Efficient integrators for the variational Gaussian wavepacket dynamics with applications to tunneling and vibronic spectra

Accurate and efficient evaluation of vibronic spectra of polyatomic molecules remains a challenge of molecular and optical physics. In the time-dependent approach to spectroscopy [1], the variational Gaussian wavepacket dynamics [2] is the most accurate method among single-trajectory Gaussian-based methods for solving the time-dependent Schrödinger equation. In contrast to Heller’s original thawed Gaussian approximation [3], which has been successfully combined with on-the-fly ab initio electronic structure [4, 5], the variational method is symplectic, conserves energy exactly, and partially takes into account 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 [6] 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 drastically speed up convergence compared to the second-order algorithm (we show an example, where this speedup is by a factor of 100 if a moderate accuracy of 10^-6 is required for the wavefunction). Moreover, in contrast to the popular fourth-order Runge-Kutta method, the proposed integrators 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 can include tunneling and, in calculation of vibronic spectra, improves accuracy over the non-variational thawed Gaussian approximation.

[1] E. J. Heller, The semiclassical way to dynamics and spectroscopy (Princeton University Press, Princeton, NJ, 2018).

[2] R. D. Coalson and M. Karplus, J. Chem. Phys. 93, 3919 (1990).

[3] E. J. Heller, J. Chem. Phys. 62, 1544 (1975).

[4] M. Wehrle, M. Šulc, and J. Vani´cek, J. Chem. Phys. 140, 244114 (2014).

[5] T. Begušic, E. Tapavicza and J. Vani´cek, J. Chem. Theory Comput. 18, 3065 (2022).

[6] E. Faou and C. Lubich, Comput. Visual. Sci. 9, 45 (2006).