ON HARTREE-FOCK EQUATIONS FOR HELIUM WITH NON-ORTHOGONAL ORBITALS

It is shown that the multiplets of helium which should have non-orthogonal orbitals in their Hartree-Fock wave functions are those which originate from configurations in which the two electrons have identical azimuthal quantum numbers and which do not become inadmissible for two equivalent electrons (i.e. identical principal as well as azimuthal quantum numbers). Integro-differential equations are derived for the general case of a multiplet which should have non-orthogonal orbitals. A consistency relation and the analogue of the Hellmann-Feynman formula are derived for the solutions of the integro-differential equations. It is shown that these formulae are of considerable use in a perturbation treatment of the equations.