Download Ab initio simulations of core level spectra Ida Josefsson

Ab initio simulations of core level spectra
Towards an atomistic understanding of the dye-sensitized solar cell
Ida Josefsson
c
Ida
Josefsson, Stockholm 2013
Distributor: Department of Physics, Stockholm University
List of Papers
This thesis is based on the following papers, referred to in the text by their Roman numerals.
PAPER I: Ab Initio Calculations of X-ray Spectra: Atomic Multiplet and Molecular
Orbital Effects in a Multiconfigurational SCF Approach to the L-Edge Spectra of Transition Metal Complexes
I. Josefsson, K. Kunnus, S. Schreck, A. Föhlisch, F. de Groot, Ph. Wernet, and
M. Odelius, The Journal of Physical Chemistry Letters 2012, 3 (23), 3565-3570.
DOI: 10.1021/jz301479j
PAPER II: Collective Hydrogen-Bond Dynamics Dictates the Electronic Structure of
Aqueous I−
3
I. Josefsson, S. K. Eriksson, N. Ottosson, G. Öhrwall, H. Siegbahn, A. Hagfeldt,
H. Rensmo, O. Björneholm, and M. Odelius, Submitted to PCCP.
PAPER III: Energy Levels of I− /I−
3 in Solution from Photoelectron Spectroscopy
S. K. Eriksson, I. Josefsson, N. Ottosson, G. Öhrwall, O. Björneholm, A. Hagfeldt,
M. Odelius, and H. Rensmo, In Manuscript.
Reprints were made with permission from the publishers.
Contents
List of Papers
iii
Abbreviations
vi
1
Introduction
7
2
Computational framework
2.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . .
2.2 Quantum chemical methods . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The independent particle model and self-consistent field theory .
2.2.2 Electron correlation . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Multiconfigurational theory – the active space . . . . . . . . . .
2.2.4 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Relativistic effects . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Solvation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Polarizable continuum . . . . . . . . . . . . . . . . . . . . . .
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4
Transition metal complexes
4.1 [Ni(H2 O)6 ]2+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
25
5
I− /I−
3 in water, ethanol, and acetonitrile solutions
5.1 Solvated I− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Solvated I−
3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3
Core-level spectroscopy
3.1 Photoionization . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Chemical shifts . . . . . . . . . . . . . . . . . . .
3.1.2 The sudden approximation and shake-up processes
3.2 X-ray absorption and emission . . . . . . . . . . . . . . .
3.3 Peak intensities . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Linewidths and shapes of x-ray spectra . . . . . .
3.3.2 Photoemission intensities . . . . . . . . . . . . . .
3.3.3 RIXS intensities . . . . . . . . . . . . . . . . . .
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Acknowledgements
xxxii
References
xxxiii
Abbreviations
AO
Atomic orbital
BO
Born-Oppenheimer
CASPT2
Second-order perturbation theory with CASSCF reference function
CASSCF
Complete active space self-consistent field
CI
Configuration interaction
DSC
Dye-sensitized solar cell
HF
Hartree-Fock
HOMO
Highest occupied molecular orbital
LMCT
Ligand-to-metal charge transfer
LUMO
Lowest unoccupied molecular orbital
MD
Molecular dynamics
MLCT
Metal-to-ligand charge transfer
MO
Molecular orbital
PCM
Polarizable continuum model
PES
Photoelectron spectroscopy
QC
Quantum chemistry
RASSCF
Restricted active space self-consistent field
RDF
Radial distribution function
RIXS
Resonant inelastic x-ray scattering
SCF
Self-consistent field
SIBES
Solvent-induced binding energy shift
UV/Vis
Ultraviolet-visible spectroscopy
XAS
X-ray absorption spectroscopy
XPS
X-ray photoelectron spectroscopy
1. Introduction
The world energy consumption is growing rapidly. Fossil fuels still provide 80 percent of the
energy consumed in the world [1]. Both for environmental reasons and because the resources
are finite, understanding and developing renewable energy sources is however desirable. The
sunlight reaching the Earth’s surface every hour would be sufficient to cover the world’s entire
need for energy over the next year [2], but solar energy still accounts only for a slight fraction
of the world energy sources [1].
The first important property of a solar cell converting light into electrical energy is that
it is able to absorb sunlight. Next, charges – electrons and holes – must be separated, and
transported through the cell and an outer circuit, where the current can be used for external work.
The amount of energy is determined by the voltage over the cell. Finally, the photogenerated
charges should be collected in order to enter a new photocycle.
The silicon device yet dominates the solar cell market, but photovoltaics based on molecular materials is a promising competitor for low cost production. One such is the dye-sensitized
solar cell (DSC) [3], shown schematically in Figure 1.1. In the DSC, an organic or transition
metal-based dye absorbs light. The dye is attached to a nanoporous wide-bandgap semiconductor surface, often TiO2 , that does not absorb in the visible spectrum. An electrolyte, traditionally
the I− /I−
3 redox couple dissolved in an organic solvent, mediates electrons between the electrodes. Other material combinations have also shown great potential for use in the DSC; recent
I3−
I−
Figure 1.1: An illustration of the components of the DSC: a dye-sensitized nanoporous semiconductor surface surrounded by an electrolyte. The arrows show the electron current when light is
converted to electricity.
developments include replacing the dye with an inorganic semiconductor and the electrolyte
with a solid molecular hole conductor [4–7]. The standard dye-electrolyte cell described here is
however the model commercially available.
7
D+ /D∗
EF
hν
VOC
−
I−
3 /I
V vs NHE
Energy
CB/CB−
D+ /D
Figure 1.2: Redox potentials for the processes in the solar cell. The dye is photoexcited from the
ground state D to D∗ (red arrow). Fast electron injection can then take place from the excited state to
the conduction band CB of the semiconductor (green arrow). The oxidized dye D+ is reduced back
to the neutral ground state D by the electrolyte (blue arrow).
As the dye in its ground state D absorbs light with the energy hν, it is excited to the state
D∗ , from which it can inject an electron into the conduction band (CB) of the semiconductor. It
is hence crucial that the energy difference between the ground and a possible excited state of the
dye matches the energy of the photons striking the surface. The charge separation is described
by the two reactions
D∗ → D+ + e−
(1.1)
and
CB + e− → CB− .
(1.2)
Finally, the dye – left in an oxidized state D+ after electron injection – is regenerated by accepting an electron from the surrounding electrolyte, while I− is oxidized to I−
3 [8]:
D + + e− → D
(1.3)
−
3I− → I−
3 + 2e .
(1.4)
From Equation 1.1 and 1.3 follows that the energy of the absorbed photon (i.e. the energy difference between D∗ and D) equals the difference between the redox potentials involving the dye,
shown in the energy scheme in Figure 1.2. The maximum voltage VOC the solar cell can generate is determined by the energy separation between the redox potential of the electrolyte (I− /I−
3
in Figure 1.2) and the semiconductor Fermi level under illumination.
For a complex system like the DSC, detailed knowledge about the electronic and molecular
structure of the materials influencing the energy conversion processes is important, in order
to optimize performance and stability. The electronic structure in working solar cells can be
probed with atomic resolution using x-ray and extreme ultraviolet radiation techniques. For a
reliable interpretation of the experimental data we need however to establish accurate theoretical
approaches for spectrum simulations, with sufficiently realistic models of the systems studied.
