Simple algorithm for evaluating SU(2) non-abelian geometric phase in the vicinity of molecular conical intersection of odd electron number

Molecular conical intersection of odd-electron number are known to give arise to non-abelian geometric phases. In this work, we present a simple algorithm for evaluating the SU(2) geometric phases given a trajectory represented as a discrete set of coordinates in the five-dimensional branching space. Motivated by the special property that such conical intersection behaves as abelian ones in three-dimensional subspace, we take two steps to establish the algorithm. We first prove that it is possible to construct an augmented trajectory consisting of spherical triangles that has the same geometric phases as the original trajectory. We then prove that the geometric phase contribution from each spherical triangle can be evaluated from its properties using Stokes' theorem in a way that closely resembles abelian conical intersections. The total geometric phases are thus evaluated by path-ordered products of contributions from each spherical triangle. Based on the augmented trajectory, we further show that the geometric phases of the upper and lower states will switch under inversion transformation of the trajectory. This relation is further generalized to show that the Wilson loop, defined as the trace of the geometric phases, will switch between the two states under any five-dimensional improper isometric transformation. Example calculations of geometric phases for trajectories generated on five-dimensional polytopes are also presented and discussed.