Approximations to Hartree—Fock Perturbation Theory

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 many-electron 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 times 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 three-electron 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.