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 the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be $\mathcal{O}(M^{-1})$, provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and $M$ is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain $\mathcal{O}(M^{-1})$ 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 trace $h:=\mathrm{Tr}(H e^{-\beta H})/\mathrm{Tr}(e^{-\beta H})$ with respect to the electron degrees of freedom and $H$ is the Weyl symbol corresponding to a quantum many body Hamiltonian $\widehat{H}$. It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy $\mathcal{O}(M^{-1}+ t\epsilon^2)$, for correlation time $t$ where $\epsilon^2$ is related to the variance of mean value approximation $h$. The proof of this estimate does not rely on diagonalizing the electron operator and consequently coinciding electron eigenvalues are allowed. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation.