Download Electronic structure of diatomic molecules

The 17th CMD workshop
Advanced study woth ES-Opt
ES-Opt”
“Advanced
Multi-reference
Multi
reference DFT
Koichi Kusakabe
Grad. Sch. Eng. Sci, Osaka University
Electronic structure of diatomic
molecules
Anti-bonding
Bonding
Non-bonding
a
Molecular orbital picture
Heitler-London picture
Known results in DFT-LDA
• Molecular orbital picture is given for any inter-atomic
inter atomic distance.
• The stable electronic state in equilibrium is well-reproduced.
Molecular orbital given by DFTLDA
0
1s bonding
2
1s ant-bonding
3 3'
2
1
1
0
3
Dependence on inter-atomic
distance
3 3'
2
1
0
& 1s,2
may be constructed.
1s,1
Non-bonding
Hitler-London picture
a
Is there any concise method to describe this dependence?
Correlation effects
„ In DFT-LDA, the Kohn-Sham orbital is
determined with correlation effects included in
the effective potential.
„ The correlation effect can behave differently
y
depending on atomic configuration.
„ A key is,
e2
1
Exc [n] = ∫ drn(r )∫ dr'
nxc (r, r')
2
r − r'
1
nxc (r, r') = n(r')∫ [gn (r, r' , λ ) − δ (r − r')]dλ
Charge fluctuation!
0
= ∫ [ Ψλ : (nˆ(r ) − n(r ))(nˆ(r') − n(r')) : Ψλ / n(r )]dλ.
1
0
H2
Anti-bonding
Bonding
Non-bonding
a
Thi iis a weak
This
k correlation
l ti li
limit!
it!
Thi is
This
i a strong
t
correlation
l ti limit!
li it!
O2
This picture is not true!
2s2 2p4
a
S=1 ground state!
This is a strong correlation limit!
S=1 ground state!
This is a strong correlation limit!
How the multi-reference density
functional theory acts?
„ The Kohn-Sham single-particle description
may be used, but may not always be reliable.
„ So, if a simplified description allowing a multiSlater determinant is introduced, we may
y use
the method as another starting point.
„ A simplest case is a DFT-LDA
DFT LDA solution for a
degenerate ground state.
S=1, Sz=1
S=0, Sz=0
Determination
D
t
i ti off GS may be
b
possible by looking at
• charge-charge correlation
or
• variational energy.
Two physical quantities specifying
correlation effects
„ Density-density correlation : FRM
(
n i = ni↑ + ni↓ − ni↑ − ni↓
2
)
2
„ Variational energy : DFVT
Fluctuation reference method
„ To have a simplified description based on
DFT, we need information given by another
accurate electronic structure calculation.
„
„
Diffusion Monte-Carlo method for electron gas
⇒ LDA, GGA
Complete-Active-Space ConfigurationInteraction method ⇒ FRM to determine U.
FRM : Fluctuation reference method
Two Hydrogen systems
H2 Molecule
Hydrogen array
Local fluctuation on φ1s of H2
(
n i = ni↑ + ni↓ − ni↑ − ni↓
2
)
2
as a function of the inter atomic distance
0.7Å
n bonding = 2 .0
n bonding = 0
2
n1 s = 1 .0
n
2
1s
= 0 .5
Small U regime
4.0Å
n bonding = 1 .0
n bonding = 1.0
2
n1 s = 1 .0
n 1s = 0
2
Large U regime
We are able to find relevant orbitals with Coulomb suppression by
1 Finding
1.
Fi di an orbital
bit l ( or a sett off orbitals
bit l ) on which
hi h fl
fluctuation
t ti becomes
b
1.
1
2. Using a unitary transformation to have localized orbitals.
Orbital fluctuation in CI calculation
CAS-CI calculation for H2 by S. Yamanaka.
Small U/t
U ～ 0.7eV
Molecular orbital limit
2
ni
Heitler-London limit
Large U/t
U ～ 10eV
In the whole range, mean occupation of 1s is 1.
A test calculation of Hydrogen
systems
„ Exc[n] is given by LDA of PW91.
„ Plane-wave expansion with the Troullier-Martins soft-
pseudopotential
d t ti l is
i introduced
i t d d for
f H.
H
„ The energy cutoff for the plane-wave is 40Ry.
