials (AIMPs) [47] with a basis set contraction scheme 13s10p7d/1s2p1d
for the valence shell.
The calculations are performed within the CASSCF approach, in which the WF is
obtained as a full CI expansion in an active orbital space [35, 36]. This WF incorporates to a different extent, depending on the active orbital space, the near-degeneracy
correlation and a part of the dynamical correlation effects in the system.
The first estimate of the energies of the d-d excitations is obtained for the cubic
geometry by constructing CASSCF WFs in an active space formed by the eg -like and
t2g -like Mn 3d orbitals and 4 active electrons. Let us denote this minimal active
space as CASSCF-d. CASSCF-d accounts for the mixing of the different d4 O2−
configurational state functions (CSFs) in the WF expansions. We extend the active
0
space by adding a set of five correlating virtuals d of the same symmetry as the
Mn 3d orbitals: CASSCF-dd’. These virtuals turn out to form a second shell of Mn
3d -like orbitals [40, 41]. Previous studies [37, 40, 41] on local excitations in TM ion
compounds have shown that the active space constructed in this manner leads to a
variational treatment of a large part of the d-d electron correlation energy. It avoids
its overestimation that was obtained within a CASPT2 treatment of this effect [40],
based on the CASSCF-d WF as a reference WF.
CASSCF-dd’ accounts for part of the dynamical correlation due to the valence
172
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
electrons that are in orbitals which have mainly Mn 3d character [37]. The remaining
part of that correlation, plus the correlation effects associated with the semi-core
electrons on the TM-ion (Mn-3s, 3p), the ligand O-2s and 2p electrons is accurately
treated by CASPT2 [42].
The non-dynamical correlation effects related to CT configurations are accounted
for only partially within the CASSCF-dd’/CASPT2 [37]. A better treatment of the
differential electron correlation effects on the Mn 3d4 manifold, due to the contribution of CT configurations can be achieved by adding occupied O -2p orbitals to the
CASSCF-dd’ active space [37]. Some earlier studies [37, 62] on local d-d excitations
in bulk NiO and a NiO(100) surface have shown that we can apply a simple criterion
in the selection of the O-2p orbitals to be included in the active space. We expect the
largest contributions to arise from CT configurations connected to excitations from
the O-2p orbitals with the same symmetry character as the Mn 3d orbitals.
The study of low lying CT states also requires extending the active space with
occupied ligand O-2p orbitals to CASSCF -(n)pdd’.
The dynamical correlation effect of the valence O-2p electrons can be treated at the
CASSCF level, if the active space contains the O-2p orbitals and a set of correlating
0
O − 2p orbitals. However, such an extended active space is too demanding to be
0
handled. Therefore, no ligand O -2p orbitals are in the active space. Moreover, most
of the doubly occupied O-2p are placed in the inactive space. The O-2p orbitals which
are relevant either to the CT configurations contributing to the WFs of the Mn 3d4
states or are involved in the CT excitations are kept in the active space. This active
space reduction leads to a perturbational treatment of the dynamical correlation of
the O-2p electrons, which was shown to be quite accurate [37].
In the study of the CT states, we make a further reduction of the active space
0
by removing the M n − 3d orbitals from the active space. The resulting active space
CASSCF -(n)pd is sufficient to account for the mixing of the various d4 O2− and
d5 O1− CSFs in the CASSCF WF of a CT state. However, the large dynamical
correlation effect of the valence Mn 3d electrons is no longer accounted for by this
CASSCF WF, but treated perturbationally and overestimated by CASPT2 [40]. The
correlation effects for the CT states are expected to be larger than for the d4 states,
because the d5 O1− states have a larger number of Mn 3d electrons. Therefore, the
CT excitation energies obtained with CASPT2 within this active space, should be
considered with care. They appear to be lower due to the overestimation of the
dynamical correlation, which is different for the ground and the CT states. Taking
into consideration all effects of the active space, we provide an estimate of the error
in the quantitative results. It is expected that the character and relative order of
those transitions is predicted correctly within the CASSCF-(n)pd/CASPT2 approach.
Furthermore, CASPT2 accounts for the main orbital relaxation effects accompanying
the CT excitations. In the following paragraph, we present a detailed description of
the active space, used in the study of the CT states.
The CASSCF active space for the CT states is denoted as CASSCF -(n)pd, where
n is the number of active O-2p orbitals. We perform the calculations in D2h symmetry.
CASSCF -(n)pd has only active O-2p orbitals in the irreducible representations, where
Crystal Structure, Material Model and Computational Information
173
the O-2p hole resides. There are n O-2p active orbitals in that symmetry species.
The value of n is determined by the number of low lying CT states in that particular
irreducible representation. For instance, the CASSCF -(3)pd for the ungerade CT
states of T1u and T2u symmetry contains 3 O-2p active orbitals in either b1u , b2u or
b3u in D2h . The configurational mixing of the d4 O2− CSF in the WFs of the CT
states is discussed in section 6.3.2. In order to obtain a balanced description of the
relative energies of the CT states, the ground state WF is expressed in the active space
of only 5 Mn 3d orbitals. The study of the cluster size effects on the CT manifold is
carried out also within CASSCF -(n)pd.
In this paragraph, we briefly comment on some details in the calculation of the
CASSCF WFs for both manifolds. Some states could not be obtained by optimizing
the orbitals for the CASSCF WF of that particular state in a ”single state” calculation.
This is due to the fact, that the character of the state for which the WF is optimized
changes in each iteration and thus no convergence can be achieved. This optimization
problem occurs when the state of interest and another state of the same spatial
symmetry and multiplicity as the state required become near-degenerate, when the
WF of the other state is expressed in the orbitals of the state of interest. In those
cases, we perform an orbital optimization for an average of states. The outcome of this
optimization are states WFs expressed in one set of the average orbitals. The lack of
a fully optimized orbital set for each state, leads to an increase in the states energies.
To obtain the closest approximation to the single state calculation, we increase the
weight of the state of interest to the maximum possible value which still leads to a
convergent solution [37, 62]. This technique has been applied successfully by Geleijns
et al. in the calculations on the local electronic (d-d and CT) transitions at the NiO
(100) surface [62].
In some calculations on the CT manifold the CASPT2 approach failed to give a
correct estimate of the dynamical electron correlation. In those cases, the weight (ω)
of the zero-order CASSCF WF in the first-order corrected WF is low compared to
the corresponding ω of the ground state. We considered ω to be too low if it showed
a deviation of more than 4 % compared to the reference ω of the ground state. This
breakdown of CASPT2 is sometimes caused by the appearance of intruder states in
the first-order WF [51]. If the Hamiltonian matrix element between this intruder
state and the reference WF is not large, the intruder state can be artificially shifted
up in energy by adding an arbitrary value to the expectation value of the zero-order
Hamiltonian of this intruder state. This approach to avoid the breakdown of CASPT2
is known as level shift technique [49]. Geleijns et. al. [62] have given a justification
of this method. However, in some cases the technique applied to shift the energy
of an intruder state can introduce another singularity in the energy denominator in
the expression for the second-order correction to the energy. N. Forsberg and P.- Å.
Malmqvist have introduced an ”imaginary” level shift technique which avoids new
singularities [50]. We apply this approach to recover the CASPT2 breakdown that
occurred in some of our results.
The Al-2s and 2p electrons were correlated by CASPT2 in clusters [B], [C] and
[D]. This has a minor effect on the results. The reason to correlate those orbitals is
174
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
technical and makes no difference for the final conclusions.
All calculations are carried out with the MOLCAS program package [48].
6.3
6.3.1
Results and Analysis
Mn d-d excitations
Attention is focused mainly on the lowest localized excitations because they constitute
the low energy optical spectra. We calculate the energies of those lowest excitations
in the Mn 3d4 spectra at three levels of distortion of the octahedra. The summary of
the results is presented in Tables 6.2, 6.3, 6.4. All states in Oh and D2h geometries are
obtained in a single state calculation. The analysis of the main electronic configuration
of a particular state is based on Mulliken (spin) population analysis (MSPA) and
Mulliken gross population analysis (MGPA) of the active natural orbitals (ACNOs).
An inspection of the composition of the CI vector of the WFs confirms as well the
electronic configurations established in the other analyses. The simplest model of the
system in the present study is cluster [A]: [MnO6 ]9− .
Cluster [MnO6 ]9−
Oh geometry - ideal cubic structure
The study in cubic symmetry is performed in parallel with that of the non-tilted and
real forms in order to clarify the importance of accounting for the JT effect and the
tilting for a realistic theoretical description of the compound. Most of the physical
properties are directly related to the valence electronic structure, hence it is relevant
to investigate the effects of the distortion on the relative energies and character of the
valence excitations.
The 5 Eg ground state of the [MnO6 ]9− in the ideal cubic structure of LaMnO3
is high spin coupled and characterized by a weak-field configuration t32g e1g . The configurational composition and relative energies of the lowest states obtained within
CASSCF-d are listed in Table 6.2. These states constitute the lowest part of the
CF spectra. The CASSCF-d WF of 5 Eg is a restricted open shell HF (ROHF) WF,
but the WFs of the triplets and the singlets incorporate some non-dynamical electron correlation. Next, we extend the initial CASSCF-d space with a set of virtual
orbitals of the same symmetry character as the Mn -3d orbitals: CASSCF-dd’. The
configurational composition of the WFs of all states, of Table 6.2, remains practically
unchanged within CASSCF-dd’. The total energies of the CASSCF-dd’ WFs are lowered compared to the CASSCF-d WFs, but the effect is not substantial. For instance,
the ground state energy is lowered by 0.6 eV. As expected, the CASPT2 energies
obtained with CASSCF -dd’ are raised compared to those obtained with CASSCF-d.
This is due to the overestimation of the Mn 3d dynamical correlation with CASSCFd/CASPT2 [40]. For all states, except the 5 T2g state, the CASPT2 corrections on
the relative CASSCF energies lead to a decrease in the energies (see Table 6.2). To
analyze whether the CASPT2 correction on the CASSCF energy of the 5 T2g state has
Results and Analysis: d-d Excitations
175
Table 6.2: [MnO6 ]9− cluster representing LaMnO3 . Cubic crystal structure. Relative energies (in eV) and configurational composition of some excited states in the d4 manifold;
CASSCF/CASPT2 results; active space CASSCF-d and CASSCF-dd’; CASPT2 correlates
the Mn- 3s, 3p, 3d and O- 2s, 2p electrons;
State
5
Eg
3
T1g
1
T2g
1
Eg
3
Eg
5
T2g
t32g e1g
100
2
9
10
99
t42g
State
5
Eg
3
T1g
1
T2g
1
Eg
3
Eg
5
T2g
t32g e1g
100
2
9
8
99
t42g
a
96
90
88
96
90
89
CASSCF -d
CASSCF-d
t22g e2g
0.00
2
0.99
1
2.46
2
2.64
1
2.84
100
2.13
CASSCF -dd’
CASSCF-dd
t22g e2g
0.00
2
0.81
1
2.29
2
2.42
1
2.70
100
2.17
Weight of the reference wavefunction
b
CASPT2
0.00
0.63
1.76
2.09
2.14
2.60b
ωa
0.684
0.686
0.683
0.684
0.678
0.685
CASPT2
0.00
0.63
1.78
2.14
2.24
2.41b
ωa
0.688
0.691
0.689
0.690
0.684
0.687
Crystal -field splitting 10 Dq
176
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
an opposite sign due to atomic dynamical correlation effects, we perform model calculations on an Mn3+ ion in vacuum. The relative CASSCF-dd’ energy of the Mn3+
ion 3 H state with respect to the relative zero energy of the 5 D ground state is 2.82
eV and increases to 3.01 eV at CASSCF-dd’/CASPT2. The CASPT2 correction on
the CASSCF energy of 3 H is of the same magnitude and has the same sign as that for
the 5 T2g state in the [MnO6 ] cluster. We conclude, that the increase in the relative
energy of the 5 T2g state at CASPT2 is mainly due to atomic correlation effects.
MGPA of the ACNOs of the ground state, 5 Eg , shows that the occupied eg -type
orbital has 92 % Mn-3d character. The t2g orbitals, compared to the eg ones are
more localized - 97 % Mn-3d. Analogous analysis for the 5 T2g state points out a
t22g e2g configuration and the same composition of the eg - and t2g - type orbitals as that
in the ground state. The occupied eg -type orbital of the 3 Eg state has a quite different
character- 88 % Mn-3d while the t2g orbitals preserve the same composition as for
5
Eg .
This MGPAs analysis indicates a low covalency of the compound. We note, however that within our cluster approximation, the localization of the occupied Mn 3d
eg orbital may be overestimated. Moreover, to obtain a reasonable estimate of the
covalency of the compound, one also needs to consider the contribution of CT configurations in the WFs of the Mn 3d4 states. A recent study on the CF excitations
of ideal and JT distorted LaM nO3 , performed by Hozoi et. al [33] have shown that
the weight of CT configurations in the CI expansion of the ground state WF is less
than 15 %. In the next section, we study the covalency of LaM nO3 by analyzing the
configurational composition of the WFs of some Mn 3d4 states.
The relative energies of the lowest d4 states obtained in cubic geometry follow
the Tanabe-Sugano diagrams [52]. We define the CF splitting parameter (10 Dq) for
local Oh geometry as the energy difference between the high spin coupled 5 Eg (t32g e1g )
and 5 T2g (t22g e2g ) states. Our calculations within CASSCF -dd’ yield 10 Dq equal to
2.4 eV. The lowest excitation involves a 5 Eg (t32g e1g ) → 3 T1g (t42g ) transition at 0.6 eV
(see Table 6.2). Furthermore, the Mn 3d exchange parameter, defined as the energy
difference between the 5 Eg (t32g e1g ) and 3 Eg (t32g e1g ) states, has a value of 2.2 eV.
Further studies on the JT distorted and tilted forms are performed within CASSCFdd’.
D2h and Ci geometry -JT distorted and tilted structures
When distortions are present in the system all degenerate in Oh states split up.
The JT splitting parameter (4JT ), defined as the energy difference between the
5
Eg (t32g e1g (d3x2 −r2 )) and 5 Eg (t32g e1g (dz2 −y2 )) states has a value of 1.23 eV in the D2h
geometry and changes to 1.32 eV when the tilting of the MnO6 octahedra is taken
into account. These values are the relative CASSCF-dd’/CASPT2 energies of the
5
Eg (t32g e1g (dz2 −y2 )) state obtained in a state average calculation over the two 5 Eg like states. The occupied eg - type orbital in the 5 Eg ground state of the JT distorted
form is directed along the longer M n − Ox bond and has predominantly d3x2 −r2
character.
Results and Analysis: d-d Excitations
177
Table 6.3: [MnO6 ]9− cluster representing LaMnO3 . JT distorted crystal structure. Relative
energies (in eV) and configurational composition of some excited states in the d4 manifold;
CASSCF/CASPT2 results; active space CASSCF-dd’; CASPT2 correlates the Mn- 3s, 3p,
3d and O- 2s, 2p electrons; Notations corresponding to the Oh symmetry are used;
State
5
Eg
3
T1g
3
Eg
5
T2g
a
t32g e1g
100
2
100,
98
t42g
t22g e2g
95
2
0, 2
CASSCF-dd’
0.00, 1.16
1.39, 1.45, 1.66
2.68, 3.08
100
2.21, 2.45, 2.53
Weight of the reference wavefunction
b
CASPT2
0.00, 1.23b
1.40, 1.47, 1.68
2.40, 2.66
ωa
0.680, 0.678
0.684
0.677, 0.675
2.59, 2.83, 2.91
0.678
Jahn -Teller splitting, 4JT
For the 5 Eg ground state obtained within CASSCF-dd’, we observe about ∼6 %
O-2pσ character mixed in the occupied Mn 3d eg - type orbital. Analogous analysis
for the 5 T2g - and 3 Eg -like states points out ∼6-9 % O-2pσ character of the occupied
Mn 3d eg -like orbital. The t2g -like orbitals have ∼2-3 % O-2pπ . These results
suggest that both t2g - and eg -like orbitals are quite localized and hardly mix with
O-2pπ(σ) .
The optical conductivity analysis of Jung et al. [21] for the real structure of the
compound pointed out a strong covalent interaction between the eg -like and O-2pσ
orbitals which led to broad eg -like bands. Our MGPAs analysis for the real structure suggests a low covalency. However as pointed out above the localization of the
occupied eg -like orbital might be overestimated within the present cluster and WF
model.
We carried out an analysis of the composition of the WFs of the 5 Eg -like ground
state and the 5 T2g -like states in order to obtain an estimate for the covalency of the
compound. The WFs of the states are constructed within an active space containing
5 Mn 3d (the t2g -like and eg -like) orbitals and 5 O 2p orbitals :CASSCF-5pd. We
found the weight of CT configurations in the CI expansions of the WFs of these Mn
3d4 states to be less than 10%.
The definition of 10 Dq in the distorted forms requires some attention. In D2h and
Ci symmetry, the 5 Eg and 5 T2g states split into two and three components, respectively. In these local symmetries, we define parameter the energy difference between
two average values. 10 Dq changes from 2.4 eV in Oh to 2.2 eV in D2h and finally
to 2.0 eV in Ci symmetry (see Figure 6.1). The values are obtained within CASSCFdd’/CASPT2. The configurational composition of the WFs and relative energies of
the states are summarized in Table 6.4. We performed for the real structure state
average calculations, averaging those states that in Oh are the degenerate components
of a quintet (5 Eg , 5 T2g ), or a triplet (3 Eg , 3 T1g ) state. This optimization maintains
0
the correct character of the active Mn-3d orbitals, i.e. 5d and 5d .
The decrease of 0.4 eV in the 5 Eg (t32g e1g ) - 5 T2g (t22g e2g ) splitting (10 Dq) brings
the energies of the 5 T2g -like states, close to those of the 3 Eg -like states. Therefore, in
Figure 6.1 we observe no separation of the two sets of states. The Mn 3d exchange
178
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
Table 6.4: [MnO6 ]9− cluster representing LaMnO3 . JT distorted and tilted crystal structure. Configurational composition and relative energies (in eV) of some excited states in the
d4 manifold; CASSCF/CASPT2 results in active space CASSCF-dd’; CASPT2 correlates
the Mn- 3s, 3p, 3d and O- 2s,2p electrons; Notations corresponding to Oh symmetry are
used;
Oh
term
5
Eg
3
T1g
3
Eg
5
T2g
a
State
in Ci
a5 Ag
b5 Ag
a3 Ag
b3 Ag
c3 Ag
d3 Ag
e3 Ag
c5 Ag
d5 Ag
e5 Ag
t32g e1g
100
100
2
2
4
90
93
t42g
96
96
92
8
6
t22g e2g
2
2
4
2
1
100
100
100
Weight of the reference function
CASSCFdd’
-1691.5828
1.24
1.56
1.71
1.89
2.68
2.96
2.22
2.45
2.59
JT
CASPT2
ωa
-1693.4333
1.32JT
1.48
1.66
1.84
2.41
2.52
2.50
2.73
2.87
0.684
0.681
0.688
0.688
0.687
0.682
0.679
0.683
0.682
0.681
Jahn -Teller splitting, 4JT
Figure 6.2: Valence electronic structure of the M n3+ ion in cubic, JT distorted and real
geometries; CASPT2 results. The lowest singlet states are not shown. Their energies are
presented in Table 6.2;
Results and Analysis: d-d Excitations
179
splitting parameter, defined as the energy difference 3 Eg (t32g e1g )-5 Eg (t32g e1g ) decreases
from 2.2 eV in Oh to 2.0 eV in D2h and finally to 1.8 eV in Ci . This trend suggests that
the prolongation of the M n−Ox bond in the distorted form allows for a delocalization
of the d3x2 −r2 orbital directed along this bond.
Although the state average calculations yield a qualitatively correct order of the
excitation energies of the states, we perform single state or weighted state average
calculations to obtain also an estimate of the extra relaxation gain when the states
WFs are expressed in relaxed orbital sets. The relaxation energy is less than 0.1 eV
and about 0.1 eV for the quintets and the triplets, respectively.
The JT splitting parameter increases by less than 0.1 eV within the average calculation, whereas the CF (10 Dq) and exchange parameters preserve their values of 2.0
eV and 1.8 eV, respectively, in both schemes. The CASSCF energies of the 3 Eg -like
and 5 T2g -like states place the triplets above the quintets. At CASPT2, the larger
dynamical correlation for the triplets leads to a larger decrease in their energies. This
results in similar energies for the sets of non-degenerate 3 Eg -like and 5 T2g -like states.
In the JT distorted form the average energy of the quintets differs from that of the
triplets by 0.3 eV, in the tilted geometry this energy difference is 0.2 eV.
Scalar relativistic effects are found to affect the relative d-d excitation energies by
less than 0.01 eV, spin-orbit coupling was not considered.
The splitting of the 5 Eg -like states upon distortion, is an expected and well known
effect, but the magnitude of that splitting is still controversial. We find 1.23 eV for the
JT splitting in D2h . When tilting is introduced in the system, this splitting increases
by less than 0.1 eV. The lower energy part of the manifold preserves its structure
upon tilting, whereas the higher energy part undergoes moderate changes. We notice
that the energy order of the 5 T2g - and 3 Eg -like states in the tilted form is similar
to that in the JT distorted structure (see Fig. 1, Ci manifold). The decrease in the
energies of the 3 Eg - and 5 T2g -like states upon tilting is less or ∼ 0.1 eV. In Ci , the
3
T1g -like states increase their relative energies by ∼0.1-0.2 eV compared to those
in D2h . This is due to the smaller value of 10 Dq in the tilted form. Note that, as
expected, the CF and Mn 3d exchange parameters undergo only a modest change
when the distortions are introduced in the structure. These model parameters can be
evaluated rather accurately using an idealized model [MnO6 ] cluster.
This summary on the change of the Mn 3d4 spectra in D2h and Ci demonstrates,
that the main features in the structure of the manifold are already present in the
idealized D2h geometry and the tilting adds only moderate corrections to the energies.
Interpretation of optical excitations determining the model parameters
Although the theoretical results available in the literature are obtained in model
Hamiltonian or DFT based band structure calculations accounting in a different manner for the electron exchange and correlation effects, we compare them with our ab
initio results in order to point out the new insights in the valence electronic structure
of the compound.
