Electron propagator poles

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

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. 

Table 9.3 Electron propagator poles at the SCF level (Koopmans) and at the EP2 level are compared with experiment for the H3B OH2 complex...

Table 9.5 Comparison of the electron propagator poles at the SCF (Koopmans) level and at the EP2 level with the same UPS spectrum as in Table 4...

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. 

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. 

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... 

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 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. 

The basis set is chosen such that around the Zth Dyson pole, the electron propagator matrix is diagonal. Consequently, the eigenstate /) can be presented as a linear expansion over atomic orbitals with coefficients ck(l) that depend on the number of the Dyson pole, Z, and the number of the atomic orbital, k, so that... 

Coefficients, ck(l), pole strengths, alr and poles, e , usually are reported as output in ab initio electron propagator calculations on molecules. 

A similar analysis may be carried out for the electron detachment poles as well and the Auger poles (Eq — E 1( )) will obviously be found in the first quadrant of the complex E-plane. With this brief discussion of the pole structure of the complex scaled electron propagator as a common background, its utility in direct and simultaneous treatment of resonances of N+l electron and N-l electron systems becomes manifest e.g., for Be both Be+ (Is-1) 2S Auger and 2P Be- shape resonances have been calculated simultaneously from a single calculation on Be/25,26/. 

While the resonant pole of the dilated electron propagator is persistent once uncovered and should be invariant to further changes in the complex... 

Figure 10. Theta trajectories for the Be+ (Is-1) Auger pole from the zeroth (bi-variational SCF), second order ( 3), quasiparticle second order (Ej), diagonal Sph-TDA ( 3pA TIM) and quasiparticle diagonal Sph-TDA (E3ph TDA) decouplings of the dilated electron propagator. The disparity between the theta trajectories for the SCF and propagator poles makes apparent the magnitude of correlation and relaxation effects attending the Auger resonance formation.

In the diagonal, second-order approximation to the self-energy of the electron propagator, solutions of the Dyson equation (with self-consistent pole energies, cvp) satisfy... 

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. 

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