# Bethe-Salpeter solver: diagonalization

In this module you learn how to obtain an optical absorption spectra within the Bethe-Salpeter equation (BSE) framework by diagonalizing a previously calculated Bethe-Salpeter (BS) kernel.

## Contents

## Prerequisites

- You must first complete the Static screening and Bethe-Salpeter kernel modules

**You will need**:

- The
`SAVE`

databases for 3D hBN - The
`3D_QP_BSE`

directory (provided) which contains the database with the GW corrections. - The
`3D_BSE`

directory containing the databases from the Static screening and Bethe-Salpeter kernel modules - The
`yambo`

executable -
`gnuplot`

for plotting spectra

## Background

The macroscopic dielectric function (from which the absorption and EEL spectra can be computed) is obtained from the eigenvalues *E*_{λ} (excitonic energies) and eigenvectors A^{λ}_{cvk} (exciton composition in terms of electron-hole pairs) of the two-particle Hamiltonian:

To get the two-particle Hamiltonian eigensolutions you need to diagonalize the two-particle Hamiltonian:

for which the *2V - W* part was evaluated in the Bethe-Salpeter kernel module.

The difference of quasiparticle energies *Δε _{cvk}= ε_{ck} - ε_{vk}* is added to the matrix just before the solver.
There are two possible choices:

- the Kohn-Sham energies (calculated at DFT level) are corrected through a Scissor and the renormalization of the conduction and valence bandwidth (linearly with respect to the conduction band minimum and the valence band maximum respectively):

*Δε _{cvk}= M_{c}(ε_{ck}^{KS} - CBM) - M_{v}(ε_{vk}^{KS} - VBM) + Scissor*

- the quasiparticle energies calculated at GW level are used. Missing energies are computed by interpolation.

## Choosing the input parameters

Invoke yambo with the "-y d" option in the command line:

$ yambo -F 03_3D_BSE_diago_solver.in -y d -V qp -J 3D_BSE

The input is open in the editor. The input variable to be changed are

% BEnRange 2.00000 | 8.00000 | eV % BEnSteps= 200

which define 200 evenly spaced points between 2 and 8 eV at which the spectrum is calculated (ω in the equation for the macroscopic dielectric function),

% BDmRange 0.10000 | 0.10000 | eV %

which defines the spectral broadening (Lorentzian model),

% BLongDir 1.000000 | 1.000000 | 0.000000 | %

which defines the direction of the perturbing electric field (in this case the in-plane direction).

Another parameter to modify is

% KfnQP_E 1.440000 | 1.000000 | 1.000000 | %

This gives the quasiparticle corrections to the Kohn-Sham eigenvalues and is deduced either from experiment or previous GW calculations. With reference to the equation in the Background, the format is

Scissor|M|_{c}M|_{v}

The alternative of directly input corrections calculated from a previous GW calculation is shown in the next section.

## Bethe-Salpeter solver runlevel

$ yambo -F 03_3D_BSE_diago_solver.in -J 3D_BSE

In the log (either in standard output or in l-3d_BSE_optics_bse_bsk_bss), after various setup/loading, the BSE is diagonalized using the linked linear algebra libraries:

<01s> [06] BSE solver(s) <01s> [LA] SERIAL linear algebra <01s> [06.01] Diagonalization solver <01s> BSK diagonalize |########################################| [100%] --(E) --(X) <01s> EPS residuals |########################################| [100%] --(E) --(X) <01s> BSK epsilon |########################################| [100%] --(E) --(X)

The report r-3d_BSE_optics_bse_bsk_bss contains information relative to this runlevel in section 6:

[06] BSE solver(s) ==================

Take some time to inspect the log and the report to check the consistency with the input variables. This run produces a new database in the 3D_BSE directory

3D_BSE/ndb.BS_diago_Q01

So if you need the spectrum on a different energy range, direction, with a different broadening or on more points the diagonalization is not repeated, just the spectrum is recalculated.

