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This tutorial aims at showing how to obtain self-energy corrections to the DFT Kohn-Sham eigenvalues within the GW approximation, in the metallic case, without the use of a plasmon-pole model. The band width and Fermi energy of Aluminum will be computed.

The user may read the papers

F. Bruneval, N. Vast, and L. Reining, Phys. Rev. B

**74**, 045102 (2006) [[cite:Bruneval2006]], for some information and results about the GW treatment of Aluminum. He will also find there an analysis of the effect of self-consistency on quasiparticles in solids (not present in this tutorial, however available in Abinit). The description of the contour deformation technique that bypasses the use of a plasmon-pole model to calculate the frequency convolution of G and W can be found inS. Lebegue, S. Arnaud, M. Alouani, P. Bloechl, Phys. Rev. B

**67**, 155208 (2003) [[cite:Lebegue2003]],

with the relevant formulas.

A brief description of the equations implemented in the code can be found in the [[theory:mbt|GW notes]] Also, it is suggested to acknowledge the efforts of developers of the GW part of ABINIT, by citing the [[cite:Gonze2005|2005 ABINIT publication]].

The user should be familiarized with the four basic tutorials of ABINIT, see the tutorial index as well as the first GW tutorial.

[TUTORIAL_README]

This tutorial should take about one hour to be completed (also including the reading of [[cite:Bruneval2006]] and [[cite:Lebegue2003]].

Visualisation tools are NOT covered in this tutorial. Powerful visualisation procedures have been developed in the Abipy context, relying on matplotlib. See the README of Abipy and the Abipy tutorials.

*Before beginning, you might consider to work in a different subdirectory as for the other tutorials. Why not Work_gw2?*

In tutorial 4, we have computed different properties of Aluminum within the LDA. Unlike for silicon, in this approximation, there is no outstanding problem in the computed band structure. Nevertheless, as you will see, the agreement of the band structure with experiment can be improved significantly if one relies on the GW approximation.

In the directory *Work_gw2*, copy the files *tgw2_x.files* and *tgw2_1.in* located in *$ABI_TUTORIAL/Input*, and modify the *tgw2_x.files* file as usual (see the first tutorial).

```
cd $ABI_TUTORIAL/Input
mkdir Work_gw2
cd Work_gw2
cp ../tgw2_x.files . # modify this file as usual (see tutorial 1)
cp ../tgw2_1.in .
```

Then issue:

`abinit < tgw2_x.files > tgw2_1.log 2> err &`

{% dialog tests/tutorial/Input/tgw2_x.files tests/tutorial/Input/tgw2_1.in %}

This run generates the WFK file for the subsequent GW computation and also provides the band width of Aluminum. Note that the simple Fermi-Dirac smearing functional is used ([[occopt]] = 3), with a large smearing ([[tsmear]] = 0.05 Ha). The **k**-point grid is quite rough, an unshifted 4x4x4 Monkhorst-Pack grid (64 **k**-points in the full Brillouin Zone, folding to 8 **k**-points in the Irreducible wedge, [[ngkpt]] = 4 4 4). Converged results would need a 4x4x4 grid with 4 shifts (256 **k**-points in the full Brillouin zone). This grid contains the *Γ* point, at which the valence band structure reaches its minimum.

The output file presents the Fermi energy

` Fermi (or HOMO) energy (eV) = 7.14774 Average Vxc (eV)= -9.35982`

as well as the lowest energy, at the *Γ* point

```
Eigenvalues ( eV ) for nkpt= 8 k points:
kpt# 1, nband= 6, wtk= 0.01563, kpt= 0.0000 0.0000 0.0000 (reduced coord)
-3.76175 19.92114 19.92114 19.92114 21.00078 21.00078
```

So, the occupied band width is 10.90 eV. More converged calculations would give 11.06 eV (see [[cite:Bruneval2006]]).

This is to be compared to the experimental value of 10.6 eV (see references in [[cite:Bruneval2006]]).

