# Approximations to Hartree-Fock perturbation theory

## Abstract

A detailed examination is made of several approximations to the first-order Hartree-Fock perturbation equation. Four distinct methods are considered: the coupled (Method a) and the uncoupled (Method c) approximation of Dalgarno, a new alternative uncoupled approximation (Method b), and the simplified uncoupled approximation of Karplus and Kolker (Method d). By a comparison of the pertinent equations, it is shown that Methods a, b, and d correspond to each other in the use of a core potential analogous to that appearing in the zeroth-order Hartree-Fock equation, while Method c differs due to the inclusion of an extraneous self-potential term. An alternative analysis based on an orbital basis set expansion of the perturbed function demonstrates that Method c has an energy denominator of the simple Hückel type, while both Methods a and b include important two-electron correction terms. Also, it is found that of the four approximations, only Method c can be obtained as the first-order correction with a zeroth-order manyelectron Hamiltonian that has the Hartree-Fock determinant as its eigenfunction; the other techniques are one electron in character, as is the Hartree-Fock method itself. The significance of the difference in the core potential is demonstrated by test calculations of dipole and quadrupole polarizabilities and shielding factors for the two-, three-, and four-electron isoelectronic series. A variational technique is used with a trial function that has the form of a polynomial tunes the unperturbed orbital. For the polarizabilities, it is found that Method b is an excellent approximation to Method a, indicating that the self-consistency condition on the Method a solutions has a very small effect. The shielding factors, however, appear to be more sensitive to the self-consistency requirement. Both Method c and Method d introduce larger errors than Method b, with Method c particularly poor for threeelectron atoms and ions. The constraint introduced by choice of trial function is shown to be unimportant for polarizabilities, but quite severe for the four-electron atom shielding factors. A comparison of the complexity of the various techniques shows that the relative computing times for Methods a, b, c, and d are in the ratio of 300 to 60 to 75 to 1, Thus, Method d is simplest by far, although its speed is achieved by some loss of accuracy with respect to Methods a and b.