Computing quantum dynamics in the semiclassical regime

Abstract

The semiclassically scaled time-dependent multi-particle Schrödinger equation describes, inter alia, quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimensions. This paper reviews and studies numerical approaches that are robust to the small semiclassical parameter. We present and analyse variationally evolving Gaussian wave packets, Hagedorn’s semiclassical wave packets, continuous superpositions of both thawed and frozen Gaussians, and Wigner function approaches to the direct computation of expectation values of observables. Making good use of classical mechanics is essential for all these approaches. The arising aspects of time integration and high-dimensional quadrature are also discussed.

Efficient Semiclassical Dynamics for Vibronic Spectroscopy beyond Harmonic, Condon, and Zero-Temperature Approximations

“…Of course, this semiclassical method is only approximate, since the true wavepacket deforms from its initial Gaussian shape after evolving for long enough in an anharmonic potential. Nevertheless, it is accurate at short times and for moderately anharmonic potentials, [41] which makes it suitable for molecular spectroscopy [38,39,[42][43][44] and ultrafast photoinduced processes. [45] In addition, the method is exact for harmonic potentials because the second-order Taylor expansion of Eqn.…”

“…[57] This would require a method that can propagate more general initial states. Hagedorn wavepackets, [41,58] which are of the "Gaussian times a general polynomial" form, are a promising solution to this problem. Similarly, the anharmonicity is only partially included in the thawed Gaussian approximation.…”

“…For strongly anharmonic potential energy surfaces or for high-resolution spectroscopy, where the wavepacket must be propagated for longer times, this approximation might fail. Single-trajectory Hagedorn wavepackets or variational Gaussian wavepacket dynamics [41,59] are two promising methods, which, given additional potential energy data, such as the third or fourth derivatives of the potential energy, could yield more accurate spectra. Importantly, following the work presented here, these wavepacket propagation methods can easily include non-Condon and finite-temperature conventional ab initio molecular dynamics.…”

Nontensorial generalised hermite spectral methods for PDEs with fractional Laplacian and Schrödinger operators

“…Differences with Hagedorn wavepackets [18] and some other generalisations [46]. The Hagedorn wavepackets first introduced in [18] are deemed as an effective numerical tool in computing quantum dynamics (see [24] for an up-to-date review). They generalise the usual Hermite functions to several dimensions that allow for flexible localisation in position and momentum.…”

Finite-Temperature, Anharmonicity, and Duschinsky Effects on the Two-Dimensional Electronic Spectra from Ab Initio Thermo-Field Gaussian Wavepacket Dynamics

“…Let us consider a Gaussian wavepacket where q t and p t are the real, D -dimensional expectation values of the position and momentum, respectively, A t is a D × D complex symmetric matrix with a positive-definite imaginary part, γ t is a complex scalar whose imaginary part ensures normalization of the wavepacket, and D is the number of coordinates. Within the thawed Gaussian approximation, 58 one replaces true potential energy V ( q ) by its local harmonic approximation about the center q t of the wavepacket, which leads to the following equations of motion for the Gaussian’s parameters: 58 , 84 where L t = p t T ·(2 m ) −1 · p t – V ( q t ) is the Lagrangian along the trajectory ( q t , p t ) and m is the symmetric mass matrix. According to eqs 23 – 26 , the position and momentum of the Gaussian wavepacket evolve classically, while matrix A t depends on the Hessians along the classical trajectory.…”