The One-Electron Energies of a Neutral Zinc Atom Evaluated Using Nonlinear Field Theoretical Methods: Comparison with Hartree–Fock Calculations

Abstract

The Method of Coherent Structures (MCS) has been used, as an example of a sufficiently complex atomic system, to determine the one-electron eigenvalues of a neutral zinc atom. An ensemble procedure is used in MCS to properly account for spatial correlations even for a relatively small number of particles. The MCS is a semiclassical approach, but the essentially Fermionic character of the electrons emerges via physical boundary conditions on the choice of classical field. We interpret the classical field as a coherent state envelope which partly determines the effective potential in which quantum fluctuations take place. We find that three distinct régimes of behaviour of the classical field exist depending on the magnitude of the effective Coulomb repulsion, μ. When μ<μ0, a critical value, those classical fields which correspond to certain discrete values of μ are physically acceptable normalisable bound charge states. When μ<μ0 and does not correspond to the discrete values, classical solutions exhibit a damped oscillatory behaviour typical of charged "ring" waves. For μ>μ0, corresponding to an excess of electronic charge, only a small fraction of the charge, represented by a few oscillations about the nucleus, remains attracted to this centre while the remainder escapes from the nucleus. Parameters representing the repulsive energy strength and energy shifts agree well with trends in the Slater parameter F 0(2s,2p) as a function of atomic number Z and also Moseley's Law. Agreement is also obtained with the first-principles form of charge density at small distances from the nucleus as given by Kato's Theorem and also at large distances as outlined by March. The usefulness and validity of this approach has been shown by comparing such energies with those determined by a Hartree–Fock–Slater (HF) method which is well known to satisfactorily account for many specific details of the level structure of atoms. Good agreement with HF has been obtained for both shell positions and energy ordering of states within a shell.