[Paper Review] Family of Gaussian wavepacket dynamics methods from the perspective of a nonlinear Schrödinger equation

Jiří Vaníček|arXiv (Cornell University)|Feb 20, 2023

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TL;DR

This paper unifies various Gaussian wavepacket dynamics methods—such as Heller's thawed Gaussian, variational Gaussian, and single-Hessian approximations—under a common framework of a nonlinear Schrödinger equation with a state-dependent quadratic potential. It derives general equations of motion, proves conservation of norm, energy, and symplectic structure, and introduces a new 'single-quartic' variational method that improves accuracy over the local cubic approximation while preserving key geometric properties at low computational cost.

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.

Motivation & Objective

• To unify diverse Gaussian wavepacket dynamics methods under a single theoretical framework based on a nonlinear Schrödinger equation with state-dependent quadratic potentials.
• To derive general equations of motion for Gaussian wavepacket parameters and prove fundamental conservation laws: norm, effective energy, and symplectic structure.
• To develop high-order geometric integrators for efficient and accurate numerical solution of the nonlinear Schrödinger equation.
• To propose and analyze a new 'single-quartic' variational Gaussian approximation that enhances accuracy over the local cubic method while preserving energy and symplectic conservation.
• To demonstrate the framework's generality by applying it to variational and non-variational thawed and frozen Gaussian approximations, including global harmonic, local harmonic, single-Hessian, local cubic, and local quartic limits.

Proposed method

• Formulates Gaussian wavepacket dynamics as exact solutions of a nonlinear time-dependent Schrödinger equation with an effective Hamiltonian whose parameters (quadratic potential coefficients) depend on the instantaneous state of the wavepacket.
• Derives general equations of motion for the Gaussian’s center, width matrix, phase, and momentum using the time-dependent variational principle applied to a Gaussian ansatz.
• Proves time reversibility, norm conservation, and conservation of effective energy and symplectic structure by analyzing the underlying geometric and algebraic properties of the nonlinear system.
• Introduces high-order geometric integrators that preserve the symplectic structure and effective energy, ensuring long-term stability and accuracy in numerical simulations.
• Applies the framework to multiple approximations: variational and non-variational thawed and frozen Gaussians, and their limits using harmonic, cubic, and quartic potential approximations.
• Proposes a new 'single-quartic' variational method by augmenting the local cubic approximation with a single fourth derivative term, maintaining computational efficiency while improving accuracy and preserving geometric invariants.

Experimental results

Research questions

• RQ1 Can various Gaussian wavepacket dynamics methods—such as Heller’s thawed Gaussian and Coalson-Karplus variational Gaussian—be unified under a single nonlinear Schrödinger equation framework with state-dependent potentials?
• RQ2 What are the general equations of motion for Gaussian wavepacket parameters in this nonlinear framework, and what geometric properties (e.g., norm, energy, symplectic structure) are conserved?
• RQ3 How can high-order geometric integrators be constructed to preserve the symplectic structure and effective energy in the numerical solution of the nonlinear Schrödinger equation?
• RQ4 Can a new variational Gaussian approximation be designed that improves accuracy over the local cubic approximation without sacrificing energy or symplectic conservation?
• RQ5 What is the impact of including higher-order potential derivatives (e.g., fourth derivative) on accuracy and conservation properties in Gaussian wavepacket dynamics?

Key findings

• The effective energy is conserved exactly in the nonlinear Schrödinger equation framework, which generalizes and unifies multiple Gaussian wavepacket methods.
• The symplectic structure is preserved by the equations of motion, ensuring long-term stability and accuracy in numerical simulations.
• The proposed 'single-quartic' variational Gaussian approximation improves accuracy over the local cubic approximation while conserving both effective energy and symplectic structure.
• The framework reveals that the single-Hessian approximation, despite its simplicity, conserves effective energy exactly and possesses a Hamiltonian structure in an augmented phase space.
• The general theory is valid in both Heller’s and Hagedorn’s parametrizations of the Gaussian wavepacket, enabling broad applicability across different formulations.
• High-order geometric integrators are derived that preserve the symplectic structure and effective energy, enabling efficient and accurate long-time propagation of Gaussian wavepackets.