This thesis focuses on modeling of core-level spectroscopy using ab initio quantum chemistry methods and correlating the models with experimental measurements. Solvation effects are
studied combining electronic structure calculations with molecular dynamics simulations of I−
and I−
3 in solution. A major challenge when studying heavy elements like iodine and transition
8
metal systems is the strong spin-orbit coupling, and particularly for transition metals also the
multiconfigurational character of the system. An important part of the computational work is
a correct treatment of these effects and accurate modeling of excited states, including charge
transfer effects.
Outline
The computational methods and models used for the calculations in Paper I-III are described in
Chapter 2 in this thesis. Chapter 3 gives an overview of the relevant core-level spectroscopic
techniques. Many of the figures illustrating the concepts in Chapter 2 and 3 are based on the
actual models and/or results in the work described in Paper I-III, although the main part of the
results is summarized in Chapter 4 and 5. Based primarily on Paper I, the topic of Chapter 4
is x-ray absorption and resonant inelastic scattering spectroscopy of transition metals, while
Chapter 5 gives a summary of photoelectron spectroscopy studies of I− and I−
3 in solution (Paper
II and III).
9
2. Computational framework
The typical picture of a molecule is a composite of charged particles: nuclei and electrons. The
fundamental interaction, determining the chemistry and properties of an atom or a molecule
in its environment, is the Coulomb interaction between these particles. Every system can be
described by equations – solving the differential equation describing a chemical system provides
information about all its properties and processes.
2.1
Born-Oppenheimer approximation
Since it is not possible to separate the degrees of freedom for the internal motion of more than
two interacting particles, the differential equations for all chemical systems larger than the hydrogen atom must be solved numerically. However, physical properties can often be used for
an approximate separation of variables. In the Born-Oppenheimer (BO) approximation [9], the
motion of the heavy nuclei and the light electrons are assumed to occur on different timescales:
In the electron’s perspective, the nuclei appear fixed.
Due to their large mass, atoms and molecules can usually be well described with classical
dynamics, whereas the wave-like character of the electrons has to be treated with quantum mechanics. Consequently, we rely on quantum chemical (QC) methods for calculating electronic
spectra. In order to capture the effect of the strong intermolecular interactions in solution, a
large number of individual solvent molecules needs to be taken into account. In practice, it is
not possible to describe the solvent molecular motion with quantum mechanics, but molecular
dynamics (MD) can be used to simulate the solution dynamics.
2.2
Quantum chemical methods
The central equation in quantum mechanics is the Schrödinger equation [10]:
H(r,t)Ψ(r,t) = i
∂ Ψ(r,t)
,
∂t
(2.1)
where the Hamiltonian H is the sum of the kinetic and potential energy operators, with the time
dependence of the Hamiltonian contained in the potential energy operator. The solution to the
Schrödinger equation gives the wavefunction Ψ, from which all information about the system
can be extracted. The time and space variables of the wavefunction can be separated and for
a time-independent potential, the total energy of the system is given by the time-independent
Schrödinger equation, an ordinary differential equation:
H(r)Ψ(r) = E(r)Ψ(r).
10
(2.2)
If the coupling between the nuclear and electronic motion is neglected (BO-approximation), the
total wavefunction Ψ separates into an electronic and a nuclear part. The electronic wavefunction
depends parametrically on the nuclear coordinates R, and the corresponding eigenvalues ε(R)
create a potential energy surface on which the nuclei move. The nuclear equation can then be
solved on the electronic surface, which yields a set of rotational and vibrational levels.
2.2.1
The independent particle model and self-consistent field theory
The potential energy operator corresponding to Coulomb repulsion between the electrons in
the electronic Hamiltonian is not separable. The independent particle approximation decouples
the complex dynamics of a many-electron system to single-particle problems, through the assumption that an individual electron moves independently of the dynamics of the others. The
corresponding part of the Hamiltonian can hence be replaced by the interaction of the electrons
with the average field in which they move.
In the independent particle model, a molecule can be described with a set of one-electron
wavefunctions. The one-electron Hamiltonian contains the kinetic energy of the electron and
the potential energy of the electron in the field of the nuclei. With each independent electron is
associated a spatial function ϕ – an orbital – and a corresponding energy eigenvalue, obtained
by solving the Schrödinger equation with this Hamiltonian.
Not only the spatial function, but also the spin, describes the electron. However, an Nelectron wavefunction cannot simply be expressed as a product one-electron space and spin
functions (spin orbitals) χ – it also has to be antisymmetric in order to obey the Pauli exclusion
principle [11]. In the Hartree-Fock method [12; 13], the approximate wavefunction is formed
by a Slater determinant [14] with the N occupied spin orbitals – for a molecule, the molecular
orbitals (MOs) – arranged in columns and the electron coordinates in rows:
χ1 (1) χ2 (1) . . . χN (1) 1 χ1 (2) χ2 (2) . . . χN (2) Ψ= √ (2.3)
.
..
..
..
..
.
N .
.
.
χ1 (N) χ2 (N) . . . χN (N) The determinant ensures that the antisymmetry criterion is fulfilled. The wavefunction expressed
as a single determinant as in Equation 2.3, can be reduced to a set of one-electron differential
equations1 , resembling single particle Schrödinger equations (the Hartree-Fock equations):
Fi χi0 = εi χi0 ,
(2.4)
where Fi is the Fock operator, containing four terms: the kinetic and electron-nuclei potential
energy contributions, which depend only on the coordinate of electron i, and two terms that
sum over all electron positions. These two-electron operators contain the electrostatic repulsion
between the electrons (the Coulomb operator Ji j ) and a nonclassical term (the exchange operator
Ki j ). The last term results from the Pauli principle and eliminates the probability of two electrons
1 Since
the determinant remains the same under an unitary transformation, we can choose χi to be an
orthonormal set of molecular orbitals χi0 .
11
with the same spin to occupy the same position in space. The solutions to the HF equations are
molecular orbitals χi0 with the corresponding orbital energies εi .
In the ground state N-electron wavefunction, N orbitals are occupied successively in increasing energy, beginning at the lowest corresponding orbital energy. The highest occupied orbital
(HOMO) and the lowest unoccupied (LUMO) are particularly important when studying photochemistry, since the lowest possible excitation from the ground state typically is an electronic
transition from the HOMO to the LUMO.
The MOs are complicated mathematical functions, but can be expanded in a set of basis
functions. Often, a set of functions φα localized at the nuclei is chosen, and each MO is then
expressed as a linear combination of these known “atomic orbitals” (AO)
M
χi =
∑ cαi φα ,
(2.5)
α=1
where M is the number of basis functions. The unknown MO coefficients cαi must be calculated
iteratively, by minimizing the total energy of the wavefunction.1 The resulting eigenfunctions χi
which are solutions to the HF equations are called self-consistent field (SCF) orbitals. With N
MOs occupied, the M − N empty functions in the basis set are referred to as virtual orbitals.2
From now on, I am going to omit the spin coordinate and refer to MOs as spatial orbitals,
containing at most two electrons with opposite spins.
2.2.2
Electron correlation
The two assumptions described in Section 2.2.1 – that each electron interacts only with the
average field created by the other electrons and that the wavefunction can be written as a single
Slater determinant – introduces an error in the wavefunction and the total energy. The difference
between the exact energy of the system and the HF energy is usually called the correlation
energy.
The electron correlation can be separated into two categories: static and dynamic. For systems where the wavefunction is not dominated by a single Slater determinant, the (long-range)
static correlation is important. This is often the case in transition-metal complexes: because of
the mixing of electronic configurations, HF fails to describe the ground state.
