Comparison of the Hartree—Fock Orbital with the First Natural Spin Orbital for Two-Electron Systems

A demonstration is given that for two-electron systems the Hartree—Fock and first natural spin orbitals are different functions. The method used is perturbation theory for which the zero-order problem is the Hartree—Fock approximation. A perturbation expansion through the second order is obtained for the first natural spin orbital. The two orbitals begin to differ in the second order, and their energies in the fourth order. An equation for the second-order part of the orbital difference function is derived. Estimates of the norms of the orbital difference functions are calculated for the ground states of the helium atom and the hydrogen molecule and are found to be small.