Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces

TL;DR

This paper provides an error estimate for Born-Oppenheimer molecular dynamics near crossing potential surfaces, linking the error to excited state probabilities and mass ratios, with a numerical method for estimating these probabilities.

Contribution

It introduces a novel error estimate for molecular dynamics with nearly crossing surfaces without assuming a spectral gap, and proposes a numerical method to estimate excited state probabilities.

Findings

Error bound depends on mass ratio and excited state probability

Error estimate is uniform across particle numbers

Numerical method for excited state probability estimation

Abstract

The difference of the values of observables for the time-independent Schroedinger equation, with matrix valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states and the electron/nuclei mass ratio. The paper first proves an error estimate (depending on the electron/nuclei mass ratio and the probability to be in excited states) for this difference of microcanonical observables, assuming that molecular dynamics space-time averages converge, with a rate related to the maximal Lyapunov exponent. The error estimate is uniform in the number of particles and the analysis does not assume a uniform lower bound on the spectral gap of the electron operator and consequently the probability to be in excited states can be large. A numerical method to determine the probability to be in excited states is then presented, based on Ehrenfest molecular dynamics and stability analysis of a perturbed eigenvalue problem.