No abstract

The equations-of-motion method is discussed as an approach to calculating excitation energies and transition moments directly. The proposed solution [T. V. McKoy, Phys. Rev. A 2, 2208 (1970)) of these equations is extended in two ways. First we include the proper renormalization of the equations with respect to the ground state particle-hole densities. We then show how to include the effects of two-particlehole components in excited states which are primarily single-particle-hole states. This is seen to be equivalent to a single-particle-hole theory with a normalized interaction. Applications to various diatomic and polyatomic molecules indicate that the theory can predict excitation energies and transition moments accurately and economically.

We have used the equations-of-motion method to study various states of N2, CO, and ethylene. In this approach one attempts to calculate excitation energies directly as opposed to solving Schrödinger's equation separately for the absolute energies and wavefunctions. We have found that by including both single particle-hole and two particle-hole components in the excitation operators we can predict the excitation frequencies of all the low-lying states of these three molecules to within about 10% of the observed values and the typical error is only half this. The calculated oscillator strengths are also in good agreement with experiment. The method is economical, requiring far less computation time than alternative procedures.

We have applied our proposed higher random-phase approximation (HRPA) to the T and V states of ethylene. In the HRPA, unlike the RPA, one solves for the excitation frequencies and the ground-state correlations self-consistently. We also develop a simplified scheme (SHRPA) for solving the equations of the HRPA, using only molecular integrals sufficient for the usual RPA calculations. The HRPA removes the triplet instability which often occurs in the RPA. The excitation energy for the N-->T transition is now in good agreement with experiment. The N-->V transition energy increases by IS% over its RPA value. TheN--> V oscillator strength changes only very slightly. These results are also useful in explaining the appearance and ordering of states obtained in recent direct open-shell SCF calculations.

Fock studied the hydrogen atom problem in momentum space by projecting the space into a 4-dimensional hyperspherical space. He found that as a consequence of the symmetry of the problem in this space the eigenfunctions are the R4 spherical harmonics and that the eigenvalues are determined only by the principal quantum number n. In this paper we note that if his method is applied to the 2-dimensional Kepler problem in momentum space, the eigenfunctions are R3 spherical harmonics, Ylm, and the eigenvalues are determined only by the quantum number l. These facts enable one to give a visualizable geometrical discussion of the dynamical degeneracy.