Perturbed Hartree–Fock Theory. I. Diagrammatic Double-Perturbation Analysis

Abstract

The coupled Hartree–Fock scheme, for calculating atomic and molecular properties which are second order in some perturbing field, is studied with the help of a double‐perturbation expansion and diagrammatic techniques. The double‐perturbation expansion, with the external field as one perturbation $({\mathcal{H}}^{\text{(1)}})$ and the difference between the true electron‐repulsion potential and the Hartree–Fock potential as the other $\mathrm{(V)}$ , is compared with an iterative solution of the set of coupled Hartree–Fock equations for the first‐order perturbed orbitals. The term in the double‐perturbation expansion that is second order in ${\mathcal{H}}^{\text{(1)}}$ and zeroth order in $V$ is shown to be the Dalgarno “uncoupled” second‐order energy. The coupled Hartree–Fock second‐order (in ${\mathcal{H}}^{\text{(1)}}$ ) energy expression includes all terms first order in $V$ and some of the terms to all orders in $V$ , up to infinite order. It is shown how to separate the diagrams contributing to the CPHF energy from those that do not; in particular, the CPHF equations are composed of the so‐called “bubble diagrams” and their corresponding exchange diagrams. It is shown that each particle–hole pair, or “bubble,” is independent of all the other particle–hole pairs, in the sense that the total energy contribution from the diagram can be factored into contributions from the individual particle–hole pairs. It is shown that the first‐order perturbed wave‐function obtained from the coupled Hartree–Fock formalism includes effects due to all singly‐excited determinants and a certain class of linear combinations of doubly‐excited determinants, namely those singlet spin eigenfunctions which correspond to two electrons being excited without spin flips from Hartree–Fock unexcited orbitals to virtual orbitals; there exists a second, linearly independent, doubly excited singlet function which is not included in the CPHF wavefunction and does not contribute to the perturbation energy. The contribution to the second‐order energy from the doubly excited determinants amounts to about 10% of the total CPHF second‐order energy in the case of some electric and magnetic properties of H₂ and 25% of the total second‐order energy corresponding to the electric dipole polarizability of the Be atom. The larger contribution of the terms involving doubly excited determinants for Be has its origin in the fact that a double excitation from the $\text{2s}$ to the $\text{2p}$ orbital gives a large fraction of the correlation energy between the $\text{2s}$ electrons and that the single excitation from the $\text{2s}$ to the $\text{2p}$ orbital is the largest single contribution to the polarizability. An alternate iterative scheme for the solution of the coupled Hartree–Fock equations is derived. This alternate iterative scheme, which includes certain terms summed to infinite order, is expected to be a considerable improvement over the simple scheme which starts with the “uncoupled” approximation as the zeroth iteration. Comparisons of the different schemes for the properties considered in this paper are made to confirm this conclusion. The CPHF energy (second order in ${\mathcal{H}}^{\text{(1)}}$ ) is partitioned approximately into a part due to correcting the unperturbed Hartree–Fock wavefunction for correlation and a part due to correcting the uncoupled Hartree–Fock perturbed wavefunction for correlation; the contribution from the former was shown to be about 10%–20% of that from the latter.