Electron propagator residues

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. 

Consider first the case where P and Q are simple creation and annihilation operators, e.g. P = af and Q — a . The residue <0 P nj> is then nonvanishing only if m > contains one electron less than 0 > and the residue of the last term vanishes unless m > contains one electron more than 0>. The poles of the first term are thus the ionization potentials for state 0> while the poles of the second term are the electron affinities. The af a, propagator is called the electron propagator and will be discussed in more detail in Section VI. 

Now assume that P = af af and Q — a a. Following the same reasoning as above we see that m> = Al + 2> if 0> = V>isan N-electron state. The poles and residues yield energy differences and overlap amplitudes of Auger spectra, respectively. Two-electron propagators of the type afia a yy have in... 

The residue corresponding to an electron propagator pole, is defined by... 

An initial approximation to the electron propagator matrix G(E) = GSS E) neglects the residual Hres and it holds then that... 

Poles and residues of the electron propagator Gv, ( j provide the spin orbital energies and the molecular orbital amplitudes. The sum of the energies of the occupied spin orbitals is used as a measure of the total energy of the 7t-orbital system and Coulson observed that this could be expressed as a contour integral in the complex energy plane... 

EVom the definition of the spectral representation, it follows that the elements of the electron propagator matrix G( ) become infinite when E equals an electron binding energy. Then, the inverse G E) has a zero eigenvalue at such an energy. This result can be used to devise iterative methods to find the electron propagator poles and residues at a given level of perturbation expansion. 

Partitioning the operator manifold can lead to efficient strategies for finding poles and residues that are based on solutions of one-electron equations with energy-dependent effective operators [16]. In equation 15, only the upper left block of the inverse matrix is relevant. After a few elementary matrix manipulations, a convenient form of the inverse-propagator matrix emerges, where... 

The admittance response at 1 kHz has also been interpreted in terms of the behavior at residual defects in anodic films. This interpretation is based on electron optical characterization, which shows that anodic films contain a distribution of preexisting defects associated with substrate inclusions and mechanical flaws (96,102). In aggressive environments, pits nucleate from these defects and propagate into the metal substrate. In this model, pits are distinct from anodic film flaws, and both can contribute to the measured admittance. Measurements of anodic films exposed to chloride solutions showed that the dissipation factor increased with time, but the capacitance remained nearly constant. Under these conditions, pit propagation at a flaw led to a pit area increase, which increased the resistive component of the admittance, resulting in an increased dissipation factor, but no increase in the capacitance. Measurements at 100 kHz were reflective of the electric double layer and not the components of the oxide film. 

Hydroxyl radical may hydroxylate tyrosine to 3,4-dihydroxyphenylalanine (DOPA). DOPAs are the main residues corresponding to protein-bound reducing moieties able to reduce cytochrome c, metal ions, nitro tetrazolium, blue and other substrates (S32). Reduction of metal ions and metalloproteins by protein-bound DOPA may propagate radical reactions by redox cycling of iron and copper ions which may participate in the Fenton reaction (G9). Abstraction of electron (by OH or peroxyl or alkoxyl radicals) leads to the formation of the tyrosyl radical, which is relatively stable due to the resonance effect (interconversion among several equivalent resonant structures). Reaction between two protein-bound tyrosyl radicals may lead to formation of a bityrosine residue which can cross-link proteins. The tyrosyl radical may also react with superoxide, forming tyrosine peroxide (W13) (see sect. 2.6). 

Both P and Q are sums of excitation operators (with weighting coefficients p and 9 )- Thus, P and Q applied to 0> create a polarization of 0> and we call P 6 a polarization propagator. In the special case where P and Q are both single particle-hole excitations, i.e. only one term in Eqs (5) and (6), we talk about the particle-hole propagator. It is important to note that only the residues of the polarization propagator and not of the particle-hole propagator determine transition moments (Oddershede, 1982). We must have the complete summations in Eqs (5) and (6) in order to represent the one-electron operator that induces the transition in question. 

---