This run produces as well human readable files (o-*). Specifically o-3D_BSE.eps_q1_diago_bse contains the real and imaginary part of the macroscopic dielectric function

$ less o-3D_BSE.eps_q1_diago_bse ... # # E/ev[1] EPS-Im[2] EPS-Re[3] EPSo-Im[4] EPSo-Re[5] # 2.00000 0.16162 5.83681 0.04959 4.14127 2.03015 0.16787 5.88612 0.05090 4.15638 ...

which is in the format

Energy in eV | Imaginary part BSE | Real part BSE |Imaginary part IPA | Real part IPA |

where real and imaginary parts refer to the macroscopic dielectric function. The imaginary part corresponds to the optical absorption. The latter (column 2) can be plotted versus the photon energy (column 1) and compared with the independent particle approximation (IPA, column 4), e.g.:

$ gnuplot ... plot 'o-3D_BSE.eps_q1_diago_bse' u 1:2 w l t 'BSE', 'o-3D_BSE.eps_q1_diago_bse' u 1:4 w l t 'IPA'

The addition of the kernel has the effect to red-shift the spectrum onset and to redistribute the oscillator strengths. Note that the convergence with respect to k-points smooths out the low energy peaks in the IPA (which are an artifact of poor convergence with k-points producing an artificial confinement) to give the shoulder corresponding to the van Hove singularity in the band structure. On the other hand the low energy peak in the BSE is genuine and it is the signature of a bound exciton.

## Reading the QP corrections from a previous *GW* calculation

In the above calculation we have used a simple scissor operator to correct the Kohn-Sam DFT energies. In this part we see how we can instead take the corrections from a previous Yambo Gw calculation. We create and edit the input:

$yambo -F 03_3D_QP_BSE.in -y d -V qp -J 3D_BSE

We set all parameters as in the previous calculation, except for the part regarding the QP correction:

KfnQPdb= "E < 3D_QP_BSE/ndb.QP" # [EXTQP BSK BSS] Database KfnQP_N= 1 # [EXTQP BSK BSS] Interpolation neighbours % KfnQP_E 0.000000 | 1.000000 | 1.000000 | # [EXTQP BSK BSS] E parameters (c/v) eV|adim|adim %

Instead of setting the values for the scissor, we give the path to a database (3D_QP_BSE/ndb.QP) which contains the QP corrections. This has been created by running a GW calculation as in the GW tutorial. Run Yambo:

$ yambo -F 03_3D_QP_BSE.in -J "3D_QP_BSE,3D_BSE"

This produces the following log in the standard output. Note Section 4 (regarding external QP corrections to the kernel):

<01s> [04.01] External QP corrections (K) <01s> [QP@K] E<3D_QP_BSE/ndb.QP[ PPA XG:39 Xb:1 40 Scb:1 40] <01s> [QP] Kpts covered exactly [o/o]: 100.0000

This tells you that the file was found, read correctly and that the **k** points found in the file matched the ones you are using for the current calculation (otherwise interpolation would be needed).
It is crucial to check that the file has been read, since if not Yambo gives a warning but continues the calculation (with no QP corrections at all!).
As in the previous calculation the final results of the calculation are the files with the spectral functions. Let's compare the results for the optical absorption spectrum with those obtained previously with a simple scissor:

$ gnuplot ... plot 'o-3D_QP_BSE.eps_q1_diago_bse' u 1:2 w l t 'Explicit QP', 'o-3D_BSE.eps_q1_diago_bse' u 1:2 w l t 'Scissor'

It is clear that this makes a difference in the peak distribution and intensity. Note that beside a simple shift you can renormalise as well the bandwidth of the valence and conduction bands in KfnQP_E (respectively the third and second value). You can try as an exercise to set up a new calculation using e.g. 1.440000 | 1.200000 | 0.900000 | for KfnQP_E.

## Summary

From this tutorial you've learned:

- How to compute the optical spectrum by using the diagonal solver within the Bethe-Salpeter equation framework
- Two alternative ways to provide QP energies (explicitly or via a scissor).

## Links

- Previous module: Bethe-Salpeter kernel
- Alternative Bethe-Salpeter equation solver: Lanczos-Haydock
- Back to Calculating optical spectra including excitonic effects: a step-by-step guide tutorial
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