[219]. The GW/BSE approach has been successfully applied to XANES calculations of solids [225–231]. Until now, GW/BSE is still a theory of heavy solid state
flavor. Adapting it to a molecular theory is currently in progress [232–234].
The many-body Green’s function techniques with algebraic diagrammatic construction (ADC) [235] was also used to study core excited states [236] and the
dynamics of core holes and particles [237, 238]. So far, most applications of the
above many-body methods have been made to XANES, and their use to calculate
core excited states is limited to small systems because of computational cost.
Nonlinear X-ray spectroscopy simulation of large systems is an important
future goal.
Other theoretical methods were designed primarily for metal L-edge calculations. These are more challenging than the ligand or metal K-edge calculations
because of multiplet effects, spin-orbital coupling, and metal-to-ligand or ligand-tometal charge transfer. Such transition metal-based systems have attracted broad
interest because of the numerous applications in biology (e.g., metallic enzyme
centers) and artificial light harvesting (e.g., dye-sensitized solar cells). Because the
2p ! 3d transitions are dipole-allowed, the metal L 2 , 3 -edge spectra can better
reflect the valence electron structure of the metal 3d orbitals which are more
essential to the chemistry. For metal K-edge spectra, the 1s ! 3d transitions are
dipole forbidden. These methods have mainly been applied to XANES and RIXS
spectra. Nonlinear X-ray spectra require accurate transition dipole moments. To
obtain these, both the valence and core-excited states must be treated with consistent accuracy.
Early theoretical efforts on transition metal L-edge X-ray spectra were based on
semi-empirical methods developed by de Groot and coworkers [239, 240], namely
the crystal field multiplet (CFM) and the charge transfer multiplet (CTM) models.
These methods start with the SO-coupled multiplets of the excited metal atom and
include the effect of ligands using ligand field theory (LFT). Adjustable parameters
include the crystal field splitting and the charge transfer energy. In recent years,
there are developments in ab initio theory including the ab initio CTM based on
DFT-CI [241], ab initio multiplet ligand-field theory (MLFT) method with Wannier
orbitals [242], and method employing the Russel–Saunders coupling [243]. These
methods are usually computationally expensive and were employed for relatively
small systems with high symmetry. Neese and co-workers [244, 245] proposed an
efficient approach by combining DFT and the restricted-open-shell configuration
interaction singles (DFT/ROCIS). It introduces global empirical parameters for the
periodic table to scale the CI matrix and includes dynamic correlation and the SO
coupling effects. Excellent agreement with experiment was obtained for most
systems. Besides RASSCF, Odelius et al. [202] further tested the influence of
dynamic correlations by using the multiconfigurational second-order perturbation
theory (RASPT2) for a [Ni
II (H 2 O) 6 ]
2+ complex. TDDFT has also been tested for
this topic. Although it is widely believed that TDDFT is only valid for transition
metal-based systems with closed-shell (e.g., the low spin form of Fe
II , S ¼ O) but
not to those open-shells (e.g., the high spin form of Fe
II , S ¼ 2), it is still necessary
to examine its performance because of its high efficiency. It was found [246] that,
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
321
flavor. Adapting it to a molecular theory is currently in progress [232–234].
The many-body Green’s function techniques with algebraic diagrammatic construction (ADC) [235] was also used to study core excited states [236] and the
dynamics of core holes and particles [237, 238]. So far, most applications of the
above many-body methods have been made to XANES, and their use to calculate
core excited states is limited to small systems because of computational cost.
Nonlinear X-ray spectroscopy simulation of large systems is an important
future goal.
Other theoretical methods were designed primarily for metal L-edge calculations. These are more challenging than the ligand or metal K-edge calculations
because of multiplet effects, spin-orbital coupling, and metal-to-ligand or ligand-tometal charge transfer. Such transition metal-based systems have attracted broad
interest because of the numerous applications in biology (e.g., metallic enzyme
centers) and artificial light harvesting (e.g., dye-sensitized solar cells). Because the
2p ! 3d transitions are dipole-allowed, the metal L 2 , 3 -edge spectra can better
reflect the valence electron structure of the metal 3d orbitals which are more
essential to the chemistry. For metal K-edge spectra, the 1s ! 3d transitions are
dipole forbidden. These methods have mainly been applied to XANES and RIXS
spectra. Nonlinear X-ray spectra require accurate transition dipole moments. To
obtain these, both the valence and core-excited states must be treated with consistent accuracy.
Early theoretical efforts on transition metal L-edge X-ray spectra were based on
semi-empirical methods developed by de Groot and coworkers [239, 240], namely
the crystal field multiplet (CFM) and the charge transfer multiplet (CTM) models.
These methods start with the SO-coupled multiplets of the excited metal atom and
include the effect of ligands using ligand field theory (LFT). Adjustable parameters
include the crystal field splitting and the charge transfer energy. In recent years,
there are developments in ab initio theory including the ab initio CTM based on
DFT-CI [241], ab initio multiplet ligand-field theory (MLFT) method with Wannier
orbitals [242], and method employing the Russel–Saunders coupling [243]. These
methods are usually computationally expensive and were employed for relatively
small systems with high symmetry. Neese and co-workers [244, 245] proposed an
efficient approach by combining DFT and the restricted-open-shell configuration
interaction singles (DFT/ROCIS). It introduces global empirical parameters for the
periodic table to scale the CI matrix and includes dynamic correlation and the SO
coupling effects. Excellent agreement with experiment was obtained for most
systems. Besides RASSCF, Odelius et al. [202] further tested the influence of
dynamic correlations by using the multiconfigurational second-order perturbation
theory (RASPT2) for a [Ni
II (H 2 O) 6 ]
2+ complex. TDDFT has also been tested for
this topic. Although it is widely believed that TDDFT is only valid for transition
metal-based systems with closed-shell (e.g., the low spin form of Fe
II , S ¼ O) but
not to those open-shells (e.g., the high spin form of Fe
II , S ¼ 2), it is still necessary
to examine its performance because of its high efficiency. It was found [246] that,
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
321