Orbitals Dyson

The propagator matrix is energy-dependent poles occur when E equals an ionization energy, Eo N)-En N-l),oi an electron affinity, Eo(N)-Em(N+1). Dyson orbitals (DOs) for ionization energies are defined by... 

Electron propagator theory generates a one-electron picture of electronic structure that includes electron correlation. One-electron energies may be obtained reliably for closed-shell molecules with the P3 method and more complex correlation effects can be treated with renormalized reference states and orbitals. To each electron binding energy, there corresponds a Dyson orbital that is a correlated generalization of a canonical molecular orbital. Electron propagator theory enables interpretation of precise ab initio calculations in terms of one-electron concepts. 

Eigenfunctions that accompany these eigenvalues have a clear physical meaning that corresponds to electron attachment or detachment. These functions are known as Dyson orbitals, Feynman-Dyson amplitudes, or generalized overlap amplitudes. For ionization energies, they are given by... 

This procedure requires analytical expressions for EPP(E) and its derivative with respect to E it usually converges in three iterations. Neglect of off-diagonal elements of the self-energy matrix also implies that the corresponding Dyson orbital is given by ... 

For many ionization energies and electron affinities, diagonal selfenergy approximations are inappropriate. Methods with nondiagonal self-energies allow Dyson orbitals to be written as linear combinations of reference-state orbitals. In most of these approximations, combinations of canonical, Hartree-Fock orbitals are used for this purpose, i.e. 

Dyson orbitals in Figs. 5.2 and 5.3 are distributed similarly in the two tautomers. 

Figure 5.2 Dyson orbitals of the keto form of 9-methylguanine.

Dyson orbitals from one isomer to the other. A higher peak at 10.86 eV, with intensity closer to that of the first one, is assigned to the 7r4 hole the Dyson orbital is approximately conserved after the proton transfer. Note that Koopmans s theorem (KT) results predict the wrong order of states for the enol form. A third a hole for both isomers is in reasonable agreement with the experimental peak at 11.32 eV. Finally, the feature at 13.3 eV is assigned to the 775 hole. 

Several review articles on the theoretical aspects of electron momentum densities of atoms and molecules were written in the 1970s by Benesch and Smith [9], Epstein [10,11], Mendelsohn and Smith [12], Epstein and Tanner [13], Lindner [14], and Kaijser and Smith [15]. Since that time (e,2e) spectroscopy and the momentum densities of Dyson orbitals have been reviewed very often [16-28]. However, to my knowledge, a review article on molecular electron momentum densities has not been written recently apart from one [29] devoted solely to the zero-momentum critical point. The purpose of this chapter is to survey what is known about the electron momentum density of atoms and molecules, and to provide an extensive, but not exhaustive, bibliography that should be sufficient to give a head start to a nonspecialist who wishes to enter the field. 

Section III. Methods for obtaining momentum densities, both experimental and computational, are reviewed in Section IV. Only a sample of representative work on the electron momentum densities of atoms and molecules is summarized in Sections V and VI because the topic is now too vast for comprehensive coverage. Electron momentum densities in solids and other condensed phases are not considered at all. The literature on electron momentum spectroscopy and Dyson orbital momentum densities is not surveyed, either. Hartree atomic units are used throughout. 

In binary (e,2e) or electron momentum spectroscopy, an incoming electron collides with a molecule and two electrons leave the molecule. The measured differential cross section is proportional to the spherically averaged momentum density of the pertinent Dyson orbital within the plane-wave impulse approximation. A Dyson orbital v[/ t is defined by... 

The principal part of the eigenvectors, Ca, contains the combination of coefficients that is required to build Feynman-Dyson amplitudes (Dyson orbitals) from the reference Hartree-Fock orbitals. Dyson orbitals as they result from (1.12) are not normalized, i.e., the sum... 

FDAs suffice for constructing Dyson orbitals (DOs) for VDEs, where... 

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