The fewest switches surface hopping as an optimisation problem

Abstract

Several ways of defining the probabilities of quantum state transitions in nonadiabatic trajectory surface hopping simulations have been proposed and successfully applied to date. Despite their success, there is a question about the uniqueness of the ways to define such probabilities – some formulations require mathematically-motivated but still ad hoc assumptions to resolve the otherwise under-determined problem. In this work, I present a new approach (termed FSSH-3) to define the hop proposal probabilities. Unlike other approaches, it does not require any ad hoc assumptions and is based on the least squares optimisation of certain functionals combined with particular choices of initial conditions for the target variables. A comprehensive comparative study of several established surface hopping methodologies is conducted using several multiple-state model Hamiltonians. The recently reported formulation of the FSSH-2 method is extended to the case of multiple states and validated. It is demonstrated that the performance of all studied methods, including the new FSSH-3 (in two variants) is robust with respect to integration time step used but only if local diabatisation approach is used. Under such conditions, all methods yield nearly identical results, in excellent agreement with fully quantum simulations, even with sufficiently large integration time steps.