Spin-polarized self-consistent-field equations for paired orbitals

Abstract

Unrestricted Hartree-Fock-like equations are proposed to find multiple spin-symmetry-broken states of the molecular systems. Developed equations are pseudo-eigenvalue-type equations for the Fock-type operators constructed in such a way to include an effective field which makes different-spin orbitals biorthogonal. The eigenvectors of these operators are noncanonical Hartree-Fock orbitals becoming L\"owdin-Amos-Hall paired (corresponding) orbitals after self-consistency is achieved. The eigenvalues of the modified Fock operators appear to be the energies of the paired orbitals. Because the paired orbitals do not follow the spatial symmetry of the molecular nuclear core, the equations allow one to obtain the broken symmetry states with relative ease as demonstrated for the model ${\mathrm{H}}_{6}$ hexagon molecule. For this molecule, the \ifmmode \check{C}\else \v{C}\fi{}\'{\i}\ifmmode \check{z}\else \v{z}\fi{}ek-Paldus instability matrix analysis predicts the existence of three spin-symmetry-broken states. All these solutions are systematically achieved by the paired equations, unlike the standard unrestricted equations which basically converge to a single solution. The proposed approach is also valid for the density functional theory in which the spin-polarized Kohn-Sham equations might be transformed to paired equations.