Franck-Condon spectra of unbound and imaginary-frequency vibrations via correlation functions: a branch-cut free, numerically stable derivation

TL;DR

This paper introduces a numerically stable, branch-cut free method to compute Franck-Condon spectra for both real and imaginary vibrational frequencies using correlation functions, enhancing accuracy for complex molecular spectra.

Contribution

It provides a novel Lie algebra-based derivation of harmonic auto-correlation functions valid for real and imaginary frequencies, improving numerical stability and extending to anharmonic effects.

Findings

Accurate simulation of electronic absorption spectra for various molecules.

Method remains stable with finite-precision arithmetic.

Extensions account for anharmonic and Herzberg-Teller effects.

Abstract

Molecular electronic spectra can be represented in the time domain as auto-correlation functions of the initial vibrational wavepacket. We present a derivation of the harmonic vibrational auto-correlation function that is valid for both real and imaginary harmonic frequencies. The derivation rests on Lie algebra techniques that map otherwise complicated exponential operator arithmetic to simpler matrix formulae. The expressions for the zero- and finite-temperature harmonic auto-correlation functions have been carefully structured both to be free of branch-cut discontinuities and to remain numerically stable with finite-precision arithmetic. Simple extensions correct the harmonic Franck-Condon approximation for the lowest-order anharmonic and Herzberg-Teller effects. Quantitative simulations are shown for several examples, including the electronic absorption spectra of F$_2$, HOCl, CH$_2$NH, and NO$_2$.