Effect of Exact Exchange on the Energy Landscape in Self-Consistent Field Theory

Density functional approximations can reduce the spin symmetry breaking observed for self-consistent field (SCF) solutions compared to Hartree-Fock theory, but the amount of exact Hartree-Fock (HF) exchange appears to be a key determinant in broken Ŝ² symmetry. To elucidate the precise role of exact exchange, we investigate the energy landscape of unrestricted Hartree-Fock and Kohn-Sham density functional theory for benzene and square cyclobutadiene, which provide paradigmatic examples of closed-shell and open-shell electronic structures, respectively. We find that increasing the amount of exact exchange leads to more local SCF minima, which can be characterized as combinatorial arrangements of unpaired electrons in the carbon π system. Furthermore, we studied the pathways connecting local minima to understand the relationships between different solutions. Our analysis reveals a subtle balance between one- and two-body interactions in determining SCF symmetry breaking, shedding new light on the physical driving forces for spin-symmetry-broken solutions in SCF approaches.