A quantum-classical Liouville formalism in a preconditioned basis and its connection with phase-space surface hopping

We revisit a recent proposal to model nonadiabatic problems with a complex-valued Hamiltonian through a phase-space surface hopping (PSSH) algorithm employing a pseudo-diabatic basis. Here, we show that such a pseudo-diabatic PSSH (PD-PSSH) ansatz is consistent with a quantum-classical Liouville equation (QCLE) that can be derived following a preconditioning process, and we demonstrate that a proper PD-PSSH algorithm is able to capture some geometric magnetic effects (whereas the standard fewest switches surface hopping approach cannot capture such effects). We also find that a preconditioned QCLE can outperform the standard QCLE in certain cases, highlighting the fact that there is no unique QCLE. Finally, we also point out that one can construct a mean-field Ehrenfest algorithm using a phase-space representation similar to what is done for PSSH. These findings would appear extremely helpful as far as understanding and simulating nonadiabatic dynamics with complex-valued Hamiltonians and/or spin degeneracy.