The dynamic (“instantaneous”) contribution to the correlation energy is short-range and reflects the decreased probability of finding electrons near each other due to the Coulomb and
Pauli repulsion. Even though HF theory accounts for the correlation between electrons with the
same spin (Pauli repulsion), it neglects the correlated motion between electrons with opposite
spins, since the Coulomb repulsion from the other electrons is only seen as an average field.
The HF single determinant is an approximation of the wavefunction Ψ, but in fact any Nelectron wavefunction can be expanded exactly in an infinite number of Slater determinants. A
better approximation of Ψ can thus be formed by adding determinants with different electron
1 The
variational theorem states that the exact ground state energy is always less than or equal to the
expectation value of the Hamiltonian calculated for an approximate wavefunction.
2 The HF wavefunction depends only on the energies of the occupied orbitals, and the virtual orbitals
have no direct physical meaning.
12
configurations, that is, Slater determinants with electrons occupying other orbitals than the reference HF determinant. The exact wavefunction would be obtained as a linear combination of
Slater determinants, where all possible ways of arranging electrons are included in a complete
AO basis set:
∞
Ψ = a0 ΦHF + ∑ aS ΦS + ∑ aD ΦD + . . . =
S
D
∑ ak Φk ,
(2.6)
k=0
where the determinants have been grouped into configurations describing single (S), double (D),
etc. excitations. ΦHF is the HF wavefunction in Equation 2.3. In the configuration interaction
method (CI) [15], a set of orthonormal MOs – the orbital coefficients cαi in Equation 2.5 – is
first obtained from a HF calculation and kept fixed. For a given set of MOs, the weight of each
determinant is calculated variationally, i.e. by finding the coefficients that minimize the energy.
In practice a complete CI is not attainable, but a finite basis set is used. Full CI (even for a
finite basis set) is computationally very expensive, and instead the series is truncated to contain
determinants up to a given excitation level (single, double, etc excitations). The coefficients
that minimize the total energy are the eigenvectors of the Hamiltonian in the same basis of
determinants.
2.2.3
Multiconfigurational theory – the active space
An alternative way of treating the multiconfigurational character is by optimizing not only the
coefficients ak for the determinants, but also the MO coefficients cαi . The complete active space
self-consistent field (CASSCF) method [16] selects the configurations included in the wavefunction by a partitioning of the MOs into an active and an inactive space. The inactive orbitals are
all doubly occupied. In the active space, all orbital permutations consistent with the number of
electrons and spin multiplicity are allowed, i.e. a full CI is performed, and for all configurations,
the orbital coefficients are optimized.
Since excited configurations are included in the CASSCF wavefunction, the calculation
gives, in addition to the ground state, all the excited states that can arise from these. When
calculating transitions between electronic states, the choice of active space can hence be used as
a direct tool to control that the excitations of interest are covered.
The main challenge with the CASSCF method is that the computational cost grows rapidly
with the number of MOs included. To keep the calculation manageable, the active space can
be further divided into three subspaces, with certain limitations in the occupation numbers,
or equivalently: excitations. This method is referred to as the restricted active space SCF
(RASSCF) method [17]. The QC calculations described in Paper I-III are multiconfigurational
calculations, performed with the Molcas-7 software [18].
The [Ni(H2 O)6 ]2+ complex in Figure 2.1 serves as an illustration of the RASSCF method.
The first subspace, RAS1 is chosen to comprise the Ni 2p core orbitals. The RAS1 orbitals would
be doubly occupied in the HF, but in this particular RASSCF calculation, one hole is allowed.
RAS2 consists of the orbitals of nominal Ni 3d character with no restrictions on the occupation.
The remaining occupied orbitals are left inactive: the orbital coefficients are optimized in the
CASSCF procedure, but the orbitals are not involved in the excitations. The total number of
electrons in RAS1 and RAS2 is fixed and we can therefore separate the possible excitations into
13
Valence excitation
2b∗
H2 O 4a2∗
1
eg
Ni 3d
t2g
RAS2
1b1
H2 O 3a1
1b2
H2 O 2a1
Ni 3p
Ni 3s
Core excitation
H2 O 1a1
Ni 2p
RAS1
Ni 2s
Ni 1s
Figure 2.1: Molecular orbital diagram of Ni2+ (H2 O)6 . The active space in the RASSCF calculation
comprises the RAS1 and RAS2 subspaces with 14 electrons in 3+5 MOs. The Ni 2p core orbitals
in RAS1 can contain at most one hole, while the occupation in the Ni 3d orbitals in RAS2 is unrestricted. With the possible electron configurations included in the RASSCF, the energy of valence
excited states (transitions indicated by red arrow) and states with a 2p core hole (blue arrow) can be
accessed (Paper I).
two types: (1) configurations with one hole in RAS1, and an “extra” electron in RAS2 and (2)
configurations with RAS1 fully occupied. The first case corresponds to core-excitations 2p→3d
(with a single core hole), and the second case includes all valence excitations corresponding
to pure d-d transitions. The dominant configuration for the ground state is the one shown in
Figure 2.1, with eight electrons populating the “3d” MOs in a high-spin fashion. At this point
it is however worthwhile to recall the two parts in the RASSCF procedure: First, that the CI
coefficents are optimized for the specific state, and therefore excited configurations also give a
small contribution to the ground state wavefunction. Second, the MO coefficents are optimized
for the selected states, which implies that the nominal 3d orbitals are to some extent mixed with
ligand orbitals, i.e. hybridized. In the calculations in this work, the energies are calculated using
a single set of orbitals, optimized for the energy average of all included states of a given spatial
and spin symmetry (state-averaging).
The example in Figure 2.1 does not allow for excitations from the occupied orbitals on
the water ligands to the metal center: ligand-to-metal charge transfer (LMCT). Neither can
excitations from the metal center to the unoccupied ligand orbitals – metal-to-ligand charge
transfer (MLCT) – take place. To model MLCT, we can however introduce a third subspace,
RAS3, of (in the HF) unoccupied ligand orbitals, containing a limited number of electrons.
Similarly, occupied ligand orbitals can be incorporated in the active space to take LMCT into
account.
14
2.2.4
Perturbation theory
The added flexibility in the wavefunction in the CAS(RAS)SCF approach mainly recovers the
static correlation. The remaining dynamical correlation can be included either perturbatively
or with subsequent multireference CI (MRCI) calculations [19; 20]. We chose to work with
many-body perturbation theory, with the general idea that the solution only differs slightly from
the simpler, already solved problem, constituting the “reference” and can be added as a perturbation. The Hamiltonian of the N-electron wavefunction is then partitioned into a reference and
a perturbation Hamiltonian [21]. In second order perturbation theory, the perturbation Hamiltonian is a two-electron operator. All combinations where two electrons from occupied (in
the reference wavefunction) orbitals are excited to virtual orbitals are taken into account in the
correction to the energy. The second order perturbation approach CASPT2(/RASPT2) uses a
CASSCF(/RASSCF) reference wavefunction. In the CASPT2 calculations of multiple states in
the present work, multistate CASPT2 [22] has been used, where the included states are allowed
to mix.
2.2.5
Relativistic effects
The Schrödinger equation (Equation 2.1) does not take the consequences of the finite speed of
light, affecting both orbital sizes and shapes, into account. Also, magnetic interactions due to
the electron spin, spin–orbit coupling, is a relativistic effect. Relativistic corrections affect the
chemistry of light elements less, but for heavier elements, the effects are important.
