Which Algorithm Best Propagates the Meyer-Miller-Stock-Thoss Mapping Hamiltonian for Non-Adiabatic Dynamics?

TL;DR

This paper compares three algorithms for propagating the Meyer-Miller-Stock-Thoss Hamiltonian in non-adiabatic quantum dynamics, evaluating their accuracy, energy conservation, and computational efficiency to guide future simulations.

Contribution

It provides a comprehensive performance comparison of three propagation algorithms for the MMST Hamiltonian, highlighting the advantages of the symplectic MInt method and the efficiency of the SL method.

Findings

MInt is the only rigorously symplectic algorithm.

SL offers comparable accuracy with lower computational cost.

The DE algorithm poorly conserves energy due to approximations.

Abstract

A common strategy to simulate mixed quantum-classical dynamics is by propagating classical trajectories with mapping variables, often using the Meyer-Miller-Stock-Thoss (MMST) Hamiltonian or the related spin-mapping approach. When mapping the quantum subsystem, the coupled dynamics reduce to a set of equations of motion to integrate. Several numerical algorithms have been proposed, but a thorough performance comparison appears to be lacking. Here, we compare three time-propagation algorithms for the MMST Hamiltonian: the Momentum Integral (MInt) (arXiv:1709.07474), the Split-Liouvillian (SL) (arXiv:1609.00644), and the algorithm in arXiv:1201.1042 that we refer to as the Degenerate Eigenvalue (DE) algorithm due to the approximation required during derivation. We analyse the accuracy of individual trajectories, correlation functions, energy conservation, symplecticity, Liouville's theorem and the computational cost. We find that the MInt algorithm is the only rigorously symplectic algorithm. However, comparable accuracy at a lower computational cost can be obtained with the SL algorithm. The approximation implicitly made within the DE algorithm conserves energy poorly, even for small timesteps, and thus leads to slightly different results. These results should guide future mapping-variable simulations.