Understanding molecular geometric phase effects with exact effective force: case study of a model system

In this work, molecular geometric phase effects are studied using the idea of exact factorization (EF) (Abedi<i>et al</i>2010<i>Phys. Rev. Lett.</i><b>105</b>123002) and exact effective force (Li<i>et al</i>2022<i>Phys. Rev. Lett.</i><b>128</b>113001). In particular, we performed dynamics simulations for a two-state vibronic coupling model, and interpreted the results in three different perspectives: the Born-Huang expansion, the exact time-dependent potential energy surface (TDPES) and the exact effective force. We find that (i) at particular moment, while the vanishing nuclear density that occurs periodically in space is conventionally attributed to destructive interference of the nuclear wave packet owing to the geometric phase, such phenomenon can be equally well interpreted through the energy perspective, as manifested in the exact TDPES in the EF scheme; (ii) when combined with trajectory-based classical dynamics, the exact effective force obtained through EF qualitatively reproduces the correct nuclear density, while the adiabatic force gives the wrong density, particularly in the interference region. Our results suggest that the exact effective force is a potential starting point for making approximations and improving trajectory-based computational methods towards an accurate description of geometric phase effects.