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The Bethe-Salpeter Kernel

A formal derivation of the Bethe-Salpeter equation can be found elsewhere (see rereferences [1-2], for example) here we want to focus on a simple physical picture in the spirit of the dielectric intepretation of the Many-Body problem.

Let's introduce a generalized notation

and, similarly

The full interacting reducible response function can be written in terms of reducible vertex function

where

As outlined in more detail in the document on the BSE solver the response function can be expressed in the basis of LDA bloch states

and the electron-hole Green's function L

here L is allready represented in this basis and satisfies the Bethe-Salpeter equation

The occupied valence and unoccupied conduction band states included in the basis for the representation of L and correpsondingly of the Bethe-Salpeter Hamiltonian (cf. BSS) is specified by the parameter BSEBands and BSEEhEny.

This Dyson-like equation contains the Green's function of the non-interacting electron-hole pairs

and the Bethe-Salpeter kernel

In L0 the single particle energies are by default the LDA Kohn-Sham eigenvalues. They reappear in the BSE Hamiltonian as the diagonal resonant terms. Techniques for the inclusion of quasi particle energies is described in GW. The variable QP_E serves to include previously calculated GW quasiparticle energies. Since a calculation of all quasiparticle energies is often costly, interpolated values (or approximated values via a scissors shift and streching parameters) can be included via QP_N and QP_E. The damping parameter is an external parameter (cf. BDmRange ).

In the Bethe-Salpeter kernel, the first term is the unscreended short ranged exchange interaction

whilest the second one manifests the screened coulomb interaction between the pairs

The evaluation of the above matrices <K|V|K'> and <K|W|K'> is naturally accomplished in G-space.

As noted already in BSS we need to consider not only the interaction of (v,c) with (v',c'), which forms the resonant part of the Bethe-Salpeter hamiltonian, but also the interaction of (v,c) with (c',v'), providing the coupling interaction.

The resonant part is given by

with V being the volume of the crystal and the symmetrized inverse dielectric function introduced in BSS

For the coupling part we obtain

Hence, similar to the case of the self-energy matrix-elements (cf. Xd ) the plane wave components included in the evaluation of <K|V|K'> are specified via BSENGexx. For <K|W|K'>, as in the case of the GW calculation (cf. Xp and Xd ), the size of the screened interaction matrix W(G,G') is specified by BSENGBlk. The parameter of the static dielectric matrix needed for the calculation of W(G,G') are idependently specified as described in Xd.

The effort of calculating the matrix elements can be reduced on the basis of physical approximations. One is reducing the energy range into which electron-hole pairs can scatter via the variable BSehWind. The other is limiting via the variable BSEClmns the number of kernel columns, i.e. the k-points, for which the kernel matrix elements will be calculated into account.

References

  1. G. Strinati Rivista del nuovo cimento bf 11,1 (1988).
  2. L. Fetter and J.D. Walecka Quantum theory of Many--Body Systems, McGraw-Hill, New York, N.Y. 1981.