The relativistic wave equation for a free electron – the Dirac equation [23] – is a fourdimensional equation: in addition to two spin degrees of freedom, the Hamiltonian contains two
components arising from the electron–positron coupling. The Hamiltonian of the N-electron system is assumed to be as sum of such one-electron terms (and potential energy operators). Solving
the full four-component equation is however computationally very demanding; in a DouglasKroll-Hess transformation [24; 25], the wavefunction is reduced to two components. The relativistic calculations in Paper I-III have been calculated according to the Douglas-Kroll-Hess
scheme implemented in Molcas, partitioning the relativistic effects into a spin-free (scalar) and
a spin–orbit part: In the first step, a set of multiconfigurational and correlated states (RASSCF
or RASPT2) is computed, via a scalar Hamiltonian. The spin–orbit Hamiltonian (approximated
by a one-electron effective Hamiltonian, within the mean-field model [26]) for this set of wavefunctions is then calculated in a subsequent step [27].
In Figure 2.2, the energies of the spin-free and spin–orbit states of neutral I3 with a single
4d core hole are shown, relative to ground-state I−
3 . At the 4d level, the orbitals are atomic like
and localized on the respective nuclei. The states can be distinguished by the site on which the
core hole is localized: center or terminal. The origin of the different levels is discussed in further
detail in Chapter 5. Calculating the spin–orbit part of the Douglas-Kroll Hamiltonian, with the
spin-free states as a basis – in other words: letting the spin and orbital momenta couple – leads
to the splitting of the 4d level of almost 1.6 eV.
15
CASPT2
CASPT2+SO
52.46 (2)
52.51 (1)
52.51 (2)
52.51 (2)
52.52 (2)
52.55 (1)
Λ=2
Λ=0
Λ=1
Λ=1
Λ=2
Λ=0
54.23 (2)
Λ=2
54.58 (2)
54.73 (1)
Λ=1
Λ=0
51.83
51.87
51.88
51.88
51.89
51.91
Ω = 5/2
Ω = 3/2
Ω = 1/2
Ω = 3/2
Ω = 5/2
Ω = 1/2
53.42
53.45
53.46
53.47
53.59
53.87
54.04
Ω = 3/2
Ω = 1/2
Ω = 3/2
Ω = 1/2
Ω = 5/2
Ω = 3/2
Ω = 1/2
55.26
Ω = 3/2
55.59
Ω = 1/2
Figure 2.2: Energies of the 4d core hole states of isolated I3 , relative to the ground state energy
of the I−
3 anion (in eV). The energies to the left are calculated in a spin-free CASPT2 framework,
including scalar relativistic effects. Since the set of 4d orbitals on each atom is associated with five ml
quantum numbers, there are in total 3·5=15 possible electron configurations with one 4d core hole,
the corresponding states characterized by Λ. Lines in blue represent states where the electron has
been removed from a 4d orbital on the central iodine atom (the degeneracy is given in parentheses),
while red denote states with a core hole on either of the two terminal atoms. The spin–orbit states in
the right column are obtained from interaction of the spin-free states. Diagram from Paper III.
2.3
Solvation models
So far, we have dealt only with the idealized, static, case with molecular geometries optimized
to an energy minimum and without any interaction with surrounding molecules. In reality, the
problem contains both dynamics and environmental effects. The experiments in Paper II and III
were done in solution, and in Chapter 3 we shall see the solute–solvent interactions described in
more detail. The solvent effects can essentially be introduced in the calculations in two ways:
by implicit or explicit modeling.
2.3.1
Molecular dynamics
Just like the QC calculations described in Section 2.2, MD simulations rely on the BO approximation, but in contrast to calculating the electronic structure as a function of the nuclear
coordinates, the electrons are accounted for as a single parametric potential energy surface. No
electrons are hence explicitly included in the calculations, and N now refers to the number of
atoms (or sometimes molecular fragments) in the system. As its name suggests, MD does not
16
just give a static description of the system, but models the time-evolution. However, the atoms
are assumed to obey classical dynamics, and by numerical integration of Newton’s equations of
motion
d
Fi = m ẋi = mẍi ,
(2.7)
dt
information about how all particles move over a period of time can be obtained. After calculating
the resulting force on each atom from its neighbors, the particles are allowed to move under the
constant force for a short timestep. The forces are then recalculated for the new positions. A
classical MD simulation requires a force field describing, first, the ion–solvent and solvent–
solvent (intermolecular) interactions and, second, the intramolecular interactions. Often, the
Lennard-Jones potential [28] describes the pairwise van der Waals interactions between atoms
and a Coulomb potential the electrostatic contribution to the intermolecular interactions, each
atom represented by a point charge. Intramolecular interactions – bending and stretching of the
molecule – can be modeled as harmonic potentials. In addition, suitable initial and boundary
conditions have to be chosen.
In the preparation of Paper I-III, ions in solution were simulated with MD and the average
solvation structure was investigated. A number of configurations of the studied ion was also
sampled during its motion in the solution. Part of the sampling was done with the purpose of
studying the pure configurational effects; then the solute geometries representing the distribution
was extracted from the MD simulations and treated with QC. For inclusion of solvent effects, a
shell of the closest solvent neighbors were also included in the QC calculation. Furthermore, in
Paper II, the real-time evolution of the bond lenghts in the triiodide ion was traced. The classical MD simulations were performed under constant volume and temperature, using periodic
boundary conditions, with the simulation box replicated in all directions to simulate bulk.
Ab initio molecular dynamics
The predefined potentials used in classical MD (usually based on empirical data or independent QC calculations), are not able to describe changes in the electronic structure and hence
no bond breaking and formation. In ab initio MD, the electronic degrees of freedom are actually taken into account, as the potential is obtained from a QC calculation in every timestep.
Instead of a suitably parametetrized force field, a method for solving the Schrödinger equation hence has to be selected. Two ab initio MD methods are Born-Oppenheimer (BOMD) and
Car-Parrinello (CPMD). In BOMD the wavefunction is optimized to the ground state in every
timestep [29]. CPMD instead includes the electrons as ficticious particles moving at a different timescale than the nuclei, avoiding computationally expensive wavefunction optimization
iterations as the wavefunction remains close to the one optimized for the initial nuclear geometry [30]. A very short timestep is however required.
Radial distribution functions
MD trajectories can be analyzed in terms of radial distribution functions (RDFs). The RDF g(r)
gives the probability of finding two different atom types separated by the distance r, relative
to a uniform distribution. Figure 2.3 shows ion–water RDFs for I− in aqueous solution. Since
17
g(r)
I−O
I−H
2
3
4
5
Distance / Å
6
Figure 2.3: I–water radial distribution functions (RDFs) for I− in aqueous solution. Thick lines
corresponds to the I–O RDF and thin lines to I–H RDF. The first iodine-hydrogen maximum at 2.6
Å interatomic separation results from hydrogen bonding.
the RDF maxima correspond to the distances from the ion at which water oxygen and hydrogen
spend the most time during the simulation, they are directly related to the hydration shells around
the ions. As expected for an anion, the I–H RDF in Figure 5.4 shows that the water hydrogen is
coordinating the iodide ion in aqueous solution, with a well-defined RDF maximum at 2.6 Å –
i.e. at a typical hydrogen bond length – and the oxygen peak about 1 Å outside this maximum. A
less ordered second hydration shell is formed outside the first solvation sphere. We shall return
to the triiodide ion in solution in Chapter 5.
