be well described by a multiple Slater determinant expansion based on this reference, and the CI coefficients of the double excitation configurations are dominant,
we call it a doubly excited state. According to this definition, different reference
states may lead to different double-excitation character [300, 301].
Double excitation remains one of the challenges in DFT and TDDFT
[301]. Maitra et al. showed that a frequency-dependent exchange-correlation kernel, which is not available in the common implementation of adiabatic TDDFT, is
essential for accessing double excitations [95]. In contrast to early expectations
[302], quadratic response theory in adiabatic TDDFT only gives double excitation
frequencies as sums of two single excitation frequencies [91, 301, 303]. Adiabatic
TDDFT implicitly assumes that the double excited state wave functions are products of single excited state wave functions. This has been shown in [303]. These
trivial double excitations behave as harmonic oscillators and result in vanishing
double-quantum-coherence signals [24]. Originally dressed TDDFT was proposed
to remedy this double excitation issue, but information about the relevant double
excitation a priori hinders its practical application. Frequency-dependent exchangecorrelation kernels based on the Bethe–Salpeter equation were proposed [224, 304–
306], but they have only been tested in simple models or small molecular systems
because of complexity. The matrix elements of unknown exchange-correlation
kernel can also be extracted from the branching ratio of the experimental
L 2,3 -edge X-Ray absorption spectra of 3d transition metals [307]. Spin-flip
TDDFT (SF-TDDFT) may access some doubly excited states through a triplet
reference state [307–312]. Very recently, the constricted variational DFT
(CV-DFT) [313–315] was developed to address double excitations [316]. Implementation and testing of this method is under way. We have also combined the
ΔSCF method with REW-TDDFT to calculate doubly core excited state and apply
it in X-ray double-quantum-coherence (XDQC) signal simulation [24]. The main
difficulty with this approach is the unbalanced treatment of the two core holes: one
core hole was obtained with ΔSCF and the other with REW-TDDFT. Althgough the
two core holes are not symmetric, different calculation order (either ΔSCF first or
REW-TDDFT first) would lead to different simulation results [24]. Other high level
ab initio methods, such as CASSCF/CASPT2 [317], coupled cluster (CC) [318],
MRCI [319, 320], symmetry-adapted cluster configuration interaction (SAC-CI)
[321], algebraic diagrammatic construction (ADC) [299, 322, 323], and
multireference Møller–Plesset perturbation theory (MRMP) [324, 325], can accurately capture double excitations, but their use is limited to small systems because
of high computational cost.
The DQC signal probes doubly excited states and strongly depends on the
coupling between single excitations. In the infrared regime, DQC signals detect
the couplings between vibrational modes, which determine their anharmonicities
[326]. In the optical regime, DQC signals were used to reveal quantitative information about electron–electron interactions, many-body wave functions, and electron correlation in excitons [327]. The X-ray variant of this technique (XDQC) is
sensitive to correlation and exciton scattering in doubly core excited states, making
it an attractive experimental test for electronic structure theories of strongly
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
329
we call it a doubly excited state. According to this definition, different reference
states may lead to different double-excitation character [300, 301].
Double excitation remains one of the challenges in DFT and TDDFT
[301]. Maitra et al. showed that a frequency-dependent exchange-correlation kernel, which is not available in the common implementation of adiabatic TDDFT, is
essential for accessing double excitations [95]. In contrast to early expectations
[302], quadratic response theory in adiabatic TDDFT only gives double excitation
frequencies as sums of two single excitation frequencies [91, 301, 303]. Adiabatic
TDDFT implicitly assumes that the double excited state wave functions are products of single excited state wave functions. This has been shown in [303]. These
trivial double excitations behave as harmonic oscillators and result in vanishing
double-quantum-coherence signals [24]. Originally dressed TDDFT was proposed
to remedy this double excitation issue, but information about the relevant double
excitation a priori hinders its practical application. Frequency-dependent exchangecorrelation kernels based on the Bethe–Salpeter equation were proposed [224, 304–
306], but they have only been tested in simple models or small molecular systems
because of complexity. The matrix elements of unknown exchange-correlation
kernel can also be extracted from the branching ratio of the experimental
L 2,3 -edge X-Ray absorption spectra of 3d transition metals [307]. Spin-flip
TDDFT (SF-TDDFT) may access some doubly excited states through a triplet
reference state [307–312]. Very recently, the constricted variational DFT
(CV-DFT) [313–315] was developed to address double excitations [316]. Implementation and testing of this method is under way. We have also combined the
ΔSCF method with REW-TDDFT to calculate doubly core excited state and apply
it in X-ray double-quantum-coherence (XDQC) signal simulation [24]. The main
difficulty with this approach is the unbalanced treatment of the two core holes: one
core hole was obtained with ΔSCF and the other with REW-TDDFT. Althgough the
two core holes are not symmetric, different calculation order (either ΔSCF first or
REW-TDDFT first) would lead to different simulation results [24]. Other high level
ab initio methods, such as CASSCF/CASPT2 [317], coupled cluster (CC) [318],
MRCI [319, 320], symmetry-adapted cluster configuration interaction (SAC-CI)
[321], algebraic diagrammatic construction (ADC) [299, 322, 323], and
multireference Møller–Plesset perturbation theory (MRMP) [324, 325], can accurately capture double excitations, but their use is limited to small systems because
of high computational cost.
The DQC signal probes doubly excited states and strongly depends on the
coupling between single excitations. In the infrared regime, DQC signals detect
the couplings between vibrational modes, which determine their anharmonicities
[326]. In the optical regime, DQC signals were used to reveal quantitative information about electron–electron interactions, many-body wave functions, and electron correlation in excitons [327]. The X-ray variant of this technique (XDQC) is
sensitive to correlation and exciton scattering in doubly core excited states, making
it an attractive experimental test for electronic structure theories of strongly
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
329