Validity criteria for rotor-vibrator and independent-particle models of atoms: overlaps, oscillator strengths, and angular deviations

Abstract

A comparison is made of Hartree-Fock, independent-particle (HF) and collective, molecule-like rotor-vibrator (RV) models for atoms with twovalenceelectrons. The two models are constructed from the same pseudopotential Hamiltonian. The criteria for the comparison are (a) overlap with well-converged configuration interaction (CI) wave functions, (b) comparison of oscillator strengths with values from well-converged CI functions (and, wherever possible, from experiment), and (c) comparison of the root-mean-square deviation of the angular distribution, Le., ((B12)2 - (812~))'/~. In the comparison of the overlaps, neither model is obviously superior to the other. The RV wave functions yield better oscillator strengths in more cases than the HF functions, despite the results of the overlap calculations. Qualitative comparisons of the spatial distributions of probability density suggest that the RV functions have angular distributions much more like the CI functions than do the HF functions. A simple model calculation of the effect of a small amount of configuration mixing elucidates this similarity. A brief comparison is made of energy optimization and overlap optimization. Many methods of modeling atoms use an independent-particle approach as a first approximation. In the independent-particle model, the electrons are assumed to maintain their individual angular momenta and energies, independent of all the other electrons in the atom. A traditional measure of how well the independent-particle model describes an atom is the correlation energy. The correlation energy in atoms, strictly the difference between the exact, nonrelativistic energy and the Hartree-Fack energy, can be fairly accurately determined by comparing the energy expectation values of well-converged CI (configuration interaction) and HF (Hartree-Fack) wave functions. Two new ways of classifying ground and low-lying excited states of pseudotwo-electron atoms have suggested some different zero-order approximations to the wave functions.Id One approach uses the hyperspherical coordinate system typically used in the adiabatic appro~imation.3.~ other uses the electron-nucleus-clectron angle and the electron-nucleus distances in rovibrational functi0ns5,~ as distinct from the functions developed by Hylleraas7q8 which include the interelectron distance explicitly and so correctly describe the cusp region. This paper shows the results of a comparison of the second of these two types of correlated wave functions, the RV functions with the independent-particle model. The HF method takes into account the mean field of the probability distribution of the charge of the electrons on each electron but does not include the individual interactions between electrons. Thus, the HF method, apart from the effects of particle symmetry, omits the correlation between the electrons present in any atom or molecule involving more than one electron. In this paper, we use newly calculated HFSCF (self-consistent field) wave functions as the independent-particle wave functions. Wave functions developed by Hunter and BerryS in the form of overlap optimized rotor-vibrator (RV) wave functions are used as the extreme of correlated wave functions. Although the CI wave functions are constructed from an independent-particle basis, the completeness of the expansion set allows a formally exact convergence, and tests performed previously6 justify the assump tion that the CI wave functions are well converged. The CI wave functions used are those calculated with Sturmian basis functions by Krause and Berry6 and use a p~eudopotential~ for the core electrons of atoms with more than two electrons. Overlaps between CI and RV wave functions have previously been calculated5 for several states, as have the oscillator strengthsl0