Molecular conical intersections with odd electron number are realizations of the topological Yang monopole

The two-level conical intersection of a molecule with an odd electron number in the absence of a magnetic field obeys time-reversal symmetry T2 = −1 (referred to as the T2 = −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 (T2 = 0) behaves as the Dirac monopole, the T2 = −1 conical intersection behaves as the Yang monopole, a mathematical generalization of the Dirac monopole with SU(2) gauge field and SO(5) symmetry. This implies that we can study the topological properties of T2 = −1 conical intersections in chemistry based on what is known about the 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 provide a natural way to utilize the T2 = −1 time reversal symmetry and the SO(5) symmetry in scaled coordinates, making it simple to derive eigenfunctions, Berry connection, and Berry curvature of T2 = −1 conical intersection. In particular, this approach provides a simple proof that when viewed from the upper or lower states, the T2 = −1 conical intersection behaves as the self-dual or self-antidual Yang monopole with second Chern number C2=+122 or −122, respectively. In addition, we propose a visualization method for the SU(2) Berry connection of T2 = −1 conical intersection. This is achieved by showing that the non-zero part of the Berry connection induces a Hopf-fibration on the S3 longitude space, which is further visualized through stereographic projection.