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.

翻译：古典分子动态可以近似于古典分子量关系。 在低温的情况下, 初始分子动态潜在能量以地面状态电子值问题为基础, 准确度被证明为$\ mathcal{O}( M ⁇ \\ -1} 美元, 但前提是第一个电子量值与给定温度相比差异足够大, 美元是核和电子质量的比例。 对于与兴奋型电子状态相对应的更高温度乙基值而言, 需要获得 $\ macal{ O} (M ⁇ -1}) 美元 分子动态潜能值。 精确度和衍生假设所有电子量值都是分开的, 例如排除了锥体交叉点 。 这项工作研究一种平均水平分子动态, 中位汉密尔顿值与给定值值之间的部分微量值 : mathr{ {Tr} (H\\\\\\\\\\\\\ betqr) 与兴奋性电子量值的数值, 当量值值的比值值值值值值值 和正值的正值的硬值 直值 。