Canonical quantum observables for molecular systems approximated by ab initio molecular dynamics

TL;DR

This paper proves that a weighted average of ab initio molecular dynamics simulations, based on electron eigenvalues, can accurately approximate quantum observables of nuclei-electron systems at any temperature, with error proportional to the electron-nuclei mass ratio.

Contribution

It introduces a new method to approximate quantum observables using a weighted average of ab initio dynamics for all electron eigenvalues, extending applicability across temperatures.

Findings

Weighted average of ab initio dynamics approximates quantum observables.

Error proportional to electron-nuclei mass ratio.

Applicable for observables with specific smoothness and eigenvalue conditions.

Abstract

It is known that ab initio molecular dynamics based on the electron ground state eigenvalue can be used to approximate quantum observables in the canonical ensemble when the temperature is low compared to the first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature. The proof uses the semiclassical Weyl law to show that canonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The result covers observables that depend on time-correlations. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's law shows that the error estimate holds for observables and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei position and the observables are in the diagonal form with respect to the electron eigenstates.