Direct Calculation of Approximate Natural Orbitals and Natural Expansion Coefficients of Atomic and Molecular Electronic Wavefunctions. II. Decoupling of the Pair Equations and Calculation of the Pair Correlation Energies for the Be and LiH Ground States

Abstract

Starting from the integro-differential equations satisfied by the natural orbitals (NO's) in the framework of the antisymmetrized-product-of-strongly-orthogonal-geminals (APSG) approximation, a set of decoupled equations for the different electron pairs is derived, which corresponds to the physical model of independent electron pairs in the Hartree-Fock field of the other electrons. These simplified equations lead to an easily practicable computational scheme which furnishes directly the correlation energy of the different pairs and a wavefunction, which is an antisymmetrized product of geminals which are, however, no longer strongly orthogonal in the rigorous sense. The total intrapair correlation energy, which is defined as the energy expectation value (with respect to this wavefunction) minus the Hartree-Fock energy, differs from the sum of the pair correlation energies by a correction term, the magnitude of which depends on the highest occupation number of a weakly occupied NO. In the Be ground state, where this highest occupation number is about 0.1, the correction term amounts to ∼2% of the correlation energy; in the LiH ground state, none of these occupation numbers is bigger than 0.01 and the correction term is much less than 1% of the correlation energy. A detailed discussion of these correction terms, which is of vital importance for all theories of decoupled electron pairs, is given in the Appendix. An alternative, simpler way to get a rigorous bound for the correlation energy, based on a configuration interaction calculated only with doubly substituted NO configurations, is also indicated. The method is applied numerically to the Be and LiH ground states, and it shows quite promising results. In Be, the dependence of the intrapair correlation energy on the “localization” of the strongly occupied orbitals is examined in detail; so is the influence of occupied outer-shell electrons on the correlation energy of the inner one. Different previous calculations on the Be ground state are compared with ours, and the formal relation between these schemes is pointed out. In the LiH calculation, it is shown that the weakly occupied approximate natural orbitals, as calculated by our scheme, are localized in the same region of space as the corresponding strongly occupied ones. The rather poor result for the correlation energy of the inner pair (0.0302 a.u. rather than 0.0414 a.u. in the Li atom) is due to the inadequacy of the basis of pure Gaussians used; the basis is much better for the outer pair; 78% of the exact overall correlation energy is accounted for. A potential curve, including correlation, is calculated for the LiH molecule. The change in correlation energy is essentially that of the valence-shell correlation energy. For this, the σ-correlation energy increases with distance, and the π-correlation energy decreases, so that near the minimum, the total correlation energy remains almost constant. Twenty minutes of IBM 7094 Model 1 were necessary for one point of the potential curve.