A systematic method is presented for deriving the Thomas-Fermi equation for an atom and the quantum corrections from the many-body description. The novel feature of the method is that it does not require any u priori assumptions about the assignment of electrons to fully occupied single-particle states or about the distribution of electrons in phase space, but shows instead that the distribution which is usually assumed, or derived from the assumption of fully occupied single particle states, is a direct consequence of specifying that the many particle system is in its ground state. The procedure used in the derivation is the expansion of the mixed position-momentum representation of the Green s function in a series of powers of A. The lowest order term is found to correspond with the Thomas-Fermi density. The form of the higher order terms, which are to be considered as corrections to zeroth order term, depends on the approximations made in the many-body equations for obtaining the Green s function. This paper deals only with the Hartree-Fock approximation, but the methods presented here allow generalization to other approximations which can include correlation eBects.
As a check on some of the terms, values of y^ (nl -> /') were also obtained by means of Eq. (23) from the functions u{ determined in II. The results are as follows: y"(2p -> p)= -1.230, y"(3s -> i) = 0.303, Yoc(3p->£)= -17.83, -7oo(3#->/) = 0.485. It is seen that these values are in good agreement with those obtained from ?;/. The maximum deviation occurs for Too (3^->p), where the difference amounts to 4%. The present results for the terms due to the radial inp -> p) modes can be compared with those of Wikner and Das 22 who used a variational method and obtain: yJ rD {2p->p) = -1.22, y" WD (3p->p)=-13.03.
This paper explores further the use of interelectron coordinates in constructing atomic wave functions. A simple method is developed for constructing wave functions of this type which yields surprisingly good values for the energy of a variety of atomic systems, in zero order. The Hamiltonian for the system is split into an unperturbed part, which is separable and which contains the interelectron potentials as well as the electronnucleus potentials, and into a perturbing term which is always finite and which vanishes whenever an electron is far from the nucleus. The zero-order energies corresponding to this splitting of the Hamiltonian are at least an order of magnitude better for the light atoms than the energies given by the usual Thomas-Fermi theory, and are considerably better than the energies calculated with hydrogenic functions alone in first order.
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