Correlation and Quantum Corrections in the Thomas-Fermi Model of the Atom

A new method is described for computing the effect of correlation, inhomogeneity, and exchange on the Thomas-Fermi model of the atom. The method makes use of the many-body point of view, rather than an independent-particle point of view, by considering the hierarchy equation linking the n-particle Green's functions. The hierarchy is truncated by a prescription equivalent to the Gell-Mann and Brueckner theory of the high-density electron gas, resulting in a description of the atom in which the exchange interaction is replaced by the effective interaction. The physical significance of this replacement is noted.The Green's function for this model is then expanded as a series in powers of . The lowest order term is found to describe the Thomas-Fermi model of the atom. The equation for the next higher term contributing to this expansion is manipulated so as to yield an ordinary differential equation for the corresponding correction to the potential. This equation contains a term which expresses the effect of inhomogeneity and another which arises from the correlation of the electrons and from exchange. The inhomogeneity term is one which has been found previously. Study of the correlation term shows that it depends on the separation energy of an electron from an infinite electron gas, which suggests a generalization by which the method might be made applicable to those outer regions of the atom for which the electron density is below that to which the Gell-Mann and Brueckner theory would apply.