The value of 1.3 eV for the JT splitting calculated within our theoretical approach differs by ∼ -0.6 eV from the value determined by Jung et al. [21] from their
180
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
optical-conductivity analysis. Jung and co-workers obtained the optical conductivity
spectrum σ(ω), using Kramers-Kronig transformation, and expressed further σ(ω) as
a sum of Lorenz oscillator functions. The first oscillator strength in their analysis is
smaller by an order of magnitude compared to the second and third ones. Therefore,
Jung et al. assigned this first oscillator to the spin allowed transition within the JT
split eg shell, a transition below the O 2p → Mn 3d eg transition. This excitation is
considered to be intra-atomic, located at ∼ 1.9 eV and electric dipole forbidden. The
authors suggested, that it becomes optically active due to the strong hybridization of
the lower Mn eg band with the O 2p band. This assignment of the optical absorption
peak at 1.9 eV to an on-site d-d electronic excitation across the JT gap was explained
by Allen and Perebeinos [18].
In our ab initio study of the JT splitting parameter we did not consider the lattice
relaxation accompanying the local excitation 5 Eg (t32g e1g (d3x2 −r2 )) → 5 Eg (t32g e1g (dz2 −y2 ))
and hence, the calculated excitation energy should, in principle, correspond to the
vertical transition in the optical spectra. A relevant question to be addressed in connection with the magnitude of the JT parameter is how sensitive is this quantity
to the asymmetric oxygen displacements which occur in the distorted structure. A
simple estimate of this effect can be made by varying the lengths of the two bonds
which undergo a major change upon distortion by a small amount. We perform this
analysis for the JT distorted LaMnO3 , discussed above. The short and long Mn -O
bonds in this JT distorted form have lengths of 1.90 Å and 2.14 Å, respectively. We
introduce a small displacement outwards of 0.04 Å of the oxygen ions which form the
long Mn-O bonds and a displacement inwards of the same magnitude of the oxygen
ions which form the short Mn-O bonds. The intermediate Mn-O bond is kept at 1.95
Å. A [MnO6 ] cluster is constructed using these bond lengths. Next the CASSCF -dd’
WFs of the 5 Eg (t32g e1g (d3x2 −r2 )) and 5 Eg (t32g e1g (dz2 −y2 )) states are constructed and
the resulting excitation energy, defining the JT parameter, is evaluated to be 1.4 eV.
Clearly, the JT parameter is sensitive to the precise value of the oxygen displacements,
but by far not sensitive enough to be able to confirm Jung’s interpretation. The three
active octahedral distortion modes which enter the expressions for the various model
Hamiltonians of the crystals in the model Hamiltonian approaches have to be optimal
in order to yield reasonable quantitative estimates of the excitation energies.
The dispersion of the exciton due to the ”hopping” of the local excitation between
nearest neighbouring Mn sites is studied in a [Mn2 O11 ]16− super-cluster, representing
a fully distorted LaMnO3 , using the OF approach, introduced in Chapter 3. The
hopping matrix elements associated with this process are found to be less than 1
meV.
As we noted in section 6.1, Quijada and co-workers suggested recently that the
feature near 2.0 eV arises from the hopping transition between the JT-split eg levels at
adjacent Mn ions [59]. According to their analysis, the phonon-assisted on -site electronic transitions, predicted by Allen and Perebeinos [16] are generally much weaker
than the optically allowed charge-transfer Mn3+ →Mn3+ interband transitions, that
can be expected in LaMnO3 . We estimated the energy cost for such a hopping transition by using CASSCF results for the smaller [MnO6 ] cluster. The relative energy
Results and Analysis: d-d Excitations
181
of the inter -site excitation is deduced from the following (approximate) relationc :
Erel = E(3d → adj.3d) ≈ {E(4 A1g ; 3d3 ) − E(5 Eg ; 3d4 )} − {E(5 Eg ; 3d4 ) −
6
E( A1g ; 3d5 )} − ECoul. (M n3+ − M n3+ )
The Mn-Mn Couloumb energy is calculated at Mn-Mn distance of 3.99 Å. The
energy of the transition is estimated to be 6.63 eV. Dynamical correlation effects are
expected to lead to a lowering in the energy of the transition, but their magnitude
will not be sufficient to reduce it to 2.0 eV. A justification for the approach adopted
here for the estimate of the charge-transfer Mn3+ →Mn3+ excitations is provided by
a recent study on the X -ray Mn K-edge absorption spectra of JT distorted LaMnO3 ,
performed by Hozoi et al [34]. Hozoi et al [34] have shown that the relative energy
of the 1s→ adjacent Mn 3d transition, estimated using the smaller [MnO6 ] cluster in
the manner described above, agree well with that obtained in a [Mn2 O11 ] cluster [34].
Note that in our cluster model delocalization of the eg orbitals might be underestimated. This delocalization might lead to a relative stabilization of the ground state,
and hence to a larger JT gap. Extending the CASSCF-dd’ active space by adding O
-2p orbitals is not impossible to lead to an increase in the energy difference between
the ground state and the first excited state, which determines the JT parameter. This
is a matter of current investigation.
The discussion above has shown that the interpretation of the peak at 1.9 eV in
the optical conductivity spectra of Jung [21] is not trivial. Our JT splitting of 1.3
eV compares well with the estimate of 1.4 eV, obtained by Elfimov and co-workers
[20] within the LSDA+U approach. Similar values for the splitting of the eg -like
levels, 1.0-1.5 eV, were obtained from band structure LDA (+U) calculations [14] and
analysis of the optical dielectric tensor for the interband transition from the lower Mn
eg to the upper Mn eg band [15]. However, the authors in reference [15] proposed
that the peak originates from [Mn dx2 −y2 , dxz ; O -2p hybridized] → [Mn 3d ; O -2p]
optical interband transition, based on the considerable amount of O -2p states present
at the top of the VB and the bottom of the CB.
The Mn 3d exchange splitting parameter in our ab inito calculations is 1.8 eV,
that is ∼ 1.0 eV lower than the estimates, listed in Table 6.1. However the theoretical
approaches listed in Table 6.1 do not incorporate important electronic relaxation
effects.
The CF splitting parameter in our ab inito approach is evaluated to be 2.0 eV in
Ci (2.41 eV in Oh ). This value compares well with the value of ∼ 2.0 eV, deduced from
DFT -based BS calculations [14] and optical transmission spectra analysis [24]. The
orbital-projected DOS from FPLAPW band calculations, performed by Ravindran et
al. [15] yielded a value of 1.1 eV.
The comparison of the values of the parameters, obtained in our ab initio study,
with the values obtained in DFT -based BS calculations showed a significant difference. In the one-electron DFT -based approaches, the electron correlation effects are
accounted for by different exchange-correlation functionals and the strong Coulomb
interactions are parametrized. In the many-electron WF based methods, the WF
type and composition allows to incorporate explicitly those effects. Thus the comc The
notations correspond to octahedral symmetry, but the actual local symmetry is Ci
182
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
parison between the results of those semi-empirical methods and our ab initio results
must be regarded as a check for the results derived within the effective one-electron
approaches.
Our calculations on the d-d excitations in orthorhombic LaMnO3 , based on a
CASSCF WF description, suggest a new quantitative picture of the CF valence electronic structure of orthorhombic LaM nO3 . The embedded cluster approach, used in
our studies, have proved to be a reasonable model for localized excitations in ionic
compounds [26,37,54]. The d-d transitions are considered to be localized [16,17,33,58],
but most of the studies [6, 15, 21] claim a strong covalency between the eg - like Mn
3d states and O -2pσ states, which in addition leads to a broad eg type VB. We
note however that in our approach the lower occupied eg -like orbital is too localized
because the WF of the 5 Eg -like ground state is close to a HF WF.
Comment on the cluster size effects on the d-d excitations in orthorhombic
LaM nO3
The cluster size effects on the d-d excitations in orthorhombic LaM nO3 are investigated by considering the changes in the excitation energies of the CF states upon extending the [MnO6 ] cluster, by including extra shells of ions. This enlargement results
in clusters [B], [C] and [D]. The cluster size effects are found negligible for the excitation which determines the JT parameter. This is the excitation which involves the
two 5 Eg -like states. The largest change of about 0.2-0.4 eV in the CASSCF/CASPT2
excitation energies of the 5 T2g - and 3 T1g -like states occur upon the cluster enlarging
[A]→[B]. The other cluster expansions [B]→[C] and [C]→[D] lead to minor changes
in the calculated CASSCF and CASPT2 excitation energies. The 3 Eg -like energies
change by less than 0.01 eV at all cluster expansions. The main effects for all states
are already present at CASSCF and almost no extra effects are observed at CASPT2.
The cluster size dependence of the d-d excitation energies is illustrated in Figure 6.3.
The d-d excitations show a modest change with the cluster size, which corresponds
to a decrease of about 0.2 eV in 10 Dq to 1.8 eV. We can conclude that the d-d
excitations are indeed quite localized and their energies are negligibly influenced by
the size of the cluster.
We also considered the possibility to associate the peak at 1.9 eV in the optical
spectra of Jung [21] with the 10 Dq like transition, the transition between the lower
5
Eg -like state (a 5 Ag in the real Ci site symmetry) and the lowest component of the
5
T2g -like state (c 5 Ag in Ci symmetry). However it is found at about 2.5 eV in [MnO6 ]
and 2.3 eV in the largest cluster [D]. This makes less probable the hypothesis that
the peak at 1.9 eV measured by Jung et al. corresponds to a 10 Dq -like excitation.
Character of the top of the valence band in LaMnO3
There have been numerous experimental and theoretical studies of the electronic
structure of LaMnO3 , addressing in particular the character of the top of the VB.
We discussed in detail some of those studies in Chapter 5. From the experimental
studies we mention here the photoemission spectra analysis of Saitoh and et al. [12]
Results and Analysis: d-d Excitations
183
Figure 6.3: Illustration of the cluster size dependence of d-d excited states in orthorhombic
LaMnO3 . The ground state 5 Ag relative energy for all clusters is taken at 0.0 eV. The
different states for the clusters are depicted with different types of lines.
which led to classifying LaMnO3 as a CT insulator with a band gap of 1.7 eV. See
for further details Chapter 5 and the references therein.
To obtain an estimate of the character of the top of the VB in LaMnO3 we performed the following simple analysis. The character of the top of the VB depends on
the ionization energies for the Mn 3d and O 2p valence electrons as well as on the
width of the bands associated with the Mn 3d and O 2p ionized states. We calculated
the relative energies of the lowest Mn 3d and O 2p hole states in [MnO6 ] cluster, representing orthorhombic LaMnO3 . The calculations for the lowest Mn 3d hole state,
the ground state 4 Ag of [MnO6 ]8− , were carried out within CASSCF-dd’, whereas the
WF’s of the two lowest O 2p hole states of symmetries 6 T1g and 6 T1u in [MnO6 ]8−
were constructed within CASSCF-1p(ag /au )dd’. The cluster Mn 3d ionization energy
as calculated from the CASSCF energies of the 5 Eg -like ground state of [MnO6 ]9−
and the 4 Ag state is 4.78 eV. The lowest O 2p ionized state in [MnO6 ]8− is the 6 T1u
state with electronic configuration t32g e1g t1u . At CASSCF level, this state is 1.0 eV
above the 4 Ag (d3 ) state. The dynamical correlation effects are expected to raise the
values of the Mn 3d and O 2p ionization energies. Taking into consideration that
the relaxation and localization effects are about equal for all O 2p ionized states [28]
and that the delocalization effects are dominant, we set the theoretical bandwidth of
the O 2p ionization larger than the range of the O 2p orbital energies in [MnO6 ]9−
cluster : 3.0 eV. The band stabilization energy Ed associated with the Mn 3d ionized
(hole) state 4 Ag is estimated to be about 1.1 eV (see Chapter 4 and Chapter 5 ). We
184
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
expect that formally the localized O 2p hole state will be close to the center of the O
2p band and hence, the band stabilization energy for this state is at least 1.5 eV.
The close values of the band stabilization energies associated with the O 2p and
Mn 3d ionized states as well as their close ionization energies, makes it difficult to
determine the character of the top of the VB with a certainty. However the results
indicate that the top of the VB contains contributions of both O 2p and Mn 3d ionized
states.
6.3.2
CT excitations
The study of the CT states is performed with the Mn 6s5p4d2f basis, used before,
and a smaller O 4s3p2d basis set, instead of the O 5s4p2d basis. The effects on the
energies of the CT excitations are minor and do not have any significance for the
outcomes. This survey was carried out with the idealized cubic structure. Although
some energetic effects are not accounted for at this level of approximation, the study
provides reliable information about the character and relative energies of the lowest
CT states at a reduced computational effort. The extra effects are estimated from
calculations on the real structure.
CT states in [MnO6 ]9− cluster in Oh geometry
This part of the present study is focused on the character and relative energies of some
local O 2p → to Mn 3d CT excitations. The CT states are distinguished as gerade and
ungerade. The main configuration of the (un) gerade CT states has an (un) gerade
singly occupied oxygen orbital. We are mainly interested in the lower energy part of
the ungerade CT manifold, because of its relevance to understanding the nature of
the lowest optical excitations. We expect the lowest excitations in the CT manifold
to arise from CT states for which the spins in the Mn 3d5 shell are high-spin coupled,
to 6 A1g , characterized by an electron configuration t32g e2g . If we assume a high-spin
coupling in the M n3d5 shell, then the O 2p single electron is coupled to the shell to
form either a septet or a quintet. Our choice to study initially those excitations in
local Oh geometry (ideal cubic structure) is based on the observation that the relative
energies of the lowest states in the Mn 3d5 and Mn 3d4 spectra follow the same order
in all three local geometries (see Figure 6.2 and 6.4). Therefore, the study of the CT
manifold in Oh provides reliable information about its structure. As a consequence
of carrying out the calculations in the ideal cubic structure, some relaxation energy
due to the structural distortion of the octahedra is not accounted for. In addition the
use of a CASSCF-(n)pd active space may raise the CASSCF excitation energies while
the effect on the CASPT2 estimates has an opposite sign. These additional energetic
effects will be estimated in order to probe their significance for the main features of
the CT manifold.
First we performed a SCF calculation for a simple [O6 ]12− cluster. The central
ion is modeled by a M n2+ TIP. The relative O -2p- ionization energies obtained for
this cluster within the FO approximation are summarized in Table 6.5 and Figure 6.5
gives a pictorial description.
Results and Analysis: CT Excitations
185
Figure 6.4: Valence electronic structure of a M n2+ ion in cubic, JT distorted and real
geometries; CASPT2 results. The lowest singlet states are not shown. The relative energies
for the cubic and JT distorted form are taken from reference [33]. Their values are obtained
with basis sets Mn-7s6p4d2f and O-6s5p1d. The basis sets in Ci are Mn-7s6p4d2f and O5s4p2d.
This simplified picture provides a first estimate of the relative order of the O -2p
hole states for a cluster with an extra electron on the Mn ion and hence, gives an
approximate estimate of the order of the CT states.
Our next studies are performed for an embedded [MnO6 ] cluster. We apply an
active space, described as CASSCF-(n)pd in section 6.2, which differs for the different
states in the symmetry character and number of active O -2p orbitals. The complete
active space in this scheme contains 5 Mn-3d and n O-2p orbitals. If there is only
one CSF in the CI expansion of the WF for a particular CT state, obtained within
this active space, this WF is in fact a ROHF type WF. The CASSCF WFs of the
lowest ungerade and all gerade CT states are obtained in single state calculations.
The optimization of the higher ungerade states is performed in a weighted average
of states (see Table 6.6). The active space CASSCF-(n)pd, incorporates some nondynamical correlation effects for the gerade and ungerade quintet states and leads to
a ROHF WF for the septets. Detailed results from the calculations for the gerade and
ungerade states in [MnO6 ] cluster within CASSCF-(n)pd are summarized in Table 6.6
and pictorially presented in Figure 6.6.
All low-lying CT states, correspond to a high-spin coupled Mn 3d5 shell to a t32g e2g
configuration combined with different O -2p hole a 2 T1(g)u states. Some of the highlying states, 2 T2(g)u and b 2 T1(g)u which are not shown in Table (6.6) correspond to
186
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
Table 6.5: Oxygen hole states, in an embedded [O6 ]12− cluster in Oh symmetry; SCF,
Frozen orbital energies;
State
a 2 T1u (σ, π)
2
T2u (π)
a 2 T1g (π)
2
Eg (σ)
b 2 T2g (π)
b 2 T1u (σ)
2
A1g (σ)
State Orbital
E (eV)
0
0.61
0.69
1.06
1.67
1.81
2.25
Figure 6.5: Illustration of the O 2p hole states, obtained in an embedded [O6 ]11− cluster in
Oh symmetry; SCF, Koopman’s energies;
Results and Analysis: CT Excitations
187
Table 6.6: Relative energies (in eV) of some low-lying CT states in [A], representing cubic
LaMnO3 ; CASSCF/CASPT2 results; active space CASSCF-(n)pd (5d np). The electrons
within the Mn 3d shell of all states, presented in the Table are high-spin coupled to 6 A1g
corresponding to the electronic configuration t32g e2g (only the symmetry of the O hole is pointed
out) ; tx=1g(u) is (x) or (y), (z); tx=2(g)u is (ξ) or (η), (ζ).
n
0
0
2 t1g
2 t1g
3 eg (, θ, a1g )
2 t2g
3
3
3
3
t1u
t1u
t2u
t2u
Gerade CT states
Main Character
State
5
Eg
Mn (t32g e1g )
PS 5 Eg
Mn t32g e1g
a O−π
a 7 T1g
a O−π
a 5 T1g
O−σ
a 7 Eg
5
PS b O − π
T2g
Ungerade CT states
a O − σ(π)
a 7 T1u
a O − σ(π)
PS a 5 T1u
7
O−π
T2u
O−π
PS 5 T2u
CASSCF
-1897.5369
-1897.5369
6.46
6.61
7.87
9.77
CASPT2
-1899.3690
-1899.3685
4.57
4.72
5.31
6.98
6.57 (6.43)
7.00 (6.91)
7.44 (7.38)
7.61
4.26 (4.49)
4.48 (4.63)
5.34 (5.34)
5.51
The WF of each a 7,5 T1(2)u state is obtained for a weighted average of the
three states of the same multiplicity in ratio (a:T2u :b) - 4:1:1 for a states,
1:4:1 for 7,5 T2u , and 1:1:4 for b states. The gerade states are optimized in
a single state calculation. The values in the parentheses are obtained in a
single state calculation. PS : Imaginary level shift of 0.3 a.u.
188
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
a Mn 3d5 shell, coupled either to 4 A1g with a t32g e2g configuration or to 4 T1(2)g with
a t42g e1g configuration. We concentrate attention in the following discussion mainly
on the low-lying CT states. A detailed inspection of the configurational composition
of the WFs shows some mixing of the different d5 O− CSFs in the CASSCF WFs.
For instance, the WF of the lowest quintet- a 5 T1u consists of a CSF, described by
t32g e2g (6 A1g ) a OT1u −σ(π) electronic configuration (see Table 6.6) with a weight in the
state CI expansion of 89 % and a CSF, described by t32g e2g (4 A1g ) a OT1u − σ(π) with a
weight of 11 %. The gerade quintets show a weight of 89 % of the CSF t32g e2g (6 A1g ) a
OT1g − π. The WFs configurational composition for the higher ungerade and gerade
quintets is analogous. A summarized analysis of the results for the [M nO6 ] cluster
points out that in Oh the CT states appear above 4.0 eV (see Figure 6.6). We
expect some decrease in the CT excitation energies in Ci due to the lattice relaxation.
Furthermore, the energy order of the states shows consistency with the approximate
energy order of the O -2p hole states (see Figure 6.5). The gerade and ungerade spin
septet states are with the lower energy in the quintet-septet pairs for each symmetry.
The CASPT2 energy difference between the states in a quintet-septet pair is only
∼0.1-0.3 eV, indicating that the exchange coupling between the spins at Mn and O
is rather weak. The magnitude of these energy differences is preserved for all states
in the larger clusters except for high-lying 7,5 T2g for which it increases by ∼0.6.
Figure 6.6: Illustration of the CT manifold in the [A] cluster; CASSCF/CASPT2 results;
In principle, the CT states and the d4 states, obtained in a CASSCF calculation
are not guaranteed to be mutual orthogonal, nor non-interacting. However the lowest
gerade quintet CT states are high above the highest gerade quintet d4 states and thus,
no overlap or interaction is expected.
Results and Analysis: CT Excitations
189
Table 6.7: Change in the CT excitation energies at CASSCF and CASPT2 upon cluster
extension
Changes in the CT excitation energies with the cluster size
State
Cluster Enlarging 4, eV CASSCF
4,
eV
CASSCF/CASPT2
[A]→[B]
∼ 0.76-1.0
∼ 0.8-1.4
7,5
a T1u(g)
[B]→[C]
∼ 0.1-0.2
∼ 0.5-0.7
[C]→[D]
∼ 0.02
∼ 0.04
[A]→[B]
∼ 1.0
∼ 1.4
7,5
T2u
[B]→[C]
∼ 0.02
∼ 0.4
[C]→[D]
∼ 0.02
∼ 0.1
Cluster size effects on the CT manifold in cubic LaMnO3
In this section, we study the constraints which the cluster model implies on the
delocalization of the CT excitations. The [M nO6 ] cluster is extended to include more
shells of ions. This cluster enlargement results in clusters [B], [C] and [D], discussed
before.
The graphs in Figure 6.7a and 6.7b illustrate the change in the excitation energies
of some lowest gerade and ungerade CT states, obtained at the CASSCF and CASPT2
level of approximation, respectively. The high-lying ungerade states in the larger
clusters could not be obtained in a single state calculation within CASSCF-(n)dp.
Therefore, we optimized the orbitals for the WFs of those CT states for a weighted
average of states, as in [MnO6 ]. The values of the weights are chosen as in [MnO6 ].
The WFs of the gerade states are obtained in a single state calculation in all four
clusters.
Clearly, there is a systematic lowering in the CT excitation energies with enlargement of the cluster. We also observe that the excitation energies of the lowest CT
states converge faster with the cluster expansion, i.e. the excitation energy of a particular low-lying CT state differs by less than 0.05 eV in clusters [C] and [D]. The
decrease in the excitation energies may be caused by a gain of electron correlation or,
more likely, electron relaxation or a change in the potential within the central regions
of the clusters.
