Rothe's Method for Quantum Dynamics in Atoms and Molecules with Gaussian Wavepackets

Capable of capturing both bound and continuum quantum dynamics, Gaussian wavepackets are highly attractive basis functions for simulating laser-driven processes in atoms and molecules. Unfortunately, fully flexible Gaussian wavepackets are exceedingly challenging to propagate in a numerically stable manner within the framework of conventional time-dependent variational principles. In this chapter, we discuss the sources of the numerical issues and review an alternative approach, Rothe's method, that offers a route to improved numerical stability. Recent proof-of-concept simulations based on Rothe's method indicate that Gaussian wavepackets provide results on par with highly accurate grid-based methods for both electronic and rovibrational quantum dynamics, including ultrafast nonlinear processes that involve the continuum such as high-harmonic generation. Remarkably few Gaussian wavepackets are needed to achieve the high accuracy of grid-based approaches, indicating that further algorithmic developments and efficient implementations may enable efficient simulations of not only electronic and rovibrational phenomena but also fully coupled electronic-nuclear quantum dynamics with significantly reduced memory demands. We also point out remaining practical challenges, including matrix elements of the squared Hamiltonian and the treatment of Coulomb cusps.