At the level of Hartree theory, which assumes independent electrons, the DQC
signal vanishes because of interference. TDHF (or TDDFT) goes one step further
and provides a picture of independent transitions (quasiparticles). Here the signal
no longer vanishes, but shows a limited number of peaks. When correlation effects
are fully incorporated, the many-electron wavefunctions become superpositions of
states with different numbers and types of e-h pairs. The Ω 2 and Ω 3 axes then
contain many more peaks corresponding to all many-body states (in the frequency
range spanned by the pulse bandwidths), which project into the doubly-excited
states. Thus, along Ω 2 the peaks are shifted, reflecting the level of theory used to
describe electron correlations. Along Ω 3 , the effect is even more dramatic and new
peaks show up corresponding to splittings between various levels. We show the
X-ray DQC signals of formamide as an example in Sect. 4.3. This highly-resolved
two-dimensional spectrum provides an invaluable direct dynamical probe of electron correlations (both energies and wavefunctions) [25, 26].
2.3 Stimulated X-Ray Raman Spectroscopy
Linear techniques contain the single-excitation spectrum whereas we have just seen
that the DQC (k III ) signal gives access to the double-excitation spectrum. Both of
these spectra thus characterize the intermanifold structure of the material (the
transitions between manifolds). We may obtain a window into the intramanifold
structure (transitions within the same manifold) by using the stimulated Raman
signal (SXRS in the X-ray regime) [2, 27, 28]. As with the DQC signal, this technique
is third-order (involving four interactions with the electromagnetic field). However,
rather than four sequential pulses, 1D-SXRS employs only two pulses, each of which
interacts twice with the material. This process is shown diagramatically in Fig. 5.
Note that, because the pair of interactions with each pulse are of opposite
Hermiticity, the overall absolute phase is ϕ 1 À ϕ 1 þ ϕ 4 À ϕ 4 ¼ 0 and this technique
therefore does not require phase control to obtain a finite signal.
The first pulse in the SXRS process creates a superposition of excited states in
the ground state manifold. After a controlled delay period, the sample interacts with
the second pulse which returns the system to the original state.
For calculating this signal, we find it more convenient to work with the actual
interaction times τ rather than the time delays t j j ¼ 1, 2, 3
ð
Þ . It is straightforward
to write down a time-domain expression for the 1D-SXRS signal directly from the
diagrams in Fig. 5. Its form is similar to (10):
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Y. Zhang et al.
signal vanishes because of interference. TDHF (or TDDFT) goes one step further
and provides a picture of independent transitions (quasiparticles). Here the signal
no longer vanishes, but shows a limited number of peaks. When correlation effects
are fully incorporated, the many-electron wavefunctions become superpositions of
states with different numbers and types of e-h pairs. The Ω 2 and Ω 3 axes then
contain many more peaks corresponding to all many-body states (in the frequency
range spanned by the pulse bandwidths), which project into the doubly-excited
states. Thus, along Ω 2 the peaks are shifted, reflecting the level of theory used to
describe electron correlations. Along Ω 3 , the effect is even more dramatic and new
peaks show up corresponding to splittings between various levels. We show the
X-ray DQC signals of formamide as an example in Sect. 4.3. This highly-resolved
two-dimensional spectrum provides an invaluable direct dynamical probe of electron correlations (both energies and wavefunctions) [25, 26].
2.3 Stimulated X-Ray Raman Spectroscopy
Linear techniques contain the single-excitation spectrum whereas we have just seen
that the DQC (k III ) signal gives access to the double-excitation spectrum. Both of
these spectra thus characterize the intermanifold structure of the material (the
transitions between manifolds). We may obtain a window into the intramanifold
structure (transitions within the same manifold) by using the stimulated Raman
signal (SXRS in the X-ray regime) [2, 27, 28]. As with the DQC signal, this technique
is third-order (involving four interactions with the electromagnetic field). However,
rather than four sequential pulses, 1D-SXRS employs only two pulses, each of which
interacts twice with the material. This process is shown diagramatically in Fig. 5.
Note that, because the pair of interactions with each pulse are of opposite
Hermiticity, the overall absolute phase is ϕ 1 À ϕ 1 þ ϕ 4 À ϕ 4 ¼ 0 and this technique
therefore does not require phase control to obtain a finite signal.
The first pulse in the SXRS process creates a superposition of excited states in
the ground state manifold. After a controlled delay period, the sample interacts with
the second pulse which returns the system to the original state.
For calculating this signal, we find it more convenient to work with the actual
interaction times τ rather than the time delays t j j ¼ 1, 2, 3
ð
Þ . It is straightforward
to write down a time-domain expression for the 1D-SXRS signal directly from the
diagrams in Fig. 5. Its form is similar to (10):
284
Y. Zhang et al.