Ab-initio wave-length dependent Raman spectra: Placzek approximation and beyond

TL;DR

This paper compares the Placzek and Albrecht approximations for calculating Raman spectra, showing Placzek's qualitative agreement and Albrecht's ability to include multiple vibrational excitations, with implications for resonant Raman spectroscopy.

Contribution

It provides a detailed derivation and comparison of Placzek and Albrecht approximations, highlighting their differences and applicability in resonance conditions.

Findings

Placzek approximation agrees qualitatively with Albrecht in non-resonant and single vibrational excitations

Albrecht approximation captures multiple vibrational excitations and matches experimental spectra

Placzek approximation omits multiple vibrational excitations, limiting its accuracy in resonance

Abstract

We analyze how to obtain non-resonant and resonant Raman spectra within the Placzek as well as the Albrecht approximation. Both approximations are derived from the matrix element for light scattering by application of the Kramers, Heisenberg and Dirac formula. It is shown that the Placzek expression results from a semi-classical approximation of the combined electronic and vibrational transition energies. Molecular hydrogen, water and butadiene are studied as test cases. It turns out that the Placzek approximation agrees qualitatively with the more accurate Albrecht formulation even in the resonant regime for the excitations of single vibrational quanta. However, multiple vibrational excitations are absent in Placzek, but can be of similar intensities as single excitations under resonance conditions. The Albrecht approximation takes multiple vibrational excitations into account and the resulting simulated spectra exhibit good agreement with experimental Raman spectra in the resonance region as well.