# Efficient geometric integrators for nonadiabatic quantum dynamics. II. The diabatic representation

## TL;DR

This paper develops high-order geometric integrators for nonadiabatic quantum dynamics in the diabatic representation, significantly improving efficiency and accuracy over standard methods while preserving key geometric properties.

## Contribution

It introduces automated recursive symmetric compositions of split-operator algorithms of arbitrary even order, reducing computational cost and memory use while maintaining geometric invariants.

## Key findings

- Achieved up to 900-fold speedup over second-order methods.
- Validated geometric property preservation analytically and numerically.
- Demonstrated effectiveness on multi-dimensional molecular models.

## Abstract

Exact nonadiabatic quantum evolution preserves many geometric properties of the molecular Hilbert space. In a companion paper [S. Choi and J. Van\'{\i}\v{c}ek, 2019], we presented numerical integrators of arbitrary-order of accuracy that preserve these geometric properties exactly even in the adiabatic representation, in which the molecular Hamiltonian is not separable into a kinetic and potential terms. Here, we focus on the separable Hamiltonian in diabatic representation, where the split-operator algorithm provides a popular alternative because it is explicit and easy to implement, while preserving most geometric invariants. Whereas the standard version has only second-order accuracy, we implemented, in an automated fashion, its recursive symmetric compositions, using the same schemes as in the companion paper, and obtained integrators of arbitrary even order that still preserve the geometric properties exactly. Because the automatically generated splitting coefficients are redundant, we reduce the computational cost by pruning these coefficients and lower memory requirements by identifying unique coefficients. The order of convergence and preservation of geometric properties are justified analytically and confirmed numerically on a one-dimensional two-surface model of NaI and a three-dimensional three-surface model of pyrazine. As for efficiency, we find that to reach a convergence error of 10$^{-10}$, a 600-fold speedup in the case of NaI and a 900-fold speedup in the case of pyrazine are obtained with the higher-order compositions instead of the second-order split-operator algorithm. The pyrazine results suggest that the efficiency gain survives in higher dimensions.