Application of symmetry-adapted pair functions in atomic structure calculations: A variational-perturbation treatment of the Ne atom

The atomic $N$-electron wave function of the independent-pair approximation is defined in terms of orbital configurations of one-electron functions and symmetry-adapted pair functions in the form of partial-wave (PW) expansions. This form is convenient for an extensive use of irreducible tensor operators. The formulation was used to set up a variational-perturbation scheme for closed-shell atoms in the case of the Rayleigh-Schrödinger perturbation theory with the symmetric sum of Hartree-Fock operators for the zerothorder Hamiltonian (RS-HFPT). A detailed study of the second- and third-order correlation energies of Ne is made in order to analyze the nature of various correlation effects. All PW's up to $l^{\prime }$,$l^{\prime \prime }\le 9$ are considered. Particular attention is given to the problem of eliminating the radial basis saturation errors. The upper bound to the second-order energy is determined by means of 13 880 nonoptimized configurations to be -0.38638 a.u., which represents 99.3% of the "experimental" correlation energy. Extrapolation of the pair energies for $l^{\prime }$, $l^{\prime \prime }>9\mathrm{}$ results in a second-order energy of -0.3879 a.u. (99.7% of the total correlation energy). The PW expansions for the ($\mathrm{ns}$,$n^{\prime }s$) pairs are compared for He, Be, and Ne. Remarkable regularities are observed, indicating that the PW formulation represents a convenient tool for the investigations of correlation effects. The third-order energy obtained for a shorter expansion of the first-order wave function amounts to 0.00245 a.u. A discussion of the relative importance of the diagonal and off-diagonal contributions is presented, and detailed comparisons with the results of many other methods are made. It turned out that the RS-HFPT approach in the present formulation has several advantages over other perturbation methods.