Speaker
Description
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 Heller's thawed Gaussian approximation, Coalson and Karplus's variational Gaussian approximation, and other Gaussian wavepacket dynamics methods fit into this framework if the effective potential is a quadratic polynomial with state-dependent coefficients. We study such a nonlinear Schrödinger equation in full generality [1]: after deriving general equations of motion for the Gaussian's parameters, we demonstrate the time reversibility and norm conservation and analyze conservation of the energy, effective energy, and symplectic structure. We also describe efficient, high-order geometric integrators for the numerical solution of this nonlinear Schrödinger equation. The general theory is illustrated by known as well as new examples of this family based on various approximations for the potential energy. In my talk, I will accompany the theory with applications of the different versions of Gaussian wavepacket dynamics to the calculation of vibrationally resolved electronic spectra as well as with numerical examples demonstrating the geometric properties and fast convergence of the integrators. Time permitting, I will mention how the method can be generalized to propagating non-Gaussian states (using Hagedorn wavepackets [2,3] ) and to nonzero temperatures (using thermofield dynamics [4]).
[1] J. J. L. Vaníček, J. Chem. Phys. 159, 014114 (2023).
[2] Z. Tong Zhang and J. J. L. Vaníček, communication, J. Chem. Phys. 161, 111101 (2024).
[3] Z. Tong Zhang, Máté Visegrádi, and J. J. L. Vaníček, letter, Phys. Rev. A 111, L010801 (2025).
[4] T. Begušić and J. Vaníček, J. Chem. Phys. 153, 024105 (2020).
