Discretising the Herman–Kluk propagator

Abstract

The Herman-Kluk propagator is a popular semi-classical approximation of the unitary evolution operator in quantum molecular dynamics. In this paper we formulate the Herman-Kluk propagator as a phase space integral and discretise it by Monte Carlo and quasi-Monte Carlo quadrature. Then, we investigate the accuracy of a symplectic time discretisation by combining backward error analysis with Fourier integral operator calculus. Numerical experiments for two-and six-dimensional model systems support our theoretical results. arXiv:1605.00588v1 [math.NA] 2 May 2016It is parametrised by a phase space point z = (q, p) ∈ R 2d . Gaussian wave packets enjoy the striking property that any square integrable function ψ ∈ L 2 (R d ) can be decomposed according toThe precise meaning of the integral is given by the inversion formula of the Fourier-Bros-Iagolnitzer (FBI) transform. From this we get the formal equationso that lim M →∞ sup z∈R 2d |D ε M (z 1 , . . . , z M ; z)| = 0 implies (34). and for the integralwhere the last equation also uses Fubini's theorem.

WavePacket: A Matlab package for numerical quantum dynamics. III. Quantum‐classical simulations and surface hopping trajectories

“…Future releases of WavePacket will also include some of the more recent developments in SHT techniques, above all the various decoherence corrections in FSSH, to keep up with the recent progress in surface hopping . Moreover, also class definitions for various semi‐classical approaches based on Gaussian packets in phase space shall be implemented. Of particular interest will be nonadiabatic extensions of Gaussian propagation methods, such as the multiple spawning technique, surface hopping Gaussian propagations, or adaptive variants thereof …”

Computing quantum dynamics in the semiclassical regime

“…The following result clarifies why the discrepancy function is crucial for equi-weighted quadrature. It gives a Koksma-Hlawka-type result as presented in Lasser and Sattlegger (2017); see also Aistleitner and Dick (2015) and Dick et al (2013), and the original papers by Koksma (1942) and Hlawka (1961).…”

“…The direct use of the wave packet transform, however, as pursued in Proposition 5.4 and Corollary 5.5, appears to be new and offers an even more elementary method for assessing the accuracy of the frozen approximation. The numerical realization of the Herman-Kluk propagator as a particle method was considered in Lasser and Sattlegger (2017).…”

“…Following Lasser and Sattlegger (2017), we briefly apply the previous results on quasi-Monte Carlo quadrature (Theorem 8.3) and plain Monte Carlo quadrature (Theorem 8.4) for the case that the initial data are a Gaussian centred in some point z 0 ∈ R 2d , i.e. ψ 0 = g z 0 .…”

Herman-Kluk propagator is free from zero-point energy leakage

“…In order to obtain expectation values, one has to deal with double phase-space integrals, which are treated using the combined sampling strategy as laid out in Ref. 86.…”