Approximate Spin Projection for Geometry Optimization of Biradical Systems: Case Studies of Through-Space and Through-Bond Systems

Molecular structures of Cr2(O2CCH3)4(OH2)2 and Fe2S2(SCH3)4 are optimized by using spin-restricted (R), spin-unrestricted broken-symmetry (BS), and approximate spin-projection (AP) methods with a B3LYP functional set, as model systems of through-space and through-bond biradical systems respectively. The effect of a spin contamination error (SCE) in the BS method and the static correlation correction involved in the AP framework are examined, based on the differences between these methods in the optimized geometry. The effective bond order and magnetic coupling values are also calculated to explain the differences between these methods. The AP method successfully corrects both the static correlation and spin contamination errors, and the spin-projected bond orders can clarify how the AP method works on them.