Family of Gaussian wavepacket dynamics methods from the perspective of a nonlinear Schrödinger equation

Abstract

Many approximate solutions of the time-dependent Schrödinger equation can be formulated as exact solutions of a nonlinear Schrödinger equation with an effective Hamiltonian operator depending on the state of the system. We show that several well-known Gaussian wavepacket dynamics methods, such as Heller's original thawed Gaussian approximation or Coalson and Karplus's variational Gaussian approximation, fit into this framework if the effective potential is a quadratic polynomial with state-dependent coefficients. We study such a nonlinear Schrödinger equation in general: in particular, we derive general equations of motion for the Gaussian's parameters, demonstrate the time reversibility and norm conservation, and analyze conservation of the energy, effective energy, and symplectic structure. We also describe efficient geometric integrators of arbitrary even orders of accuracy in the time step for the numerical solution of this nonlinear Schrödinger equation. The general presentation is illustrated by examples of this family of Gaussian wavepacket dynamics, including the variational and nonvariational thawed and frozen Gaussian approximations, and special limits of these methods based on the global harmonic, local harmonic, single-Hessian, local cubic, and single quartic approximations for the potential energy. Without substantially increasing the cost, the proposed single quartic variational thawed Gaussian wavepacket dynamics improves the accuracy over the local cubic approximation and, at the same time, conserves both the effective energy and symplectic structure, in contrast to the much more expensive local quartic approximation. Most results are presented in both Heller's and Hagedorn's parametrizations of the Gaussian wavepacket.

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• Isotope effects in the electronic spectra of ammonia from ab initio semiclassical dynamics
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• On Hagedorn wavepackets associated with different Gaussians
• Can increasing the size and flexibility of a molecule reduce decoherence and prolong charge migration?
• Thawed Gaussian wavepacket dynamics with $\Delta$-machine learned potentials
• On-the-Fly Ab Initio Hagedorn Wavepacket Dynamics: Single Vibronic Level Fluorescence Spectra of Difluorocarbene
• Simulating Molecular Single Vibronic Level Fluorescence Spectra with ab initio Hagedorn Wavepacket Dynamics
• On the single-Hessian Gaussian wavepacket dynamics
• Refined approach to cellularization: going from Heller's thawed Gaussian approximation to the Herman--Kluk propagator