The variational Gaussian approximation combined with high-order geometric integrators with applications to quantum tunneling and vibronic spectra

Among the single trajectory Gaussian-based methods for solving the time-dependent Schrödinger equation, the variational Gaussian approximation (VGA) [1,2] is the most accurate one. However, the equations of motion for the parameters of the Gaussian wavepacket require expectation values of the potential and its first two derivatives, making the method much more expensive than original Heller’s thawed Gaussian approximation (TGA) [3], which requires these potential energy properties only at the center of the wavepacket. To improve the efficiency of the VGA, we describe geometric integrators, which can achieve an arbitrary even order of convergence in the time step and are obtained by symmetrically composing the second-order symplectic integrator of Faou and Lubich [4]. We demonstrate that the high-order integrators can drastically speed up convergence compared to the second-order algorithm and, in contrast to the Runge-Kutta method, are time-reversible and conserve the norm exactly. To avoid making further approximations, we demonstrate the properties of the VGA and of the geometric integrators on several systems, in which the expectation values of the potential energy can be evaluated analytically. We show that the VGA, in contrast to the TGA, conserves energy exactly and takes into account tunneling at least qualitatively and, in calculation of vibronic spectra, it agrees better than the TGA and global harmonic approximation with benchmark exact quantum calculations. Finally, to show that the method is not restricted to low-dimensional systems, we also applied it to a nonseparable twenty-dimensional model of coupled Morse oscillators.

Reference

[1] E. Heller, The Journal of Chemical Physics, 64, 63-73 (1976)

[2] R. Coalson, M. Karplus, The Journal of Chemical Physics, 93, 3919-3930 (1990)

[3] E. Heller, The Journal of Chemical Physics, 62, 1544-1555 (1975)

[4] E. Faou, C. Lubich, Comput. Visual Sci., 9, 45-55 (2006)