Vibrational effects in x-ray absorption and resonant inelastic x-ray scattering using a semiclassical scheme

A method is presented for describing vibrational effects in x-ray absorption spectroscopy and resonant inelastic x-ray scattering (RIXS) using a combination of the classical Franck-Condon (FC) approximation and classical trajectories run on the core-excited state. The formulation of RIXS is an extension of the semiclassical Kramers-Heisenberg formalism of Ljungberg et al. [Phys. Rev. B 82, 245115 (2010)] to the resonant case, retaining approximately the same computational cost. To overcome difficulties with connecting the absorption and emission processes in RIXS, the classical FC approximation is used for the absorption, which is seen to work well provided that a zero-point-energy correction is included. In the case of core-excited states with dissociative character, the method is capable of closely reproducing the main features for one-dimensional test systems, compared to the quantum-mechanical formulation. Due to the good accuracy combined with the relatively low computational cost, the method has great potential of being used for complex systems with many degrees of freedom, such as liquids and surface adsorbates.

Figure 1

XAS spectra for the high-frequency ground state. Left: the classical FC method. Blue: quantum XAS; red: classical FC without ZPE; yellow: classic FC with ZPE. Right: semiclassical XAS. Blue: quantum XAS; red: dynamics on ground state; yellow: dynamics on excited state; purple: dynamics on the average between ground and excited states. The spectra are normalized.

Figure 2

XAS spectra for the low-frequency ground state. Left: the classical FC method. Blue: quantum XAS; red: classical FC without ZPE; yellow: classic FC with ZPE. Right: semiclassical XAS. Blue: quantum XAS; red: dynamics on ground state. The spectra are normalized.

Figure 3

RIXS maps for HCl. Upper: KH; lower: SCKH.

Figure 4

Averaged spectra for HCl. Upper frame: averaged spectrum over outgoing frequency. Blue: KH; red: SCKH. Lower frame: spectrum averaged over incoming frequency. Blue: KH; red: SCKH. The spectra are normalized.

Figure 5

Normalized emission spectra for HCl. Blue: KH; red: SCKH. The incoming energy goes from 194 to 212 eV, bottom to top.

Figure 6

RIXS map for the methanol dimer. Upper frame: KH; middle frame: SCKH with individual dynamics; lower frame: SCKH with dynamics on the first core-excited state.

Figure 7

Averaged spectra for the methanol dimer. Upper frame: averaged spectrum over outgoing frequency. Blue: KH; red: SCKH. Lower frame: spectrum averaged over incoming frequency. Blue: KH; red: SCKH with individual dynamics; yellow: SCKH with dynamics on first excited state. The spectra are normalized.

Figure 8

Normalized emission spectra for the methanol dimer. Blue: KH; red: SCKH with individual dynamics; yellow: SCKH with dynamics on first excited state. The incoming frequency goes from 526 to 544 eV, bottom to top.