Self-energy approximations

The usual initial guess, Cp -I- Epp(cp), usually leads to convergence in three iterations. Relationships between diagonal self-energy approximations, the transition operator method, the ASCF approximation and perturbative treatments of electron binding energies have been analyzed in detail [17, 18]. 

In the latter expression, the derivative is evaluated at the converged energy. Diagonal self-energy approximations therefore subject a frozen Hartree-Fock orbital < F(x) to an energy-dependent correlation potential Epp(E). 

Diagonal matrix elements of the P3 self-energy approximation may be expressed in terms of canonical Hartree-Fock orbital energies and electron repulsion integrals in this basis. For ionization energies, where the index p pertains to an occupied spinorbital in the Hartree-Fock determinant,... 

Theta trajectories of the resonant root for the Ca 2P shape resonance from the various self-energy approximants are plotted in fig. 6. The results for resonance energy and width of the 2P shape resonance in e-Ca scattering are collected in table III. 

Both classes of self-energy approximations yield useful data for C60. Nondiagonal, renormalized methods reveal the presence of correlation states in photoelectron spectra. 

Approximate propagators may be designed to provide perturbative corrections to Koopmans results or may have the flexibility to account for correlation states where Ih or Ip descriptions of final states are qualitatively invalid. For the first case, in which the one-electron picture of electron detachment or attachment is adequate, quasiparticle self-energy approximations are suitable. In other cases, nondiagonal self-energy approximations are necessary. 

The P3 method is generally implemented in the diagonal self-energy approximation. Here, off-diagonal elements of the self-energy matrix in the canonical, Haruee-Fock orbital basis are set to zero. The pseudoeigenvalue problem therefore reduces to separate equations for each canonical, Hartree-Fock orbital ... 

Only energy iterations are needed in the diagonal self-energy approximation. For example, lLpp E) may be evaluated at = to obtain a new guess for E. The... 

Within this self-energy approximation, the Green s operator G" " is diagonal on the atomic orbital basis set. Its matrix elements are equal to ... 

The expressions of the local and total densities of states, derived in Equations (1.4.45) and (1.4.46), and of the ionic charges, derived in Equation (1.4.48), are then recovered. When the anion (cation) orbitals are degenerate, the self-energy approximation derived here is thus equivalent to writing M F) as a single delta function peaked at the position of its first moment. 

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