Self energy static part

According to Eq. (42) the extended self energy can be written as the sum of a uniquely determined static, energy independent part (oo) and the so-called dynamic part M(u>) ... 

According to Eqs. (41) and (34) the static self energy (oo) is given by the primary block of the matrix of Eq. (33) minus its zeroth order part. An explicit expression can be found with the help of the super-operator formalism and using the definition of the extended states (47) ... 

Concluding, we have shown that the present approximation to the matrix leads to the RPA. In terms of the self energy, the present approach consists of a first order static part of the self energy like in the FOSEP approximation and additionally to that part of the dynamic self energy comprising the AT-states which are in zeroth order degenerate to parts of the primary (static) block. 

This two-body interaction part of the static self energy becomes even more intuitive, when we consider a local interaction potential and transform the formulas into position space. Take, e. g.. Coulomb interacting electrons where the matrix elements read... 

In position space the interaction part of the static Green s function optical Hamiltonian (or the static self energy) reads... 

Thus the kinetic energy and the influence of external forces is already described exactly by the static part of the optical Hamiltonian and independent of approximations for the target wavefunction. Note that the possibility for two particles labelled by r and s to exchange can be seen explicitely in the kinetic energy and external field parts of Eqs. (119) and (120) by the terms that antisymmetrise the matrix with respect to interchange of these labels. An expression for the interaction part contained in the static self energy is... 

In this appendix we present the interaction part (at,a, y o, a,<) of the static particle-hole self energy discussed in Sec. VIC. We assume Coulomb interacting electrons with the usual position space representation (106) of the two-body interaction V. The expression for the interaction part of the static self particle-hole self energy can then be readily evaluated, either from the definitions of the extended states (1) or from Eq. (68) ... 

Interestingly, the static part can be evaluated once the dynamic self-energy is known. ... 

Here, e denotes the diagonal matrix of all one-particle (orbital) energies and 2(oo) stands for the static self-energy matrix. We recall that the latter can be computed exactly once the dynamic self-energy part is known. The one-particle GF is determined as the upper left block of the inverse of the matrix T(l — A. More precisely... 

In such an approach, the correlation energy is partitioned into a complete active space self-consistent field (CASSCF) part which describes the static correlations and a MR configuration interaction part for the dynamic correlations. In the case of metals we select an active space for the CASSCF calculation including the important bands around the Fermi level. The dynamic correlations are treated on top of the CASSCF wavefunction with an approximately size-extensive MR correlation method, i.e. an MR averaged coupled pair functional (MR-ACPF). 

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