2.3.2
Polarizable continuum
Rather than considering explicit solvent molecules, continuum models take the configurational
averaging implicitly into account through macroscopic properties of the solvent, while the solute
is treated quantum mechanically. In the Polarizable continuum model (PCM) [31], the solute is
placed in a cavity in a uniform polarizable medium with a dielectric constant ε characterizing
the solvent. Point charges on the cavity surface represent the reaction field created due to the
polarization of the solvent in the presence of the solute. This reaction field in turn perturbs
the solute itself. By adding the reaction field perturbation in the Hamiltonian, the Schrödinger
equation can be solved self-consistently, thus including the effect of the environment.
18
3. Core-level spectroscopy
The core electrons of a molecule do not participate in the chemical bonding; unlike valence
orbitals, they are localized on the atomic nuclei and have well separated binding energies associated with them, typically in the x-ray region of the electromagnetic spectrum. In contrast,
the molecular bonds are – in a molecular orbital description – formed by valence electrons in
orbitals delocalized over the entire molecule. The valence binding energies lie closer than the
core levels and often in the range of visible or ultraviolet light.
Since each chemical element has its characteristic pattern of wavelengths at which it absorbs
light, x-ray spectroscopy can be used to gain element-specific information of the material. This
section gives a brief description of characterization techniques employing x-ray radiation. The
techniques can be classified according to what is detected: in the first case, the material is
ionized and the ejected electrons detected (photoelectron spectroscopy) and in the second case,
the absorption and possibly subsequent emission of x-rays is measured (x-ray absorption and
resonant inelastic x-ray scattering spectroscopy). [32–37]
3.1
Photoionization
The basis of photoelectron spectroscopy (PES) is the photoelectric effect [38; 39]. If the energy
hν of photons illuminating a surface is sufficiently high – at least equal to the binding energy
BE of a specific electron – an electron can be released from the material. According to the
photoelectric law, the photoelectron is emitted with kinetic energy
KE = hν − BE.
(3.1)
In x-ray photoelectron spectroscopy (XPS), the amount of electrons escaping the sample after
interaction with x-rays is measured as a function of their kinetic energy. The resulting spectrum
is represented as a graph of the intensity against the binding energy. XPS is thus used to probe
the occupied levels.
If the electronic transition from the initial state |0i (with N electrons) to the final state | f i
(with N − 1 electrons + a non-interacting photoelectron) takes place on a timescale much shorter
than the typical timescale for molecular vibrations, the molecular geometry can be assumed
not to change during the transition. However, the sudden decrease in screening of the nuclear
charge upon ionization “immediately” causes relaxation of the outer orbitals. In such a vertical
transition, the binding energy, expressed as the energy difference between the final and initial
states
BE = E (N−1) − E (N) ,
(3.2)
includes electronic, but not nuclear, relaxation.
19
The experimental photoelectron spectra in Paper II and III were measured by collaborators,
using synchrotron radiation at MAX-lab in Lund and BESSY at the Helmholtz-Zentrum Berlin.
The liquid samples were injected into the experimental chamber as a micro jet [40–42].
3.1.1
Chemical shifts
Even though core electrons do not participate directly in the chemical bonding, the core binding energies are affected by the chemical environment. These chemical shifts, in the order of
a few eV, add site-specific information to the spectra of chemical compounds: Two chemically
inequivalent atoms (of the same element) have different core binding energies, due to the differences in nuclear screening depending on the valence charge distribution on the surrounding
atoms.
In addition, the spectra of solutions provide information of the solute–solvent interaction.
Ionization of the solute changes both the solvent polarization and the solvent-induced polarization of the solute, which leads to a shift of the binding energy – solvent-induced binding
energy shift (SIBES) – with respect to photoionization in the gas phase. When the photoelectron
leaves an anion, the negative ion charge decreases, thus weakening the electrostatic interaction
between solute and solvent. In contrast, the positive charge of a cation increases upon photoemission. Figure 3.1 shows that this results in a positive SIBES for an anion and negative for a
cation.
gas
Intensity / Arb. units
LiI(aq)
62
60
58
56
54
52
gas
solvated
solvated
2+
I
Li
I−
Li+
50
Gas (theo.)
I4d
Li1s
77
75
73
52
50
Binding energy (eV)
48
46
Figure 3.1: The solvent-induced binding energy shifts of the Li+ and I− peaks in the photoelectron
spectra (left figure) have opposite signs. The scheme to the right illustrates that this is due to that
the electrostatic interaction either vanishes (anion) or increases (cation) upon electron removal. The
black arrows in the right figure represent the binding energies, and the SIBES corresponds to the
change in binding energy when going from gas phase to solution. (Spectra from Paper III.)
3.1.2
The sudden approximation and shake-up processes
For rapid photoionization processes, the use of the sudden approximation [43] is justified. The
electronic relaxation is then neglected during the ionization process itself: the abrupt change
20
in the effective potential leaves the system in the same state. This state |Ψ∗ iN , is simply what
remains of the N-electron wavefunction after pulling out an electron effectively away from the
electrostatic well. It is hence not an eigenstate |ki f of the new Hamiltonian, but can be expressed
as a linear combination of the new, electronically relaxed, eigenstates:
|Ψ∗ iN = ∑ ck |ki f ,
(3.3)
k
where the (real) coefficients ck are given by
ck = f hk|Ψ∗ iN .
(3.4)
Among the possible eigenstates, states with an unbound electron are found, with the correspond(N−1)
ing energy eigenvalues Ek
.
The sudden change in electrostatic potential may give rise to shake-up satellites in the spectrum, as the departing core electron interacts with the valence. In this case, the resulting final
state, which is a proper eigenstate |ki f , can be considered a simultaneous core ionization and
valence excitation. The kinetic energy of the photoelectron is thus slightly (a few eV) reduced
in the shake-up process.
3.2
X-ray absorption and emission
In x-ray absorption spectroscopy (XAS) the sample is irradiated with x-rays and excited resonantly from the initial state |ii to the state | f i with a core electron occupying previously empty
valence orbitals. Just as the core-electron binding energies can be attributed to specific atomic
sites in the system (Section 3.1), so can core-excitation energies in the XAS process. Consequently, measurements of the x-ray absorption intensity as a function of the incident photon
energy map the uncoccupied valence states onto a particular atom in the system.
The short-lived core-excited state can decay either through emission of a photon or an electron. Resonant inelastic x-ray scattering spectroscopy (RIXS), is applied on the first decay
type; the incident energy is scanned across an absorption edge and the intensity of the subsequently emitted energies is measured over a certain energy range. The RIXS process is described
schematically in Figure 3.2. From the law of energy conservation, the emitted photon is seen to
have energy
ω = Ω − ω f 0,
(3.5)
where Ω is the energy of the incident photon and ω f 0 the energy loss: the difference in energy
between the final and initial states.
Decay of the core-excited state back into the ground state gives rise to an elastically scattered
peak, where the emitted photon has the same energy as the absorbed photon. There are however
other possible decay channels of the core hole, in which the core hole is filled and the system is
left in a valence excited state. This is observed as a fluorescence line at an energy a few eV lower
than the incident energy. The energy loss corresponds precisely to the valence excitation energy.
The resulting two-dimensional RIXS spectrum can hence be used to probe the unoccupied levels
with chemical specificity, as a function of the incident energy.
