Complete basis set correlation energies. I. The asymptotic convergence of pair natural orbital expansions

An expression for the ’’correlation energy’’ of a multiconfiguration wave function is developed using perturbation theory. The asymptotic form of this expression for an N-configuration pair natural orbital expansion is Error(N×N)?(Σμ = 1NCμ)2 (−225/4608)N−1. The asymptotic form attributes the dominant variation in multiconfiguration pair correlation errors to an interference effect between low-lying natural orbitals. Three levels of extrapolation based on the asymptotic convergence of pair natural orbital expansions are examined. The first requires separate calculations with 5 and 14 natural orbitals. When applied to the helium atom, for which E(5) = −2.897 484 and E(14) = −2.901 697, the extrapolated value, E = −2.903 724, is accurate to within 0.05% of the error from the 14 natural orbital wave function (i.e., the absolute accuracy is ≲0.000 001 hartree). The second extrapolation requires separate calculations with 5 and 14 pair MCSCF configurations and is accurate to within 2% of the MCSCF (14) error (i.e., the absolute accuracy is ≲0.000 05 hartree) for the helium isoelectronic series. The third extrapolation requires only the 5-configuration MCSCF calculation. This extrapolation is accurate to ∼10% of the MCSCF (5) error (i.e., the absolute accuracy is ∼0.0005 hartree) for the cases examined, including CH2, Ne, He, and H2. This is comparable to the accuracy of an MCSCF calculation including ten times as many natural orbitals (which would require a factor of ∼104 more computing time).