High-order geometric integrators for the local cubic variational Gaussian wavepacket dynamics

Abstract

Gaussian wavepacket dynamics has proven to be a useful semiclassical approximation for quantum simulations of high-dimensional systems with low anharmonicity. Compared to Heller's original local harmonic method, the variational Gaussian wavepacket dynamics is more accurate, but much more difficult to apply in practice because it requires evaluating the expectation values of the potential energy, gradient, and Hessian. If the variational approach is applied to the local cubic approximation of the potential, these expectation values can be evaluated analytically, but still require the costly third derivative of the potential. To reduce the cost of the resulting local cubic variational Gaussian wavepacket dynamics, we describe efficient high-order geometric integrators, which are symplectic, time-reversible, and norm-conserving. For small time steps, they also conserve the effective energy. We demonstrate the efficiency and geometric properties of these integrators numerically on a multi-dimensional, nonseparable coupled Morse potential.

Figures (11)

• Figure 1: Dynamics of a wavepacket $\psi_{t}$ in a one-dimensional Morse potential $V$. Panel (a) shows the effective potential $V_{\textrm{eff}}$ of the local cubic variational GWD. Panels (b) and (c) compare the effective $(E_{\textrm{eff}})$ and exact $(E)$ energies of the wavepacket propagated with different methods. For clarity, the insets of panels (b) and (c) do not show the local harmonic results.
• Figure 2: Top panel compares the exact and effective potential energies of a Gaussian wavepacket propagated in a one-dimensional Morse potential with the local cubic variational GWD. These two energies are almost equal whenever the position covariance, shown in the bottom panel, is small.
• Figure 3: Top panel compares the exact and effective potential energies of a Gaussian wavepacket propagated in a one-dimensional Morse potential with the local harmonic GWD. These two energies are almost identical whenever the position covariance, shown in the bottom panel, is small.
• Figure 4: Absorption spectrum of a one-dimensional Morse potential with anharmonicity $\chi=0.01$. The spectra obtained with the (a) variational, (b) local cubic variational, and (c) local harmonic GWD are compared to the exact quantum spectrum.
• Figure 5: Dynamics of a wavepacket propagated in a two-dimensional coupled Morse potential with different methods. Effective energy [panel (a)], energy [panel (b)], and position of the Gaussian's center along two different coordinates [panels (c) and (d)] are shown. For clarity, the insets of panels (a) and (b) do not show the local harmonic results.