### 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

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