In Table 6.7 we have summarized the changes in the CT excitation energies at
CASSCF and CASPT2 which occur upon a cluster enlargement. For the lowest CT
states the main changes occur at the CASSCF level and for [A]→[B]. The dynamical
correlation gain at [A]→[B] is moderate and hence, the energy lowering is determined
mainly by extra relaxation effects and/or changes in the external potential. The cluster [C] appears to be the optimal cluster model for the lowest CT excitations. It
allows for incorporating the relevant relaxation and electron correlation effects.
We concluded above that the main changes in the CT excitation energies occur
at CASSCF and for [A]→[B]. To clarify their origin, we consider the effect of several
190
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
Figure 6.7: Illustration of the cluster size dependence of CT excitation energies in eV in
cubic LaMnO3 at a) CASSCF and b) CASPT2. The ground state 5 Eg relative energy for
all clusters is taken as 0.0. The order of the states in the right part of the graphics is the
order, found in the [M nO6 ] cluster. The different states for the clusters are depicted with
different types of lines. The second array (from the left to the right site) of states for the
[M nO6 Al6 La8 ] is obtained, when the La 5p shell is included in the correlation at CASPT2.
Results and Analysis: CT Excitations
191
model representations of the nearest embedding shell M n3+
6 of [A] on the CT excitation energies. Previous studies [26] on the ionization and excitation energies in CuCl
and NiO have shown significant effects of the various embeddings.
First, we represent the nearest embedding shell of 6 Mn3+ ions by bare effective
core potentials (ECPs) [44] for the Al3+ core shell. The ECPs for Al3+ account satisfactory for the Pauli repulsion between the cluster ions and the nearest neighboring
Mn3+ cations. This embedding accounts also for the finite size of the ions around
the cluster. The exchange and orthogonality interactions are (partially) included in
the electronic structure description [26]. The differences in the excitation energies
obtained with AlECP and MnT IP s with CASSCF are less than 0.05 eV (see Figure
6.8). The use of MnT IP s instead of bare AlECP leads to a decrease of 0.2-0.3 eV in
the CASPT2 energies.
If the material is an ideal ionic system, the bare ECPs [44] for the Al3+ core
electrons or the TIPs for the Mn3+ ions consitute a reasonable representation of
the ions, external to the cluster. The strong orthogonality condition, imposed by
TES [60], between the cluster electrons and the electrons of the ions in the nearest
embedding, is maintained for these ionic systems. When the cluster WF has nonnegligible values outside the cluster region, the strong orthogonality condition can still
be maintained by adding basis functions to the ions represented by model potentials
[61]. Thus to ensure orthogonality we add (1s1p1d) basis functions to the bare AlECP s
to describe the Al valence shell. This embedding brings a decrease in the CASSCF
excitation energies by at most 1.0 eV compared to the energies obtained using bare
AlECP s . The dynamical correlation contribution to the overall decrease in the energies
is in the range of 0.2-0.5 eV.
Finally, we check whether the orthogonality is fully recovered by adding the basis
functions to the bare AlECP s . We do so, by including explicitly the electrons associated with the Al63+ in the quantum mechanical treatmentd This results in a further
decrease in the excitation energies of the CT states at CASSCF and CASPT2 by less
than 0.08 eV. Thus we can conclude that the orthogonality conditions are maintained
by the augmented AlECP s .
Next, to distinguish the effects of the relaxation and the change in the potential
on the lowest CT energies at [A]→[B], we consider the changes in the core orbital
energies of the ground state of [A] when the nearest ions around the Mn3+ ion in [A]
are modeled either by bare AlECP s , MnT IP s , augmented AlECP s , or Al3+ ions. The
changes in those orbital energies indicate changes in the potential close to the nuclei.
The difference in the decrease in the energies of the Mn and O core orbitals at different
embeddings is related to the change in the external potential. This difference is ∼
0.3 eV for [A]→[B] and for the augmented AlECP s . The replacement of the MnT IP s
by bare AlECP s introduces only a negligible effect. Thus we can conclude that the
total decrease in the CASSCF excitation energies at [A]→[B] involves a decrease of
∼ 0.3 eV, caused by a change in the potential and of ∼ 0.7 eV, due to a relaxation
energy gain. Similar analysis for the lower-lying CT states at [B]→[C] indicates that
the change in the external potential contributes at most to the overall decrease in the
d in
fact this representation of the embedding is equivalent to enlarging the cluster [A]→[B]
192
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
Figure 6.8: Dependence of the CT excitation energies in cubic LaMnO3 on the embedding
representation; a) CASSCF and b) CASPT2
Results and Analysis: CT Excitations
193
CASSCF excitation energies.
An inspection of the charge effects for the states, based on MPA, shows almost
no real charge transfer between the Mn3+ and O2− ions. The atomic gross charges
obtained from MPA point out a transfer of charge between the Mn3+ and O2− in the
CT states, compared to that in the ground state of ∼0.4 for all clusters.
In summary, the excitation energies of the CT states in cubic LaMnO3 are significantly influenced by the cluster model. For the low-lying gerade and ungerade states,
cluster [C] is found to be large enough for an accurate description of the CT states.
Furthermore comparing the estimated energies of the d-d excitations and the CT
excitation energies obtained in the largest cluster, [D], reveals a small energy gap of
∼ 0.4 eV between the lowest CT state and the highest d4 state, considered in this
study, 5 T2g . There is no low-lying 5 T2g state in the CT manifold, the lowest such
state is at about 5.5 eV (CASPT2 result for cluster [C]).
Estimate of the effects of structural distortion and electron correlation
The study of the cubic LaMnO3 elucidated the character and excitation energies of
the lowest CT states as well as their cluster size dependence. Next to estimate the
effects of structural distortion and Mn 3d dynamical electron correlation on the CT
energies, we focus on the lowest CT states, a 7,5 T1u and a 7,5 T1g , but now obtained
in the orthorhombic LaMnO3 . We first consider cluster [A].
CASSCF WFs were constructed for the lowest a 7,5 T1u -like states a, b, c 7,5 Au ,
and the a 7,5 T1g -like states, a, b, c 7,5 Ag , within an active space containing 5 Mn 3d
orbitals and 9 au or ag O -2p orbitals, respectively : CASSCF-9pd. We performed
a state average calculation over the three components a, b, c 7,5 Au or a, b, c 7,5 Ag , of
each a 7,5 T1u -like or a 7,5 T1g -like state. In Figure 6.9 the changes are illustrated
when the lattice distortions are introduced in the system.
Figure 6.9: Lowest CT states. The a 7,5 T1u -like and a 7,5 T1g -like CT states in Oh and
Ci local geometry. The three components a, b, c 7,5 Au(g) in Ci of each state are depicted with
the same type of lines as the T states.
The decrease in the CASSCF energies of the a 7,5 Ag states, i.e. the lowest compo-
194
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
nents of the a 7,5 T1g -like states, when the distortions are introduced, is ∼ 0.3-0.5 eV.
The energy splittings of the a 7,5 T1u -like states in Ci appear to be somewhat larger
and thus, the lowest CT state is the a 7 T1u -like state, i.e. a 7 Au . The energy gain for
the a 7,5 T1u -like states is in the range ∼ 0.6-0.9 eV. At CASPT2 the energy gains for
the gerade and ungerade states are vice versa and the lowest CT state appears to be
a 7 Ag . Clearly, the lattice distortions introduce a decrease in the relative excitation
energies of the lowest CT states. This is mainly a localization effect.
The configurational composition of the WFs of the septets and ungerade quintets,
obtained within CASSCF- 9pd shows a mixing of different 3d5 O− CSFs.
The WFs of the gerade quintets show some mixing of the different 3d5 O− and
4 2−
3d O CSFs in addition to the mixing of the various O hole CSFs.
The scalar relativistic corrections to the CASSCF values within CASSCF-9pd for
all states are 0.2-0.3 eV. The scalar relativistic corrections are obtained as the energy
difference between the non-relativistic energy and the energy with the mass-velocity
and Darwin relativistic terms included.
To estimate the extra electron correlation within the d5 shell, CASSCF WFs were
constructed for the lowest a 7,5 T1g - and a 7,5 T1u -like CT states in [A] within an
0
active space containing 5 Mn 3d orbitals, a second shell of correlating virtuals d and
3 ag or 3 au O -2p orbitals, respectively :CASSCF-3pdd’. CASSCF-3pdd’ accounts
for the mixing of the different 3d5 O− and 3d4 O2− CSFs in the WFs of the CT states
as well as for the large Mn 3d dynamical correlation effects [37]. The results are listed
in Table 6.8, together with the results (in brackets) obtained within CASSCF-9pd.
An inspection of the CASSCF excitation energies obtained within CASSCF-3pdd’
reveals a decrease of about 0.3-0.4 eV compared to the excitation energies within
CASSCF-9pd. As expected, the CASPT2 energies are underestimated within CASSCF9pd due to an overestimation of the Mn 3d dynamical correlation (see also [40]). The
underestimation is ∼ 0.2 eV.
Finally we estimated the extra orbital relaxation energy associated with expressing
each state in its own optimized orbital basis set. The results from these single state
calculations are listed in Table 6.9. Note that results, obtained within CASSCF-3pd
and within an active space containing 5 Mn3d and only one O -2p (CASSCF -1pd),
differ by 3 meV using CASSCF. CASSCF -1pd raises the CASPT2 energy by ∼0.05
eV, compared to that obtained within CASSCF-3pd.
Comparing the excitation energies of 5 Au obtained from the state average calculation
(Table 6.8) and the single state calculation (Table 6.9) reveals that the extra relaxation
energy is ∼ 0.2 eV.
Analogous to the analysis, performed for the cubic LaMnO3 , we studied the cluster
size dependence of the a 7,5 T1u and a 7,5 T1g - like CT states in [A], [B] and [D] as well
as the effect of several model embedding representations of the nearest ions around
[A] on those CT states. Because the results of this analysis do not introduce new
insights in the effect of the cluster size on the CT manifold, we do not discuss them
explicitly here.
Results and Analysis: CT Excitations
195
Table 6.8: Relative energies (in eV) of the lowest CT states in cluster [A], representing
orthorhombic LaMnO3 ; CASSCF/CASPT2 results; active space CASSCF-3p(ag or au )dd’.
The spins in the Mn 3d shell are coupled to 6 A1g . In brackets are listed the energies obtained
within CASSCF -9pd.
Main Character
CASSCF
ωa
CASPT2
Gerade CT states
-1691.5828
-1693.4332
−π
5.98 (6.35)
4.64 (4.15)
−π
6.04 (6.41)
4.29 (4.48)
−π
6.29 (6.71)
4.96 (4.74)
−π
5.49 (5.92)
4.17 (4.02)
−π
5.99 (6.41)
4.57 (4.41)
−π
6.07 (6.50)
4.76 (4.57)
Ungerade CT states
a 5 Au
5.70 (6.01)
4.71 (4.47)
OT1u -σ(π)
T1u
O -σ(π)
b 5 Au
5.76 (6.20)
4.71 (4.66)
OT1u -σ(π)
c 5 Au
6.16 (6.46)
4.77 (4.66)
T1u
7
O -σ(π)
a Au
5.41 (5.76)
4.47 (4.28)
OT1u -σ(π)
b 7 Au
5.56 (5.97)
4.69 (4.40)
OT1u -σ(π)
c 7 Au
5.81 (6.12)
4.70 (4.57)
The CASPT2 values are obtained with Imaginary shift of 0.1
t32g e1g
a OT1g
a OT1g
a OT1g
a OT1g
a OT1g
a OT1g
a
a
a
a
a
a
State
5
Ag
a 5 Ag
b 5 Ag
c 5 Ag
a 7 Ag
b 7 Ag
c 7 Ag
0.686
0.655
0.652
0.656
0.661
0.658
0.660
0.660
0.661
0.655
0.663
0.665
0.662
a.u.
Table 6.9: Relative energy in (eV) for a 5 T1u in [A] and [B] clusters, representing cubic and orthorhombic LaMnO3 ; The results for the orthorhombic form refer to the lowest
component of the a 5 T1u state in Ci : a 5 Au ; CASSCF/CASPT2 results; The results in the
parentheses are values with scalar relativistic corrections; The last row lists the result for the
corresponding a 7 T1u .
Local
Symmetry
Active space
Ci
CASSCF1pdd’
CASSCF -3pd
CASSCF
-3pdd’
CASSCF
-3pdd’
Oh
Oh
Oh
Cluster [A]
Cluster [B]
CASSCF
5.55 (5.78)
CASPT2
4.87
CASSCF
4.72 (4.94)
CASPT2
3.82
6.90
6.38
4.63
5.04
6.11
-
3.49
-
6.12
4.48
-
-
196
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
Introducing lattice distortions in the system does not change the finding that the
lower part of the valence spectra of LaMnO3 is constituted of the d-d excitations. The
CT states constitute the higher part of the valence spectra. The energies of the lowest
CT states are expected to become very close, even somewhat lower, to the energies of
the highest quintet d4 states of T2g symmetry in the [D] cluster of the orthorhombic
LaMnO3 . However the lowest gerade quintet CT states will remain still high above
the highest 5 T2g -like states in the d4 manifold.
6.4
Conclusions
We performed a detailed investigation of the d-d and CT excitations which constitute
the valence electronic structure of LaMnO3 . This study provides an ab initio evaluation of model parameters, such as JT, CF and exchange splitting parameters, which
magnitudes appear to be a question of controversy in the theoretical and experimental literature. We studied separately the effects of the JT distortion and tilting of
the [MnO6 ] octahedra on the CF excitations. When the tilting is introduced in the
system only moderate changes in the d-d spectra occur compared to the d-d spectra
of the system with only JT distortions. Hence, we concluded that the main features
of the d4 manifold are already well described in an embedded cluster model which
incorporates only the JT distortion. We find JT splitting parameter of about 1.3 eV
in the tilted form compared to 1.2 eV in the JT distorted structure. Our value is
close to some other theoretical results by Elfimov and co -workers [20] and Ravindran
et al. [15]. The peak at 1.9 eV in the optical conductivity spectra of Jung et al. [21]
has been interpreted by these authors as determining the JT splitting parameter.
However, the assignment of this peak turns out to be also controversial. The optical
conductivity analyses of Quijada et al. [59] and Jung et al. [21] predict either a hopping CT between the JT splitt eg -like orbitals at adjacent Mn ions or an on-site eg
→ eg transition. We considered with attention the assignment of the peak at 1.9 eV
by Quijada et al. [59] to an inter-site CT hopping transition. Our ab initio estimate
disregarded this alternative because of the large energy, required for CT hopping. The
local d-d excitation at 1.3 eV in the ab initio CF spectra can not explain the origin of
the controversial peak which imposes that this optical transition does not correspond
directly to the JT splitting parameter, characterizing a localized excitation.
Our calculated CF parameter, 1.8 eV, in the orthorhombic LaMnO3 compares well
with the values obtained by Satpathy, Popovic and Vukajlovic [14] within the LDA
(+U) framework and by Lawler et al. [24] from optical transmission spectra. The Mn
3d exchange parameter is evaluated to be 1.8 eV. We considered the possibility the
peak at 1.9 eV to originate from the 5 Eg → 5 T2g transition, which determines the
CF parameter. We found however that this excitation is too high in energy to be
associated with the transition at 1.9 eV.
The d-d excitations showed only a modest change in the energies with the cluster
size. The excitation energies converge fast with the expansion of the cluster and
hence, even the smaller clusters provide a reasonable estimate of the CF spectra.
These modest changes do lead to a decrease in 10 Dq by about 0.2 eV. This cluster
6.4 Conclusions
197
size dependence indicates that the d-d excitations are rather localized. A [MnO6 Mn6 ]
cluster already incorporates the most significant energetic effects and further cluster
expansions do not introduce new insights in the CF spectra of the system.
Based on the analyses of the CT states, considered in this study, we conclude that
the lowest CT excitation energies also converge relatively fast with the cluster size. For
these lowest CT excitations, cluster [MnO6 Mn6 La8 ] is an adequate embedded cluster
to incorporate the important electron correlation and relaxation effects. The effect of
adding an O24 shell of ions to the latter cluster on the change in the relaxation energy
and external potential is minor for the low-lying CT states. The main contribution
to the energy decrease of the CT excitations on increasing cluster size is due to a
relaxation energy gain.
We found a small energy gap of 0.4 eV between the highest significant d4 state,
5
T2g , considered in this study, and the lowest CT state, in cubic LaMnO3 . Hence, we
expect that the lower part of the valence spectra of LaM nO3 is constituted mainly of
CF excitations. Introducing relevant distortions in the system does not lead to another
interpretation of the valence electronic structure of LaMnO3 . Our analysis pointed
out that the CT manifold is located at and above ∼3.0 eV in the cubic structure
which is in agreement with the result obtained in previous ab initio embedded cluster
calculations on the valence spectra of ideal and JT distorted LaM nO3 [33].
198
Chapter 6, d-d Excitations and Charge Transfer States in LaM nO3
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7
Delocalization of excited, hole and
added-electron states in NiO
ABSTRACT — A new theoretical approach was formulated in Chapter 3 for
the generation of delocalized and correlated many-electron wave functions for excited states in extended systems. Using this method, we study the dispersion of the
energy of excitations in NiO which are considered to be predominantly localized.
The excited state associated with the crystal field splitting of the system is discussed in some detail. Furthermore, the lowest hole and added-electron states are
addressed in relation with the top of the valence and the bottom of the conduction
band.
7.1
Introduction
We introduced in Chapter 3 a new wave function based method for describing excited
states in extended systems with strong electron correlation effects. One aspect of
this approach is the computation of the Hamiltonian matrix elements and overlap
integrals between localized states in extended systems. This was addressed in Chapter
4 where we studied the double exchange parameters in lightly hole- and electron-doped
manganites.
In this chapter, we probe the energy (K) dispersion of hole and added-electron
states in NiO by means of the new method, designed to take into consideration explicitly the electron correlation and relaxation effects. The lowest hole and added-electron
states are associated with the top of the valence and the bottom of the conduction
bands and hence, the method allows for obtaining an estimate of the fundamental
band gap of the compound. Before addressing these states we direct our attention
towards the K-dependence of the lowest excited state, which is associated with the
crystal field splitting of the Ni 3d orbitals in NiO.
NiO has a cubic face-centered (FCC) NaCl (rock salt) crystal structure with a
Chapter 7, Delocalization of excited, hole and added-electron states in
204
NiO
crystal space group Fm3m above its Néel temperature, TN =523 K. In this T regime
the crystal is paramagnetic [51]. Below the Néel temperature, NiO is a type-II FCC
antiferromagnetic compound (AF2 ) [51] with a ferromagnetic alignment of the magnetic moments of Ni ions on every (111) plane and an antiferromagnetic alignment for
adjacent planes (see Figure 7.1). The transition from the paramagnetic state to the
antiferromagnetic state is accompanied by a slight rhombohedral distortion from the
original cubic unit cell along one of the h111 i directions, (R3m). The lattice constant
is 4.17 Å below TN and increases slightly up to 4.20 Å between 7 and 700 K [52]. In
a simple ionic ansatz NiO is thought to be built from Ni2+ (...3d8 ) and O2− (...2p6 )
ions.
Embedded-cluster models have been routinely applied in the calculation of the d-d
excitations in NiO (see for example [32, 33]). These excitations have been referred to
as predominantly localized and hence, the local cluster approach has been considered
as an adequate model. A relevant question is whether indeed the cluster approach
is justified for the d-d excitations. This question has been addressed by Fink [34]
by means of the periodic CI method, which allows the author to describe transitions
which belong to different K-vectors. See also Chapter 3 for somewhat more detail on
the periodic CI method.
The conventional band structure calculations based on the effective one-electron
picture have provided valuable information for compounds with weakly correlated
electrons. Most often the effective potentials employed in this one-electron picture are
derived within the LDA or LSDA to DFT [1]. The theoretical foundations of ground
state- DFT do not provide a justification to relate the eigenvalues of the Kohn-Sham
equations to the one-electron excitation energies of a system. Nevertheless, the band
structures obtained within this approximation, are often in a quantitative agreement
with results obtained in the angle-resolved photoemission spectroscopy [6]. In TimeDependent DFT however the differences between the eigenvalues of occupied and
unoccupied one-electron orbitals are considered to be approximations to the oneelectron excitation energies.
It is well known that in some classes of solids the LDA or LSDA-DFT fails to
describe correctly the band gaps of the compounds. The transition metal compounds
(TMO) with partially filled d - and f - shells with strong electron Coulomb interactions
are representative for those classes of solids. These solids are referred to as strongly
correlated materials.
NiO belongs to the family of the strongly correlated materials. The LSDA band
structure calculations predict a value for the band gap of only a fraction of an eV [2].
NiO is known experimentally as an antiferromagnetic insulator with a band gap of
4.3 eV [3].
Although LSDA-DFT performs usually well for the ground state properties of the
compounds, the excitations related to the band gaps can not be derived correctly from
the Kohn-Sham eigenvalues obtained from a ground state calculation. The effective
potentials, felt by the valence and conduction states are different, but within the
Kohn-Sham theory the same potential is employed in both cases. We mention in
this context the unoccupied-states potential correction within the LSDA formalism
Introduction
205
proposed by Anisimov et al [9] which is based on using different potentials for the
electrons in the valence and conduction bands. The method produces a band gap of
the compound of 3.9 eV.
Norman and Freeman [5] have shown, that if the band gap can be expressed as
the difference of the total energies of the ground state and excited states (the latter are considered to be the states with Ni d7 and Ni d9 configurations) instead of
being read off from the LSDA band structure, it is indeed possible to obtain a reasonable agreement within the framework of the LD super-cell impurity approach with
angle-resolved-photoemission spectra (PES) [6], x-ray-PES [3] and Bremsstrahlung
isochromat spectroscopy (BIS) [3].
