Many-Body Approach to Electron Correlation in Atoms and Molecules

Many-body perturbation theory (MBPT) is used to investigate the energy, wave function, and two-electron density matrix of a closed-shell atom or molecule when the initial approximation to the wave function is taken in the Hartree-Fock approximation. Since, as Kelly has found in his calculations using MBPT, certain classes of terms dominate in all orders of perterbation theory, models are investigated in which certain types of terms are summed to all orders of perturbation theory. The diagrams which are summed are explicitly chosen to provide results in the form of inhomogeneous integrodifferential equations which are as similar as possible to the models used by Kelly, Sinano {g} {g}{}lu, and Nesbet, thereby enabling some comparison to be made between these theories and indicating some of the corrections to them.It is proven that the diagrams approximated by Kelly to all orders of perturbation theory, which have only two electrons excited at a given time, could be calculated from a knowledge of what Sinano {g} {g}{}lu calls Bethe-Goldstone (BG) pairs, which represent the perturbative corrections for correlation for a pair of electrons moving in the field of the remaining electrons, which are in their ground-state Hartree-Fock (HF) orbitals. The uncoupled linear inhomogeneous equations for the pairs could be solved by the use of a correlated basis set, and their solution can be found variationally, along with the individual pair correlation energies, by varying part of the energy of a simple cluster-type wave function. However, as in the case of the other models presented in this work, certain variations of part of the energy are shown to be equivalent to the summation of particular classes of diagrams to all orders of perturbation theory.The doubly excited diagrams omitted by the BG pairs arise from the fact that when a pair of electrons are excited out of the HF "sea," the Pauli principle no longer prohibits the remaining electrons from occupying the space left behind by the excited electrons. The inclusion of these exclusion effects results in coupled linear inhomogeneous equations for the new pair-correlation correction functions. Single and triple excitations modify the pair equations by the inclusion of a two-body potential, respectively, describing the virtual de-excitation of one of the two correlating electrons and virtual excitations of a third electron in the field of the two correlating electrons. Alternatively, coupled linear inhomogeneous equations can be written for the one-, two-, and three-electron cluster functions, and these equations may be solved variationally.The generalization of the above results to cover the case in which the zeroth-order wave function is a single Slater determinant of arbitrary orthonormal spin orbitals (as well as a treatment of the higher excitations) is presented in the following paper. In the appendix, a discussion is presented to indicate the connection between what Sinano {g} {g}{}lu calls "unliked clusters" and the diagrams of MBPT.