Multiconfigurational Approach to X-ray Spectroscopy …
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energy, for example 3p instead of 2p. Unless this is prevented using e.g., a restricted
step algorithm [40], the core orbitals have to be frozen during the orbital optimization.
A downside to this is that this prevents the expected contraction of those orbitals upon
core excitation, but the main effect is expected to be a global shift of the final states
to higher energy. Often the relative edge position is more important than the absolute
one, and the frozen-core approximation has been used to predict oxidation state
shifts with errors of 0.3 eV when using the same active space and number of states
[33, 49].
3.4 Simulating Light-Matter Interaction
The last step to obtain a spectrum is to compute transition intensities at different
wavelengths. For bound states, the intensity can be calculated from the matrix element of the operator representing the light-matter interaction between the initial and
final states. Most often, the plane wave representing the light is approximated by an
electric dipole. If core-hole states have been optimized separately from the valence
states, transition intensities can still be calculated correctly by taking advantage of a
biorthogonalization scheme [61]. From the computed intensities, the final spectrum
is generated using a Lorentzian lifetime broadening convoluted with a Gaussian
broadening to account for the experimental resolution. If only absorption intensities
are considered, this corresponds to a spectrum collected in transmission mode. However, the transmission spectra and those obtained by measuring the photoelectron
current, the total electron yield, are similar. RIXS spectra can be calculated from
the transition intensities for both absorption and emission processes according to the
Kramers–Heisenberg formula [25]. L-edge XAS spectra of metal complexes in solution are in many cases collected by measuring the fluorescence from the core-excited
states, the partial fluorescence yield (PFY) mode. The PFY-XAS spectra can be calculated from the RIXS cross sections by integrating the relevant emission channels
for each incident energy [27, 28, 47].
Results obtained for L-edge XAS of ferric chloride and ferricyanide are shown in
Fig. 4 [36, 73, 91]. The spectra are divided into two separate regions, L 3 and L 2 , split
by the 2p spin–orbit coupling, as will be discussed in detail below. RAS calculations
can be used to correlate spectra and electronic structure. In ferric chloride, the ligand
field is weak and different configurations mix strongly, making it difficult to assign
transitions to specific 3d orbitals. One exception is the high-energy peak in the L 3
edge that has been identified as a 2p → 3d transition, combined with a σ → 3d
LMCT [71, 91]. In ferricyanide the ligand field is strong, and a molecular orbital
picture becomes more relevant. The sharp first peak corresponds to a 2p electron
filling the t 2g hole and the second peak is an excitation to e g . The third peak is a
signature of π -backbonding and is typically labeled after the π
∗ molecular orbital,
see Fig. 3, but is in reality a strong mix of e g and π
∗ contributions [36, 71].
The 1s → 3d transitions of K pre-edge are dipole forbidden in centrosymmetric complexes and still relatively weak in many other systems. In the former case,
193
energy, for example 3p instead of 2p. Unless this is prevented using e.g., a restricted
step algorithm [40], the core orbitals have to be frozen during the orbital optimization.
A downside to this is that this prevents the expected contraction of those orbitals upon
core excitation, but the main effect is expected to be a global shift of the final states
to higher energy. Often the relative edge position is more important than the absolute
one, and the frozen-core approximation has been used to predict oxidation state
shifts with errors of 0.3 eV when using the same active space and number of states
[33, 49].
3.4 Simulating Light-Matter Interaction
The last step to obtain a spectrum is to compute transition intensities at different
wavelengths. For bound states, the intensity can be calculated from the matrix element of the operator representing the light-matter interaction between the initial and
final states. Most often, the plane wave representing the light is approximated by an
electric dipole. If core-hole states have been optimized separately from the valence
states, transition intensities can still be calculated correctly by taking advantage of a
biorthogonalization scheme [61]. From the computed intensities, the final spectrum
is generated using a Lorentzian lifetime broadening convoluted with a Gaussian
broadening to account for the experimental resolution. If only absorption intensities
are considered, this corresponds to a spectrum collected in transmission mode. However, the transmission spectra and those obtained by measuring the photoelectron
current, the total electron yield, are similar. RIXS spectra can be calculated from
the transition intensities for both absorption and emission processes according to the
Kramers–Heisenberg formula [25]. L-edge XAS spectra of metal complexes in solution are in many cases collected by measuring the fluorescence from the core-excited
states, the partial fluorescence yield (PFY) mode. The PFY-XAS spectra can be calculated from the RIXS cross sections by integrating the relevant emission channels
for each incident energy [27, 28, 47].
Results obtained for L-edge XAS of ferric chloride and ferricyanide are shown in
Fig. 4 [36, 73, 91]. The spectra are divided into two separate regions, L 3 and L 2 , split
by the 2p spin–orbit coupling, as will be discussed in detail below. RAS calculations
can be used to correlate spectra and electronic structure. In ferric chloride, the ligand
field is weak and different configurations mix strongly, making it difficult to assign
transitions to specific 3d orbitals. One exception is the high-energy peak in the L 3
edge that has been identified as a 2p → 3d transition, combined with a σ → 3d
LMCT [71, 91]. In ferricyanide the ligand field is strong, and a molecular orbital
picture becomes more relevant. The sharp first peak corresponds to a 2p electron
filling the t 2g hole and the second peak is an excitation to e g . The third peak is a
signature of π -backbonding and is typically labeled after the π
∗ molecular orbital,
see Fig. 3, but is in reality a strong mix of e g and π
∗ contributions [36, 71].
The 1s → 3d transitions of K pre-edge are dipole forbidden in centrosymmetric complexes and still relatively weak in many other systems. In the former case,