A general explanation for the discrepancy is the strong Coulomb interaction between the electrons in the Ni- 3d shell. Due to these interactions, the differential
electron correlation effects between dn configurations with a different n are rather
substantial and hence, lead also to a substantial contribution to the energy splitting
between dn configurations with different n. The dn configurations of the initial and
final states, involved in transitions across the band gap, are characterized usually by
different n. The complex dn multiplet structure of the TM compounds is hard to
describe starting from the Kohn-Sham equations. The valence band PES [12] spectra
as well as x-ray absorption spectra (XAS) [13] of various TMO are well reproduced
by configuration interaction (CI) calculations on small clusters, that contain one TM
ion and the neighbouring shell of ligands.
Within an ab initio cluster study, the intra-shell Coulomb and exchange interactions are treated explicitly. The advantages with respect to the LD approaches are
that an extension beyond the effective one-electron models is straightforward and the
dn multiplet structure can be described with a high level of accuracy [11]. The dn
multiplet structure of the d -shell, modified by the crystal field splitting, is proved by
valence band PES [12, 21] as well as valence band XAS [13] to persist in TM solids.
While the cluster CI studies of Fujimori et al [12] on the valence band PES employ
a semi-empirical determination of parameters, other cluster studies are based on a
fully ab initio treatment of the cluster. The localized d-d excitations observed in the
optical spectra of NiO have been studied in detail using cluster models [32]. Early ab
initio embedded cluster calculations on the [NiO6 ]10− cluster have been performed
using SCF and limited CI expansions by Bagus and Wahlgren [69] who have demonstrated also that the localized approach is appropriate as a starting point for the
description of the 3d states. Most recently, Satitkovitchai et al [14] have performed
ab initio embedded cluster calculations on the ground and low-lying exited states of
the bulk and of the (001) surface of NiO in relation with the second-harmonic and
optical absorption spectra of the compound.
In this context, we also mention the CI approach with model Hamiltonians applied
to Anderson impurities and clusters [27, 28]. These models provide many-electron
solutions for the system by taking into account the multiplet splitting of the single
d -shell [31].
When one discusses the valence hole (ionization) spectra and the electron addition
spectra in transition metal oxides one must consider the effect of the d-d correlation
Chapter 7, Delocalization of excited, hole and added-electron states in
206
NiO
and, in addition, also the TM 3d -O 2p interactions on the ionization and electronaddition processes. Those effects can be incorporated in the quantum mechanical
treatment in a straightforward manner if the systems are modeled within the embedded cluster approach. We discussed in the previous chapters applications of this
approach in the calculation of localized excited states, ionized and added-electron
states as well as magnetic couplings in transition metal compounds. In this Chapter
we consider the ab initio embedded cluster study of the band gap in NiO performed
by Janssen and Nieuwpoort [11].
The nature of the fundamental absorption edge, charge transfer or Mott-Hubbard
type, has been a question of controversy for a long time. Initially NiO was considered
as a representative Mott insulator in which the insulating gap arises from the strong
on-site Coulomb d-d repulsion [20]. This point of view however was not supported
by PES and Bremsstrahlung-isochromat-spectroscopy measurements [3]. Sawatzky et
al [3] proposed that NiO should be classified as CT insulator with a fundamental band
gap resulting from the energy difference between an ionized state of oxygen character
and an added-electron state of Ni d9 character. However the oxygen hole state in
their interpretation of the valence band structure lies inside the large correlation gap
in the Ni 3d band and is thus considered to be a 3d hole strongly screened by an
oxygen electron.
Recent X-ray emission and absorption spectroscopy studies accompanied by Xray photoelectron spectroscopy measurements of the valence and conduction band
states of NiO by Schuler et al [4] indicated that the valence and conduction states
immediately below and above the Fermi energy are primarily of Ni 3d -character
hybridized with a small amount of O 2p -character states. These authors suggested
that the character of the fundamental band gap of NiO may be better considered as
a mixture of CT and Mott-Hubbard type.
The optical band gap in NiO is usually assumed to be of CT type. The photoelectron spectroscopy measurements [3] have estimated the conductivity band gap to
be 4.3 eV which is slightly larger than the optical absorption edge at 3.8 eV [22]. The
optical band gap is regarded by Hüfner et al. [23] as the energy difference between
the onset of the O 2p ionization considered to be near the Fermi level and the first d9
state, and its magnitude is almost the same as that of the fundamental band gap. On
the other hand some revised experimental results obtained by Hüfner [24] re-invoked
the possibility that the optical band gap in NiO corresponds to a transition between
lower and higher Ni 3d bands.
Despite the fact that NiO is now widely accepted to be best characterized as CT
insulator, the character of the fundamental band gap is still somewhat questionable.
While the bottom of the conduction band regarded as arising mainly from the lowest
Ni d9 added-electron state is not questionable, the top of the valence band appears
to be a subject of debate.
Along with the experimental results some theoretical model Hamiltonian calculations on impurities Ni ions in an O 2p band [27, 28] have predicted the lowest ionized
state to be predominantly of d8 O− character. Fujimori et al [12] performed a model
CI study on an embedded [NiO]6 9− cluster and proposed the interatomic 3d-3d charge
Introduction
207
transfer transitions to be the origin of the NiO fundamental absorption edge. However
they also considered the possibility that the 3d hole state associated with the lowest
ionization energy is accompanied by an oxygen to metal 3d charge transfer.
Janssen and Nieuwpoort have addressed as well this controversial subject using the
cluster approach [11]. Cluster models have been employed in a combination with the
LDA-DFT formalism [25], but they reproduce to some extent the problems existing
in the conventional LDA band structure calculations, the band gaps are found to be
less than an 1 eV.
Janssen and Nieuwpoort obtained an estimate for the Mott-Hubbard gap U and
the CT gap 4 of NiO using a model [NiO6 ]10− cluster in an electrostatic field of optimized point charges. They performed HF and first-order CI (FOCI) [11] calculations
to investigate the ionization and electron-addition spectra of [NiO6 ]10− which allowed
them to obtain an insight in the character of the states involved in the band gap
transition. The authors also adopt the widely accepted point of view concerning the
Ni-Ni interactions in the bulk solid as being negligible. They justify their choice with
the resemblance of the low-energy optical spectrum of MgO:Ni and that of NiO [47].
The values obtained for U and 4 within the cluster model are rather similar and no
definite answer could be given as to which quantity 4 or U determines the band gap.
This was due to the large uncertainty in the relative positions of the first ionization
energies obtained from the cluster study. Nevertheless the authors have shown the importance of the inclusion of the local electron correlation and relaxation effects as well
as the extra-cluster bulk polarization corrections for the accurate determination of U
and 4. Janssen and Nieuwpoort comment on the inclusion of the Ni-Ni interactions
but they do not expect a large effect on the energies.
The purely localized approach explains the experimental observations in the X-ray
PES and Bremsstrahlung-isochromat -spectroscopy [3]. Band calculations based on
the self-interaction-corrected DFT [27] and the LDA+U method [7,8] were performed
to compute the energy gaps. Indeed the energy gaps improved compared to the LDA
and LSDA results however, the usual implementation of LDA+U [8] introduces two
semi-empirical parameters and the strong on-site electron correlation effects, which
are of importance for the Ni 3d states, are parametrized.
Another post-LDA approach which handles the self-energy error, is the many-body
GW approximation [35]. The self-consistent GW approximation has been found
able by Massidda et al [37] to obtain a magnitude of the band gap in reasonable
agreement with experiment, ∼ 3.7 eV. Non-self-consistent GW calculations, carried
out by Aryasetiawan and Gunnarsson [36], have yielded a band gap significantly larger
(∼ 5.5) than the experimental band gap. Recent self-consistent GW calculations
within the all-electron full potential linear muffin-tin orbital method [38], carried out
by Faleev et al, have led to a band gap of 4.8 eV in a very good agreement with the
experimental gap of 4.3 eV [3].
A detailed study of the electronic structure of NiO performed by Moreira et al [39]
using three different periodic approaches, UHF, LDA and exploiting several hybrid
DFT functionals, such as the B3LYP functional, has revealed the potentials and
drawbacks of the three approaches for the description of the band gap, magnetic
Chapter 7, Delocalization of excited, hole and added-electron states in
208
NiO
coupling, lattice constant and other properties. DOS obtained within UHF, LDA,
B3LYP and other hybrid functionals with 35% and 50% Fock exchange, predicts an
insulating character with a large band gap while, LDA predicts NiO to be a metal.
They show that the band gap is rather sensitive to the amount of the Fock exchange
and varies from 15.1 eV to almost 0.0 eV. The B3LYP method predicts a band gap in
excellent agreement with the experiment. The authors point out that increasing the
fraction of the Fock exchange leads to an increase in the O (2p) density at the Fermi
energy and a decrease in the Ni 3d density, thus resulting into a charge transfer type
band gap. Dovesi et al [43] performed also periodic UHF and L(S)DA calculations
on the electronic structure of NiO and their findings are essentially the same.
Within the framework of the UHF, UHF+LYP, B3LYP, Perdew-Wang GGA DFT
approaches to periodic calculations, Bredow and Gerson [40] discussed the nature of
the upper part of the valence band in NiO. These authors also found that the most
important difference between the methods based on the pure HF exchange and those
based on a hybrid DFT approach is that in the DFT approaches the contribution of
the Ni 3d orbitals in the upper part of the valence band is increased [40]. We mention
also the analogous periodic HF study performed earlier by Towler et al [42].
Takahashi et al [41] carried out UHF calculations within the local cluster approach
employing a [NiO6 ] cluster embedded in a point charge representation of the ions
surrounding the cluster. They noted that the main discrepancy between the band
gap obtained from their study, 9.9 eV, and that reported by Zaanen et al [30], 4.3 eV,
is due to the fact that electron correlation has not been taken into account within their
analysis and that the point charge representation of the ions, surrounding the cluster,
is only a rough approximation of the Coulomb short-range interactions between cluster
and surrounding crystal. The latter results into not sufficiently accurate ionization
energy and electron affinity. As we see later in the section 7.4 the ionization energies
and electron-addition energies are indeed very sensitive to the model representation
of the nearest embedding ions around the cluster. Furthermore, compared to Janssen
and Nieuwpoort [11], Takahashi et al [41] obtained distinguishable values of the MottHubbard gap, U, and the CT gap, 4, from which they concluded that the band gap
of the compound is a CT type. However, Janssen and Nieuwpoort [11] carried out
ab initio calculations of a higher accuracy accounting in a rigorous manner for the
electron correlation and relaxation effects (see above).
As discussed above, angle-resolved photo-emission studies on NiO by Shen et al [6]
have revealed that neither the conventional one-electron band structure theory nor
the purely localized (cluster) approaches can yield complete and accurate picture of
the lowest holes states. A semi-empirical model of Bala et al [15], which includes
the multiplet structure of the electronic excitations in NiO and the interaction of the
O 2p holes in the O bands with the spins localized on Ni ions, agrees well with the
experimental data of Shen et al [6]. Bala et al have found that the low-energy hole
states in NiO are O 2p hole states and they show a considerable k -dependence.
Despite the success of the cluster approach to reproduce accurately angle-integrated
quantities in different photoemission and photoabsorption spectroscopies and to allow for a proper account for the electron correlation and relaxation effects, its local
7.2 Theory
209
character makes it impossible to obtain k-resolved electron ionizations and additions.
It disregards the periodicity of the correlated sites and the dispersion of the bands in
the solid.
The necessity of a new ab initio approach which incorporates both the local manyelectron and band-like effects in one becomes evident from the discussion above. In
Chapter 3, we mentioned a number of such approaches that make use of the so-called
incremental scheme [48] which allows for incorporating the electron correlation effects
in the band structure calculations [49, 50]. We introduced as well the method developed in this thesis to allow for combining the advantages of the local methods with
those of the band approaches. The study of the top of the valence band and the bands
at the bottom of the conduction band is based on this new method and demonstrates
its advantages above the conventional local and one-electron band approaches.
The rest of the chapter is organized in the following manner. We outline in a
concise form the formalism of the method, that was introduced in Chapter 3. Next
we consider the K-dependence of the excitation related to the 10 Dq parameter. Then,
we report our results for the dispersion of the energy (K) with the momentum K of
hole and added- electron states in NiO. The role of the magnitude of the Hamiltonian
matrix elements and overlap integrals, related to this process is analyzed.
7.2
Theory
As described in Chapter 3, one can express the crystal wave functions ΨnK of excited,
ionized or added-electron states in an extended system in terms of linear combinations
of Bloch sums of local many-electron (ME) basis functions {Φa (r1 , r2 , ...., rN+M )}.
The Bloch-type ME basis functions are given as:
ΘaK (r1 , r2 , ...., rN+M ) = √
X
1
eiK.R Φa (r1 − R, r2 − R, ...., rN+M − R),(7.1)
N1 N2 N3 R
where Φa (r1 , r2 , ...., rN+M ) describes a localized excited, ionized or added-electron
state a and N1 , N2 and N3 are the number of unit cells in each of the three crystal
dimensions, respectively. The summation runs over all lattice vectors R. Using these
Bloch-type ME basis functions, one needs to apply the variational procedure to obtain
the best approximations to the crystal wave functions ΨnK ,
X
ΨnK (r1 , r2 , ...., rN+M ) =
κna (K)ΘaK (r1 , r2 , ...., rN+M )
(7.2)
a
The ΨnK can be derived by solving a generalized eigenvalue problem. Furthermore,
the matrix elements of H(K) and S(K) between the Bloch-type ME basis functions
are expressed in terms of the matrix elements between the localized ME basis functions
from the basis set {Φa (r1 , r2 , ...., rN+M )}.
Next the localized ME basis functions are chosen to be antisymmetrized product
wave functions, constructed from a correlated MCSCF wave function for a relevant Nelectron super-cluster and an M-electron embedding wave function, where the latter
Chapter 7, Delocalization of excited, hole and added-electron states in
210
NiO
represents a frozen electron distribution for the environment. This approximation to
the local ME basis functions allows one to approximate the H(R) and S(R) matrix
elements between the localized ME basis functions by the matrix elements of the
super-cluster.
The Hamiltonian matrix elements and overlap integrals obtained within the embedded super-cluster calculations are employed to obtain the eigenvalues nK associated with the eigenfunctions ΨnK . We referred in Chapter 3 to the distribution of
the nK for a given n with respect to K as to a ”many-body” energy band.
We employ this approach below to study the K- dependence of hole and addedelectron states in NiO associated with the top of the valence and the bottom of the
conduction bands. The localized orbital sets in which the CASCI wave functions of
the super-clusters, representing the localized ME basis functions, are expressed, are
derived from CASSCF calculations on fragments. The study of different overlapping
fragment schemes applied to hole and electron states in hole- and electron-doped
manganites (Chapter 4 ) has shown, that the optimal scheme for those states is OF1.
This OF1 scheme is employed throughout this study to obtain the localized orbital
sets of the super-clusters.
Before addressing in detail the hole and added-electron states in NiO, we study
the K- dependence of d-d excitations, regarded usually as localized excitations. We
consider in particular the excitation which determines the 10 Dq parameter in the
compound. While for the hole and added-electron states the OF1 scheme was established to be optimal, this may not be true for localized excitations. We discuss this
issue in section 7.4.1.
7.3
Material model and Computational information
To obtain the dispersion of the bands at the top of the valence band and at the bottom of the conduction band, the AF2 state of NiO without the small rhombohedral
distortion is considered and the bands are calculated along selected symmetry directions and at high symmetry points within the first Brillouin zone corresponding to
the cubic AF compound. The Ni-O distances are fixed to the experimental values of
2.085 Å or 3.9343 bohr [53]. In the rock salts structures, each ion is surrounded by
its six nearest neighbour counterions so the NiO6 units form regular octahedra and
have a local symmetry, characterized by the Oh symmetry point group. The formal
ionic charges are +2 for Ni and -2 for O which leads, omitting the inner shell orbitals,
to the Ni2+ (... 3d8 ) O2− ( ... 2p6 ) valence shell configuration for the ground state.
The ground state of Ni2+ in NiO is a 3 A2g state in terms of the Oh symmetry point
group.
Making use of the concept of a super-cluster built from fragments, we designed
[NiO6 ] fragments and various super-clusters along the h100 i direction of the cube,
[Ni2 O11 ] and [Ni2 O11 O12 ] and within the (110 ) plane, [Ni2 O10 ] and [Ni2 O10 O12 ].
These super-clusters allow us to access the Hamiltonian and overlap matrix elements
between CASCI wave functions representing Ni hole or added-electron states localized
around two distinct Ni lattice sites.
Material model and Computational information
211
Figure 7.1: Crystal and magnetic structure of AF2 NiO; The up- and down- arrows denote
Ni ions with spin up and spin down, respectively. The tiny rhombohedral distortion is not
depicted;
Furthermore a super-cluster [Ni6 O6 ] was designed to allow for a balanced treatment the two different hole states at the Ni or O ions. The treatment is not balanced
if one employs the [NiO6 ] cluster for both Ni and O hole states, because of the nonequivalent nearest environment of the Ni and O ions. Finally, to obtain an estimate
for the matrix elements between the localized O hole states, a linear super-cluster
[Ox2 Ni3 ], built along the h100 i direction, is employed which is constructed from two
overlapping [ONi2 ] fragments, built also along the h100 i direction of the cube. The
model super-cluster [Ox2 Ni3 ] provides an access to O-O σσ -type interactions along the
h100 i direction. In order to access also the O-O ππ - and σπ -type interactions along
the h100 i direction and within the (110 ) plane, respectively, three other super-clusters
were designed, namely [Ox2 Ni5 ] with the O-Ni-O bonds along h100 i, and [Oxy
2 Ni3 ] and
[Oxy
2 Ni4 ] within the (110 ) plane. The four super-clusters can be viewed as constructed
from two overlapping [ONi2 ] or [ONi3 ] fragments with one shared Ni ion for [Oxy
2 Ni3 ],
[Ox2 Ni3 ] and [Ox2 Ni5 ], and two shared Ni ions for [Oxy
Ni
],
respectively.
4
2
The fragments and super-clusters are embedded in effective model potentials [54].
The short-range Coulomb interactions are accounted for by representing the nearest
neighbour ions by either bare ab initio model potentials [54] or AIMPs augmented
with 1s1p basis functions (for the case of super-cluster [Ni6 O6 ]). The latter functions
are used to maintain the strong orthogonality condition between the cluster and the
ions in the direct environment [55, 56].
The long range electrostatic interactions are described by an array of optimized
point charges placed at lattice positions to reproduce the external Madelung potential
on a fine grid within the fragment/super-cluster region. The orbitals are expanded
in atom centered ANO-type basis functions [65]: for Ni a primitive set of [21s, 15p,
Chapter 7, Delocalization of excited, hole and added-electron states in
212
NiO
Figure 7.2: Super-cluster [Ni6 O6 ]. The ions in the super-cluster are depicted as white (O)
and black (Ni) balls. The hollow equatorial balls represent embedding Ni ions (AIMPs-1s1p)
while the hollow striped balls represent the embedding O ions (AIMPs-1s1p).
10d] Gaussians is contracted to (6s, 5p, 4d) and for O a [14s, 9p]/(4s, 3p) Gauss type
basis set is used.
The specification of the active spaces, employed for fragments and super-clusters
in the construction of the CASCI and CASSCF wave functions for the hole and
added-electron states, is provided in the sections, where those states are discussed.
7.4
7.4.1
Results and Discussion
Delocalization of d-d excitations
Employing the new method we studied the delocalization of an excited state in NiO
associated with the crystal field splitting 10 Dq of the compound. Earlier investigations have been performed within the conventional cluster models (see e. g. [32,33,55])
and have shown that the excitations involved in the determination of the crystal field
splitting (10 Dq) have a rather local character. Moreover, de Graaf et. al. [55] have
established that the details of the embedding influence the 10 Dq by a small amount
of +0.15 eV when replacing the nearest point charges for Ni and O ions around a
[NiO6 ] cluster by AIMPs augmented with 1s1p basis functions.
In a simple crystal-field model the five Ni 3d levels split into three t2g and two
eg levels. The 3 A2g ground state has a ....t62g e2g configuration; the first excited state,
3
T2g , has one electron promoted from t2g to eg . : ....t52g e3g . The energy difference
between the two states is to first approximation equal to the t2g -eg energy splitting.
This energy difference is traditionally called 10 Dq. For the 10 Dq excitation, a value
Results and Discussion
213
of 1.13 eV was deduced experimentally [19]. Furthermore, de Graaf et. al. [32, 55]
have found that an accurate account for the static and dynamic electron correlation
effects via the CASSCF/CASPT2 method is necessary for obtaining a quantitative
agreement with experiment. Their best estimate of 10 Dq of ∼ 1.15 eV is obtained
within the CASSCF-pdd/CASPT2 method and an embedding representation of the
nearest 18 Ni ions around the basic [NiO6 ] cluster by either AIMPs+1s1p or by a
frozen charge distribution [55].
The delocalization effects on the excitation have been probed explicitly by Fink
[34], using a newly designed periodic CI method which combines wave function based
approaches for excited states with accounting for the translational symmetry of the
crystal. The method allows Fink to calculate energy bands associated with the excitations, i.e. K- dependent excitation energies. The periodic CI approach incorporates a
large amount of static electron correlation effects but it gives only limited account of
orbital relaxation and dynamic correlation. The latter can be obtained in a straightforward manner within the cluster and within the method, presented in this thesis.
To obtain an insight into the delocalization effects on the excitation involved in
the 10 Dq transition, we calculated the effective hopping matrix elements associated
with the hopping of the exciton between two neighbouring Ni lattice sites along the
h100 i, h001 i and h110 i directions. For this purpose, we employed two-Ni-center
super-clusters, [Nix2 O11 ], [Niz2 O11 ] and [Nixy
2 O10 ], built along h100 i, h001 i and h110 i,
respectively from two overlapping [NiO6 ] fragments. The [NiO6 ] fragments, as well
as the super-clusters, are embedded in bare AIMPs, representing the nearest Ni ions
around the basic clusters, and a set of optimized point charges describing the distant
Ni and O ions. Compared to the ionized and added-electron states which energies are
very sensitive to the details of the embedding, the localized excitations are affected
little by the presence of orthogonalizing 1s1p basis functions on the bare AIMPs.
Using a [NiO6 ] cluster, de Graaf et. al. [55] have found a decrease of 0.1 eV in the
excitation energy when the bare AIMPs are augmented with 1s1p basis functions.
