A multi-state mapping approach to surface hopping

We describe a multiple electronic state adaptation of the mapping approach to surface hopping introduced recently by Mannouch and Richardson [J. Chem. Phys. 158, 104111 (2023)]. Our modification treats populations and coherences on an equal footing and is guaranteed to give populations in any electronic basis that tend to the correct quantum-classical equilibrium values in the long-time limit (assuming ergodicity). We demonstrate its accuracy by comparison with exact benchmark results for three- and seven-state models of the Fenna-Matthews-Olson complex, obtaining electronic populations and coherences that are significantly more accurate than those of fewest switches surface hopping and at least as good as those of any other semiclassical method we are aware of. Since these results were obtained by adapting the scheme of Mannouch and Richardson, we go on to compare our results with theirs for a variety of problems with two electronic states. We find that their method is sometimes more accurate, especially in the Marcus inverted regime. However, in other situations, the accuracies are comparable, and since our scheme can be used with multiple electronic states it can be applied to a wider variety of electronically nonadiabatic systems.