Gy Csanak

Green's Function Technique in Atomic and Molecular Physics

Publisher Summary This chapter seeks to familiarize the quantum chemist and molecular physicist with some of the ways one can apply the Greens function technique to the problems of calculating excitation energies, ionization energies, ground state energies, and transition matrix elements. Density matrices and natural orbitals can be calculated directly without prior calculation of the wavefunctions and therefore the Greens function technique is closely related to the density matrix methods already developed in quantum chemistry. The Greens function method is clearly the way to calculate the density matrix and the natural orbitals directly. For finite atomic and molecular systems, the branch cuts are replaced by a set of poles and adjoining branch cuts. Poles can exist on the physical sheet, as well as on the nonphysical sheets reached by continuing across the cut.

The application of first order many-body theory to the calculation of the differential and integral cross sections for the electron impact excitation of the 21S, 21P, 23S, 23P states of helium

The first order form of the many-body theory for inelastic scattering (called random phase approximation) has been applied in the 29.6 eV<or=E<or=81.63 eV energy region for the electron impact excitation of the 21S, 21P, 23S, 23P states of helium. Differential and integral cross sections are calculated and compared with recent experiments. The results for each state are discussed and compared with other calculations.

Many-body theory of the generalized optical potential for electron-atom (molecule) inelastic scattering

Using the Martin-Schwinger variational technique (1959), a new theory is obtained for electron-atom (molecule) inelastic scattering. This theory constitutes a higher order approximation than the GRPA elastic theory. In analogy with elastic scattering, an equivalent optical potential, named, the transition potential, is introduced for inelastic scattering. In the GRPA this transition potential is of the first-order static-exchange type. The present theory adds three new effects to the GRPA transition potential. One of them represents the screening of the first-order exchange term, another one the transition polarization effect and the third the effect of the final state field. In contrast to close-coupled or Kohn-type variational theories, the present theory has the advantage of decoupling the scattering channels through the use of the energy-dependent transition potential.

Quantum mechanical theory of translation‐internal mode energy transfer in collisions of atoms with diatomic systems

The field theoretical Greens function technique has been used to define an elastic scattering optical and inelastic scattering transition potential for the structureless atom‐diatom scattering problem. The functional differentiation technique of Schwinger has been utilized to generate approximations for these potentials. First and second order approximations are given and their physical interpretation discussed.

The application of the time-dependent Hartree-Fock method to the calculation of excited state-excited state transition densities of atoms /molecules/

The time-dependent Hartree-Fock method (or random phase approximation) has been used to derive a coupled system of integrodifferential equations which can be used to calculate excited state-excited transition densities. The physical interpretation of these equations is also given.

The calculation of orientation and alignment parameters for the electron impact excitation of the 21P state of He in the first-order many-body theory

An attempt has been made to resolve an apparent disagreement between two very similar calculations, by Thomas et al. (1974) and Madison and Shelton (1973), for the electron impact excitation of He 21P at 80 eV impact energy. The present results are in good agreement with those of Thomas et al. Another parameter is calculated which has not been calculated previously in the first-order many-body theory, and the angular rate of the calculation is extended to compare it with newly obtained experimental data.

A time-dependent first-order Bethe-Goldstone theory

The time-dependent first-order Bethe-Goldstone theory is defined by using the Hartree-Fock approximation in the Bethe-Goldstone equation with the external field fully coupled in. Starting from this equation and using the Gell-Mann and Low projection technique various amplitude equations are obtained. Finally an equation is obtained in differential form for the excited-state pair amplitude. This can be used to calculate transition moments between the ground and excited states of atomic or molecular systems which incorporate short-range correlation effects.

The orthogonality condition in many-body scattering theory

It is shown that the orthogonality condition referring to total scattering states with identical boundary conditions is satisfied in the random phase approximation. This orthogonality condition is also discussed for elastic scattering.

The application of the Martin-Schwinger functional differentiation technique to spin-degenerate systems

It is shown that Martin-Schwinger functional differentiation technique can be extended with certain limitations to systems with spin-degenerate ground states. The Hartree-Fock and the time-dependent Hartree-Fock approximations are discussed for these systems.

Analytical approach to the first-order Bethe-Goldstone equation

The first-order Bethe-Goldstone theory (BG1) is defined as the Bethe-Goldstone equation for the two-particle Greens function with the one-particle Greens function calculated in the Hartree-Fock approximation. It is shown that this equation is closely related to the fundamental equations of pair theories developed for electronic systems in atoms and molecules. With the aid of an analytical technique it is shown that this theory can be formulated in terms of coupled integrodifferential equations.