CASSCF calculations are carried out for the ground state, 3 A2g , and excited state,
T2g , of the [NiO6 ] fragment within an active space of 8 electrons in 5 Ni 3d orbitals
(CASSCF-d). Within this active space, the configurational composition of the wave
function of 3 A2g is 100 % (t2g ξ)2 (t2g η)2 (t2g ζ)2 (eg )1 (eg θ)1 . Here we have made use
of the conventional notation [70]. In this notation t2g ξ, t2g η and t2g ζ are the three
components, yz, xz and xy, of T2g , and eg θ and eg are the two components, 2z2 -x2 y2 and (x2 -y2 ), of Eg in the point group Oh . The CASSCF energy of 3 T2g within
CASSCF-d is at 1.07 eV above the ground state. In the present study we concentrate
attention on the component of the 3 T2g excited state which is represented by a single
Slater determinant within the formalism of the crystal-field theory. Using conventional
notation [70] we write the dominant electronic configuration associated with this
component (3 T2g ζ) of the 3 T2g excited state as (t2g ξ)2 (t2g η)2 (t2g ζ)1 (eg )2 (eg θ)1 .
The contribution of this configuration in the wave function of the 3 T2g excited state
is practically 100 %. We compute t of this component, 3 T2g ζ, with the same excited
state, but now translated to neighbouring Ni ions in the h100 i, h001 i and h110 i
directions. Interactions of 3 T2g ζ with other 3 T2g components at these neighbouring
3
Chapter 7, Delocalization of excited, hole and added-electron states in
214
NiO
Table 7.1: Comparison of OF1 and OF2 for the description of the localized excited state,
5
16−
Ez, of the [Nix2 O11 ]18− super-cluster and the localized excited state, 5 B2 , of the [Nixy
2 O10 ]
super-cluster; CASCI energies, H11 =H22 , in hartrees; Effective hopping integrals tx (along
h100i) and txy (along h110i) are listed in meV.
5
Ez
-tx
Hx11
5
B2
-txy
Hxy
11
OF1
1.35
-3959.927628
OF2
2.86
-3959.904043
1.30
-3687.328499
0.81
-3687.267739
Ni ions as well as interactions with other sites are zero by symmetry or negligible. The
composition of the singly occupied eg θ, or 2z2 -x2 -y2 orbital for this state according to
Mulliken population analysis constitutes of 91 % Ni 3d character. The singly occupied
t2g ζ or xy orbital is rather localized: 99 % Ni 3d.
In constructing the localized orbital basis for the CASCI wave functions of the
super-clusters, representing the same excited state, but now localized around one or
the other Ni ions in the super-clusters, we compared both OF1 and OF2 schemes.
In the OF1 scheme the doubly occupied orbitals for the O ion, shared between the
fragment with the excitation, [NiO6 ]∗ , and the fragment in the ground state configuration, [NiO6 ], are constructed as linear combinations, with equal coefficients, of the
unitary transformed doubly occupied orbitals of that O ion, derived for the fragments
[NiO6 ]∗ and [NiO6 ]. In the OF2 scheme, the doubly occupied orbitals for the shared
O ion are chosen to be those derived for the excited [NiO6 ]∗ fragment.
The [Nix2 O11 ]18− super-cluster has D4h symmetry with the C4 axis along the h100 i
direction, but the localized CASCI wave functions have only C4v symmetry. They
transform as Ez, Ey and A2 . The localized CASCI excited state, that we concentrate
on has symmetry 5 Ez. Analogously, the symmetry of this state in the [Niz2 O11 ]18−
super-cluster is 5 B2 .
16−
The symmetry of the [Nixy
super-cluster is D2h with the C2 axis along the
2 O10 ]
h110 i direction. The localized CASCI excited state transforms according to 5 B2 in
C2v . In Table 7.1 we summarize for both OF1 and OF2 schemes the CASCI energies
of the localized excited states as well as the effective hopping integrals associated
with the hopping of the exciton between two neighbouring Ni ions. As expected,
the effective hopping integrals are small. An inspection of the values of tx and txy ,
the hopping integrals along h100 i and along h110 i, respectively, shows that they are
affected by the chosen overlapping scheme. The tz integrals are not listed in the
Table because they were found to be negligible in agreement with the mutual δ- type
orientation of the singly occupied t2g ζ orbitals in the two localized excited states.
Comparing the total CASCI energies of the localized excited states of the superclusters, obtained within OF1 and OF2 (in Table 7.1) reveals that the OF1 scheme
yields lower total energies. These energies are at about 1.0 eV above the super-clusters
Delocalization of hole and added-electron states
215
ground state in both OF1 and OF2, which is essentially the same as the excitation
energy for the [NiO6 ] fragment. Although OF1 and OF2 yield different magnitudes
for the effective hopping integrals, in both schemes these effective matrix elements
are very small. The OF1 scheme appears to be the best scheme for this study of the
excited states because it produces lower CASCI energies.
The small hopping integrals lead to very narrow energy bands associated with the
localized excited states. The band stabilization energy 4E related to the delocalization of, for example, the localized excited CASCI state which has the e2g θ and t2g ζ orbitals singly occupied, is obtained using the simple relation, 4E ∼ 4txy +2tx +2ty +2tz
≈ -0.01 eV. Clearly, the excitation energy bands will have a width of at most 0.02
eV. This finding is in reasonable agreement with the results of Fink [34] who obtained
the energy of the localized d-d excitation to differ from the excitation energy at the Γ
point by about 0.05 eV. The explicit derivation of the many-body excitation energy
bands, associated with the localized excited states, 5 Ez, does not bring new insights
and thus, it is not presented.
7.4.2
Delocalized hole and added-electron states
Key ingredients to the correct description of valence hole states and added-electron
states in these strongly correlated TM oxides are an accurate account for the electron correlation and relaxation effects as well as the polarization effects that occur in
the crystal upon ionization or electron addition. A delocalized one-electron approach
does not allow for incorporating these effects. The local effects stabilize the localized
electron or hole state and in case their contribution is dominant, the localized states
provide a better description. For extensive studies on localized states in molecules
and solids see [60–62] and [57–59], respectively. Wave functions which describe hole
states localized at a particular lattice site are not in general adequate approximations
to the electronic wave functions of the crystal because of the relaxed symmetry restrictions. The localized wave functions do not transform according to the irreducible
representations of the space group of the crystalline system. Therefore, they have
to be re-symmetrized. While in the case of core hole states the on-site effects are
predominant [60, 61] and the energy associated with the localized wave function may
be (almost) the same as the energy associated with the resymmetrized wave function,
it has been shown by Janssen [57–59] et al. and Broer [60, 62] that the delocalizing
and localizing effects are of equal importance in the case of valence hole states of
TM oxides and thus, they must be treated on an equal footing. It is clear that the
effective one-electron model, on which the conventional band theory is based, is not
sufficient for the proper treatment of those states. The delocalized description must
incorporate relaxation and polarization contributions, which is not feasible within the
one-electron approach. The (band) stabilization energy for the lowest (translational)
symmetry-adapted states may be comparable to the relaxation energy for the localized states. In that case, as pointed out by Janssen [57], resymmetrization of the
localized states is necessary.
The discussion above concerns the hole states but similar considerations hold for
Chapter 7, Delocalization of excited, hole and added-electron states in
216
NiO
the added-electron states for which the delocalization effects are expected to have
also a contribution comparable to that of the localization effects. Within the new
approach both the localization and delocalization effects can be included.
7.4.3
Localized valence hole states
In order to incorporate the relevant electronic relaxation effects at the same level of
approximation for both Ni and O hole states, one must design fragments and superclusters for which the nearest neighbour environment is equivalent. The relative
energy difference between localized hole states at Ni and O ions respectively, is obtained employing a neutral super-cluster [Ni6 O6 ], that treats the electronic relaxation
effects for the two different types of holes on an equal footing. The super-cluster’s
topology is illustrated in Figure 7.2. The local symmetry of the super-cluster is D4h ,
the symmetry of the localized hole states is C4v . The central Ni ions is denoted Nic ,
the central oxygen ion Oc .
In analogy with the studies in Chapter 4 we can choose the starting fragments to
be [ONi6 ] and [NiO6 ] embedded clusters. In that case the procedure for constructing
the localized orbital sets for the wave functions of the hole states in the super-cluster
[Ni6 O6 ] would involve corresponding orbital analysis between the inactive or doubly
occupied orbitals of the two fragments as well as between the inactive and active or
singly occupied orbitals. The general approach was already discussed in Chapter 3
and applied in Chapter 4 only between the inactive orbitals of two or more [MnO6 ]
fragments. The corresponding orbital analysis between the inactive and active occupied orbitals is straightforward but we choose in this case to obtain the relative
energies of the Ni and O hole states by performing CASSCF calculations directly for
the Ni6 O6 super-cluster, because it is simpler.
The dependence of the Ni 3s ionization energies in NiO on different embedding
schemes has been established by de Graaf et al [55]. The authors have shown that the
description of the Pauli repulsion between the ions in the basic cluster and the ions
in its nearest surrounding is of a significant importance for the ionization energies.
In particular the use of bare AIMPs leads to an overestimation of the effect of the
Pauli repulsion and a consequent underestimation of the ionization energies. The
AIMPs augmented with 1s1p basis functions, used here, produce results close to
those, obtained by representing the ions in the nearest cluster environment by frozen
charge distributions [55].
For the sake of interest we computed the first Ni 3d ionization energy for a [NiO6 ]
fragment, using three different embeddings, using CASSCF results with an active
space of 5 Ni 3d orbitals. The first Ni 3d ionization energy is taken as the relative
energy of the lowest state of the [NiO6 ]9− fragment, with symmetry 4 T1g , with respect to the ground state 3 A2g energy of [NiO6 ]10− . This means that the hole is
predominantly in Ni 3d (t2g ). The configurational composition of the CASSCF wave
function of 4 T1g shows 92 % t52g e2g and 8% t42g e3g character. The CASSCF energy of
3
A2g is obtained within an active space containing 8 electrons in 5 Ni 3d orbitals.
The CASSCF wave function of the ground state is 100 % t62g e2g . As expected, the
Delocalization of hole and added-electron states
217
three different embedding schemes, optimized point charges, bare AIMPs and AIMPs
augmented with 1s1p basis functions for the nearest Ni embedding ions of [NiO6 ], produced diverse ionization energies. The ionization energy was largely overestimated
using point charges (11.16 eV) and largely underestimated using bare AIMPs (3.09
eV). The AIMPs+1s1p1d yielded for the ionization energy 6.14 eV.
It is clear that in order to obtain the relative energy difference for the Ni and
O hole states, one needs to choose the optimal description of the embedding ions.
Preliminary CASSCF calculations on the [ONi6 ] fragment, within an active space
containing 17 electrons in 12 Ni -3d (eg ) -like and 3 O -2p orbitals, have indicated
that the ionization energy of the lowest high-spin coupled O ionized state, 14 T1u of the
[ONi6 ] fragment decreases by only about 0.1 eV when the point charges description
of the nearest embedding O ions is replaced by bare AIMPs.
The O ionization energy, associated with the creation of an O hole, is much less
sensitive to the representation of the closest environment of the [ONi6 ] fragment than
the ionization energy for the Ni hole to the nearest embedding of the [NiO6 ] fragment.
The ground state of [ONi6 ] is denoted as 1 A2g in the symmetry species of the Oh
symmetry point group. The magnetic structure of NiO (see Figure 7.1) indicates low
spin couplings between the high-spin coupled spins at each of the distinct Ni ions in
the [ONi6 ] fragment. Note however that at this level of approximation (CASSCF) we
estimated the exchange coupling constant J between two neighbouring Ni ions along
each of the three directions x, y, z to be -3.1 meV.
Because of the pronounced dependence of the Ni ionization energy on the embedding, the nearest Ni and O ions around [Ni6 O6 ] have to be described by AIMPs+1s1p.
P MP
This requirement leads to the following symmetrical configuration [Ni6 O6 ]NiM
12 O12
MP MP
Ni5 O5 . Constructing this super-cluster from two overlapping [ONi6 ] and [NiO6 ]
fragments would imply the use of embeddings for the fragments analogous to that of
MP MP
P
P
P MP
O6 .
NiM
and [ONi6 ]OM
the super-cluster, [NiO6 ]NiM
6
12 Ni8
12 O8
Using AIMPs+1s1p fragment embeddings for ions that are present in the supercluster leads to some technical problems. To circumvent this technical difficulty, in
this particular case, we performed the CASSCF calculations directly for the supercluster.
CASSCF calculations within an active space containing 24 electrons in 12 Ni 3d eg -like orbitals, 3 Nic -t2g -like orbitals, and 3 Oc -2p are performed for the lowest
high-spin coupled state 13 A1g of [Ni6 O6 ]. This active space is denoted as CASSCF-pd.
The absolute CASSCF energy of 13 A1g changes by about 1 meV when an active space
consisting of only twelve Ni 3d (eg ) orbitals and 12 electrons is employed (minimum
active space: CASSCF-d). 13 A1g lies above the ground state 1 A1g of [Ni6 O6 ] by about
0.02 eV. It differs from 1 A1g by the high-spin coupling between the spins at distinct
Ni ions. To a very good approximation this state may be regarded as a relative zero
energy when evaluating the relative energies of the lowest Oc and Nic hole states with
respect to the ground state of [Ni6 O6 ].
An estimate of the effect of the different representation of the embedding on the
ionization energies of the high-spin coupled Ni and O holes is obtained by comparing their relative energies yielded within the frozen orbital approximation for three
Chapter 7, Delocalization of excited, hole and added-electron states in
218
NiO
different embeddings of [Ni6 O6 ], optimized point charges, bare AIMPs and AIMPs
augmented with 1s1p basis functions. Although the energies obtained within the
frozen orbital approximation are not reliable in a quantitative sense, the change in
the relative energies of the Nic and Oc hole states with the embedding is instructive
for the embedding effect. The relative energies of the Nic and Oc hole states obtained
from the energy of the 13 A1g state of [Ni6 O6 ] within the three embeddings and within
CASSCF -d, are depicted in Figure 7.3. The Nic hole states, 14 B2 and 14 E, in C4v
notation have the hole in a Nic -3d (t2g ) -like orbital. Within the three embeddings,
the hole states associated with the central Oc ion, 14 A1 and 14 E in C4v notation,
are found below those associated with the central Nic ion. The threefold degeneracy
of the hole states is lifted because the site symmetry of Nic and Oc in the [Ni6 O6 ]
cluster is not strictly Oh . The relative energy difference between the Ni and O hole
states changes significantly from PC (∼ 5.17 eV) to bare AIMPs (∼ 1.97 eV) and
AIMPs+1s1p (∼ 3.74 eV).
E. eV
18
Ni holes
14B
14E
2
16
14
O holes
14A 14E
1
14E
12
14A
1
14A
1
14E
PC
bare AIMPs
AIMPs+1s1p
Figure 7.3: [Ni6 O6 ] cluster. Relative energies (in eV) of Ni and O hole states obtained
within the frozen orbital approximation for three embeddings: point charges (PC), bare
AIMPs and AIMPs+1s1p.
Clearly, there is an initial state effect of the embedding on the relative energy
difference between the Ni and O hole states. This is due to a compression of the electron distribution when AIMPs are used. While the point charges description largely
overestimates the FO ionization energies, the bare AIMPs overestimate the Pauli
repulsion and consequently underestimate the FO ionization energies. The AIMPs
augmented with 1s1p functions yield ionization potentials which are close to those
obtained within the frozen electron distribution approximation for the nearest embedding ions. Figure 7.3 shows clearly that the embedding effect is more pronounced
Delocalization of hole and added-electron states
219
Table 7.2: Super-cluster ([Ni6 O6 ]) relative energies (in eV) of some low lying hole states,
associated with an ionization from the central Ni and O ions; The relative energies are given
with respect to the lowest Ni-Ni-high-spin coupled state 13 A1 .
hole character
Ni-3d (eg )
Ni-3d (t2g )
O-2p
State
12
A1 , 12 B1
14
B2 , 14 E
14
A1 , 14 E
12
A1
2
A1
Energy (eV)
8.60, 8.67
7.02, 7.05
8.36, 8.39
7.47
7.43
for the Ni hole states than for the O hole states.
This simple analysis confirms our initial statement that the relative energy difference between the Ni and O hole states must be obtained using [Ni6 O6 ], embedded in
AIMPs for the nearest embedding Ni and O ions, augmented with 1s1p basis functions. This cluster model was employed in the next set of CASSCF calculations, the
results are listed in Table 7.2.
Next, CASSCF calculations were carried out for the Nic and Oc hole states, with
an active space containing the Ni 3d (eg ) -like orbitals plus, depending on the hole
state considered, one Nic - 3d (t2g ) and one Oc -(2p -t1u ). The results in Table 7.2
show that at this level of approximation (no dynamical correlation effects and only
intra- cluster relaxation included), the Ni-Ni high-spin coupled hole states localized at
the central Nic ion are about 1.3 eV lower in energy than the lowest Ni-Ni high-spin
coupled hole states localized at the central Oc ion. The nearly degenerate 14 B2 and
14
E Nic hole states have the additional hole in a Nic 3d (t2g ) -like orbital. They can
be related to the three 4 T1g states in the [NiO6 ] fragment. The nearly degenerate
localized Nic hole states 12 A1 and 12 B1 have the additional hole in Nic 3d (eg ). They
are associated with the 2 Eg states in [NiO6 ] and are found at about 1.6 eV higher
energies. The lowest energies of the Ni-Ni-high-spin coupled Ni and O hole states and
the corresponding Ni-Ni -low-spin coupled Ni and O hole states with total spin S= 12
are anticipated to differ by about 0.02 eV in [Ni6 O6 ].
Furthermore we constructed the CASSCF-pd wave function of an O hole in [Ni6 O6 ]
12
with a total spin S= 11
A1 wave function is rather complex
2 . The composition of this
but using Mulliken Population Analysis in combination with Mulliken Spin Population
Analysis we identified that the main electronic configuration has a hole at O. This
electronic configuration is analogous to that of 14 A1 , except for the low-spin coupling
between the unpaired electron at the central Oc and the 3d (eg ) -type electrons at
Nic . The total energy gain for 12 A1 is about 0.9 eV compared to the relative energy
of the Ni-Ni-high-spin-coupled O hole state 14 A1 .
To complete the analysis we constructed the CASSCF wave function of the lowest O hole state with total spin S= 12 . The composition of this wave function shows
contributions of different CSFs and in order to determine the leading electronic configurations we made use of Mulliken (spin) population analysis. This analysis showed
that the spin couplings between the spins residing at different Ni ions are antiferro-
Chapter 7, Delocalization of excited, hole and added-electron states in
220
NiO
magnetic. The energy gain for 2 A1 compared to
12
A1 is about 0.04 eV.
The energy of the O hole states is lowest for low-spin Ni-O couplings. To estimate
the energy difference between high- and low-spin coupling between an O hole and
one Ni2+ ion, we computed the energies of O hole states of a Ni-O cluster with total
spin S= 32 and S= 12 . The Ni-O cluster was embedded in AIMPs augmented with 1s1p
basis functions. CASSCF wave functions were constructed for 4 Π and 2 Π states (in
C∞v point symmetry group) within an active space of 2 Ni- 3d (eg ) and 1 O- 2p
orbitals. We find the lowest O hole state 2 Π to be below the high-spin coupled O hole
state 4 Π by about 0.44 eV. The energy difference between high and low spin coupling
of the O hole with the Ni ions in [Ni6 O6 ] is 0.9 eV, i.e. twice as large as in [NiO],
because there are now two relevant Ni-O spin couplings instead of one. The 2 Π state
which has a low spin coupling between the electrons within the Ni 3d shell is ∼ 3.0
eV higher. The relative energy of the high-spin coupled O hole state 4 Π with respect
to the ground state 3 Π is 8.53 eV which is approximately the same as the relative
energy of the lowest Ni-Ni-high-spin coupled O hole state 14 A1 found in [Ni6 O6 ] (see
Table 7.2).
An important question to be addressed is also the interaction between localized O
2p hole states and localized Ni 3d hole states. To obtain an estimate of the effective
hopping integral parametrizing such an interaction, we allowed for the two localized
hole states of the [Ni6 O6 ] super-cluster, which have the hole localized at either a Nic 3d
eg - like orbital or the Oc 2px orbital, to interact. The Oc 2p hole state with symmetry
12
A1 has the hole localized at the Oc 2px orbital. The Nic hole state 12 A1 has the
hole localized mainly at 3d 2x2 −y2 −z2 , taking the x axis along the h100 i direction.
The energy difference between these two localized hole states is 1.2 eV. When the two
localized hole states are allowed to interact the resulting eigenstates differ by 3.87
eV. The wave function of the lower eigenstate is a linear combination of the localized
hole wave functions with a largest coefficient for the localized O hole wave function.
The energy of this eigenstate is about 0.3 eV below the energy of the localized O
hole state. After constructing also the proper normalized linear combination of the
Nic 3d holes localized at the nearest neighbour Ni ions along h100 i of the O ion
with the hole, we obtain an energy gain of 0.42 eV with respect to the energy of the
localized O hole state. The effective hopping integral associated with the hopping
of the hole between the nearest neighbour Ni and O lattice sites is evaluated to be
1.8 eV. If in addition we take into consideration also the correction to the energies of
the localized Ni and O hole states due to the dynamical electron-correlation effects,
the energy difference between the two localized hole states becomes ∼ 3.0 eV. The
correlation energy correction for the Nic 3d (eg ) -like hole is estimated to be +2.1
eV using the method by Janssen et al. [11], described below. Even if we assume that
the effective hopping matrix element between the nearest neighbouring Nic and Oc
lattice sites, is not affected by dynamical electron correlation, the overall energy gain
upon the formation of the mixed states is expected to decrease. We do not consider
the interactions between the Nic 3d (eg ) -like hole states and Oc 2p hole states at this
level of approximation. Including those interactions will lead to an additional lowering
in the energies of the states at the top of the valence band (a shift to lower binding
Delocalization of hole and added-electron states
221
energies), but the character of the top of the valence band will remain predominantly
O.
