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

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 they 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 multidimensional, nonseparable coupled Morse potential.

Finite-temperature vibronic spectra from the split-operator coherence thermofield dynamics.

We present a numerically exact approach for evaluating vibrationally resolved electronic spectra at finite temperatures using the coherence thermofield dynamics. In this method, which avoids implementing an algorithm for solving the von Neumann equation for coherence, the thermal vibrational ensemble is first mapped to a pure-state wavepacket in an augmented space, and this wavepacket is then propagated by solving the standard, zero-temperature Schrödinger equation with the split-operator Fourier method. We show that the finite-temperature spectra obtained with the coherence thermofield dynamics in a Morse potential agree exactly with those computed by Boltzmann-averaging the spectra of individual vibrational levels. Because the split-operator thermofield dynamics on a full tensor-product grid is restricted to low-dimensional systems, we briefly discuss how the accessible dimensionality can be increased by various techniques developed for the zero-temperature split-operator Fourier method.

Isotope Effects on the Electronic Spectra of Ammonia from Ab Initio Semiclassical Dynamics.

Despite its simplicity, the single-trajectory thawed Gaussian approximation has proven useful for calculating the vibrationally resolved electronic spectra of molecules with weakly anharmonic potential energy surfaces. Here, we show that the thawed Gaussian approximation can capture surprisingly well even more subtle observables, such as the isotope effects in the absorption spectra, and we demonstrate it on the four isotopologues of ammonia (NH3, NDH2, ND2H, and ND3). The differences in their computed spectra are due to the differences in the semiclassical trajectories followed by the four isotopologues, and the isotope effects─narrowing of the transition band and reduction of the peak spacing─are accurately described by this semiclassical method. In contrast, the adiabatic harmonic model shows a double progression instead of the single progression seen in the experimental spectra. The vertical harmonic model correctly shows only a single progression but fails to describe the anharmonic peak spacing. Analysis of the normal-mode activation upon excitation provides insight into the elusiveness of the symmetric stretching progression in the spectra.

High-order geometric integrators for the variational Gaussian approximation.

Among the single-trajectory Gaussian-based methods for solving the time-dependent Schrödinger equation, the variational Gaussian approximation is the most accurate one. In contrast to Heller's original thawed Gaussian approximation, it is symplectic, conserves energy exactly, and may partially account for tunneling. However, the variational method is also much more expensive. To improve its efficiency, we symmetrically compose the second-order symplectic integrator of Faou and Lubich and obtain geometric integrators that can achieve an arbitrary even order of convergence in the time step. We demonstrate that the high-order integrators can speed up convergence drastically compared to the second-order algorithm and, in contrast to the popular fourth-order Runge-Kutta method, are time-reversible and conserve the norm and the symplectic structure exactly, regardless of the time step. To show that the method is not restricted to low-dimensional systems, we perform most of the analysis on a non-separable twenty-dimensional model of coupled Morse oscillators. We also show that the variational method may capture tunneling and, in general, improves accuracy over the non-variational thawed Gaussian approximation.