Nonorthogonal Formulation of Hartree-Fock Perturbation Theory

Perturbation theory, up to first order in the wave function and second order in the energy, is formulated for a many-electron system without requiring the perturbed one-electron states to be orthogonal. The most general self-consistent coupled equations, referred to as Method 1, form the counterpart of Langhoff, Karplus, and Hurst's (LKH) Method $a$ for orthogonal orbitals. The uncoupling of the perturbations $\delta {\psi }_i$, $\delta {\psi }_j$ to the zero order wave functions ${{\psi }_i}^0$ and ${{\psi }_j}^0$ produces equations referred to as Method 2. Further approximation in the Method-2 equations yields a set of equations called Method 3. Methods 2 and 3 are counterparts of LKH's Method $b$, but have computational advantages over Method $b$ in that normalization and orthogonalization are accomplished in a particularly simple fashion. In comparing the uncoupled Method-3 equations with Dalgarno's equations, an additional difference is found involving the overlap integral between perturbed states, besides the difficulty pointed out by LKH. Application of the Method-2 and -3 equations is made to the spin-polarization problem of the ${\mathrm{Fe}}^{+3\mathrm{}}$ ion, leading to a hyperfine constant in reasonable agreement with earlier unrestricted Hartree-Fock (UHF) calculations. A comparison between results obtained by Methods 2 and 3 and Dalgarno's equations permits a relative evaluation of these methods. We have also studied the effect of indirect spin polarization of the $s$ electrons through the action of the $p$ electrons which are in turn polarized by the unpaired $d$ electrons. This contribution is found to be about 10% of the direct effect.