The extra-cluster polarization effects related to the polarization of bands and ions
outside the [Ni6 O6 ] cluster, discussed by Janssen and Nieuwpoort [11], will be of
the same magnitude for both Ni and O hole states obtained employing the [Ni6 O6 ]
super-cluster and therefore, accounting for them is not expected to change the relative energy difference between the lowest Ni and O hole states. Another important
contribution to the relative energies of the hole states, however, is due to the dynamical correlation effects. In this study we used estimates of the dynamical correlation
effects analogous to those, made by Janssen and Nieuwpoort [11], of the change in
the electronic correlation energy due to the removal or addition of an electron in the
system. The estimates concerning the Ni hole and added-electron states have been
deduced by those authors by comparing the atomic experimental ionization energies
and electron affinities with those obtained from numerical Hartree-Fock calculations.
Including the dynamical correlation effects leads, to higher ionization energies and
lower electron affinities: +2.4 eV for the ionization of a 3d electron from a free Ni2+
ion and -3.0 eV for the addition of a 3d electron to Ni2+ . These estimates include not
only the dynamical correlation effects but also relativistic effects, though the latter
constitute a smaller fraction of the total energy correction. The electron correlation
effect for the ionization of an O 2p electron from O2− ion is estimated to be 0.4 eV
based on HF and CI calculations on O ions surrounded by a set of point charges to
mimic the NiO electrostatic field, and using a large basis set [11]. Accounting for the
dynamical electron correlation effects by using similar estimates leads to an increase
in the lowest Ni ionization energy to about 9.4 eV and brings the lowest O ionization
energy to about 7.8 eV.
7.4.4
Added-electron states
Finally, the lowest added-electron state of the [Ni6 O6 ] super-cluster was considered.
CASSCF calculations for a [NiO6 ] fragment within an active space of 5 Ni -3d orbitals
were carried out using three embeddings, point charges, AIMPs and AIMPs+1s1p1d.
The configuration of the lowest 2 Eg states within the given active space is t62g e3g . While
the point charges embedding provides a low value of the lowest Ni -3d electron affinity
(-2.3 eV), the bare AIMPs gives -11.2 eV. The AIMPs+1s1p embedding yields a value
of -7.8 eV. Thus the difference between the ionization potential I and electron affinity
A, I -A, depends on the embedding. This simple analysis illustrates the importance
of model potentials augmented with orthogonalizing basis functions.
Next, CASSCF wave functions for the Nic 3d (t32g e2g ) states 12 A1 and 12 B1 of the
[Ni6 O6 ] super-cluster were constructed within an active space of all Ni 3d (eg ) orbitals.
The lowest added-electron states, 12 A1 and 12 B1 , which have the extra electron on
the central Nic ion are found at 7.05 and 7.08 eV above the ground state. Introducing
low-spin couplings between the spins at the distinct Ni ions leads to an energy gain
of 0.02 eV. Accounting for the dynamical electron correlation effects, by using the
estimates made by Janssen and Nieuwpoort [11], leads to a decrease in energy, to
Chapter 7, Delocalization of excited, hole and added-electron states in
222
NiO
about 4.1 eV above the ground state.
7.4.5
Effective hopping matrix elements for hole states
At this level of approximation the lowest ionization potential is associated with the
O hole states. The question which of them will mainly constitute the top of the
valence band can be addressed with a higher certainty after a consideration of the
band-like effects for the different hole states. The relevant quantities are the band
stabilization energies associated with the lowest O and Ni hole states. Since those
quantities are directly related to the effective hopping matrix elements for Ni holes
and O holes, respectively, the next study is concerned with obtaining those effective hopping integrals. The effective hopping matrix elements associated with the
hopping of a hole between two neighbouring lattice Ni sites are calculated in four
super-cluster models. Here, we analyze the hopping matrix elements related to the
two lowest Ni hole states of the two (Ni)-center super-clusters. The super-clusters,
used to access the unique hopping matrix elements for the Ni hole states are the
[Nix2 O11 ] and [Nix2 O11 O12 ] super-clusters, built along the h100i crystal direction and
the [Nix2 O10 ] and [Nix2 O10 O12 ], built along h110i. The overlapping fragments are
[NiO6 ] and [NiO6 O8 ] embedded clusters.
The localized orbital sets, in which the wave functions of those hole states are
expressed, are derived from CASSCF calculations on the fragments. The embedding
of fragments and super-clusters in this set of calculations consists of bare AIMPs
for the nearest embedding Ni ions. Since the hopping matrix elements concern two
localized hole states of the same type, i. e. Ni holes, we do not anticipate that the
AIMPs augmented with basis functions will have an impact on the magnitude of the
hopping integrals. It has been demonstrated in Chapter 5 that the hopping integrals
for Mn-Mn holes change by only few meV when the bare AIMPs are replaced by
AIMPs augmented with 1s1p basis functions. The use of bare AIMPs will lead to an
underestimation of the energies, E1 and E2 , for the two hole states localized at either
one of the two Ni ions in the super-clusters. However, the hopping integrals will not
be significantly affected by the introduction of the 1s1p basis functions.
We consider first the hopping integrals obtained using the model [Nix2 O11 ] supercluster built along the h100 i direction. The super-cluster is viewed as constructed
from two overlapping [NiO6 ] fragments. It has a D4h symmetry but the localized
CASCI wave functions of the hole states have only C4v symmetry. The orbitals
of the fragments are optimized within a CASSCF active space containing either 8
or 7 electrons in 5 Ni 3d orbitals, for either the 3 A2g ground state of the [NiO6 ]10−
fragment, or for one of the two lowest states, 4 T1g and 2 Eg of [NiO6 ]9− . The orbitals of
the [NiO6 ]10− fragment are optimized accounting for the presence of the neighbouring
hole. This is done by increasing the effective nuclear charge at the neighbouring Ni ion
by one, just as in the case of the doped manganites. The configurational composition
of the 4 T1g states is 92 % t52g e2g and 8% t42g e3g . The CASSCF wave functions of the
2
Eg states consist of 95 % t62g e1g , 4 % t42g e3g and about 1% t32g e4g .
A corresponding orbital analysis on the fragments [NiO6 ]10− and [NiO6 ]9− orbitals
Delocalization of hole and added-electron states
223
Table 7.3: Hopping matrix elements thN i−N i for a hole state of symmetry 6 E, and 6 B2 . The
hole is localized mostly in a t2g orbital; CASCI 5d+5d; The presence of the neighbouring hole
is accounted for in the calculations for the [NiO6 ] fragment in a ground state configuration;
First row: thN i−N i for a high-spin coupling between the Ni ions; Second row: thN i−N i for a
low spin Ni-Ni coupling.
Cluster
t2g hole
- thN i−N i (meV)
- thN i−N i (meV)
Nix2 O11
dxy dxz dyz
63, 62, < 1
22, 21, < 1
Nixy
2 O10
Nix2 O11 O12
dxy dxz dyz
dxy dxz dyz
dxy dxz dyz
72, 24, 24
24, 7, 7
95, 95, 1
32, 32, < 1
117, 27, 19
39, 9, 6
Nixy
2 O10 O12
is performed in order to determine which doubly occupied orbitals of both fragments,
associated with the shared O ion, should be replaced by normalized linear combinations of those doubly occupied orbitals (OF1 ). All remaining [NiO6 ]10− and [NiO6 ]9−
orbitals are combined to form the localized orbitals for [Nix2 O11 ]. Using those localized
orbitals, we constructed the CASCI wave functions for the localized hole states, 6 E,
6
B2 , 4 A1 and 4 B1 of [Nix2 O11 ] and computed the Hamiltonian matrix elements and
overlap integrals between these wave functions. The configuration composition of the
localized CASCI 6 E and 6 B2 states is given for the composite Ni 3d- Ni 3d shell: 92
% t52g e2g - t62g e2g and 8% t42g e3g - t62g e2g . The 4 A1 and 4 B1 states are composed of 96 %
t62g e1g -t62g e2g and 4 % t42g e3g -t62g e2g . Here the occupied orbitals in the electronic configurations are given for each Ni -3d shell separately and they belong to the irreducible
representations of the local point symmetry group, Oh at a distinct Ni ion.
The relevant two-center Ni-Ni hopping integrals are calculated as follows,
t=
H12 − H11 S12
,
2
1 − S12
(7.3)
where H22 =H11 . The results for the Ni-Ni hopping integrals, associated with the
localized hole states 6 E, 6 B2 and 4 A1 , 4 B1 , are summarized in Tables 7.3 and 7.4. We
have also listed in Table 7.3 the t2g -like orbital, where the hole resides, for each of
these states 6 E and 6 B2 . In the irreducible representations of the C4v symmetry point
group these t2g -like orbitals are ez, ey and b2 corresponding to the components xy,
xz and yz of t2g .
The contribution of the 6 E and 6 B2 states to the delocalization of the hole along
the h100 i direction is rather small taking into consideration the magnitude of the
relevant hopping integral thN i−N i (Table 7.3). In Table 7.3, we have listed the thN i−N i
integrals corresponding to the high-spin coupling between two nearest neighbour Ni
ions along h100 i as well as those corresponding to the low-spin coupling. The interactions between the nearest neighbour Ni ions along h100 i, h010 i or h001 i are
antiferromagnetic and thus the relevant thN i−N i parameters are those corresponding
to the low-spin coupling for the ions, i.e. 2 E and 2 B2 .
We found the spin dependence of thN i−N i to be analogous to that of t for the lightly
doped manganites, i.e. it follows the simple Anderson-Hasegawa model [64]. Taking
into consideration the spin dependence of thN i−N i , the low-spin thN i−N i between two
Chapter 7, Delocalization of excited, hole and added-electron states in
224
NiO
Table 7.4: Hopping matrix elements thN i−N i for an hole state of symmetry 4 A1 and 4 B1 .
The hole is mostly in a eg orbital; CASCI 5d+5d; The presence of the neighbouring hole
is accounted for in the calculations for the [NiO6 ] fragment in a ground state configuration;
First row: thN i−N i for a high-spin coupling between the Ni ions; Second row: thN i−N i for a
low spin Ni-Ni coupling.
Cluster
eg hole
- thN i−N i (meV)
- thN i−N i (meV)
Nix2 O11
Nixy
2 O10
d2x2 −y2 −z2
d2x2 −y2 −z2
d2x2 −y2 −z2
d2x2 −y2 −z2
dy2 −z2
dy2 −z2
dy2 −z2
dy2 −z2
362, <1
182, <1
1, 1
<1, <1
367, <1
184, <1
22, 5
11, 3
Nix2 O11 O12
Nixy
2 O10 O12
localized hole states of either symmetry 2 E is 22 meV. The effective hopping parameter
associated with the hole state of symmetry 2 B2 is negligible and therefore, it is not
considered in the following analysis. In a simple model, this hopping parameter
corresponds to a ππ interaction between the b2 (yz) - type orbitals in the hole states
2
B2 , localized around either one or the other Ni ions in [Nix2 O11 ]. The interactions
between localized hole states with a different singly occupied orbital, for example b2
(yz) for 2 B2 and e (xy) for 2 E, are not considered because we expect them to be
negligibly small.
The first column in Table 7.4 lists thN i−N i associated with the localized 4 A1 and 4 B1
states of [Nix2 O11 ]. The Table lists only the interaction between the 4 A1 states which
have the hole at the a1 -orbital corresponding to the Ni 3d (eg ) orbital 3d2x2 −y2 −z2 a .
The interaction between those states is considerable, while the interaction between
the 4 B1 states, which have the hole at the other b1 -orbital corresponding to the
Ni 3d (eg ) orbital 3d√3(y2 −z2 ) , is smaller than 1 meV. The interaction between a
4
A1 state localized at one of the Ni ions and a 4 B1 state localized at the other Ni
ion is also less than 1 meV and it will not be considered in the following studies.
The degeneracy of the states 4 A1 and 4 B1 , localized at the same Ni ion, is lifted
(4E(4 A1 -4 B1 )=0.19 eV) due to the different super-cluster environment experienced
by the hole at 3d√3(y2 −z2 ) or 3d2x2 −y2 −z2 . The value of thN i−N i in the first row of
Table 7.4 is for the high-spin coupling between the adjacent Ni ions along h100 i.
The value of thN i−N i for the low-spin coupling is the relevant parameter for NiO. As
expected from the simple Anderson-Hasegawa spin dependence, this value is 182 meV,
half the magnitude of thN i−N i for the high-spin coupling. Clearly this interaction will
contribute considerably to the delocalization of the hole along the h100 i direction.
It is well known, that when an extra charge is introduced at a lattice site in the
crystalline system the surrounding medium polarizes [57]. In NiO, one may expect
both O2− and Ni2+ ions to polarize, but the polarization effects on the O2− ions are
found to be largest when the extra polarizing charge is set at a Ni lattice site [57].
The polarization energy is also found to be proportional to r−4 with r the distance
between the polarizing charge at the Ni site and the O2− ions at the other lattice sites.
ax
and y axes are chosen along h100 i and h010 i, respectively
Delocalization of hole and added-electron states
225
The polarization effects have been found to be of a significant importance for the first
ionization energies of the Ni and O ionized states as well as for the first electron
addition energy of the lowest added-electron state with Ni d9 electron configuration
and finally, for the magnitude and character of the fundamental band gap [11].
Within our approach, we study the effect of the polarization, induced by a hole
within the super-cluster on the surrounding O ions and the effective hopping matrix
elements. Assuming that the effect is most significant only for the first layer of O ions
around the [NiO6 ] fragments and [Nix2 O11 ] super-cluster, respectively we extended
the fragments and super-cluster by including those 8 and 12 O ions, respectively in
the quantum mechanical treatment. This results into the super-cluster [Nix2 O11 O12 ]
built from two overlapping extended fragments [NiO6 O8 ]. These 8 and 12 O ions are
represented with the same O [14s, 9p]/(4s, 3p) Gaussian type basis set used for the
initial fragments and super-cluster O ions.
The results for the effective hopping integrals obtained using the model supercluster [Nix2 O11 O12 ] are listed in the third columns of Tables 7.3 and 7.4 for the 6,2 E
and 6,2 B2 states and the 4,2 A1 and 4,2 B1 states, respectively. The hopping integrals
associated with the 4 A1 and 4 B1 hole states are slightly affected by the presence of
the additional O ions in [Nix2 O11 O12 ] partly, because of the orientation of the relevant
a1 or a2 orbital, where the hole resides, with respect to those polarizable O ions. A
slightly larger increase in t of at most 0.03 eV is observed for the 6,2 E states.
Next, analogous to the super-clusters built along the h100i crystal direction, two
xy
other super-clusters, [Nixy
2 O10 ] and [Ni2 O10 O12 ], within the xy plane are considered.
Localized hole states in these super-clusters have lower symmetry than C4v , namely
C2v with the C2 axis along the h110i direction. The 6,2 E, 6,2 B2 and 4,2 A1 , 4,2 B1 states
belong to different irreducible representations in C2v . To avoid new notations for the
same states but denoted according to the symmetry species of C2v , we preserved the
C4v symmetry notations for the states.
In the [Nixy
2 O10 O12 ] super-cluster, the hopping integral within the xy plane associated with the 4 A1 hole state is comparable to the hopping integrals associated with
the 6 B2 and 6 Ey states. The presence of the polarizable O ions around the superclusters affects at most the magnitudes of those hopping integrals t between localized
states for which the orbitals φ1 and φ2 , where the holes reside, are oriented within the
xy plane. Moreover the overlap integral hΦrelax
|Φrelax
i becomes smaller due to the
1
2
. The Hamilopposite direction of the orbital polarization effects in Φrelax
and Φrelax
1
2
relax
relax
tonian matrix elements hΦ1
|H|Φ2
i also decrease and thus t becomes smaller.
Despite the fact that the two orbitals φ1 and φ2 , where the holes reside in Φrelax
1
and Φrelax
, are also allowed to relax and delocalize, in this case the delocalization is
2
limited, and it does not compensate the reduction of t. The latter is due to relaxation
of the other super-cluster orbitals. The delocalization of the relaxed φrelax
=3dxy and
1
φrelax
=3dxy is limited because they are oriented between the axes x and y, i.e. they
2
are not directed along any Ni-O-Ni bond. The overall effect is a decrease in t compared to the t that was obtained in the same super-cluster [Nixy
2 O10 O12 ] but without
accounting for the presence of the extra charge at the neighbouring Ni ion while deriving the orbitals of the fragment in the ground state configuration (these results
Chapter 7, Delocalization of excited, hole and added-electron states in
226
NiO
are not discussed explicitly here). The interactions between the nearest neighbour Ni
ions within the xy, or xz or yz planes are either ferromagnetic or antiferromagnetic
depending on whether both ions lay in the (111) plane or one of them lays in a plane
parallel to the (111) plane (see Figure 7.1). Therefore the effective hopping matrix
elements for high- and low- spin coupling between the neighbouring Ni ions are both
relevant parameters.
Finally, the effective hopping matrix elements associated with the O hole states
were considered. As explained in the section 7.3, to access the different type of
interactions between hole states, localized at neighbouring O ions along the h100 i
x
and h110 i directions, various super-clusters are designed, [Ox2 Ni3 ], [Oxy
2 Ni3 ], [O2 Ni5 ]
xy
and [O2 Ni4 ]. The corresponding fragments are [ONi2 ] and [ONi3 ] embedded clusters.
The [ONi2 ] fragments are built along the h100 i or h010 i direction with an angle NiO-Ni of 180o . The [ONi3 ] fragments are constructed in such a manner that the two Ni
ions which have an angle Ni-O-Ni of 180o with the O ion are along the h100 i direction
while the third Ni ion lays on the h110 i or h1-10 i direction. The different superclusters have different point symmetry groups. For all super-clusters we consider the
same type O hole states and for simplicity we adopt symmetry notations for these
states corresponding to the point symmetry group C1 .
The CASSCF wave functions of O hole states of symmetry 6 A1 (in the C1 ) in
the [ONi2 ] fragments are constructed within an active space containing 9 electrons
in 4 Ni- 3d (eg -like) orbitals and 3 O- 2p orbitals. For the [ONi3 ] fragments, the
CASSCF wave functions are constructed for the 8 A1 O hole states within an active
space of 11 electrons in 6 Ni- 3d (eg -like) orbitals and 3 O- 2p orbitals. The CASSCF
wave functions of the fragments [ONi2 ] and [ONi3 ] in the ground state configurations,
described by the 5 A1 and 7 A1 states, are defined within an active space of 10 or 12
electron in 4 or 6 Ni- 3d (eg -like) orbitals, respectively, and 3 O- 2p orbitals. The
orbitals in which these wave functions are expressed incorporate information about
the presence of the nearby hole, localized at one or the other O ions.
The localized orbital sets for the CASCI wave functions of the super-clusters, representing localized hole states, are derived within the OF1 scheme. In this particular
case any two fragments share a Ni ion and hence, the corresponding orbital analysis
involves not only the doubly occupied orbitals of the two fragments, associated with
the shared Ni ion, but also the singly occupied Ni- 3d (eg -like) orbitals, localized at
this ion. The shared Ni ion contributes with eg -like orbitals to the active spaces of the
fragments and thus, to eliminate the double counted electrons, associated with those
double counted eg -like orbitals, the activeN i1 -activeN i2 block of the orbital overlap
matrix between the fragments orbitals is also transformed via a bi-orthogonalization.
The double counted active or inactive orbitals are eliminated according to the OF1
scheme. The corresponding orbital analysis is done separately for the inactive and
active orbitals.
In Table 7.5 are listed the effective hopping matrix elements associated with the
hopping of the O hole between two nearest neighbourng O ions along the h100 i
and h110 i directions. Note that for the hopping integrals for the O holes as well
as for the Ni holes, we have provided only the symmetry unique parameters in two
Delocalization of hole and added-electron states
227
Table 7.5: Hopping matrix elements thO−O for an O hole state of symmetry 2 A1 .
Cluster
S12
H22 -H11 (meV)
- thO−O (meV)
Ox2 Ni3
0.0577
0
107
Oxy
2 Ni3
0.0643
17
193
Oxy
2 Ni4
0.0110
2
31
crystal directions h100 i and h110 i, chosen arbitrary. All other parameters in the other
directions of the crystal can be derived by symmetry. Table 7.5 lists only the hopping
matrix elements associated with the Ni-Ni low-spin coupled O hole states, which are
the relevant hopping integrals for total spin of the hole states wave functions S= 12 .
In the first column are reported the hopping integrals associated with O hole states
of [Ox2 Ni3 ], which have the hole at O orbitals, a1 , oriented along the h100 i direction.
These a1 orbitals correspond to O 2px orbitals. The mutual orientation of those O
2p orbitals in the two localized CASCI oxygen hole states is denoted as σσ.
In the second column are summarized the relevant effective hopping integrals
associated with O hole states of [Oxy
2 Ni3 ]. In this super-cluster, the CASCI hole
state, localized around the O ion, lying along h100 i, has the hole at O 2px orbital,
whereas the other CASCI hole state, localized around the O ion, lying along h010 i,
has the hole at O 2py orbital. This mutual orientation of the O 2p orbitals is denoted
as σπ -type.
Finally the last two columns report the interactions between two O hole states
localized at nearest neighbouring O ions along h100 i and h110 i, respectively, for which
the O 2p orbitals, where the hole resides have a mutual orientation of ππ -type. In
[Ox2 Ni5 ], the two localized CASCI hole states have the hole at O 2py orbitals, whereas
in [Oxy
2 Ni4 ], the holes for both localized CASCI states reside at O 2px orbitals.
An inspection of the effective hopping integrals between two localized CASCI
states, reveals that the interactions, denoted above as σπ -type interactions contribute
at most to the delocalization of the O hole. The relevant Ni-Ni low-spin coupled O
hole states involved in the ππ -type interactions along h100 i and h110 i, respectively
are an order of magnitude smaller than the σπ and σσ -type interactions. The effective
hopping integrals for the low spin couplings are not related via a simple AndersonHasegawa model to those for the highest spin couplings (not shown in the Table).
The effective hopping matrix elements obtained in this study determine clearly a
larger width of the bands associated with the O holes than the band width for the Ni
holes.