„ The
Th solver
l
off single-particle
i l
i l part off the
h extended
d d KohnK h
Sham equation is given by the “opt” code.
„ Effective
Eff ti many-body
b d problem
bl is
i solved
l d by
b the
th exactt
diagonalization with e.g. the Lanczos method.
The value of U for H2 determined by
FRM
R f
Reference
calculation:
l l ti
CASCI method
th d ffor H2
R 2 0Å
R=2.0Å
R=3 0Å
R=3.0Å
R=1.2Å
The 1st method to have U by the fluctuation reference method.
K. Kusakabe, et al. J. Phys.: Condens. Matter (2007).
Charge density of H2 given by MRWeak correlation regime
DFT
R=0.6Å
U is not
needed.
This section is
Shown in the
right panels.
Strong correlation regime
To reproduce bonding charge in
correlated electron regime, we need to
overcome difficulty in the single-reference
description. It is achieved by using the
multi-reference
multi
reference description as known in
the Hybrid-DFT.
(Cf. S. Yamanaka, et al.)
R=2.0Å
U is need.
Variational method
DFVT for MR-DFT
DFT model formation
PBE0+GW, PBE0+U will be refined.
-
It is possible to evaluate ΔE[Ψ].
Note: If DFVT is used in a space of single Slater
determinant the Hartree
determinant,
Hartree-Fock
Fock approx.
approx is given
given.
This is natural, but not so trivial.
Thanks to Dr. Maruyama & Dr. Friedlich.
To understand MR
MR-DFT
DFT
This functional contains
the universal energyd
density
it functional
f
ti
l F[n]
F[ ]
This functional should be
evaluated
l t d using
i
a multilti
Slater description.
Heisenberg exchange:
Introduction to the localized spin model
„ Heisenberg
H i
b
and
d Di
Dirac considered
id d quantum
t
mechanical origin of magnetic interaction
↑
↓
-J Si ・Sj
J>0 : Parallel spin configuration
Ferromagnetic
J<0
J
0 : Anti-parallel
t pa a e spin
sp co
configuration
gu at o
Antiferromagnetic
whose order of magnitude can be of the order
of the Coulomb interaction.
Where does J come from?
Idea of restriction of the pphase space
p
„ For low-energy states, we may consider only
states in a restricted phase space
space.
The 1s orbital of
the Hydrogen atom
Let’s consider the phase space spanned
by spin states |1 s , , |1 s , ↑ .
↑
Atomic orbitals and molecular orbitals
„ Atomic orbitals
r1
φa (r )ξ↑ (1)
†
ca↑ 0
a
„ Molecular orbitals (in LCAO)
r1
r2
a
b
1
(φa (r ) + φb (r ) )ξ↑ (1)
2
1 †
(ca↑ + cb†↑ ) 0
2
The Heitler
Heitler-London
London theory of H2
A picture in the strong correlation limit!!
r1
r2
r2
r1
a
b
a
b
Note that there is no doubly occupied 1s orbital in this theory.
theory
(3)
(4)
Energy of spin states
Evaluation
va ua o of
o J: Direct
ec exchange
e c a ge
Ferromagnetic direct exchange I.
I
*
*
*
*
Ferromagnetic direct exchange II.
„ Exchange integral of two orthogonal orbitals with S=0
becomes ferromagnetic.
A proof: Consider the exchange integral when S=0.
I = ∫ dr1dr2
e2
φa * (r1 )φb (r1 )φb * (r2 )φa (r2 )
r1 − r2
Note that the expression below.
(
)
(
)
φa * (r1 )φb (r1 ) = Re φa * (r1 )φb (r1 ) + i Im φa * (r1 )φb (r1 )
I = ∫ dr1dr2
(
2
)
*
e
φa * (r1 )φb (r1 ) φa * (r2 )φb (r2 )
r1 − r2
(
)
(
)
(
)
(
)
e2
*
*
= ∫ dr1dr2
Re φa (r1 )φb (r1 ) × Re φa (r2 )φb (r2 )
r1 − r2
e2
*
*
+ ∫ dr1dr2
Im φa (r1 )φb (r1 ) × Im φa (r2 )φb (r2 )
r1 − r2
To show the positivity of I, it is enough to consider an integral of real functions,
e2
I = ∫ dr1dr2
Φ (r1 )Φ (r2 )
r1 − r2
Ferromagnetic direct exchange III
III.
Φ
Φ
-q).
Slater determinants for a system with two electrons I.
a
b
Slater determinants for a system with two electrons II.
Multi Slater determinants
Spin states in the second quantization
2-electrons in molecular orbitals
The Heitler-London wf.
Ionic states!
The Heitler-London state v.s. the molecular
orbitals
bit l
The true w.f. is close to a variational state with 0<α<1.
α is dependent on the inter-electron interaction or choice of
the model (interaction parameters etc.).
The Hubbard model : I.
1. Consider a system with atomic sites. Each site is assumed
to have an orbital for the conduction electrons
2 Electrons
2.
El t
can hop
h
between
b t
neighboring
i hb i
sites
it by
b a transfer
t
f
integral t.
3. Two electrons coming across at a site feels a repulsive
interaction U.
↑
↑ ↓
↓
↑
U
t
↓
The Hubbard model : II.
↑ ↓
↑
↓
The Hubbard model : III.
The Hubbard model : IV.
The Hubbard model : V.
The Hubbard model : VI.
2 site Hubbard model : II.
2-site
t
U
a
2-site Hubbard model : II.
b
2-site Hubbard model : III.
Crossover from weak
coupling regime to strong
coupling
li
regime
i
2-site Hubbard model : IV.
Note: GS is always
a singlet
i l state.
The Hubbard gap
Energy scale of the
AF kinetic exchange
Strong coupling regime
(HL)
｝Singlet states
Weak coupling regime (MO)
T i l t states
Triplet
t t
Concept
p of the Mott insulator
„ Electron transfer from an orbital to an occupied
orbital is prohibited by the Coulomb repulsion
repulsion.
This leads to an insulating state.
↓
↑
↑
↑
↑
e↓
↑
↓
↓
↑
↓
↑
Mott gap
or
Hubbard gap
U
J
The half-filled Hubbard model
shows
h
the
th Mott
M tt insulating
i
l ti
ground
d
state.
Formation of the Mott gap in
th single
the
i l particle
ti l excitation.
it ti
The kinetic exchange
„ Effective interaction at around U=∞
P
↑
↓
Q
↑
P
↑
H eff
↓
2t 2
=
U
t
↑
↓
U
↑ ↓
↓
↑
↓
1
(S i ⋅ S j − )
∑
4
i, j
We have another
process starting from
transfer of the up
spin moving to the
neighboring site.
Antiferromagnetic
Heisenberg model.
Superexchange
p
g interaction
„ In the transition-metal oxides, there is a structure of M-O-M
(M: transition metal, O: oxygen）. Along this structure, two
localized spins couples via the superexchane through nonmagnetic oxygen atom.
Often, Mott insulator is formed.
→ M has an open d-shell.
M
O
M
dx2-y2 @ M
px @ O
dx2-y2 @ M
JAF
The Kanamori-Goodenough
g rule
„ Consider description by non-orthogonalized atomic
orbitals (LCAO picture)
„ The electron transfer is possible from O to M when two
neiboring orbitals are not orthogonal.
„ The sign of exchange interaction between d and p is,
„ Ferromagnetic if two orbitals are orthogonal with each
other,
„ Antiferromagnetic if two orbitals are not orthogonal
„ Occupation of d-orbitals are determined by the crystal
field splitting and the Hund rule.
Following the above rule, we can determine the sign off
superexchange in a qualitative manner.
P.W. Anderson reformulated the rule using the orthogonalized
Wannier basis allowing the second quantization scheme.
Exanple of antiferromagnetic
exchange
Non-orthogonal:
Non
orthogonal: finite transfer
J: antiferromagnetic
pπ
pσ
(2,3)
Mn2+ : 6S5/2
d
5
Non-perturbative
States in excitation
JAF
JF
(1,4)
U
(1)
Mn2+ : 6S5/2
Non-perturbative
ground
d state
t t
d
5
M
O
M
Various types of exchange
interaction
„ Direct exchange : (Cf. Heitler-London theory)
„ Kinetic exchange : (Cf. the Hubbard model)
„ Super
S
exchange
h
„ Double exchange
„ RKKY interaction
„ Anisotropic exchange interaction
„ Dzyaloshinsky-Moriya interaction
1. Exchange interactions except for the 1st one
are effective interactions.
2. Effective interactions can be derived from a
model with local interactions.