In order not to lose time, let us start the calculation of the screening file before the examination of the corresponding input file. So, copy the file *tgw2_2.in*, and modify the *tgw2_x.files* file as usual (replace occurrences of twg2_x by tgw2_2). Also, copy the WFK file (*tgw2_1o_WFK*) to *tgw2_2i_WFK*. Then run the calculation (it should take about 30 seconds).

{% dialog tests/tutorial/Input/tgw2_2.in %}

We now have to consider starting a GW calculation. However, unlike in the case of Silicon in the previous GW tutorial, where we were focussing on quantities close to the Fermi energy (spanning a range of a few eV), here we need to consider a much wider range of energy: the bottom of the valence band lies around -11 eV below the Fermi level. Unfortunately, this energy is of the same order of magnitude as the plasmon excitations. With a rough evaluation, the classical plasma frequency for a homogeneous electron gas with a density equal to the average valence density of Aluminum is 15.77 eV. Hence, using plasmon-pole models may be not really appropriate.

In what follows, one will compute the GW band structure without a plasmon-pole model, by performing explicitly the numerical frequency convolution. In practice, it is convenient to extend all the functions of frequency to the full complex plane. And then, making use of the residue theorem, the integration path can be deformed: one transforms an integral along the real axis into an integral along the imaginary axis plus residues enclosed in the new contour of integration. The method is extensively described in [[cite:Lebegue2003]].

Examine the input file *tgw2_2.in*.

{% dialog tests/tutorial/Input/tgw2_2.in %}

The ten first lines contain the important information. There, you find some input variables that you are already familiarized with, like [[optdriver]], [[ecuteps]], but also new input variables: [[gwcalctyp]], [[nfreqim]], [[nfreqre]], and [[freqremax]]. The purpose of this run is simply to generate the screening matrices. Unlike for the plasmon-pole models, one needs to compute these at many different frequencies. This is the purpose of the new input variables. The main variable [[gwcalctyp]] is set to 2 in order to specify a *non* plasmon-pole model calculation. Note that the number of frequencies along the imaginary axis governed by [[nfreqim]] can be chosen quite small, since all functions are smooth in this direction. In contrast, the number of frequencies needed along the real axis set with the variable [[nfreqre]] is usually larger.

In order not to lose time, let us start the calculation of the band width before the study of the input file. So, copy the file *tgw2_3.in*, and modify the *tgw3_x.files* file as usual (replace occurrences of twg2_x by tgw2_3). Also, copy the WFK file (*tgw2_1o_WFK*) to *tgw2_3i_WFK*, and the screening file (*tgw2_2o_SCR*) to *tgw2_3i_SCR*. Then run the calculation (it should take about 2 minutes on a 3 GHz PC).

The computation of the GW quasiparticle energy at the *Γ* point of Aluminum does not differ from the one of quasiparticle in Silicon. However, the determination of the Fermi energy raises a completely new problem: one should sample the Brillouin Zone, to get new energies (quasiparticle energies) and then determine the Fermi energy. This is actually the first step towards a self-consistency!

Examine the input file *tgw2_3.in*:

{% dialog tests/tutorial/Input/tgw2_3.in %}

The first thirty lines contain the important information. There, you find some input variables with values that you are already familiarized with, like [[optdriver]], [[ecutsigx]], [[ecutwfn]]. Then, comes the input variable [[gwcalctyp]] = 12. The value *x2* corresponds to a contour integration. The value *1x* corresponds to a self-consistent calculation with update of the energies only. Then, one finds the list of k points and bands for which a quasiparticle correction will be computed: [[nkptgw]], [[kptgw]], and [[bdgw]]. The number and list of **k**-points is simply the same as [[nkpt]] and [[kpt]]. One might have specified less **k**-points, though (only those needing an update). The list of band ranges [[bdgw]] has been generated on the basis of the LDA eigenenergies. We considered only the bands in the vicinity of the Fermi level: bands much below or much above are likely to remain much or much above the Fermi region. In the present run, we are just interested in the states that may cross the Fermi level, when going from LDA to GW. Of course, it would have been easier to select an homogeneous range for the whole Brillouin zone, e.g. from 1 to 5, but this would have been more time-consuming.

