The Hartree-Fock equations for continuous states with applications to electron excitation of the ground configuration terms of O I

Abstract

Part I is concerned with the general theory of anti-symmetric wave functions for continuous states of atomic systems. For an ( N + 1) -electron system the complete wave function is expressed in terms of an expansion involving products of the N -electron core functions multiplied by free electron orbitals, the equations satisfied by the latter being obtained from the Schrödinger equation. It is shown that the only consistent means of obtaining anti-symmetric wave functions in approximate solutions is to make the expansion explicitly anti-symmetric. This procedure gives equations for the free-electron orbitals which are similar to bound-state Hartree-Fock equations. The further approximation of using Hartree-Fock wave functions for the core states is then discussed. Certain nlqkl configurations are analyzed in detail using a total angular momentum representation. It is shown that the equations may be uncoupled if the energy differences between the nl q terms are neglected (exact resonance approximation), and that approximate solutions of the full coupled equations may be obtained in terms of the exact resonance solutions provided that a suitable normalization condition is used. Part II is concerned with applications to electron excitation of the ground configuration terms of OI. Distorted wave approximations show that other effects are insignificant compared to the contribution from the p angular momentum component of the free orbitals, but give for this results which are too large by several orders of magnitude. The coupled equations for the p -wave are solved in an exact resonance approximation, with neglect of 1 s , 2 s exchange interactions. At a check point an exact resonance solution including 1 s , 2 s exchange terms is obtained, and finally a complete solution of the coupled equations. Inelastic collision cross-sections calculated from the exact resonance solutions are found to be 72% (without 1 s , 2 s exchange) and 95% (with 1 s , 2 s exchange) of the result from the complete solution. Final curves for the collision parameters, which rise to within 70 % of the limit set by charge conservation, are considered to be of an accuracy approaching that of the Hartree-Fock method for bound-state problems. A final section is concerned with the contributions of the p -wave to elastic scattering of slow electrons by OI.