Electron propagators

Show that Eq. (4.36) follows from Eq. (4.15) for the electron propagator in the geometric approximation. 

Zubarev has written an interesting account of the theory and application of general double-time Green s functions, published in Soviet Physics Uspekhi 3, 320 (1960) (English translation). 

Some of the consequences for the Green s function of a spherically symmetric potential energy are explored. The WKB (Wentzel-Kramers-Brillouin) electron propagator for this case is introduced. 

The electron propagator or Green s function describes an electron scattering from a spherically symmetric center or equivalently from another particle (electron or proton) in the center-of-mass coordinate system. The physics of such a scattering process possesses cylindrical symmetry about the polar axis and is independent of the polar angle (j . 

The hamiltonian is separable in the variables r and 7, and we obtain the total Green s function as a convolution  

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Good early overviews of the electron propagator (that is used to obtain IP and EA data) and of the polarization propagator are given in ... 

Ortiz J V 1997 The electron propagator picture of molecular electronic structure Computational Chemistry Reviews of Current Trends vo 2, ed J Leszczynski (Singapore World Scientific) pp 1-61... 

If the P/Q operators correspond to removal or addition of an electron, the propagator is called an electron propagator. The poles of the propagator (where the denonainator is zero) correspond to ionization potentials and electron affinities. 

Hartree-Fock (HF), molecular orbital theory satisfies most of the criteria, but qualitative failures and quantitative discrepancies with experiment often render it useless. Methods that systematically account for electron correlation, employed in pursuit of more accurate predictions, often lack a consistent, interpretive apparatus. Among these methods, electron propagator theory [1] is distinguished by its retention of many conceptual advantages that facilitate interpretation of molecular structure and spectra [2, 3, 4, 5, 6, 7, 8, 9]. 

The physical meaning of the electron propagator rests chiefiy in its poles (energies where singularities lie) and residues (coefficients of the terms responsible for the singularities) [1]. In its spectral form, the r,s element of the electron... 

It is possible to use full or limited configuration interaction wavefunctions to construct poles and residues of the electron propagator. However, in practical propagator calculations, generation of this intermediate information is avoided in favor of direct evaluation of electron binding energies and DOs. 

Thus the matrix elements of the electron propagator are related to field operator products arising from the superoperator resolvent, El — H), that are evaluated with respect to N). In this sense, electron binding energies and DOs are properties of the reference state. 

According to equation 15, eigenvalues of the superoperator Hamiltonian matrix, H, are poles (electron binding energies) of the electron propagator. Several renormalized methods can be defined in terms of approximate H matrices. The... 

Applications of electron propagator methods with a single-determinant reference state seldom have been attempted for biradicals such as ozone, for operator space partitionings and perturbative corrections therein assume the dominance of a lone configuration in the reference state. Assignments of the three lowest cationic states were inferred from asymmetry parameters measured with Ne I, He I and He II radiation sources [43]. 

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. 

Spin Density Properties from the Electron Propagator Hyperfine and Nuclear Spin-Spin Couplings... 

When the operators A and B in Eq. (2.7) are sin q)le creation and annihilation operators the resulting propagator is called electron (nopagator or one-particte Green s function, and = -t-1. Collecting all these creation and amiihi-lation operators in a row vector a, the electron propagator can be expressed as. 

Thus, the electron propagator matrix elements can be written as,... 

The complete Hamiltonian of the molecular system can be wrihen as H +H or H =H +H for the commutator being linear, where is the Hamiltonian corresponding to the spin contribution(s) such as, Fermi contact term, dipolar term, spin-orbit coupling, etc. (5). As a result, H ° would correspond to the spin free part of the Hamiltonian, which is usually employed in the electron propagator implementation. Accordingly, the k -th pole associated with the complete Hamiltonian H is , so that El is the A -th pole of the electron propagator for the spin free Hamiltonian H . 

The matrix Hfj would be the transpose of Hf, if it were Hermitian. The Hermiticity of the superoperator Hamiltonian has been a concern since the beginnings of the electron propagator theory (46,129). For a Hermitian spin ftee Hamiltonian (// ) the following relation can be written describing the Hermiticity problem,... 

The Hi jCf tl-Hjj)" matrix is usually computed in the iterative pole search of the electron propagator (31,130). Thus, the implementation of the above expression for the splitting, Eq. (7.1.1), becomes simple, since it requires only the additional calculation of the Hf, and Hfj matrices. 

It is still necessary to perform an order analysis of the correlation potential in the calculation of. The usual implementation of the electron propagator is performed up to the third or partial fourth orders (31,32,129,130), which needs... 

The inclusion of the correlation corrections to the spin-spin coupling calculation via electron propagator is quite straightforward since at third order it can be written as... 

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