Volume 56, Number 6, November-December 2022
|Page(s)||2197 - 2238|
|Published online||08 December 2022|
Canonical mean-field molecular dynamics derived from quantum mechanics
Institutionen för Matematik, Kungl. Tekniska Högskolan, 10044 Stockholm, Sweden
2 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
* Corresponding author: firstname.lastname@example.org
Accepted: 20 September 2022
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 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 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 := Tr(He−βH)/Tr(e−βH) with respect to the electron degrees of freedom and H is 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 accuracy O(M-1 + tϵ2), for correlation time t where ϵ2 is related to the variance of mean value approximation h. 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.
Mathematics Subject Classification: 35Q40 / 81Q20 / 82C10
Key words: Quantum canonical ensemble / correlation observables / molecular dynamics / excited states / mean-field approximation / semi-classical analysis / Weyl calculus / path integral
© The authors. Published by EDP Sciences, SMAI 2022
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