be represented by loop diagrams where the ket moves first forward and then
backward to account for the bra [168]. The second protocol uses the density matrix
and can be represented by ladder diagrams [1]. The first protocol does not maintain
the bookkeeping of relative time ordering of bra and ket interactions and results in
n þ 1 basic terms for the nth order response. The second protocol fully keeps track
of time ordering. Both ket and bra move forward, yielding more 2
n terms [1, 23].
2
Even though the Casida (or CEO) equations of motion represent the reduced single
electron density matrix, this density matrix is simply used for parameterizing the
many electron wavefunction given by a single Slater determinant. The response
predicted by the equations of motion for the density matrix turns out to correspond
to the many-electron wavefunction rather than the density matrix [86, 89].
RT-TDDFT should have many advantages in nonlinear X-ray spectroscopy
simulation. Because it does not calculate individual states, it saves computing
time when many excited states are involved, which is the case for ultrashort
broadband X-ray pulse excitation. Direct propagation of the density matrix, only
involves occupied orbitals, so the computational scaling of RT-TDDFT is much
better than any SOS method [112], which involves a large number of virtual
orbitals. There are already many linear scaling algorithms both in both time [169]
and frequency domains [170] for RT-TDDFT in excited state calculations. Both
methods rely on the diagonal dominance of single-electron density matrices or
transition density matrices. The key issue is to calculate the highly nonlocal
exchange components in the popular hybrid density functionals efficiently, which
has only recently been addressed [171]. Unlike the perturbation method (to be
discussed in the next section), high order functional derivatives are not necessary in
RT-TDDFT calculations, so the well-behaved but complicated energy functionals,
such as the orbital-dependent functionals or the optimized effective potential (OEP)
functionals, can be readily used. In addition, RT-TDDFT has advantages for
nonlinear response because the calculations are no more difficult than for linear
response, whereas for the frequency domain methods such as SOS, they become
increasingly more complex for higher order response. Nuclear motions can also be
accounted for by Ehrenfest dynamics [159, 172]. RT-TDDFT offers a direct
simulation of nonlinear spectroscopy experiments with short pulses. However, we
still have some tradeoffs in using RT-TDDFT. Because individual excited states are
not available in RT-TDDFT, it is hard to interpret the spectral features. So far,
RT-TDDFT applications have been restricted to calculating standard dynamical
hyperpolarizabilities. Simulations of the signals presented in Sect. 2 constitute
challenges.
2 Each electronic oscillator which parameterizes the evolution of a single-electron density matrix
corresponds to a single electronic excited state appearing in the linear response regime. Nonlinear
SOS response calculations are then reformulated as sum-over-oscillator expressions, where multiply excited oscillators appear in the higher order responses leading to 2
n terms. It is interesting
that the expressions for the TDHF CEO [89] (or equivalently the Casida TDDFT [86]) response
obtained from the equations of motion correspond to the wavefunction, not the density matrix.
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
313
backward to account for the bra [168]. The second protocol uses the density matrix
and can be represented by ladder diagrams [1]. The first protocol does not maintain
the bookkeeping of relative time ordering of bra and ket interactions and results in
n þ 1 basic terms for the nth order response. The second protocol fully keeps track
of time ordering. Both ket and bra move forward, yielding more 2
n terms [1, 23].
2
Even though the Casida (or CEO) equations of motion represent the reduced single
electron density matrix, this density matrix is simply used for parameterizing the
many electron wavefunction given by a single Slater determinant. The response
predicted by the equations of motion for the density matrix turns out to correspond
to the many-electron wavefunction rather than the density matrix [86, 89].
RT-TDDFT should have many advantages in nonlinear X-ray spectroscopy
simulation. Because it does not calculate individual states, it saves computing
time when many excited states are involved, which is the case for ultrashort
broadband X-ray pulse excitation. Direct propagation of the density matrix, only
involves occupied orbitals, so the computational scaling of RT-TDDFT is much
better than any SOS method [112], which involves a large number of virtual
orbitals. There are already many linear scaling algorithms both in both time [169]
and frequency domains [170] for RT-TDDFT in excited state calculations. Both
methods rely on the diagonal dominance of single-electron density matrices or
transition density matrices. The key issue is to calculate the highly nonlocal
exchange components in the popular hybrid density functionals efficiently, which
has only recently been addressed [171]. Unlike the perturbation method (to be
discussed in the next section), high order functional derivatives are not necessary in
RT-TDDFT calculations, so the well-behaved but complicated energy functionals,
such as the orbital-dependent functionals or the optimized effective potential (OEP)
functionals, can be readily used. In addition, RT-TDDFT has advantages for
nonlinear response because the calculations are no more difficult than for linear
response, whereas for the frequency domain methods such as SOS, they become
increasingly more complex for higher order response. Nuclear motions can also be
accounted for by Ehrenfest dynamics [159, 172]. RT-TDDFT offers a direct
simulation of nonlinear spectroscopy experiments with short pulses. However, we
still have some tradeoffs in using RT-TDDFT. Because individual excited states are
not available in RT-TDDFT, it is hard to interpret the spectral features. So far,
RT-TDDFT applications have been restricted to calculating standard dynamical
hyperpolarizabilities. Simulations of the signals presented in Sect. 2 constitute
challenges.
2 Each electronic oscillator which parameterizes the evolution of a single-electron density matrix
corresponds to a single electronic excited state appearing in the linear response regime. Nonlinear
SOS response calculations are then reformulated as sum-over-oscillator expressions, where multiply excited oscillators appear in the higher order responses leading to 2
n terms. It is interesting
that the expressions for the TDHF CEO [89] (or equivalently the Casida TDDFT [86]) response
obtained from the equations of motion correspond to the wavefunction, not the density matrix.
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
313