Effective hopping matrix elements for added-electron states
Next, we analyzed the effective hopping matrix elements associated with the hopping
of an added electron between two neighbouring Ni lattice sites. The super-clusters
are the same as those used in the study of the Ni hole states. CASSCF wave functions
are constructed for the lowest 2 Eg -like states of the [NiO6 ]11− or [NiO6 O8 ]27− fragment within an active space containing 9 electrons in 5 Ni- 3d orbitals. The CASSCF
Chapter 7, Delocalization of excited, hole and added-electron states in
228
NiO
Table 7.6: Hopping matrix elements teN i−N i for an added-electron state of symmetry 4 A1
and 4 B1 . The electron is localized mostly in a eg orbital; CASCI 5d+5d; The presence of the
neighbouring added-electron is accounted for in the calculations for the [NiO6 ] fragment in a
ground state configuration; First row: thN i−N i for a high-spin coupling between the Ni ions;
Last row: thN i−N i for a low spin Ni-Ni coupling.
Cluster
added eg electron
- teN i−N i (meV)
- teN i−N i (meV)
Nix2 O11
Nixy
2 O10
d2x2 −y2 −z2
dy2 −z2
240, < 1
121, < 1
Nix2 O11 O12
Nixy
2 O10 O12
d2x2 −y2 −z2
d2x2 −y2 −z2
d2x2 −y2 −z2
dy2 −z2
dy2 −z2
dy2 −z2
53, 18
37, 10
233, < 2
117, < 1
64, 11
42, 7
calculations for the [NiO6 ]10− or [NiO6 O8 ]26− fragments, i.e. the fragments in the
ground state configuration, are carried out within the same active space as that used
for the hole states, namely, 8 electrons in 5 Ni- 3d orbitals. Analogous to the hole
states the orbitals of the [NiO6 ]10− or [NiO6 O8 ]26− fragments are optimized accounting for the presence of the nearby added-electron by decreasing the effective nuclear
charge of the relevant Ni ion by one. A corresponding orbital analysis between the
doubly occupied orbitals of two fragments, [NiO6 ]10− and [NiO6 ]11− or [NiO6 O8 ]26−
and [NiO6 O8 ]27− is performed in order to determine the localized orbital bases for
the localized CASCI wave functions of the super-clusters. The configurational composition of the CASCI wave functions of the 4 A1 and 4 B1 states of, for example, the
[Nix2 O11 O12 ] super-cluster is 100 % t62g e3g - t62g e2g (for the Ni 3d -Ni 3d shell).
In Table 7.6 we have listed the effective hopping integrals associated with the
hopping of an extra electron between two neighbouring Ni lattice sites. As expected,
the hopping integrals between the Ni ions along the h100 i cube direction are an order
of magnitude larger compared to those between the Ni ions in the h110 i direction.
This is due to the fact that the lobes of the orbitals involved in the interaction, between the Ni ions aligned at h100 i, are oriented along the x and y axes and hence,
the interaction along h100 i is a σσ -type. Along the h110 i direction, the interaction
involves the same type of orbitals but now their mutual orientation is a δδ -type. The
magnitudes of the hopping integrals, obtained from the [Nix2 O11 ] and [Nix2 O11 O12 ]
super-clusters, are considerable only between those CASCI states, localized at either
one or the other Ni ions, which have the eg -like orbitals 3d3r2 −z2 doubly occupied.
The other two CASCI states with the 3d2r2 −x2 −y2 -like orbitals doubly occupied contribute negligibly to the delocalization of the added electron along the h100 i direction.
The contribution of those localized CASCI states to the delocalization of the extra
electron along the h110 i direction is smaller. Note that while the hopping integrals
for high- and low-spin couplings along h100 i are in agreement with the AndersonHasegawa model, this is not the case for the hopping integrals along h110 i. The
magnitudes of the effective matrix elements in all directions are not affected by the
presence of the electron clouds on the nearest polarizable O ions.
Delocalization of hole and added-electron states
7.4.6
229
Many-body hole and electron bands
The many-body O 2p and Ni 3d hole bands are derived within the framework of the
method described in Chapter 3. We construct only the energy bands derived from
the lowest localized hole states because they may contribute to the top of the valence
band which character appears to be controversial in the literature.
In Figures 7.4 and 7.5, we have plotted the bands along the high symmetry directions, Γ → X and Γ → L in the first Brillouin zone. We have used 50 k points in each
symmetry direction. The energies are given with respect to the ground state energy,
approximated by the energy of the ground state of the [Ni6 O6 ] super-cluster. These
many-body hole bands reflect the dispersion of the energy of the many-electron N-1
ionized states in the different symmetry directions. The ionized states are associated
with the localized Ni and O hole states, considered in the previous section. We have
considered explicitly the bands derived from the localized CASSCF Ni hole states with
symmetry 12 A1 , and 14 E in the [Ni6 O6 ] super-cluster. The many-body Ni hole band
derived from the localized hole state with symmetry 14 B2 is degenerate with the 14 E
band by symmetry. The band associated with the Ni hole state with symmetry 12 B1 is
degenerate with the 12 A1 band. In constructing the Ni hole bands we have employed
the Hamiltonian matrix elements and overlap integrals, computed from the ab initio
embedded super-cluster calculations. Since these matrix elements were derived from
different super-clusters, they were transformed to the corresponding matrix elements
between localized states, obtained from the super-cluster [Ni6 O6 ] and localized states,
obtained by translating the former on the nearest and next-nearest neighbouring Ni
lattice sites. The ferro- and antiferromagnetic couplings in different crystal planes are
taken into consideration.
The construction of the many-body hole bands associated with the O holes requires
some further attention. In Figure 7.6 the O ions in the reference unit cell are illustrated
as black balls and the Ni ions are represented schematically by grey equatorial balls.
The reference coordinate system is positioned at the O ion which lays along the h100 i
direction. This O ion is denoted as O1. The ions O2, O3 and O4 lay along the h110 i,
h101 i and h011 i directions, respectively. For each O lattice site in the unit cell, we
have considered three degenerate localized ME basis functions, corresponding to the
three localized O 2p hole states with symmetries 2 A1 , 2 Ey and 2 Ez in [Ni6 O6 ]. These
states have the hole residing on a O 2px , O 2py and O 2pz orbital, respectively. In
the [Ni6 O6 ] super-cluster these three localized hole states were only nearly degenerate
because of the slightly different cluster environment in the three crystal directions. We
considered them degenerate in the band calculation. The ferro- and antiferromagneitc
couplings between the Ni ions in the lattice are also taken into consideration by
employing the Hamiltonian matrix elements and overlap integrals between the Ni-Ni
low spin coupled localized O hole states.
The ab initio many-body bands in Figures 7.4 and 7.5 show that the bands associated with the O 2p hole states have the lowest electron binding energies, or lowest
ionization energies. The band stabilization energy, Ed , obtained as the difference in
the energy of the localized O hole CASSCF states and the lowest band energy, (K)
which is at the Γ point, is 1.4 eV. The width W of the O bands is the difference in
Chapter 7, Delocalization of excited, hole and added-electron states in
230
NiO
E, eV
10
9
8
7
Γ
Δ
Χ
Figure 7.4: Energy bands associated with the lowest Ni and O hole states in AF2 cubic
NiO, along the symmetry direction Γ → X. Solid lines: O 2p bands; Dashed thick lines: Ni
3d (t2g ) bands; Dashed thin lines: Ni 3d (eg ) bands.
Delocalization of hole and added-electron states
231
E, eV 11
10
9
8
7
Γ
Λ
L
Figure 7.5: Energy bands associated with the lowest Ni and O hole states in AF2 cubic
NiO along the symmetry direction Γ → L. Solid lines: O 2p bands; Dashed thick lines: Ni
3d (t2g ) bands; Dashed thin lines: Ni 3d (eg ) bands.
Chapter 7, Delocalization of excited, hole and added-electron states in
232
NiO
Figure 7.6: The black balls represent the O lattice sites and the grey equatorial balls represent the Ni lattice sites in the unit cell of NiO.
the energy of the top and bottom bands at the Γ point and is 3.1 eV. In this case
the estimates of those quantities, obtained using effective hopping matrix elements,
are not accurate enough because of the more significant overlap integrals between the
localized O hole states. This is reflected also in the low symmetry of the O bands.
Our computed band width corresponds well to the band width of 3.8 ± 0.2 eV derived
from photoemission measurements, performed by McKay and Henrich [71]. Janssen
and Nieuwpoort [11] have also obtained a theoretical band width of the O 2p ionized
states equal to 3.5 eV by setting it equal to the range of O 2p orbital energies of their
[NiO6 ]10− cluster.
The t2g type Ni hole bands, derived from the 14 E localized Ni hole states, have
a width of 0.79 eV at the Γ point and the band stabilization energy, associated with
the formation of the bands is 0.31 eV, computed again at the Γ point. Although the
overlap integrals between those localized hole states are smaller than those between
the localized O hole states, their magnitude is sufficient to introduce asymmetry in
the bands. In this case an estimate based on the effective hopping matrix elements
yields about 0.38 eV for Ed and about 0.76 eV for W.
The eg type Ni hole bands, derived from the 12 A1 localized Ni hole states have
a width of 0.6 eV which is the difference in the energy of the lowest band at the Γ
point and the highest band at the L point. The band stabilization energy is about
0.5 eV. Using effective hopping matrix elements we estimated about 0.42 eV for Ed
and about 0.84 eV for W. Clearly also in this case the overlap integrals between the
localized ME basis functions, approximated by the super-cluster localized CASSCF
wave functions, can not be neglected.
We have not considered explicitly the interaction between the localized Ni and O
hole states. This interaction will lead to O -hole wave functions with Ni-hole character
mixed in which will give rise to modified O hopping matrix elements and consequently,
Delocalization of hole and added-electron states
233
to modified O bands. However due to the large energy difference between the localized
Ni and O hole states, the additional energy gain for the lowest mixed hole states is
expected to be smaller. The character of the top of the valence band, formed by
those mixed hole states, will have still predominantly O character. Because those
interactions are not explicitly included, Figures 7.4 and 7.5 represent a super-position
of the Ni and O many-body hole bands. Introducing the interactions between the
Ni and O hole states in the non-orthogonal tight-binding calculation will lead to a
change in the bands at the higher electron binding energies, however the states at
the lowest binding energies, i.e. at the top of the valence band, will remain mostly
the many-body O hole states with some Ni 3d character mixed in. Thus our ab
initio many-body hole bands indicate that NiO is a CT type insulator. This finding
is analogous to the result, obtained in the HF band structure calculations [39, 42].
However our many-body hole bands incorporate electron correlation and relaxation
effects which are not accounted for in the HF band calculations.
In principle, if we are concerned with the lowest ionization energy of the crystal,
then using a very large embedded cluster combined with a MC wave function description of the that ionized state, will provide an ionization energy very close to the lowest
ionization energy which would be obtained if we would consider also the additional
stabilization energy from the formation of the corresponding band states. As long as
one is concerned only with the magnitude of this lowest ionization energy, such an
embedded cluster calculation provides a reasonable estimate, particularly when the
width of the band is very small, i.e. when the ionization is rather localized. In the
case when the band stabilization energy is more substantial one can still obtain a
reasonable lowest ionization energy within the local approach, however a very large
cluster is needed. In any case, the check of the convergence in the energy with the
cluster size is compulsory.
Instead, we considered using a smaller cluster with an accurate MC wave function description of the lowest ionized state. Then we obtained the additional stabilization energy contribution to the lowest ionization energy by calculating explicitly
the Hamiltonian matrix elements and overlap integrals between such ionized wave
functions, centered around different lattice sites. Thus we started with Wannier-like
functions and then we re-symmetrized them in order to obtain Bloch-like many-body
functions. Hence, we obtained a continuous spectra of many-body hole states. The
same considerations hold for the added-electron states. Because we optimized the
localized wave functions for the hole- or added-electron states, we also incorporated
electronic relaxation. The important non-dynamical energy contributions connected
with the presence of CT configurations in the wave function CI expansion of the Ni
hole states can be accounted for by considering the hopping of the hole between the
Ni and O lattice sites. In this case we have not explicitly included the corresponding
hopping matrix elements.
An important effect, left out of the present study, are the bulk polarization effects.
These effects arise from the change in the electron density in the crystal upon the
creation of the hole or added electron state. The bulk polarization effects associated
with the removal or the addition of an electron to a [NiO6 ] cluster were evaluated by
Chapter 7, Delocalization of excited, hole and added-electron states in
234
NiO
Janssen and Nieuwpoort, using classical methods [11]. They have found that the polarization energies arising from introducing the hole and added-electron in the [NiO6 ]
cluster are about 3-5 eV. We did not consider explicitly these effects. Accounting for
them will only lead to a shift in the many-body bands to even lower binding energies
and we expect that the lowering for both Ni and O hole states will be of the same
magnitude.
The many-body electron-bands are not explicitly derived because, taking into consideration the hopping integrals associated with the localized added-electron states,
they will be similar to the Ni 3d hole bands.
7.5
Summary and Conclusions
We performed a detailed investigation of the lowest hole and added-electron states
in NiO which form the top of the valence and the bottom of the conduction bands
in NiO. To do so, we considered all unique interactions between Ni or O hole states
localized at nearest and next-nearest Ni or O lattice sites. In addition, the interactions
between the hole states, localized at nearest neighbour Ni and O lattice sites were also
investigated but they were found to be of less importance for deducing the character
of the states with the lowest electron binding energies. The latter form the top of the
valence band.
To compute the ionization energies associated with the Ni and O hole states, we
employed a large embedded cluster, [Ni6 O6 ], which allows for treating of the Ni and
O hole states on the same footing. We found that the energy difference between the
lowest localized O and Ni hole states is only 0.4 eV and in the lowest hole state the hole
resides at Ni t2g -like orbital. Applying corrections, which include atomic dynamical
electron correlation and relativistic effects, to the ionization energies, we deduced for
the lowest Ni hole state a relative energy of 9.5 eV and for the lowest O hole state a
relative energy of 7.8 eV. Thus after accounting for the relevant electron correlation
effects, the lowest localized hole state is an O hole state, i.e. a state in which the
leading configuration has a singly occupied orbital, which is mainly localized on one
oxygen ion. Next, we computed the interactions between the hole states localized
at nearest and next-nearest lattice sites, using smaller two-center (Ni-Ni) or (O-O)
super-clusters of different shape and size in order to access all unique interactions.
Using the computed Hamiltonian matrix elements and overlap integrals, modified
so to correspond to the hole states obtained in the [Ni6 O6 ] cluster, we carried out
a non-orthogonal tight-binding calculation to construct the many-body hole bands.
These bands include explicitly the electron correlation and relaxation effects which
accompany the ionization processes. We obtained that the band states which form
the top of the valence band originate mainly from the localized O hole states. The
bottom of the conduction band is formed by Ni 3d 9 added electron states. Hence,
we conclude that NiO is a CT insulator. The O hole bands showed a significant
dispersion along relevant symmetry directions and a large band stabilization energy
of about 1.4 eV and a band width of 3.1 eV. The bands derived from the localized
Ni hole states are characterized with a band width of less than 1.0 eV and a band
7.5 Summary and Conclusions
235
stabilization energy of at most 0.5 eV.
Furthermore we considered the lowest localized added-electron states with Ni 3d9
character, the states 12 A1 and 12 B1 in the [Ni6 O6 ] cluster. Analogous to the Ni
and O hole states, we computed all unique interactions between such added electron
states, localized at nearest and next-nearest Ni lattice sites. To do so, we made use
of the same two-center Ni-Ni clusters as for the Ni hole states. We found the effective
matrix elements associated with the hopping of the added-electron for these lowest
localized states to be of about the same magnitude as those for the localized Ni hole
states of 12 A1 and 12 B1 symmetries. These effective hopping matrix elements lead
to an estimate of the band stabilization energy of about 0.5 eV which is the same as
the band stabilization energy for the Ni hole states of symmetry 12 A1 . We did not
derive explicitly the bands originating from the added-electron states because they
are expected to be similar to the bands derived from the Ni hole states, 12 A1 .
Finally, we also considered the K -dependence of one of the lowest excited states
in NiO, denoted as 3 T1g in the octahedral Ni2+ site symmetry, which determines the
10 Dq of NiO. We found the effective matrix elements, associated with the hopping
of the exciton between two nearest or next-nearest neighbouring Ni lattice sites to
be less than 2 meV. Thus, we concluded that indeed those excited states are very
localized and their treatment within the embedded cluster approach is fully justified.
Chapter 7, Delocalization of excited, hole and added-electron states in
236
NiO
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A
Matrix elements in the R- space for
LaMnO3
Matrix elements in the R- space for LaMnO3
242
Localized state
Φa1
Φb2
Φc3
Φd4
Φa1
1.00000
-4.33898 10−3
0.337731 10−3
0.097190 10−3
Φa1
-6400.90394497
27.78318733
-2.163746264
-0.622121337
1.00000
0.0952825 10−3
0.778773 10−3
Φb2
-6400.90551468
-0.609916221
-4.987168396
Φb2
1.00000
-4.34108 10−3
Φc3
-6400.90336388
27.79666089
Φc3
1.0000
Φd4
-6400.90577637
Φd4
Table A.1: Hamiltonian matrix elements and overlap integrals between CASCI wavefunctions representing localized doped electron-hole
states within the reference unit cell of LaMnO3 ; The CASCI wavefunctions are derived from calculations on four-center clusters.
Localized state
Φa1
Φb2
Φc3
Φd4
Localized
state
Φ1a
Φ2b
Φ3c
Φ4d
Φ2b
-6400.90551468
-0.021955267 10−3
-2.318739963 10−3
Φ1a
-6400.90394497
9.7899110−3
-1.965127377 10−3
-0.017299094 10−3
-6400.90336388
9.82226 10−3
Φ3c
-6400.90577637
Φ4d
Table A.2: Effective hopping matrix elements between CASCI wavefunctions representing localized doped electron-hole states within the
reference unit cell of LaMnO3 ; The CASCI wavefunctions are derived from calculations on four-center clusters.
243
244
Matrix elements in the R- space for LaMnO3
A
246
List Of Abbreviations
List Of Abbreviations
AIEMP
AIMP
AFM
ANO
CASCI
CASPT2
CASSCF
CASSCF-d
CASSCF-dd
CASSCFpdd
GFT
CI
CSF
CT
DE
DDCI
DFT
DRF
DMFT
FM
FOCI
GGA
GWA
HFA
KS DFT
LDA
LSDA
MBPT
MCSCF
ME
MP2
MRCI
OF
OF1
OF2
OF3
Ab Initio Embedding Model Potential
Ab Initio Model Potential
Antiferromagnetic
Atomic Natural Orbital
Complete Active Space Configuration Interaction
Complete Active Space Second-Order Perturbation Theory
Complete Active Space Self-Consistent Field
CASSCF with five active orbitals of mainly TM -3d character
CASSCF with ten active orbitals; 5 TM -3d orbitals + 5
correlating orbitals of the same symmetry character; the
TM-3d’ orbitals
CASSCF with an active space of the TM -3d and TM-3d’
orbitals, extended with some orbitals with mainly O -2p
character
Group-Function Theory
Configuration Interaction
Configuration State Function
Charge Transfer
Double Exchange
Difference Dedicated Configuration Interaction
Density Functional Theory
Direct Reaction Field
Dynamical Mean-Field Theory
Ferromagnetic
First-Order Configuration Interaction
Generalized Gradient Approximation
GW approximation
Hartree-Fock Approximation
Kohn-Sham DFT
Local Density Approximation
Local Spin Density Approximation
Many-body Perturbation Theory
Multi-Configurational Self-Consistent Field
Many-Electron
Second-Order Møller-Plesset Perturbation Theory
Multi-Reference Configuration Interaction
Overlapping Fragment
Overlapping Fragment Scheme 1
Overlapping Fragment Scheme 2
Overlapping Fragment Scheme 3
247
RASSCF
RAS1, 2, 3
RASSI
RHF
SCF
SD
TDDFT
TES
TIP
TM
UHF
VB
Restricted Active Space Self-Consistent Field
Restricted Active Space 1, 2, 3
Restricted Active Space State Interaction
Restricted Hartree-Fock
Self-Consistent Field
Slater Determinant
Time-Dependent DFT
Theory of Electron Separability
Total Ion Potential
Transition Metal
Unrestricted Hartree-Fock
Valence band
Samenvatting
Ongeveer 15 miljard jaren geleden, onstonden de materie, de ruimte en de tijd in wat
momenteel wordt geloofd een explosie te zijn geweest die voor de mens onbegrijpelijk
is. Tegenwoordig ontwikkelen de mensen theorieën en experimenten om de puzzel van
de creatie van het universum op te lossen. Nieuwe ontdekkingen zijn voortgekomen uit
het onderzoek naar de compositie en de eigenschappen van de materie en de krachten
die samen de ingewikkelde structuur van de wereld opbouwen. Mensen ontdekten dat
de materie uit atomen bestaat, die zelf uit elementaire deeltjes bestaan, nl. elektronen,
protonen en neutronen. De drie toestanden van materie, vaste stof, vloeistof en
gas zijn intensief bestudeerd door fysici en chemici wegens hun eigenschappen, zoals
faseovergangen die afhankelijk zijn van de temperatuur, samendrukbaarheid, korteen lange-afstands structurele regelmatigheid, thermische eigenschappen, geleidings
eigenschappen, enz. Deze eigenschappen zijn theoretisch beschreven door de klassieke
en kwantum fysica en chemie.
In dit proefschrift worden de eigenschappen van vaste stoffen met een langeafstands structurele regelmatigheid, die als kristallijne materialen bekend staan, bestudeerd
door middel van kwantumchemische methodes. In de kwantumchemie worden de
elektronische structuur en daarmee de eigenschappen van atomen, moleculen en vaste
stoffen verkregen door de Schrödinger vergelijking op te lossen, veelal binnen de BornOppenheimer benadering. In deze benadering worden de bewegingen van kernen en
elektronen losgekoppeld en kan de Schrödingervergelijking voor de elektronen worden
opgelost bij elke kernenconfiguratie. De oplossingen van deze Schrödingervergelijking,
de golffuncties, beschrijven het veelelektronensysteem. Echter, de Schrödinger vergelijking kan slechts voor eenvoudige éénelektron systemen exact worden opgelost. Dit
is zo omdat in het geval van veelelektronensystemen, de golffunctie de ingewikkelde
gecorreleerde beweging van alle elektronen beschrijft. Dus het is slechts mogelijk
benaderde oplossingen van de Schrödingervergelijking te verkrijgen.