In the output file, one finds the quasiparticle energy at *Γ*, for the lowest band:

```
k = 0.000 0.000 0.000
Band E_lda <Vxclda> E(N-1) <Hhartree> SigX SigC[E(N-1)] Z dSigC/dE Sig[E(N)] DeltaE E(N)_pert E(N)_diago
1 -3.762 -9.451 -3.762 5.689 -15.049 5.676 0.777 -0.287 -9.390 0.060 -3.701 -3.684
```

(the last column is the relevant quantity). The updated Fermi energy is also mentioned:

` New Fermi energy: 2.469501E-01 Ha , 6.719854E+00 eV`

The last information is not printed in case of [[gwcalctyp]] lower than 10.

Combining the quasiparticle energy at *Γ* and the Fermi energy, gives the band width, 10.404 eV. Using converged parameters, the band width will be 10.54 eV (see [[cite:Bruneval2006]]). This is in excellent agreement with the experimental value of 10.6 eV.

The access to the non-plasmon-pole-model self-energy (real and imaginary part) has additional benefit, e.g. an accurate spectral function can be computed, see [[cite:Lebegue2003]]. You may be interested to see the plasmon satellite of Aluminum, which can be accounted for within the GW approximation.

Remember that the spectral function is proportional to (with some multiplicative matrix elements) the spectrum which is measured in photoemission spectroscopy (PES). In PES, a photon impinges the sample and extracts an electron from the material. The difference of energy between the incoming photon and the obtained electron gives the binding energy of the electron in the solid, or in other words the quasiparticle energy or the band structure. In simple metals, an additional process can take place easily: the impinging photon can *extract an electron together with a global charge oscillation in the sample*. The extracted electron will have a kinetic energy lower than in the direct process, because a part of the energy has gone to the plasmon. The electron will appear to have a larger binding energy… You will see that the spectral function of Aluminum consists of a main peak which corresponds to the quasiparticle excitation and some additional peaks which correspond to quasiparticle and plasmon excitations together.

In order not to lose time, this calculation can be started before the examination of the input file. So, copy the file *tgw2_4.in*, and modify the *tgw4_x.files* file as usual (replace occurrences of twg2_x by tgw2_4). Also, copy the WFK file (*tgw2_1o_WFK*) to *tgw2_4i_WFK*, and the screening file (*tgw2_2o_SCR*) to *tgw2_4i_SCR*. Then run the calculation (it should take about 2 minutes on a 3 GHz PC).

{% dialog tests/tutorial/Input/tgw2_4.in %}

Compared to the previous file (*tgw2_3.in*), the input file contains two additional keywords: [[nfreqsp]], and [[freqspmax]]. Also, the computation of the GW self-energy is done only at the *Γ* point.

The spectral function is written in the file *tgw2_4o_SIG*. It is a simple text file. It contains, as a function of the frequency (eV), the real part of the self-energy, the imaginary part of the self-energy, and the spectral function. You can visualize it using your preferred software. For instance, start |gnuplot| and issue

`p 'tgw2_4o_SIG' u 1:4 w l`

You should be able to distinguish the main quasiparticle peak located at the GW energy (-3.7 eV) and some additional features in the vicinity of the GW eigenvalue minus a plasmon energy (-3.7 eV - 15.8 eV = -19.5 eV).

!!! tip

```
If |AbiPy| is installed on your machine, you can use the |abiopen| script
with the `--expose` option to visualize the results:
abiopen.py tgw2_4o_SIGRES.nc -e -sns
![](gw2_assets/al_spectra_b0_gamma.png)
```

Another file, *tgw2_4o_GW*, is worth to mention: it contains information to be used for the subsequent calculation of excitonic effects within the Bethe-Salpeter Equation with ABINIT or with other codes from the ETSF software page.