A discontinuous energy–density functional

The properties of the integrand in the Foldy–Wilson formula E(Z̄) = −∫0Z̄ R(Z)dZ are studied. Here, E(Z̄) denotes the total, nonrelativistic, electronic energy of an atom with nuclear charge Z̄, and R(Z) = ∫dr r−1ρ(r,Z), where ρ(r,Z) is the electron density corresponding to nuclear charge Z. From studies of two-electron atoms and an analysis of the Z−1 perturbation expansion, it is postulated that, in small atoms at least, R(Z) generally resembles a series of discontinuous straight lines in the range 0⩽Z⩽Z̄, at constant number of electron N. The discontinuities in R(Z) occur at critical points in Z, which are close to intergral values. Each critical point corresponds to the ejection of an electron from the confines of the nucleus. For two-electron atoms, the discontinuity is compatible with Stillinger’s observation of a two-electron localized bound state in the continuum at Z<1. The properties of R(Z) are then used to construct equations to generate total energies that incorporate electron–electron correlation from Hartree–Fock densities by means of ’’trapezoid recursion relations.’’ Use is made of sums of the R(Z) and sums of chemical potentials (electronegativities) as defined by Parr et al. Near-exact total nonrelativistic energies are obtained for all neutral atoms through Ar.