The principle-quantum-number (and the radial-quantum-number) expansion of the correlation energy of two-electron atoms

The second-order correlation energy of two-electron ions is studied in terms of an expansion in minimal approximations to the first-order natural orbitals (NOs). The non-linear parameters of these NOs are determined by minimization of the second-order energy. An approximation to the total second-order correlation energy is obtained as a sum of increments e(lp), depending on the angular quantum number l and the radial quantum number p. (Either l or p can be eliminated in favor of the principal quantum number n = l + p.) Closed expressions for these energy increments are derived. For fixed p the increments go as (l + 1)(-5). This is consistent with the behavior of the exact partial wave increments (that depend on the parameter l only) as (l + 1/2)(-4). While the partial wave increments correspond to a summation of elp over p, other partial summations of the two-parameter increments lead to either the principal-quantum-number expansion (PQNE) with energy increments similar to n(-4), or the radial-quantum-number expansion, with a less transparent convergence pattern. Unfortunately these partial summations can neither be done in closed form nor from the asymptotic expansion, but some insight is obtained from a numerical summation. The hope to find a rigorous derivation of the PQNE has not been fulfilled.