Canonical Quantum Observables for Molecular Systems Approximated by Ab Initio Molecular Dynamics

Aku Kammonen,Anders Szepessy,Mattias Sandberg,Petr Plecháč

doi:10.1007/s00023-018-0699-x

Aku Kammonen, Anders Szepessy + Show 2 more

Open Access

https://doi.org/10.1007/s00023-018-0699-x

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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.

Highlights

Given a quantum system defined by the Hamiltonian Hs(zs) that (Hs) (x, −i ∇) acting on L2(RN ), the quantum canonical ensemble at the inverse temperature β = 1/(kBT ) is described by the density operator ρ = e−βH
A quantum observable is defined by a Hermitian, densely defined operator Aon L2(RN ), and the quantum canonical ensemble average is obtained from the normalized trace of the product as
We study molecular dynamics approximations of canonical ensemble averages for quantum observables of the nuclei–electrons system

Summary

Background

By including the electron part as a matrix-valued operator, one can derive the limit as the electron–nuclei mass ratio 1/M tends to zero, see [15] which, in Section 6, includes an overview of previous results. This limit can be approximated by ab initio molecular dynamics simulations for nuclei, with the potential generated by the electron eigenvalue problem, see [10,12], based on the nuclei and electron scale separation using the Born–Oppenheimer approximation [13,14]. When the temperature is not small compared to this electron eigenvalue gap, the probability to be in excited states is substantial and the molecular dynamics associated with the electronic groundstate energy will not yield accurate approximation of quantum observables

Overview of Results

Problem Formulation and Weyl Quantization

Weyl Quantization Estimates