21
Total energy
Em
Ω
ω
Ef
E0
Figure 3.2: The RIXS scattering process. By absorption of a photon with energy Ω, the molecule is
excited from the initial state |0i to a metastable intermediate state |mi with a core hole. The system
decays to the final state | f i under emission of a photon with energy ω. The dashed arrow represents
the elastic transition with Ω = ω, and the solid decay to a valence excited final state. The energy left
in the system (loss) is given by Ω − ω. Adapted from Glatzel et al. [36].
3.3
Peak intensities
The intensity of a peak in a spectrum is directly related to the probability for photon absorption
or emission. The most important type of interaction between electromagnetic radiation (such as
x-rays) and an atom or molecule is via electric dipole interaction, where the oscillating electric
field of the radiation perturbs the Hamiltonian due to the interaction with the charge distribution
in the system.
Within the dipole approximation, the probability that linearly polarized light induces the
electronic transition |ii → | f i depends on the transition (dipole) moment between the initial and
final states:
Mi f = h f |µ|ii,
(3.6)
where µ is the electric dipole moment operator. Given that Mi f is nonzero, absorption of a
photon with energy ωi f can occur if its energy matches the difference between the initial and
final state energies (Ei and E f ) as expressed in Fermi’s Golden Rule [44]:
w f i ∝ |Mi f |2 δ (E f − Ei − ωi f ),
(3.7)
where w f i is the transition probability and the delta function takes care of the energy conservation. The RIXS process is more involved than XPS and XAS and addressed in Section 3.3.3.
It is important to note that not only the amplitude of Mi f , but also its direction (the polarization direction) relative to the electric field vector of the incident light, determines the absorption
probability: absorption is maximal if the vectors are parallel and zero if they are perpendicular
to each other. In a kinetic viewpoint, the probability of absorption or scattering is usually given
in terms of a cross section, proportional to Mi2f .
22
3.3.1
Linewidths and shapes of x-ray spectra
One important factor limiting resolution of x-ray spectra is the intrinsic lifetime: due to the timeenergy uncertainty relation, the final state energy is only known up to the accuracy determined
by the lifetime of the final core-excited state. The lifetime broadening leads to a Lorentzian
lineshape. In addition, configurational broadening and experimental resolution (in the energy of
the x-ray beam and the analyzer resolution) give rise to a Gaussian distribution of the energies.
The configurational broadening can be understood as small chemical shifts caused by geometry
fluctuations.
Vibronic coupling and fine structure may also give a small Gaussian contribution to the
spectrum. The total peak typically has a Voigt profile, namely a convolution of the different
broadening mechanisms of Gaussian and Lorentzian character.
3.3.2
Photoemission intensities
A fast XPS process, induced by high energy photons, can be considered a sudden annihilation
of an electron from the initial state. The initial state wavefunction with an unbound electron in
Equation 3.3 can then be expressed as an annihilation operator aq acting on the photoelectron
(the qth electron) in the initial state: |Ψ∗ iN = aq |ΨiN . From the square of the coefficients in
Equation 3.4 we obtain the relative XPS intensities Pk within the sudden approximation:
Pk ∝ |Nhaq Ψ|ki f |2 .
(3.8)
Pk thus gives the probability for transitions from the ground state of the N-electron system
to the eigenstates |ki f for the core-ionized (N − 1)-electron system – including final states with
collective core and valence excitations. In the theoretical XPS spectra in the present work, we
have however only calculated the binding energies (not the intensities), and added a broadening
according to section 3.3.1.
3.3.3
RIXS intensities
Since RIXS is a second order process, where the molecule absorbs a photon with energy Ω and
emits a photon with energy ω, the first order perturbation term in Equation 3.6 does not provide
a sufficient description. Rather than separable absorption and emission processes, the process
should be considered a single-step scattering event. The correlation is treated by the KramersHeisenberg formula [45], which gives an expression for the scattering amplitude Ff (Ω), containing a sum over all intermediate (core-excited) states. If the intermediate states lie close in
energy, the decay channels may interfere, and the interference effects are taken into account in
the square modulus of the scattering amplitude:
2
2 Ω
−
ω
−
iΓ
i0
i
0†
Ff (Ω) = ∑ ωi f ωi0 D Di0
(3.9)
,
fi
2
2
i
(Ω − ωi0 ) + Γi where ωi f and ωi0 are the intermediate-final and intermediate-initial state energy differences
(Em − E f and E f − E0 in the scheme in Figure 3.2). D0†f i and Di0 correspond to the probabilities for the transitions and are defined as the projection of the electronic dipole moment µ of
23
the molecule onto the electric field vectors of the incident and emitted light respectively. The
remaining complex Lorenzian function adds a broadening of the line due to the finite lifetime Γi
of the core-excited state. In the calculated RIXS spectra in Paper I, summarized in Chapter 4,
we have however neglected interference and the polarization dependence in D f i and Di0 .
The total RIXS intensity is obtained by summation over all possible final states, and depends
on the energies of both the incident and emitted photons:
2
f 0)
Γf
Ω 2
1 − (ω−Ω+ω
γ2
√
I(Ω, ω) ∝ ∑ Ff ·
·
e
ω f
πγ
π((ω − Ω + ω f 0 )2 + Γ2f )
.
(3.10)
The intrinsic width in the energy loss direction of the spectrum is determined by the final state
lifetime Γ f in the Lorentzian in the expression above. Experimental and configurational broadening sets the value of the Gaussian broadening parameter γ.
24
4. Transition metal complexes
Transition metals, with their partially filled d (or f) valence orbitals, often form coordination
complexes that absorb in the visible region and are used as the active site in many energyrelated applications. XA and RIXS spectroscopy (Section 3.2) provide rich information about
the electronic valence in the presence of a core hole. Calculations that reproduce – and hence
can serve as an appropriate basis for interpretation of – the experimental transition metal spectra
must handle multiplet effects and chemical interactions properly. Multiplets arise in the spectra
due to the coupling between the spin and orbital angular momenta of the core hole and the
partly filled d orbitals. Excitations in the valence are mainly of either of the types referred to
in Section 2.2.3: charge transfer (LMCT or MLCT), involving transitions from/to the ligands
to/from the metal center, or d-d transitions within the d orbitals on the metal center.
Paper I presents ab initio1 2p XAS and RIXS calculations, based on a RASSCF wavefunction, of hydrated Ni2+ . Direct comparison of the calculated XA spectra with experiment shows
that the approach is highly reliable for 2p spectroscopy of 3d complexes.
4.1
[Ni(H2 O)6 ]2+
The Ni2+ (H2 O)6 complex has a high-spin d8 ground state electron configuration and is octahedrally coordinated by the water ligands. The XA spectra in Figure 4.1 were calculated with
the RAS partitioning described in Section 2.2.3 (Figure 2.1) and spin-free relativistic effects
EXP
XA intensity
RASPT2
RASSCF
850
855
860
865
870
Excitation energy [eV]
875
Figure 4.1: RASSCF and RASPT2 calculated XA spectra of Ni2+ (H2 O)6 including all [(2p)5 ,(3d)9 ]
core-excited configurations, in comparison with the experimental spectrum of NiCl2 (aq).
1 Ab
initio calculations are based entirely on first principles, in contrast to methods that introduce
empirical or fitting parameters.
