Fock\u27s representation for molecular orbitals

V. Fock studied the hydrogen atom problem in momentum space by projecting the space onto a 4-dimensional hyper-sphere. He found that as a consequence of the symmetry of the problem in this space the eigen-functions are the R4 spherical harmonics and that the eigenvalues are determined only by the principal quantum number n. In this chapter we note that if his method is applied to the 2-dimensional Kepler problem in momentum space, the eigenfunctions are the R3 spherical harmonics, Y1m, and the eigenvalues are determined only by the quantum number 1. These facts enable one to give a visualizable geometric discussion of the dynamical degeneracy

Equations-of-motion method including renormalization and double-excitation mixing

The equations‐of‐motion method is discussed as an approach to calculating excitation energies and transition moments directly. The proposed solution [T. Shibuya and 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‐particle‐hole 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

Electronic excitations of benzene from the equations of motion method

We have used the equations of motion method to calculate the excitation energies and intensities of several transitions in benzene. The ordering of the singlet and triplet states of B_(2u), B_(1u), and E_(1u) symmetry agrees with experiment and the error in the calculated frequencies ranges from 3% to 25%. This error range is reasonable considering the relatively small basis set used. The most extensive calculation included 10 hole and 28 particle states and shows the effect of changes in the sigma core for each transition. The calculated transition moment of 1.74 a.u. for the ^1A_(1g)→^1E_(1u) transition agrees well with the experimental value of 1.61 a.u

Higher Random-Phase Approximation as an Approximation to the Equations of Motion

Starting from the equations of motion expressed as ground-state expectation values, we have derived a higher-order random-phase approximation (RPA) for excitation frequencies of low-lying states. The matrix elements in the expectation value are obtained up to terms linear in the ground-state correlation coefficients. We represent the ground state as eU|HF〉, where U is a linear combination of two particle-hole operators, and |HF〉 is the Hartree-Fock ground state. We then retain terms only up to those linear in the correlation coefficients in the equation determining the ground state. This equation and that for the excitation energy are then solved self-consistently. We do not make the quasiboson approximation in this procedure, and explicitly discuss the overcounting characteristics of this approximation. The resulting equations have the same form as those of the RPA, but this higher RPA removes many deficiencies of the RPA

Application of a higher RPA to a model pi-electron system

We have applied a proposed higher-order random phase approximation (RPA) to the simple model system of two pi electrons in the double bond of ethylene. The higher-order RPA removes some difficulties involved in the usual RPA, but retains the form of the RPA equations. To derive the higher RPA we retain all terms linear in the ground state correlation coefficients in the equations of motion for the excitation energy and in the equation determining the ground state. These equations are solved self-consistently and are simpler to handle than a configuration interaction solution in a realistic example. We do not use the quasiboson approximation

Application of the equations-of-motion method to the excited states of N2, CO, and C2H4

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

Application of the RPA and Higher RPA to the V and T States of Ethylene

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 15% over its RPA Value. The N → 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

Equations-of-motion method: Potential energy curves for N2, CO, and C2H4

We have applied the equations-of-motion method to various states of N2, CO, and ethylene at nuclear configurations slightly distorted from the ground equilibrium geometry. This approach attempts to calculate energy differences instead of absolute energies and is thus relatively insensitive to the accuracy of the assumed ground state wavefunction. By using the experimental behavior of the ground state on distortion, we can generate accurate potential energy curves for the excited states in the region of spectroscopic interest. These curves confirm the spectroscopic behavior of the 1∑+ states of N2 and the 1∑+ states of CO where valence and Rydberg states of the same symmetry interact. The results for the T and V states of ethylene agree with experiment and show that the V state is predominantly a highly correlated valence state. Oscillator strengths across an absorption band are also accurately determined in this method. We report the dependence of the transition moment on bond length for the X1∑+→A1II transition in CO, which is in excellent agreement with experiment

Symmetry-adapted HAM/3 method and its application to some symmetric molecules

The semiempirical HAM/3 method developed by Lindholm and coworkers about two decades ago has been known to have a deficiency that splits energies for the degenerate energy states. We have recently proposed a group-theoretical approach to remedy the internally broken symmetry of the HAM/3 Hamiltonians. In this paper, we present some results of its application to various small molecules with symmetry Td, C3v, and D3h. The proposed scheme gives correct degeneracy for these molecules.O método semi-empírico HAM/3, desenvolvido por Lindholm e colaboradores há mais de duas décadas, tem uma deficiência. As energias de excitação calculadas por HAM/3 para estados degenerados são desdobradas. Recentemente foi proposto um método para corrigir esta deficiência. Apresentamos aqui resultados de aplicações deste novo método para algumas moléculas com simetria Td, C3v e D3h. O esquema proposto apresenta a degenerescência correta para as moléculas estudadas.445449Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP