Direct minimization of the energy by simultaneous variation of parameters in nonorthogonal basis functions I. Method

A direct method to optimize parameters in nonorthogonal basis orbitals is discussed. The partial derivatives of the energy of the state of interest, not necessarily the ground state, with respect to the orbital parameters are calculated analytically. The required cofactors up to third order of the overlap matrices over spin-space orbitals in pairs of Slater determinants are calculated by biorthogonalization. All parameters are varied simultaneously in the direction of the negative gradient (steepest descent). A search logic with dynamically adjusted step-size is shown, in the example of the minimum basis Ne ground state, to find the energy minimum in an efficient manner