25
2b∗2
4a∗1
eg
t2g
RAS2
Energy
loss
1b1
H2 O 3a1
1b2
H2 O 2a1
Ni 3p
Ni 3s
Ni 2s
860
865
870
875
a)
0
XAS
b)
1
c)850
855
860
865
870
875
0
15
X-ray
absorption
H2 O 1a1
Ni 2p
855
RAS1
-1
10
-2
-3
5
-4
-5
Ni 1s
Logarithmic RIXS Intensity
Ni 3d
850
5
XA Intensity
H2 O
Energy loss [eV]
Valence
excitation
Energy loss [eV]
treated by a scalar Douglas-Kroll Hamiltonian combined with a relativistic all-electron (ANORCC) VTZP basis set [46; 47]. The spin–orbit coupling was calculated between all the singlet and triplet states resulting from this active space (allowing all core excitations leading to
a [(2p)5 ,(3d)9 ] electron configuration). The large multiplet splitting of the 2p x-ray absorption
edge (2p → 3d) is due to the strong spin–orbit coupling of the 2p5 configuration, i.e. the core
hole. The method reproduces the features in each of the edges and the dominating core-level
spin–orbit splitting in the 2p XA spectrum agrees well with experiment. The additional multiplets within the respective edge arise due to core–valence (2p3d) electron–electron interactions.
In order to compensate for limitations in the RASSCF wavefunction, the spectrum (green
line) is shifted by -3.70 eV for comparison with experiment. A large part of the ad hoc correction
is due to dynamical correlation: The RASPT2 energies (red line) are shifted by the smaller value
-1.85 eV.
In the low-energy (L3 ) peak, final core-excited states with the total core hole angular momentum 23 dominate, while states with 2p angular momentum 21 give rise to the (L2 ) peak above
870 eV. Upon closer examination of each edge, we find that the XA resonance at the low-binding
energy side has mainly triplet character. In other words, final states with parallel 2p core hole
and 3d electron spins are stabilized with respect to singlet states (opposite spins), which are
found in the peak at higher binding energy.
We can analyze the valence further, by studying the states resulting from fluorescent decay
of the core hole. Equation 3.5 is illustrated schematically in Figure 4.2: the energy difference
between the x-rays absorbed and emitted in the RIXS process corresponds to the energy of the
final, possibly valence excited state relative to the ground state. The calculations give explicit
0
-6
850
855
860
865
870
Excitation energy [eV]
875
Figure 4.2: Each core-excitation has its particular set of decay channels from the intermediate
state in the RIXS process to final valence excited states. (a) RIXS map, with the intensity as a
function of the x-ray absorption energy and the associated energy loss, and (b) the corresponding
XA spectrum from RASPT2 calculation including [(2p)6 ,(3d)8 ] and [(2p)5 ,(3d)9 ] configurations. (c)
Charge-transfer states are included by increasing the number of orbitals in the active space.
26
access to each electronic state; guided by the extracted information, we can both assign the lines
in the XAS spectrum, and unearth the electronic character of every final valence state resulting
from RIXS at a specific absorbed photon energy. The RASPT2 spectra in panel a and b in
Figure 4.2 are obtained from the same calculation as the previous XA spectra, namely without
including ligand orbitals in the active space. The possible electron permutations in the 3d orbitals
correspond to three different configurations: (t2g )6 (eg )2 , (t2g )5 (eg )3 , and (t2g )4 (eg )4 .
Analysis of the L3 edge RIXS shows that the XA resonance at 853 eV (triplet) – apart from
the peak at zero energy loss, corresponding to resonant elastic decay from the core-excited intermediate back to the ground state (3 A2g ) – decays with high intensity to two peaks at 0.9 eV and
2.9 eV loss. The peaks originate from decay to the 3 T2g valence excited state (essentially described by the (t2g )5 (eg )3 configuration) and to 3 T1g , respectively. These two states are formally
equivalent to the final states resulting from the optical transitions 3 A2g →3 T2g and 3 A2g →3 T1g
in UV/Vis spectroscopy [48]. Since RIXS is a second-order process and the spin multiplicites
in the valence mix in the intermediate core-excited state, final valence states that are optically
spin-forbidden may also be reached; the XA peak at 866 eV (singlet character) decays with high
probability to singlet final states.
In the RIXS spectrum in panel c, the active space has been extended, and the contribution
from charge-transfer excitations arise at an energy loss above 10 eV. Here, two occupied water
orbitals have been added to RAS2, thus enabling transfer of electrons to the metal centered 3d
orbitals (LMCT). A third subspace (RAS3), consisting of three unoccupied ligand orbitals, is
also created. Since one electron is now allowed in RAS3, MLCT as single excitations to the
unoccupied ligand orbitals is taken into account.
Information about photochemical processes, such as those in the DSC, is found in the time
evolution of the nuclear geometry and valence electronic structure. With this method, also XA
and RIXS spectra of distorted geometries and transient electronic states can be modeled, and
used in the analysis of time-resolved experiments – probing changes in the electronic structure
with atomic site selectivity on the femtosecond timescale.
27
5. I−/I−
3 in water, ethanol, and acetonitrile
solutions
The I− /I−
3 redox couple has been used as the hole conducting medium in the DSC for a long time,
but the redox chemistry is still only partially understood. Information about the fundamental
solvation phenomena and the energies involved is useful for the understanding of the redox
mechanisms. Paper II and III describe the solvation of I− and I−
3 in different solvents, where we
have used core-level PES to study the electronic structure. Multiconfigurational QC calculations
combined with MD simulations give an insight in the microscopic details, supported by and
supporting the analysis of liquid jet experiments. We observe a large dependece on the local
solvent structure in the solvation energies of I− and I−
3 . The short-range solvent interactions are
also important for the I−
molecular
structure,
even
leading to symmetry breaking in aqueous
3
solution.
5.1
Solvated I−
The magnitude of the binding energy shift in solution varies with the solvent: Through the series
water → ethanol → acetonitrile, the I4d photoelectron spectra in Figure 5.1 show a decreased
SIBES. The trend is connected with the type of interaction – and hence amount of stabilization
– in the solution; water and ethanol form hydrogen bonds to the ion, while acetonitrile is a polar
solvent without hydrogen bonds. Strong hydrogen bonding leads to a large solvation energy, and
consequently large core-level SIBES to lower binding energy for the anion (see Section 3.1.1).
The solvent structure is important for the stabilization of the ion: The I4d SIBES of I− dissolved
in ethanol (forming hydrogen-bonded chains or clusters) is weaker than in aqueous solution.
Static continuum models, such as PCM, are therefore not able to capture the trend; a microscopic solvation description is needed. The SIBES computed using PCM in Table 5.1 is slightly
overestimated and essentially the same for the three different solvents. Calculations of clusters
with explicit solvent molecules take the short-range solute–solvent interactions such as hydrogen
bonding into account and reproduces the relative energy ordering of the shifts. The main challenge with this approach is however to treat clusters sufficiently large to describe the solvation
in bulk at the high QC level.
The experimental SIBES is derived from the difference between the binding energy measured in liquid experiments and gas-phase CASPT2+SO calculated values. The active space in
the CASSCF/CASPT2 calculations consisted of the 4d orbitals and the 5s, 5p valence, with the
orbitals optimized for the energy average of the 4d orbitals.
28
Intensity / arb. units
AcCN
EtOH
Water
Gas (theo.)
SIBES
58
56
54
52
50
Binding energy / eV
48
Figure 5.1: Experimental 4d core level photoelectron spectra of I− in different solvents and the
spin-orbit coupled CASPT2 spectrum of the isolated ion. The SIBES is defined as the difference
between the gas and solvated peak positions. Figure from Paper III.
