Spin dependent properties of perturbed wave functions: An analytic comparison of the exact, UHF, and spin-projected UHF states

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Abstract
A generally applicable diagrammatic representation for spin‐annihilated wave functions is developed; the diagrammatic approach avoids the complex algebra usually associated with the application of spin‐annihilation operators. The diagrammatic formulation is first applied to the perturbation expansion for the exact wave function to elucidate the diagrammatic origin of the (correct) spin eigenfunction properties of the exact perturbed state. Employing a previously derived perturbation expansion for the unrestricted Hartree–Fock (UHF) wave function, the diagrammatic spin‐annihilation formalism is then used to analyze the effect of projecting unwanted spin states from the UHF wave function. Results obtained for the projected UHF state are compared to those appropriate to both the exact perturbed wave function and the unprojected UHF wave function. It is shown that results obtained by annihilating only the lowest unwanted spin multiplicity are expected, in a perturbation theory sense, to be very similar to those obtained by annihilating all unwanted spin states. However, it is further shown that additional terms are introduced into the UHF wave function by the spin projection procedure and that these terms are in general unrelated to the exact perturbative corrections to the UHF state. In particular, the wave function, energy, one and two‐electron density matrices, and spin densities obtained for the projected state are all shown to be in error in the lowest order of perturbation theory; in contrast, the UHF state leads to the correct one‐electron density matrix and spin densities in lowest order.