Canonical mean-field molecular dynamics derived from quantum mechanics

Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature theab initiomolecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to beO(M-1), provided the first electron eigenvalue gap is sufficiently large compared to the given temperature andMis the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtainO(M-1) accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial traceh := Tr(He−βH)/Tr(e−βH) with respect to the electron degrees of freedom andHis the Weyl symbol corresponding to a quantum many body Hamiltonian ̂H. It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracyO(M-1+tϵ2), for correlation timetwhereϵ2is related to the variance of mean value approximationh. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue.