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

Abstract

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.

Classical Langevin dynamics derived from quantum mechanics

“…as verified in [10,Lemma 3.1]. Therefore, the aim is to choose the unitary matrix Ψ so that it becomes an approximate solution to the nonlinear eigenvalue problem…”

“…while the classical Gibbs density e −H/T is a time-independent solution to the classical Liouville equation However, it is shown in [10] that a solution to the quantum Liouville equationρ t with initial data ρ 0 = e −H/T generates only a small time dependent perturbation on observables up to time t < M , which motivates our use of e −H/T . The following result for approximating non equilibrium quantum observables by classical molecular dynamics observables is proved in [10].…”

“…Ab initio molecular dynamics based on the electron ground state eigenvalue can be used when the temperature is low compared to the first electron eigenvalue gap. A certain weighted average of different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature, see [10], also in the case of observables including time correlation and many particles. The elimination of the electrons provides a substantial computational reduction, making it possible to simulate large molecular systems, cf.…”

“…Therefore, we seek an equilibrium density that has the property that the marginal distribution for a subsystem has the same density as the whole system. In [10] it is motivated how this assumption leads to the Gibbs measure, i.e. the canonical ensemble; this is also the motivation to use the canonical ensemble for the composite system in this work, although some studies on heat bath models use the microcanonical ensemble for the composite whole system.…”

“…When the temperature is larger, excited electron states influence the nuclei dynamics. The work [10] derives a molecular dynamics approximation of quantum observables, including time correlations, as a certain weighted average of ab initio observables in the excited states, and these ideas are put into the context of this work in Section 5.…”