Independent trajectory mixed quantum-classical approaches based on the exact factorization

Mixed quantum-classical dynamics based on the exact factorization exploits the "derived" electron-nuclear correlation (ENC) term, aiming for the description of quantum coherences. The ENC contains interactions between the phase of electronic states and nuclear quantum momenta, which depend on the spatial shape of the nuclear density. The original surface hopping based on the exact factorization (SHXF) [Ha et al., J. Phys. Chem. Lett. 9, 1097 (2018)] exploits frozen Gaussian functions to construct the nuclear density in the ENC term, while the phase of electronic states is approximated as a fictitious nuclear momentum change. However, in reality, the width of nuclear wave packets varies in time depending on the shape of potential energy surfaces. In this work, we present a modified SHXF approach and a newly developed Ehrenfest dynamics based on the exact factorization (EhXF) with time-dependent Gaussian functions and phases by enforcing total energy conservation. We perform numerical tests for various one-dimensional two-state model Hamiltonians. Overall, the time-dependent width of Gaussian functions and the energy conserving phase show a reliable decoherence compared to the original frozen Gaussian-based SHXF and the exact quantum mechanical calculation. In particular, the energy conserving phase is crucial for EhXF to reproduce the correct quantum dynamics.