core excited states [14]. The long-range corrected density functionals are discussed
in Sect. 4.1. In addition, within the adiabatic (frequency-independent) exchangecorrelation kernels, linear-response TDDFT cannot properly describe double excitations [95]. This is a major obstacle for simulating nonlinear spectroscopy experiments, which directly access double or multiple excited states. Double excitations
and frequency-dependent exchange-correlation kernels are discussed in Sect. 4.3.
3.2.2 Restricted Excitation Window TDDFT
Calculating core excited states directly using (52) is prohibitively expensive
because there are numerous low energy excited states below the target high energy
core excited state. Any bottom-up matrix eigenvalue numerical algorithm becomes
very tedious. This difficulty can be circumvented by allowing electrons to move
only between a certain set of relevant occupied and virtual orbitals. This is the basis
for the restricted excitation window (REW) or restricted excitation channel
approach. This method was proposed by Stener and co-workers [96], and followed
by other authors [97–100]. One can select the orbitals in the restricted excitation
window by their orbital indices or energies. In the first scheme all molecular orbitals
(MO) are examined and then the relevant orbitals (e.g., the MOs dominated by the
target oxygen 1s atomic orbitals) are selected out. Alternatively, an orbital energy
or energy difference cutoff is used to filter out all relevant orbitals or transition
orbital pairs. Orbital index selection is intuitive but becomes cumbersome if there
are too many relevant orbitals. The orbital energy (energy difference) selection
scheme is convenient for building a large REW. If there are multiple target atoms of
the same type in the molecule, the target MOs would become degenerate or neardegenerate. The orbital index selection scheme can explore the contribution of a
single target atom to the core excitation and the spectroscopy signal, whereas the
orbital energy selection scheme can study hole-mixing effects. Another method
with the same effect of building a REW is to shift the core excitation energy
difference. This was proposed by Schmidt et al. [101] very recently and is very
similar to Stener and co-workers’ early implementation in the ADF package. After
the REW is determined, trial excitation vectors are prepared in this REW and a
Davidson-type iterative solver [102] is usually employed to find the relevant matrix
eigenvalues and eigenvectors. REW-TDDFT has been implemented in standard
quantum chemistry packages such as ADF [103], Q-Chem [104], ORCA [105],
NWChem [106], and Gaussian [107].
Minimum inputs (relevant orbitals, number of excited states) are needed for
running a REW-TDDFT calculation. It is almost black-box and robust and can
handle all types of excited states with deep as well as shallow holes. It is a response
method and avoids the state-specific SCF convergence problem. Hole-mixing can
be observed, which is not possible with STEX or ΔSCF-DFT. Electron correlation
can be considered in the exchange-correlation functional. Moreover, REW-TDDFT
can easily calculate many core excited states. If unrelaxed CIS-type wave functions
(Tamm–Dancoff approximation, TDA) are used to represent the excited states, the
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
303
in Sect. 4.1. In addition, within the adiabatic (frequency-independent) exchangecorrelation kernels, linear-response TDDFT cannot properly describe double excitations [95]. This is a major obstacle for simulating nonlinear spectroscopy experiments, which directly access double or multiple excited states. Double excitations
and frequency-dependent exchange-correlation kernels are discussed in Sect. 4.3.
3.2.2 Restricted Excitation Window TDDFT
Calculating core excited states directly using (52) is prohibitively expensive
because there are numerous low energy excited states below the target high energy
core excited state. Any bottom-up matrix eigenvalue numerical algorithm becomes
very tedious. This difficulty can be circumvented by allowing electrons to move
only between a certain set of relevant occupied and virtual orbitals. This is the basis
for the restricted excitation window (REW) or restricted excitation channel
approach. This method was proposed by Stener and co-workers [96], and followed
by other authors [97–100]. One can select the orbitals in the restricted excitation
window by their orbital indices or energies. In the first scheme all molecular orbitals
(MO) are examined and then the relevant orbitals (e.g., the MOs dominated by the
target oxygen 1s atomic orbitals) are selected out. Alternatively, an orbital energy
or energy difference cutoff is used to filter out all relevant orbitals or transition
orbital pairs. Orbital index selection is intuitive but becomes cumbersome if there
are too many relevant orbitals. The orbital energy (energy difference) selection
scheme is convenient for building a large REW. If there are multiple target atoms of
the same type in the molecule, the target MOs would become degenerate or neardegenerate. The orbital index selection scheme can explore the contribution of a
single target atom to the core excitation and the spectroscopy signal, whereas the
orbital energy selection scheme can study hole-mixing effects. Another method
with the same effect of building a REW is to shift the core excitation energy
difference. This was proposed by Schmidt et al. [101] very recently and is very
similar to Stener and co-workers’ early implementation in the ADF package. After
the REW is determined, trial excitation vectors are prepared in this REW and a
Davidson-type iterative solver [102] is usually employed to find the relevant matrix
eigenvalues and eigenvectors. REW-TDDFT has been implemented in standard
quantum chemistry packages such as ADF [103], Q-Chem [104], ORCA [105],
NWChem [106], and Gaussian [107].
Minimum inputs (relevant orbitals, number of excited states) are needed for
running a REW-TDDFT calculation. It is almost black-box and robust and can
handle all types of excited states with deep as well as shallow holes. It is a response
method and avoids the state-specific SCF convergence problem. Hole-mixing can
be observed, which is not possible with STEX or ΔSCF-DFT. Electron correlation
can be considered in the exchange-correlation functional. Moreover, REW-TDDFT
can easily calculate many core excited states. If unrelaxed CIS-type wave functions
(Tamm–Dancoff approximation, TDA) are used to represent the excited states, the
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
303