Approximate Hartree—Fock Wavefunction for the Helium Atom

The function φ=Aexp(−ζr)+B(1/ζ′′r)exp(−ζ′r)[1−exp(−ζ′′r)] is shown to be an excellent analytical approximation to the Hartree—Fock orbital of the helium atom, giving an energy of —2.86167 a.u., versus the exact Hartree—Fock energy —2.86168 a.u. Differences between this and other approximate orbitals, and the exact Hartree—Fock orbital, are discussed. The orbital form due to Green is recovered as the special case ξ″=0. It also yields the energy —2.86167 a.u., but it gives less accurate values for 〈r〉, 〈r2〉, 〈r3〉, and 〈r4〉 than are given by the generalized form. Reasons are given for including, in atomic and molecular variational calculations as new s-type orbitals, functions of the form, 0s=[exp(−αr)−exp(−βr)]/r.