Application of Wave Functions Containing Interelectron Coordinates. II. Approximate Energy Levels of Atoms

Abstract

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.

Hylleraas method for many‐electron atoms. I. The Hamiltonian

“…In addition, they are mutually independent of each other and of each coordinate of the initial subset. The derivatives of the wave function with the 3n coordinates and of the wave function with the redundant set of 3n ϩ ( 2 n ) coordinates, respectively, are the same due to the use of the chain rule of derivation [13].…”

“…Pluvinage proposed a perturbation method using a wave function with the r 12 coordinate in the orbital exponent and got excellent results for helium-like systems. Walsh and Borowitz [13] modified the Pluvinage method so that it would be applicable for more complex systems. These investigators demonstrated that the introduction of additional coordinates is equivalent to the use of the chain rule of differentiation.…”

“…A similar one holds for i . The Hamiltonian [13] cannot be used directly for calculations, because the cross-terms are expressed as scalar products of vectors (r ជ ij in Cartesian coordinates). Walsh and Borowitz calculated the ground state of atoms up to aluminum without using the exact Hamiltonian, but an approximated perturbed (angular) part of the Hamiltonian.…”