On transition rates in surface hopping

TL;DR

This paper provides a rigorous theoretical derivation of transition rates in surface hopping algorithms, clarifying the physics behind nonadiabatic transitions and validating key features of the standard method.

Contribution

It derives the transition rates in surface hopping from a Gaussian wave packet limit, elucidating the physical basis of the hopping rules and their key features.

Findings

Transition rates are derived at avoided crossings.

Hopping probabilities are non-zero at avoided crossings.

Electronic transition rates are linear in nonadiabatic coupling vectors.

Abstract

Trajectory surface hopping (TSH) is one of the most widely used quantum-classical algorithms for nonadiabatic molecular dynamics. Despite its empirical effectiveness and popularity, a rigorous derivation of TSH as the classical limit of a combined quantum electron-nuclear dynamics is still missing. In this work we aim to elucidate the theoretical basis for the widely used hopping rules. Naturally, we concentrate thereby on the formal aspects of the TSH. Using a Gaussian wave packet limit, we derive the transition rates governing the hopping process at a simple avoided level crossing. In this derivation, which gives insight into the physics underlying the hopping process, some essential features of the standard TSH algorithm are retrieved, namely i) non-zero electronic transition rate ("hopping probability") at avoided crossings; ii) rescaling of the nuclear velocities to conserve total energy; iii) electronic transition rates linear in the nonadiabatic coupling vectors. The well-known Landau-Zener model is then used for illustration.