De meest gebruikte aanpak om de oplossingen van de Schrödingervergelijking te
benaderen is om de veelelektronengolffunctie als lineaire combinatie van Slaterdeterminanten te schrijven: genormeerde en geantisymmetriseerde spinorbitaal producten
uitgedrukt in een orthonormale spinorbitaal basisset. In de limiet van een volledige
orbitaal basis en oneindig veel determinanten kan de resulterende golffunctie de exacte oplossing goed benaderen. Het gebruik van een volledige basis en een oneindige
set Slaterdeterminanten om de golffunctie te construeren is in de praktijk onmogelijk.
Dus zijn verschillende methodes ontwikkeld om de golffunctie binnen een eindige basis
en een beperkte lineaire combinatie van Slaterdeterminanten te produceren. De eenvoudigste onder hen is de Hartree-Fock methode waarin de golffunctie gewoonlijk door
één Slaterdeterminant (en indien dit om symmetrieredenen nodig is, door enkele Slaterdeterminanten) wordt benaderd, die één bijzondere elektronenconfiguratie beschrijft. De HF golffuncties en bijbehorende energieën worden verkregen door het variatieprincipe toe te passen: de beste golffunctie is die waarvoor de verwachtingswaarde
van de Hamiltoniaan van het veelelektronensysteem minimaal is. Dit leidt tot een set
van de Hartree-Fock vergelijkingen die in een zelfconsistente procedure moeten worden opgelost. Binnen de Hartree-Fock benadering zijn de zogenaamde exhange en
Coulomb elektroneninteractie opgenomen maar wordt de zogenaamde elektronencorrelatie niet in rekening gebracht. De energie van de elektronencorrelatie wordt dan
ook gedefinieerd als het energieverschil tussen de exacte niet-relativistische energie en
de Hartree-Fock energie. Wanneer de golffunctie van een bepaalde elektronentoestand
van het veelelektronensysteem meer dan één belangrijke elektronische configuratie bevat, kan de éénconfiguratie Hartree-Fock methode het systeem niet beschrijven. De
zogenaamde multiconfiguratie methode is het meest geschikt om deze systemen behoorlijk te beschrijven. De multiconfiguratie golffunctie wordt geëxpandeerd in spinen symmetrieaangepaste lineaire combinaties van Slaterdeterminanten.
Een representatieve klasse van kristallijne vaste stoffen waarvoor de grondtoestand en de aangeslagen elektronentoestanden door multiconfiguratiegolffuncties worden beschreven wordt gevormd door ionische overgangsmetaal verbindingen. Deze
vaste materialen bevatten één overgangsmetaal (Fe, Ni enz.) en minstens één ander
element, dat gewoonlijk chalcogeen (O, S, enz.) of halogeen (F, Cl, enz.) is.
Voor deze systemen moeten de veeldeeltjeseffecten in de elektronenbeschrijving
worden opgenomen. In principe, kan een multiconfiguratie golffunctie die wordt geconstrueerd uit relevante elektronenconfiguraties de sterke elektronencorrelatie beschrijven. Het construeren van een dergelijke golffunctie is echter moeilijk uitvoerbaar: het
impliceert de opnemen van een oneindig aantal elektronenconfiguraties in de expansie
van de golffunctie.
De elektronenstructuur structuur van de kristallijne materialen wordt het vaakst
beschreven binnen de bandtheorie. Net als de Hartree-Fock benadering impliceert
deze theorie het gebruik van een benaderde éénelektron Hamiltoniaan, met dezelfde
symmetrie-eigenschappen als de veel-elektron Hamiltoniaan. De eigenfuncties van
zón benaderde éénelektron kristal Hamiltoniaan zijn ook eigenfuncties van de translatieoperatoren voor het kristal. De praktische implementaties van de bandtheorie
worden gewoonlijk uitgevoerd binnen de dichtheidsfunctionaaltheorie (DFT) of HF
theorie. In DFT is de fundamentele grootheid de elektronendichtheid in plaats van
de golffunctie en alle fysieke observabelen worden uitgedrukt als functionalen van de
elektronendichtheid van de grondtoestand. Een aantrekkelijke eigenschap van DFT
is dat er geen behoefte is om de ingewikkelde golffunctie van het systeem te construeren. Maar toch kunnen de veeldeeltjes effecten door de exchange-correlatie potentiaal worden opgenomen. Hoewel DFT een exacte theorie voor de eigenschappen
van de grondtoestand van een elektronensysteem is, moeten exchange-correlatie functionalen die in de vergelijkingen worden geintroduceerd worden benaderd. Het DFT
formalisme wordt veel gebruikt in studies van elektroneneigenschappen van kristallijne
materie. In het geval van een sterk gecorreleerd systeem zijn verschillende correcties
aan de bestaande functionalen voorgesteld, zoals de Hubbard-achtige Coulomb on-site
parameters voor de elektroneninteractie. De effecten van de elektronencorrelatie kunnen ook geı̈ntroduceerd worden binnen de Greense functietheorie waar zij door een
niet-lokale en energieafhankelijke zelfenergieoperator in rekening gebracht worden.
Een gedetailleerd inzicht in de verschillende bijdragen tot de elektronencorrelatie
kan worden verkregen door het gebruik van een benadering, die gebaseerd is op een
multiconfiguratiegolffunctie. Binnen een dergelijke benadering kunnen de correlatieeffecten in de elektronenbeschrijving op een systematische en gecontroleerde manier
worden opgenomen door de golffunctie-expansie en de orbitaal basis aan te passen.
Aangezien de multiconfiguratie methode niet op eenvoudige wijze toegepast kan worden op een oneindig groot kristallijn systeem, kan een andere benadering voor de
modellering van het materiaal worden gebruikt dat een ongecompliceerde combinatie
met de multiconfiguratie methode wel toestaat. Dit is de ingebedde clusterbenadering.
Binnen deze methode wordt een klein maar relevant gedeelte van het kristal
genomen en de electronen structuur van dit kristalgedeelte, het cluster, wordt beschreven
op een zeer gedetailleerde manier. Het cluster wordt in een effectieve potentiaal ingebed om het effect van de rest van het kristal te vertegenwoordigen. Deze potentiaal
wordt ontworpen om de elektrostatische interactie te beschrijven maar de kwantummechanische effecten kunnen ook worden meegenomen.
Het clustermodel is een aangewezen uitgangspunt in de beschrijving van, bijvoorbeeld, overgangsmetaal d-d exitaties en binnenschil ionisaties, die gewoonlijk een behoorlijk gelokaliseerd karakter hebben. In het geval van geı̈oniseerde of toegevoegd
elektron toestanden zijn de zogenaamde elektronische relaxatieeffecten aanzienlijk.
De elektronische relaxatieeffecten worden gedefinieerd als de veranderingen die in de
distributie van de elektronen optreden, na het verwijderen van een elektron uit de
golffunctie van de grondtoestand, of na het toevoegen van een elektron.
Het clustermodel vergemakkelijkt de beschrijving van die elektronische relaxatieeffecten, omdat het gat of het toegevoegde elektron gelokaliseerd zijn. Daarnaast kunnen de effecten van de elektronencorrelatie op een ongecompliceerde manier worden
meegenomen met een golffunctie, die relevante elektron configuraties bevat. Ondanks
de voordelen van het ingebedde clustermodel in de behandeling van elektronische toestanden met hoofdzakelijk gelokaliseerd karakter, negeert dit model de translatiesymmetrie van het kristal. De juiste translatiesymmetrie-aangepaste golffuncties voor het
uitgebreide systeem moeten achteraf worden geconstrueerd.
In het huidige onderzoek worden dergelijke translatie symmetrie-aangepaste golffuncties geconstrueerd voor aangeslagen, geionı̈seerde of toegevoegde elektron toestanden in overgangsmetaal verbindingen. Wij hebben de voordelen van de clusterbenadering gebruikt om de elektronische relaxatie-en elektronencorrelatie- effecten mee
te nemen. Bovendien hebben wij de bandeffecten in rekening gebracht die genegeerd
worden in de conventionele ingebedde clusterstudies.
In Hoofdstuk 1 wordt een overzicht gegeven van de conventionele bandbenadering
omwille van de vergelijking met de zogenaamde veeldeeltjesbanden, die binnen onze
nieuwe theoretische benadering worden verkregen. Deze inleiding wordt gevolgd door
een korte bespreking van verschillende éénelektron benaderingen (HF en DFT) en het
Hubbard model en hun relevantie voor de studie van sterk gecorreleerde kristallijne
systemen.
Hoofdstuk 2 behandelt het concept van de clusterbenadering, gevolgd door een
bespreking van inbeddingstechnieken om het omringende kristal door een effectieve
potentiaal te vertegenwoordigen. Verder worden verschillende methodes besproken om
de N-elektronclustergolffunctie te benaderen. Daarna wordt de methode beschreven,
die veelvuldig in het huidige proefschrift wordt gebruikt om de matrixelementen van de
Hamiltoniaan en de overlapintegralen tussen onderling niet-orthogonale golffuncties
van een groot ingebed cluster te berekenen.
Hoofdstuk 3 beschrijft een efficiënte methode voor de generatie van gedelocalizeerde en gecorreleerde veelelektronengolffuncties voor aangeslagen, geioniseerde of
toegevoegd-elektron toestanden in uitgebreide systemen. Wij maken gebruik van een
niet-orthogonale veelelektron tight-binding aanpak om gedelocalizeerde golffuncties
te verkrijgen. Gedelocalizeerde golffuncties van het kristal worden geconstrueerd als
de lineaire combinaties van geantisymmetriseerde producten van de gelokaliseerde
CASCI (complete active space configuration interaction) clustergolffuncties en de
golffuncties voor de rest van het materiaal. De clusters worden opgebouwd uit twee
of meer kleinere clusters, ”fragmenten”. De fragmenten zijn gecentreerd rond een
bepaald ion en worden ingebed in een effectieve modelpotentiaal. Zij zijn groot genoeg gekozen om de gelokaliseerde aangeslagen, geioniseerde en toegevoegd-elektron
toestanden redelijk te beschrijven. De Hamiltoniaan en overlap matrixelementen
tussen dergelijke gelokaliseerde toestanden op twee verschillende roosterplaatsen in
het kristal worden benaderd door de matrixelementen van ingebedde clusters. De constructie van de CASCI golffuncties wordt uitgevoerd gebruik makend van onze nieuwe
overlappende fragmenten aanpak. Hierin wordt het cluster opgebouwd gedacht te zijn
uit twee of meer overlappende fragmenten. Aldus wordt de orbitaal basis waarin de
CASCI golffuncties worden uitgedrukt verkregen uit het combineren van de orbitaal
bases van CASSCF golffuncties voor de fragmenten. Verschillende manieren om dit
te doen worden vergeleken.
In hoofdstuk 4 het formalisme toegepast op de studie van de zogenaamde double exchange in manganaten, LaMnO3 CaMnO3 and La0.75 Ca0.25 MnO3 . De double
exchange wordt beschouwd als een van de belangrijke mechanismen die de transporteigenschappen in manganaten bepalen. De clusters die hiervoor worden gebruikt
zijn [Mn2 O11 ] en [Mn4 O20 ]. De orbitaal bases voor de CASCI golffuncties van deze
clusters worden verkregen uit de [MnO6 ] fragmenten.
In gat-gedoopt LaMnO3 blijken de double exchange parameters groter te zijn in
het ab vlak dan langs de c-as (kristal ruimtegroep Pbnm). Dit is zo omdat de exchange
interacties in het ab vlak ferromagnetisch zijn en langs de c-as antiferromagnetisch.
De double exchange parameters voor elektron-gedoopt CaMnO3 in het ab vlak zijn
kleiner dan die voor gat-gedoopt LaMnO3 . Dit komt omdat de exchange interacties
in alle kristalrichtingen in CaMnO3 antiferromagnetisch zijn. In La0.75 Ca0.25 MnO3
is deze anisotropie minder uitgesproken dan in LaMnO3 en CaMnO3 en de double
exchange parameters in het ab vlak en langs de c-as zijn van de zelfde omvang.
Zij zijn groter in alle kristalrichtingen dan voor de gedoopt LaMnO3 en CaMnO3 ,
overeenkomstig met de ferromagnetische interactie. Dit resultaat is in overeenstemming met het eenvoudige model van Zener waarin de geleiding en het magnetisme
van het gedoopte materiaal met elkaar verbonden zijn. Voorts blijkt dat in alle drie
verbindingen de spinafhankelijkheid van de double exchange parameters in bijna perfecte overeenkomst is met die voorspeld door een eenvoudige model van Anderson en
Hasegawa. De parameters tussen niet-naaste buren zijn een orde kleiner dan die voor
de naaste buren. Wij vinden dat de energie stabilisatie, verbonden aan delokalisatie
van het toegevoegde elektron in CaMnO3 , groot genoeg is om de antiferromagnetisch
interactie te overwinnen en een zogenaamd magnetische polaron te veroorzaken, d.w.z.
een lokaal ferromagnetisch cluster waarin de energie verlaagd wordt door delokalisatie
van het toegevoegde elektron over het cluster.
In het eerste deel van Hoofdstuk 5, worden de prestaties van een paar verschillende
overlappende fragment aanpakken vergeleken, voor het construeren van gelokaliseerde
orbitalen voor de CASCI golffuncties van de [Mn2 O11 ] clusters. Wij vinden dat slechts
één van de aanpakken een balanceerde beschrijving levert van twee gelokaliseerde
toestanden. Daarna wordt de kennis van de gelokaliseerde gat en elektron toestanden in manganaten van Hoofdstuk 4 gebruikt om de energiebanden te berekenen verbonden aan die toestanden. Wij doen dit door een ”non-orthogonal tightbinding”benadering te gebruiken waarin de Hamiltoniaan en overlap matrixelementen
tussen de gelokaliseerde toestanden verkregen worden uit ab initio berekeningen aan
[Mn4 O20 ] clusters. De banden nemen de elektronische relaxatie en elektronencorrelatie effekten mee die de ionisatie of de elektronadditie begeleiden. Wij duiden
deze banden aan als veeldeeltjes gat en elektron banden. Ze worden vergeleken met
conventionele één-elektron banden die uit DFT en HF bandberekeningen komen.
Wij vinden voor LaMnO3 een bandbreedte van ongeveer 2.4 eV voor de laagste
Mn 3d geioniseerde toestand. De band is nogmal symmetrisch wegens de zeer kleine
overlapintegralen tussen de gelokaliseerde Mn 3d gat-toestanden. In CaMnO3 dragen
twee gelokaliseerde toegevoegd-elektron toestanden per Mn bij aan de delocalisatie van
het toegevoegde elektron. Dit leidt tot een bandbreedte van 2.3 eV. Ook in CaMnO3
zijn de banden nogal symmetrisch. In CaMnO3 blijkt nochtans dat de vorming van
een ferromagnetisch polaron door de grotere stabilisatieenergie van ongeveer 2.1 eV
energetisch voordeliger is dan delokalisatie van het elektron over het hele kristal. In
LaMnO3 levert geometrische relaxatie van de omringende ionen een energiewinst van
de zelfde grootte-orde als die van delokalisatie van het gat over het hele kristal. Deze
bevinding leidt tot de conclusie dat er concurrentie is tussen de localisatie van het
gat rond een bepaalde Mn ion, en delocalisatie van het gat.
In Hoofdstuk 6 worden Mn kristalveld excitaties en ladingsoverdracht excitaties in
LaMnO3 bestudeerd. Een motivatie voor deze studie is het huidige debat over sommige grootheden zoals de Mn 3d Jahn-Teller (JT) parameter, exchange parameter,
de kristalveld parameter. Wij onderzoeken het effect van de JT vervorming en verandering van de hoek tussen de MnO6 octaëders op de laagste Mn d-d excitaties. De JT
parameter is slechts licht afhankelijk van deze hoek. De Mn 3d exchange parameter
wordt slechts matig beinvloed door vervorming van de octaëders. De belangrijkste
eigenschappen van het Mn 3d4 kristalveld spectrum worden al goed beschreven als
slechts de JT vervorming in rekening wordt gebracht. Daarnaast werd de convergentie
van het berekende energie spectrum met de clustergrootte gecontroleerd. Wij vinden
snelle convergentie en besluiten dat de Mn d-d exitaties inderdaad gelokaliseerd zijn,
en dus is de clusterbenadering gerechtvaardigd voor hun beschrijving.
De laagste ladingsverdracht toestanden werden ook bestudeerd. De bijbehorende
energieen tonen grotere afhankelijkheid van de clustergrootte dan de Mn d-d excitaties
omdat zij meer gedelokaliseerd zijn. De ladingsoverdracht is excitaties komen allen
voor in het hogere energiedeel van het valentiespectrum, terwijl het lagere deel van
het energiespectrum door de Mn 3d-3d excitaties wordt gevormd.
Tot slot wordt in Hoofdstuk 7 delocalisatie van de laagste aangeslagen, gat en
toegevoegd-elektron toestanden in NiO onderzocht door middel van de nieuwe theoretische benadering. De hopping matrixelementen van laagste aangeslagen toestand
zijn bijna verwaardloosbaar klein. Daarom levert een band beschrijving geen nieuwe
inzichten. De cluster benadering is voor deze en vergelijkbare excitaties een goede
aanpak. Daarna werden Ni 3d en O 2p gat toestanden in NiO onderzocht. Deze toestanden zijn van belang voor het bepalen van het karakter van de top van valentieband
in NiO, een controversieel onderwerp. Gelokaliseerde Ni 3d en O 2p gattoestanden
werden bestuderd in een [Ni6 O6 ] cluster. In dit cluster worden de twee soorten gat
toestanden behandeld op een evenwichtige manier, wat voor het verkrijgen van het
correcte energieverschil tussen hen belangrijk is. De Hamilton en overlap matrixelementen tussen gelokaliseerde gat toestanden van het zelfde type worden vervolgens
berekend, gebruikmakend van verschillende clustermodellen. Uit berekening van de
veel-elektronbanden blijkt dat de laagste gat toestanden O gat toestanden zijn met
wat Ni karakter bijgemengd. Dit betekent dat de top van de valentieband in NiO
hoofdzakelijk O karakter heeft. De bodem van de geleidingsband heeft Ni 3d 9 karakter en daarom, kan NiO het best als ladingsoverdracht isolator worden geclassificeerd.
Daarnaast word delocalisatie van de laagste toegevoegd-elektron toestand in NiO onderzocht. Dehopping matrixelementen zijn ongeveer even groot als die voor de laagste
gelokaliseerde Ni 3d gat toestanden.
Acknowledgments
After almost a ’heroic’ effort this book could come into existence and hereafter i would
like to thank everybody who made this possible.
First and foremost, I would like to express my gratitude to Ria who granted me the
project and scientific support and made it possible for me in so many different ways
to complete it. Ria, thank you first for the willingness to reach the final goal once
the start was made and second, for your patience and the time you spent to teach
me science. Thank you that you always found some time for me even when your
agenda was overbooked. Last but not at least i am very grateful for your criticism
on scientific questions which has been necessary and i could benefit a lot of it. The
scientific mind must be kept always (self-) aware of what one does not know so that
one learns it.
I would like to thank Jaap for supporting this project in its early stages.
To Coen de Graaf goes my gratitude for helping me these almost 5 years with many
valuable scientific advices, explanations and suggestions which always brought me to
a successful solution. Coen thank you for sharing with me your scientific expertise
and for your valuable suggestions which improved a lot the content of this thesis.
I would like to thank you and Carmen for contributing to the study on the double
exchange and the valence excitations in manganites.
I am very grateful to you, Paul for all your help and concern, for not being tired
to answer always my questions which might have seemed sometimes very naive and
may be not that bright in scientific and other aspects. Thank you for all the time you
stole from your busy schedule to help me understand different physics concepts which
were not a subject of my research. I greatly acknowledge your help for clarifying for
me some formulations in the work on the many-body bands.
I would like to thank you Robert, for answering all my questions on DFT and
DFT-functionals clearly and completely. Thank you for your valuable help for my
last scientific presentation in Luntereen. I keep very nice memories from our work
together on the course in Quantum Mechanics. I feel to say that i myself learnt a lot
from this course although i was supposed to teach the others.
I feel very grateful to Prof. Coen de Graaf, Prof. Michael Filatov and Prof. Thom
Palstra, my kind reading committee, for the very useful corrections and suggestions,
for helping me to improve the content of the thesis, and last but not at least for the
unbelievable speed with which they read my thesis.
I would like to thank all of you dear colleagues for being always nice and friendly
with me, for all our discussions, scientific or not that scientific, for the nice time spent
together at conferences, schools, scientific meetings and not that scientific meetings.
To all of you goes my THANKS.
Pina, I think you are one of the best and most tolerant listeners i have ever met.
I would not dare to thank you here, the space is not enough.
Adrjan I am very glad i met you. Besides the great sense of humor which reaches
in an asymptotic manner some decent limits, I learned from you, among many other
things, one of the most morbid and truest axiom: People never change!
Arjan i am very grateful that instead of going home on Saturday evening you had
a fight with my erroneous Samenvatting. I appreciate.
I would not dare to forget my colleagues who welcomed me in the group when i
arrived almost 5 years ago, Liviu, Lasse, Thomas, Rossana, Marcel, Meta.
Liviu thank you for being so helpful with my first steps in the Dutch scientific
society. Thank you for all your very useful scientific and friendly advices. It was nice
and still it is to share with you my ideas and thoughts.
Johan without you I would have been lost with the computer systems. Thank you
for negotiating with my computer in the cases of my computer-human crisis.
My bulgarian friends have always had a special place for me. Nonna, thank you
first for being still my friend, for being always so kind and patient and supportive.
Now that this thesis is done i will not have more excuses that i am too busy. And here
finishes my english essay which is by no means exhaustive. A lots of other people have
contributed in different ways to this thesis and to all of them I express my gratitude.
Alexandrina Stoyanova,
Groningen, October 2006
Spécialement pour toi,
Il y a peu je t’avais dit que la personne qui passerait le reste de sa vie
à tes côtés serait tout simplement la femme la plus chanceuse de cette
planète. Me voilà aujourd’hui à cette place. Qu’est ce que je pourrais
espérer de plus ? Ta patience, ton attention à mon égard et ton amour
m’ont amenée là où je suis. Mon souhait est qu’un jour tu puisses me
dire que toi aussi, tu es au moins la seconde personne la plus chanceuse
de ce monde.