• Core resonant spectroscopy offers a fast and versatile way to trigger valence
excitations at selected positions and times via a stimulated Raman process [6]
and to study their dynamics.
• Multiple cores can be excited at various delays, allowing the study of nonlinear
response of valence electrons.
The complex nature of excited state correlations and dynamics leads to characteristic patterns in nonlinear spectroscopy signals, whose interpretation calls for
state-of-the-art theoretical simulation methods. The simulation of time-domain
nonlinear X-ray spectroscopy signals poses numerous challenges to quantum
chemists. First, resonant X-ray spectroscopy involves core excited state. Most
computational molecular electronic structure activity had focused on the ground
state and valence excitations. Core excitations received much less attention because
they do not participate in typical chemical processes. Core excited states lie well
above many valence excited states, and calculating them directly with bottom-up
algorithms is not practical. Core electrons also have special asymptotic behavior
(cusp condition) close to the nuclei and large relativistic effects. Second, signals
obtained by broadband X-ray pulses require many excited states. A state-by-state
calculation scheme is tedious and it is better to obtain all excited states with an
energy range in one shot. Third, resonant X-ray signals require not only the energies
of excited states but also the transition dipoles between them. It is usually necessary
to calculate high order excited state energy gradients to determine these quantities,
which complicates the simulation. Furthermore, multiple X-ray pulses can easily
create excited states with multiple core holes, which are not well described by
single-reference-based excited state quantum chemistry methods such as adiabatic
time-dependent density functional theory (TDDFT). Most of the discussions in this
chapter are based on adiabatic TDDFT. Non-adiabatic frequency-dependent kernels
are discussed in Sect. 4.3. Finally, many electrons may respond to the core hole
created by the X-ray pulses (e.g., shake-up and shake-off processes) [13], so that
many-body effects are very important in these signals. The single-particle picture
may break down and high level methods such as multireference configuration
interaction (MRCI) or multireference perturbation theory (MRPT) are often necessary to account for electron correlation. These challenges are addressed in the
following sections. We focus on the theoretical methods (mainly DFT/TDDFT)
which have been extensively used in X-ray spectroscopy simulation. There are
excellent reviews on using TDDFT to simulate linear X-ray spectroscopy signals
[13–15]. Here we emphasize the specific issues associated with nonlinear X-ray
spectroscopy simulations and mainly discuss the methods applied to molecules.
This chapter is organized as follows. We first briefly describe the calculation of
various nonlinear X-ray spectroscopy signals, and then review existing quantum
chemistry simulation methods. We then discuss several key issues in nonlinear
X-ray spectroscopy simulation. Finally, conclusions and future directions are
outlined.
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
277
excitations at selected positions and times via a stimulated Raman process [6]
and to study their dynamics.
• Multiple cores can be excited at various delays, allowing the study of nonlinear
response of valence electrons.
The complex nature of excited state correlations and dynamics leads to characteristic patterns in nonlinear spectroscopy signals, whose interpretation calls for
state-of-the-art theoretical simulation methods. The simulation of time-domain
nonlinear X-ray spectroscopy signals poses numerous challenges to quantum
chemists. First, resonant X-ray spectroscopy involves core excited state. Most
computational molecular electronic structure activity had focused on the ground
state and valence excitations. Core excitations received much less attention because
they do not participate in typical chemical processes. Core excited states lie well
above many valence excited states, and calculating them directly with bottom-up
algorithms is not practical. Core electrons also have special asymptotic behavior
(cusp condition) close to the nuclei and large relativistic effects. Second, signals
obtained by broadband X-ray pulses require many excited states. A state-by-state
calculation scheme is tedious and it is better to obtain all excited states with an
energy range in one shot. Third, resonant X-ray signals require not only the energies
of excited states but also the transition dipoles between them. It is usually necessary
to calculate high order excited state energy gradients to determine these quantities,
which complicates the simulation. Furthermore, multiple X-ray pulses can easily
create excited states with multiple core holes, which are not well described by
single-reference-based excited state quantum chemistry methods such as adiabatic
time-dependent density functional theory (TDDFT). Most of the discussions in this
chapter are based on adiabatic TDDFT. Non-adiabatic frequency-dependent kernels
are discussed in Sect. 4.3. Finally, many electrons may respond to the core hole
created by the X-ray pulses (e.g., shake-up and shake-off processes) [13], so that
many-body effects are very important in these signals. The single-particle picture
may break down and high level methods such as multireference configuration
interaction (MRCI) or multireference perturbation theory (MRPT) are often necessary to account for electron correlation. These challenges are addressed in the
following sections. We focus on the theoretical methods (mainly DFT/TDDFT)
which have been extensively used in X-ray spectroscopy simulation. There are
excellent reviews on using TDDFT to simulate linear X-ray spectroscopy signals
[13–15]. Here we emphasize the specific issues associated with nonlinear X-ray
spectroscopy simulations and mainly discuss the methods applied to molecules.
This chapter is organized as follows. We first briefly describe the calculation of
various nonlinear X-ray spectroscopy signals, and then review existing quantum
chemistry simulation methods. We then discuss several key issues in nonlinear
X-ray spectroscopy simulation. Finally, conclusions and future directions are
outlined.
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
277