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 $\pi$ phase, leading to a change in sign. Consequently, the nuclear wave function must also change its sign to preserve the single-valueless 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, the geometric phase effects and the non-adiabatic couplings between nuclei and electrons can be implicitly handled.
In this paper, based on the quantum kinematic approach to the geometric phase, a method for directly extracting the geometric phase in molecular dynamics is proposed. To demonstrate the unique features of this method, the $E \otimes e$ Jahn-Teller model, which is a standard model incorporating a conical intersection, is employed. 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 to the geometric phase is utilized to extract the geometric phase during the evolution. In contrast to the typical topological effect of a quantized geometric phase of $\pi$, this extracted geometric phase in this case varies in a continuous manner. It is a representation-independent and a gauge-invariant formulation of the geometric phase when a quantum system performs a path in its projected Hilbert space. This research offers a new perspective for exploring molecular geometric phases and the geometric phase effects. It may also provide a possible observable for experimental studies on geometric phases in molecular dynamics.
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