Piecewise polynomial configuration interaction natural orbital study of 1 s2 helium

Abstract

We report here an analysis of extensive configuration interaction (CI) wave functions for the 1s² ground state of the helium atom using piecewise polynomial basis functions. Large numbers of natural radial orbitals (NROs) with l ranging from 0 to 11 have been treated accurately and analyzed systematically. The contribution of each NRO to the total energy is found to follow the formula ΔE∼−0.42(l+1/2) × (n−1/2)⁻⁶a.u., where n is the principal quantum number, and the expansion coefficient of the NRO configuration is found to follow the formula, c∼−0.35(l+1/2)^2/3 (n−1/2)⁻⁴. (The constants for l=0 are a little different). From an examination of the tails of the NROs, we are able to suggest an l‐dependent universal asymptotic formula, r^{β−l−1+δ}e^−ζr, where β is a constant, where δ=δ_l0, and where ζ²/2 is the ionization potential. The nodes of the NROs are also found to behave in a systematic way that yields valuable information on the choice of basis functions. So‐called ’’L’’‐limit energies E_L, more accurate than any we could find in the literature, have been determined for L=0 through 11, with a final CI energy calculated here of −2.90370 a.u. from a wave function containing 118 NROs. The increments E_L−E_L−1 are found to follow the formula −0.0740(L+1/2)⁻⁴−0.031 (L+1/2)⁻⁵ a.u., which is useful for extrapolation and extension of the CI energy by two significant digits. The n‐ and l‐dependent formulas for the energy contributions make it possible to estimate the size of a CI calculation required for a given accuracy. The high accuracy of these CI calculations is made possible by the flexibility and numerical stability of piecewise polynomial basis functions.