4.5 Vibronic Coupling
In the Frank–Condon (FC) region, nuclear motion acts as a bath for electronic
transitions. Including this effect introduces high-resolution fine structure to the
recorded spectra. Theoretical simulations are usually under the Born–Oppenheimer
(BO) and the harmonic oscillator approximations, which are usually good in the FC
region. The potential energy surfaces (PESs) of the ground and excited states are
well separated. The simplest approach to account for vibronic coupling is the linear
coupling model (LCM). It assumes the mode-k PES of excited state has the same
curvature as that of the ground state but only shifted by a displacement. This
approach is efficient and can be applied to medium and large molecules. It has
been well illustrated in various linear and nonlinear X-ray spectroscopy calculations including the XPS, XANES, RIXS, and SXRS spectra (see, e.g., [246,
382–388]). A more rigorous method is to include the Duschinsky rotation and/or
the non-Condon effects. For time-domain nonlinear spectroscopy, the nuclear and
electronic coordinates are mixed together. The response function can be evaluated
via the cumulant expansion [1] till truncated order. The vibrationally resolved
SXRS spectra were studied by Hua et al. [387] combining the LCM and cumulant
expansion till the second-order. With the inclusion of vibronic coupling, a faster
decay in the time domain signals, and new splitting and shoulder structures in the
frequency-domain were observed.
5 Conclusions and Perspectives
In this chapter we have surveyed some typical nonlinear X-ray spectroscopy signals
and the quantum chemistry methods used for their simulation. Because of their
balance in accuracy and computational cost, DFT/TDDFT methods are commonly
used in excited state calculations. With the fast development of new exchangecorrelation functionals and linear scaling algorithms, these methods provide a most
valuable quantum chemistry tool for nonlinear X-ray spectroscopy simulation.
DFT/TDDFT often provides an adequate zero order electronic structure at reasonable cost, which paves the way for the application of high level methods. DFT/
TDDFT-based semiempirical methods such as density functional tight binding
(DFTB) [389–392] or time-dependent density functional tight binding (TDDFTB)
[393, 394], and their linear scaling forms [395] have been shown useful in spectroscopy simulations of large systems.
DFT/TDDFT works well in many cases but fails for double excitations, longrange charge transfer excitations, and conical intersections. Much effort has been
made to address these difficulties by designing more elaborate functionals and
schemes. We believe that rather than putting the burden on the functionals, it
makes more sense to use DFT/TDDFT results as fast zero-order inputs to high
level wave-function approaches and many-body techniques. The recent
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
335
In the Frank–Condon (FC) region, nuclear motion acts as a bath for electronic
transitions. Including this effect introduces high-resolution fine structure to the
recorded spectra. Theoretical simulations are usually under the Born–Oppenheimer
(BO) and the harmonic oscillator approximations, which are usually good in the FC
region. The potential energy surfaces (PESs) of the ground and excited states are
well separated. The simplest approach to account for vibronic coupling is the linear
coupling model (LCM). It assumes the mode-k PES of excited state has the same
curvature as that of the ground state but only shifted by a displacement. This
approach is efficient and can be applied to medium and large molecules. It has
been well illustrated in various linear and nonlinear X-ray spectroscopy calculations including the XPS, XANES, RIXS, and SXRS spectra (see, e.g., [246,
382–388]). A more rigorous method is to include the Duschinsky rotation and/or
the non-Condon effects. For time-domain nonlinear spectroscopy, the nuclear and
electronic coordinates are mixed together. The response function can be evaluated
via the cumulant expansion [1] till truncated order. The vibrationally resolved
SXRS spectra were studied by Hua et al. [387] combining the LCM and cumulant
expansion till the second-order. With the inclusion of vibronic coupling, a faster
decay in the time domain signals, and new splitting and shoulder structures in the
frequency-domain were observed.
5 Conclusions and Perspectives
In this chapter we have surveyed some typical nonlinear X-ray spectroscopy signals
and the quantum chemistry methods used for their simulation. Because of their
balance in accuracy and computational cost, DFT/TDDFT methods are commonly
used in excited state calculations. With the fast development of new exchangecorrelation functionals and linear scaling algorithms, these methods provide a most
valuable quantum chemistry tool for nonlinear X-ray spectroscopy simulation.
DFT/TDDFT often provides an adequate zero order electronic structure at reasonable cost, which paves the way for the application of high level methods. DFT/
TDDFT-based semiempirical methods such as density functional tight binding
(DFTB) [389–392] or time-dependent density functional tight binding (TDDFTB)
[393, 394], and their linear scaling forms [395] have been shown useful in spectroscopy simulations of large systems.
DFT/TDDFT works well in many cases but fails for double excitations, longrange charge transfer excitations, and conical intersections. Much effort has been
made to address these difficulties by designing more elaborate functionals and
schemes. We believe that rather than putting the burden on the functionals, it
makes more sense to use DFT/TDDFT results as fast zero-order inputs to high
level wave-function approaches and many-body techniques. The recent
Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray. . .
335