Implications from Geometric Phase for Circulating Far Away Conical Intersection(s)

People rarely associate a geometric phase with nuclear configurations that lie far away from a conical intersection, where the ground state is energetically well-separated from excited states. However, the geometric phase is a nonlocal impact of a conical intersection that should emerge whenever a nuclear configuration travels on a closed loop trajectory that encloses the intersection in the branching plane, regardless of the distance from the intersection. In particular, if a trajectory starts at a structure with a stable closed-shell electronic configuration, how could the continuous closed-shell configuration possibly gain a sign change on finishing the loop? In this work, inspired by a recent paper by Kjønstad; ; Koch ( J. Chem. Phys. 2025, 163, 194104)., we performed full configuration interaction calculations and closed-shell single-reference calculations for the toy models of H₄ and LiH₃ to elucidate this problem. The investigation leads to the conjecture that restricted Hartree-Fock solutions necessarily encounter a point of degeneracy along closed loops that encircle a (or an odd number of) conical intersection(s). Single-reference wave functions need to change character close to such points and cusps or discontinuities in a potential energy surface and multireference characters are a likely result, even when the energy gap between ground and excited states is large.