The Application of Basis Set Methods and Many-Body Theory in Electron-Molecule Scattering

Chapter I The difficulty encountered in electron-molecule scattering, which is absent from the atomic case, is due to the nonspherical nature of the potential describing the interaction of an electron with a molecule. Therefore a prerequisite to the development of an accurate description of electron-molecule collisions is an efficient method for treating single-particle scattering from nonspherical potentials. This chapter describes such a method. The essential idea is the approximation of the potential by a separable representation in terms of a finite set of square-integrable basis functions. This approximation gives the kernel of the Lippmann-Schwinger equation a separable form and allows us to write it as a finite matrix equation. The solution of this equation by ordinary matrix methods yields a basis set representation of the transition operator, and the scattering amplitude is easily calculated using this operator. This method is applied to the calculation of scattering amplitudes for a model two-center Gaussian potential and to e- - H2 elastic scattering at low energies in the static exchange approximation. The e- - H2 elastic scattering calculation is compared with the results of other workers. An entirely different approach to low-energy, elastic electron-molecule scattering from homonuclear diatomic molecules is also discussed in this chapter. It is based on the observation that, for electron scattering from homonuclear diatomic molecules at low energies, angular momentum is approximately conserved in spite of the nonspherical nature of the molecule. This observation allows the extraction of approximate phase shifts for electron scattering in the static exchange approximation from the results of diagonalizing the Hartree-Fock Hamiltonian in a large basis. This method is applied to e- - H2 and e- - N2 elastic scattering and the results compared with those of other workers. Chapter II A discussion of the application of many-body theory to electron-atom and electron-molecule scattering is presented. First a previous many-body theory of inelastic scattering is shown in first order to be a form of the distorted wave approximation. The target atom or molecule is described in the random phase approximation (RPA), while the distorting potential for the incident and scattered waves is the Hartree-Fock potential of the ground state. The application of many-body theory to elastic scattering is also discussed using a many-body formulation of the Feshbach optical potential. The difficulty of constructing Feshbach projection operators for a system of identical particles is overcome by the use of second quantized operators. Therelationship between this theory and the optical model of Bell and Squires, in which the many-body self-energy serves as an optical potential, is discussed in detail. Chapter III The application of the RP A form of the distorted wave approximation to inelastic e- - H2 scattering is described. Differential cross sections for excitation of the b 3∑+u and a 3∑+g states are calculated for a range of incident energies from 13 eV to 20 eV. Good agreement is obtained with the limited experimental data available for the differential cross section for excitation of the b 3∑+u state. The sum of the total cross sections for the two states is compared with the experimental cross section for electron impact dissociation of H2, and good agreement is found. Extensive discussion of the details of the calculation is included. Appendix Cross sections in the Born approximation for electron impact excitation of CO2 and CO using an RP A description of the target are presented. Generalized oscillator strengths for several states of CO2 are found to be in agreement with experiment. For CO, substantial disagreement exists between theory and experiment for the B 1∑+ state. A possible explanation involving a breakdown of the Born-Oppenheimer approximation resulting from an avoided crossing is proposed.