Table 5.1: Experimental and calculated solvent-induced I4d electron binding energy shifts (SIBES)
in water, acetonitrile, and ethanol. The theoretical values are obtained within the PCM model at the
CASPT2+SO level of theory and with CASSCF+SO calculations of cluster configurations sampled
from MD simulations of LiI in solution. Table adapted from Paper III.
SIBES
Experimental
PCM (CASPT2+SO)
Cluster: 1st shell (CASSCF+SO)
Cluster: 2nd shell (CASSCF+SO)
5.2
Water
4.18
5.33
2.80 ± 0.40
3.84 ± 0.27
Ethanol
3.66
5.18
1.47 ± 0.37
Acetonitrile
3.39
5.25
1.36 ± 0.14
Solvated I−
3
In its ground state, I−
3 is symmetric, with the bond length of the isolated molecular ion calculated to 2.91 Å (CASPT2+SO). Most of the electron density is localized at the terminal sites,
which essentially share the net negative charge (-1). Recall now the binding energy diagram
in Figure 2.2, where photoelectrons emitted from core orbitals on the central site could be distinguished from those originating from one of the terminal atoms. The reason is the excess of
valence charge on the terminal iodine atoms, that results in less strongly bound core electrons
on the terminal iodine sites. The 4d core hole on the two equvialent terminal atoms give rise to
degenerate states, but with the hole localized on either on them.
What is then the effect of solvation on I−
3 ? Ethanol and acetonitrile solutions containing
−
I3 are studied with XPS in Paper III. The local structure of the SIBES not only affects the
magnitude of the SIBES, but also the I−
3 geometry; Paper II lays a particular focus on the strong
coupling between the structural rearrangement of the solvent and the solute dynamics in aqueous
I−
3 , leading to molecular symmetry breaking. The figures in this section are found in Paper II.
Other studies (e.g. [49]) show that I−
3 is linear and symmetric when dissolved in polar solvents, such as acetonitrile, but is distorted in hydrogen-bonding solvents. The experimental ace29
tonitrile I4d spectrum can easily be decomposed into terminal and center contributions, consistent with the respective contributions calculated for a symmetric molecular ion (CASSCF+SO),
in the middle figure in panel b. The spectrum from ethanol solution can be deconvoluted in the
same manner and is shown under the theoretical spectrum in Figure 5.2b.
Intensity / Arb. units
a)
MD/CASSCF
∆r
0.55 Å
c)
b)
EXP Total
I3−
LiI
I2(g)
−
I3 (g) MD
−
I3
(aq) MD
CASSCF
0.52 Å
H2O
0.32 Å
0.29 Å
0.25 Å
CASSCF
I3− (g) OPT
AcCN
0.16 Å
0.14 Å
0.10 Å Terminal
Center
0.10 Å
EXP
EtOH
EtOH
0.05 Å
60 58 56 54 52
60 58 56 54 52
60 58 56 54 52
Energy / eV
Energy / eV
Energy / eV
Figure 5.2: (a) CASSCF+SO I4d photoelectron spectra of I−
3 in 10 different configurations, ordered
according to bond length symmetry. (b) Top: Average of the spectra in (a), of the isolated ion
(red) and surrounded by a layer of explicit water molecules (violet). Middle: Calculated spectrum of
optimized I−
3 , showing lines arising from photoemission on the center and terminal atoms separately.
Bottom: Experimental spectrum of I−
3 dissolved in ethanol, decomposed into terminal and center
contributions. (c) Experimental I−
spectra
in water (top), acetonitrile (middle) and ethanol (bottom).
3
Water’s strong hydrogen bonding ability, however, leads to large bond length fluctuations of
the I−
3 ion in aqueous solution. Figure 5.3 shows the bond length distribution during a CPMD
simulation of LiI3 (aq) and includes extremely distorted geometries, with I–I distances well over
the typical covalent bond. The geometry distortions are associated with polarization of the I−
3
ion: in the observed partial dissociation, the major fraction of the negative charge is localized
at the partially detached iodine (with very little change in the center atom charge). Hence, we
should expect shifts of the core electron binding energies of a strongly distorted triiodide ion
with respect to the gas-phase optimized configuration. The spectra in panel a in Figure 5.2 are
calculated for isolated I−
3 , with geometries sampled from the MD solution simulation, and show
that the degeneracy of the terminal I4d levels is lifted, shifting the peaks in opposite directions
with increasing bond-length asymmetry.
In Figure 5.4, water–triiodide RDFs are decomposed into separate RDFs for each of the
iodine atoms: the center atom and the iodine atoms at the longest and shortest distance from
the center iodine in each timestep. The hydrogen RDF for the iodine at the longest separation
(which is also the more negative side of the molecule) has a pronounced maximum that coincides
30
4.0
left right
long short
3.8
I−I distance / Å
3.6
3.4
3.2
3.0
2.8
2.6
0
10
20
30
Time / ps
40
Distribution
Figure 5.3: The time evolution in the I–I bond lengths contains periods of a pronounced asymmetry,
highlighted by sampling of the longest and shortest I–I bond.
with the first maximum of the iodide ion. This signal of hydrogen bonding between iodine and
water is absent in the coresponding RDFs for the central atom and the closest iodine. From this
we understand that the charge localization is also correlated with the solvent polarization. The
geometry distortions, linked with the rearrangement of the water molecules in the hydrogen bond
network, can explain the fact that the experimental photoelectron spectrum of aqueous triiodide,
shown in panel c in Figure 5.3 (top), is not easily decomposed as in the other solvents.
I−O
I−H
Iterminal,short
Icenter
Iterminal,long
g(r)
I−(aq)
I3−(aq)
I3−(aq)
2
3
4
5
6
Distance / Å
Figure 5.4: I− (aq) (top) and I−
3 (aq) (middle+bottom) RDFs. Thick lines corresponds to I-O and
thin lines to I-H RDFs. The bottom figure distinguishes between the iodine atoms with respect to
their distance to the central atom (“short” or “long”). Only the iodine the farthest from the center
atom (red line) shows a distinct iodine-hydrogen maximum, at close to 3 Å interatomic separation.
The asymmetry of the I−
3 ion may favor iodide polymerization in aqueous solution, and
hence be important for the stability of the DSC: Water impurities could speed up the aging
process, due to polymerization. The coupled solute–solvent dynamics could also influence the
DSC performance through the effect of large geometry distortions on the interactions between
I−
3 and dye or semiconductor surface.
31
Acknowledgements
Everything described here was done with the support from others. First, my supervisor Michael
Odelius: thank you for your genuine help and constructive advice from the very beginning, and
Håkan Rensmo for continuous encouragement. This thesis is really a result of working closely
together with people doing the real liquid experiments – Susanna Kaufmann Eriksson, I sure
enjoy our teamwork. That pertains to everybody involved in the I− /I−
3 story; I am grateful for the
enlightening discussions and nice collaboration with you in Uppsala (and at new address). Many
people have been helpful through different theory and technical discussions. A significant part
of my time at work is spent staring at a Molcas input or output – in particular, Ulf Wahlgren has
been taking the time to explain many a RASSCF-related matter. And, Roland Lindh: Thanks
for helping us tame those bizarre PCM calculations. Considering another great experimental
collaboration, with you at Helmholtz-Zentrum Berlin: I learned a lot (and still do) from your
engaged input to the Ni(II)-project. Finally, to all of you comprising the Chemical Physics
Division at Fysikum: I’m glad that you do.
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