Bethe-Salpeter kernel

In this module you will learn how to calculate the matrix elements of the Bethe-Salpeter kernels. This is a fundamental step in the calculation of optical spectra within the Bethe-Salpeter equation framework.

Background

To solve the Bethe-Salpeter equation, the latter is usually rewritten in the space of transitions between valence and conduction states as the (pseudo)eigenvalue problem for a two-particle Hamiltonian. For a non spin-polarized system and in the q=0 limit, the two-particle Hamiltonian matrix elements are given by

where vck indicates the pair of quasiparticle states vk and ck. The first term on the RHS is the quasiparticle energy differences (diagonal only). The second term is the kernel which is the sum of the electron-hole exchange part V (which stems from the Hartree potential) and the electron-hole attraction part W (which stems from the screened exchange potential). The kernel both shifts (diagonal contributions) and couples (off-diagonal contributions) the quasiparticle energy differences:

Here ε_GG'^-1 is the inverse of the dielectric screening matrix which was evaluated in the Static screening module.

Prerequisites

Cheatsheet on BSE Kernel

You must first complete the static screening module

You will need:

The SAVE databases for 3D hBN
The 3D_BSE directory with the databases generated in the static screening module
The yambo executable

Choosing input parameters

Generate the input file by invoking yambo with the option -o b -k sex from the command line:

$ yambo -o b -k sex -F 02_3D_BSE_kernel.in -J 3D_BSE

The input opens in the standard editor. The relevant input parameters to be changed are

BSENGexx = 30000 mRy
BSENGBlk = 4000 mRy
% BSEBands
 6 | 10 |       
%

The first two are the cutoff for the summations on the reciprocal lattice vectors which appear in the expressions for V (BSENGexx) and W (BSENGBlk) here above.

The third parameter gives the range of quasiparticle states (their band index) that define the basis of quasiparticle pairs that we need to describe the excitons (in the energy range of interest). For example since we have 8 occupied bands, here we consider the pairs of bands: 6-9, 6-10, 7-9, 7-10, 8-9, 8-10 (those are 6 pairs of bands = 3 valence times 2 conduction bands).

For each of those pairs we then consider all k points in the Brillouin zone available from the DFT calculations. Since we have a 6 x 6 x 2 grid, this amounts to 72 k points in the Brillouin zone.

Then finally we are using a basis of 6 x 72 = 432 vck pairs to represent the excitons of the system.

The values for these 3 parameters are chosen from convergence studies, as discussed in the next tutorial.