state-to-state transition dipoles in REW-TDDFT reduce to sums of transition
dipoles between certain MOs because all MOs are mutually orthogonal. This
drastically reduces the computational cost compared to (42).
In order to calculate nonlinear X-ray spectroscopy signals, we developed a
computational approach based on REW-TDDFT implemented in the quantum
chemistry package NWChem. In a series of publications [6, 69, 108], we had
extended the conventional optical Raman spectroscopy techniques into the X-ray
regime. One- and two-dimensional stimulated X-ray Raman spectroscopy (1D- and
2D-SXRS) signals of the small amino acid cysteine were simulated and compared
to the conventional resonant inelastic X-ray scattering (RIXS) signals. Compared to
RIXS, which is a frequency domain technique, multi-color time domain SXRS
provide a better window to the electronic coupling dynamics in a molecule. We also
calculate the X-ray four-wave mixing k I ¼ Àk 1 þ k 2 þ k 3 and k II ¼ k 1 À k 2 þ k 3
signals. To compare these with the SXRS signals, we took the time delay t 2 between
k 2 and k 3 longer than the lifetimes (<10 fs) of the core excited states in this system,
so that only the ground-state-bleach (GSB) terms in the signals survive. k I,II signals
have three frequency variables Ω j ( j ¼ 1, 2, 3), where Ω 1 and Ω 3 correspond to core
excitations and Ω 2 corresponds to valence excitations. We can cut some slices of
these 3D signals to interpret them. In Fig. 10 we show slices of the two-color k II
(OOSS) signal with constant Ω 2 at different peaks in the two-color integrated
two-pulse SXRS signal. These plots show the correlation between core excitations
at different places in the molecule. We can find both the valence excitations at
Ω 2 ¼ 6.6 and 8.9 eV are coupled to the S1s core excitation at Ω 3 ¼ 2475.5 eV, but
they are coupled to different O1s core excitations at Ω 1 ¼ 532.2 and 536.1 eV,
respectively. The valence excitation at Ω 2 ¼ 11.4 is coupled to S1s core excitation
with higher energies. Moreover, the frequency dispersed two-pulse SXRS signals
can give the same information about electron correlation as do projected photon
echo signals [108], whereas the SXRS experiment is much simpler than the photon
echo. However, photo echo experiments have more control variables and can reveal
the correlation between core excitations directly (see Fig. 10), although SXRS can
only infer them through valence excitations.
The same simulation approach was applied to porphyrin dimers. Multiporphyrin
systems are good candidates for artificial photosynthesis or molecular electronics
applications, so understanding the detailed excitation energy transfer (EET) mechanisms in these systems becomes very important. Simulated SXRS signals of various
porphyrin heterodimer systems were obtained [92, 93] using REW-TDDFT. In
Fig. 11 we show the time-domain 1D SXRS signals and the corresponding evolving
electron and hole densities in the Zn and Ni porphyrin heterodimer (structure shown
on the top of Fig. 11). We found an almost constant π/2 phase difference between
the one-color Zn2p pump and Zn2p probe (Zn2p/Zn2p) signal and the two-color
Zn2p pump and Ni2p probe (Zn2p/Ni2p) signal ((c) at bottom left in Fig. 11).
Because the SXRS signal can be considered as an overlap between the timedependent doorway wavepacket created by the pump pulse and the timeindependent window wavepacket created by the probe pulse [92], this phase
difference corresponds to a back-and-forth motion of the doorway wavepacket.
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Y. Zhang et al.
dipoles between certain MOs because all MOs are mutually orthogonal. This
drastically reduces the computational cost compared to (42).
In order to calculate nonlinear X-ray spectroscopy signals, we developed a
computational approach based on REW-TDDFT implemented in the quantum
chemistry package NWChem. In a series of publications [6, 69, 108], we had
extended the conventional optical Raman spectroscopy techniques into the X-ray
regime. One- and two-dimensional stimulated X-ray Raman spectroscopy (1D- and
2D-SXRS) signals of the small amino acid cysteine were simulated and compared
to the conventional resonant inelastic X-ray scattering (RIXS) signals. Compared to
RIXS, which is a frequency domain technique, multi-color time domain SXRS
provide a better window to the electronic coupling dynamics in a molecule. We also
calculate the X-ray four-wave mixing k I ¼ Àk 1 þ k 2 þ k 3 and k II ¼ k 1 À k 2 þ k 3
signals. To compare these with the SXRS signals, we took the time delay t 2 between
k 2 and k 3 longer than the lifetimes (<10 fs) of the core excited states in this system,
so that only the ground-state-bleach (GSB) terms in the signals survive. k I,II signals
have three frequency variables Ω j ( j ¼ 1, 2, 3), where Ω 1 and Ω 3 correspond to core
excitations and Ω 2 corresponds to valence excitations. We can cut some slices of
these 3D signals to interpret them. In Fig. 10 we show slices of the two-color k II
(OOSS) signal with constant Ω 2 at different peaks in the two-color integrated
two-pulse SXRS signal. These plots show the correlation between core excitations
at different places in the molecule. We can find both the valence excitations at
Ω 2 ¼ 6.6 and 8.9 eV are coupled to the S1s core excitation at Ω 3 ¼ 2475.5 eV, but
they are coupled to different O1s core excitations at Ω 1 ¼ 532.2 and 536.1 eV,
respectively. The valence excitation at Ω 2 ¼ 11.4 is coupled to S1s core excitation
with higher energies. Moreover, the frequency dispersed two-pulse SXRS signals
can give the same information about electron correlation as do projected photon
echo signals [108], whereas the SXRS experiment is much simpler than the photon
echo. However, photo echo experiments have more control variables and can reveal
the correlation between core excitations directly (see Fig. 10), although SXRS can
only infer them through valence excitations.
The same simulation approach was applied to porphyrin dimers. Multiporphyrin
systems are good candidates for artificial photosynthesis or molecular electronics
applications, so understanding the detailed excitation energy transfer (EET) mechanisms in these systems becomes very important. Simulated SXRS signals of various
porphyrin heterodimer systems were obtained [92, 93] using REW-TDDFT. In
Fig. 11 we show the time-domain 1D SXRS signals and the corresponding evolving
electron and hole densities in the Zn and Ni porphyrin heterodimer (structure shown
on the top of Fig. 11). We found an almost constant π/2 phase difference between
the one-color Zn2p pump and Zn2p probe (Zn2p/Zn2p) signal and the two-color
Zn2p pump and Ni2p probe (Zn2p/Ni2p) signal ((c) at bottom left in Fig. 11).
Because the SXRS signal can be considered as an overlap between the timedependent doorway wavepacket created by the pump pulse and the timeindependent window wavepacket created by the probe pulse [92], this phase
difference corresponds to a back-and-forth motion of the doorway wavepacket.
304
Y. Zhang et al.