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

Abstract The two-level conical intersection of a molecule with odd electron number in the absence of 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, an mathematical generalization of Dirac monopole with SU(2) gauge field and SO(5) symmetry. This implies that we can study topological properties of T2=-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 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 C_2=+1/4 or -1/4 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 nonzero part of the Berry connection induces Hopf-fibration on the S3 longitude space, and is further visualized through stereographic projection.