Geometric phase in molecular dynamics

<sec>The geometric phase effect of molecules, also known as the molecular Aharonov-Bohm effect, arises from the study of the conical intersections of potential energy surfaces. When encircling a conical intersection in the nuclear configuration space, the adiabatic electronic wave function acquires a π phase, leading to a change in sign. Consequently, the nuclear wave function must also change its sign to maintain the single-valued nature of the total wave function. This phase is topologically related to the conical intersection structure. Only by appropriately introducing the molecular geometric phase can the quantum dynamical behavior in the adiabatic representation be accurately described. In the diabatic representation, both the geometric phase effects and the non-adiabatic couplings between nuclei and electrons can be implicitly handled.In this paper, according to the quantum kinematic approach to the geometric phase, we propose a method for directly extracting the geometric phase in molecular dynamics. To demonstrate the unique features of this method, we adopt the <inline-formula><tex-math id="Z-20250702085333">\begin{document}$E \otimes e $\end{document}</tex-math></inline-formula> Jahn-Teller model, which is a standard model that includes a cone intersection point. This model comprises two diabatic electronic states coupled with two vibrational modes. The initial wave function is designed in such a way that it can circumnavigate the conical intersection in an almost adiabatic manner within approximately 2.4 ms. Subsequently, the quantum kinematic approach is utilized to extract the geometric phase during the evolution. In contrast to the typical topological effect of a quantized geometric phase of π, this extracted geometric phase in this case varies in a continuous manner. When a quantum system performs a path in its projected Hilbert space, it is a representation-independent and gauge-invariant formula of the geometric phase. This research provides a new perspective for exploring molecular geometric phases and the geometric phase effects. It may also provide a possible observable for experimentally studying geometric phases in molecular dynamics.</sec>