The two-level conical intersection of a molecule with odd electron number in the absence of magnetic field obeys time-reversal symmetry T^{2}=-1 (referred to as the T^{2}=-1 conical intersection), and has a five-dimensional branching space due to the Kramer degeneracy. Similar to how the conical intersection of a molecule in a magnetic field (T^{2}=0) behaves as the Dirac monopole, the T^{2}=-1 conical intersection behaves as the Yang monopole, an mathematical generalization of Dirac monopole with SU(2) gauge field and SO(5) symmetry. This implies that we can study topological properties of T^{2}=-1 conical intersections in chemistry based on what is known about Yang monopole in high energy physics. In this work, we present a few mathematical tools to study this connection. First, we show that geometric algebra and quaternion numbers together provides a natural way to utilize the T^{2}=-1 time reversal symmetry and the SO(5) symmetry in scaled coordinates, making it simple to derive eigenfunctions, Berry connection and Berry curvature of T^{2}=-1 conical intersection. In particular, this approach provides a simple proof that when viewed from the upper or lower states, the T^{2}=-1 conical intersection behaves as the self-dual or self-antidual Yang monopole with second Chern number C_{2}=+\frac{1}{4} or -\frac{1}{4} respectively. In addition, we propose a visualization method for the SU(2) Berry connection of T^{2}=-1 conical intersection. This is achieved by showing that the nonzero part of the Berry connection induces Hopf-fibration on the S^{3} longitude space, and is further visualized through stereographic projection. 

Abstract 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 5-dimensional branching space. Motivated by the special property that such conical intersection behaves as abelian ones in 3-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 5-dimensional improper isometric transformation. Example calculations of geometric phases for trajectories generated on 5-dimensional polytopes are also presented and discussed.