Exact Exchange Calculations

by Ute Werner for exciting oxygen

Purpose: In this tutorial, you will learn how to perform band-structure calculations using the optimized effective potential - exact-exchange (EXX) method. As an example, the band structure of diamond is calculated.

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1. Introduction

A typical exact exchange (EXX) calculation consists of two parts. Initially a ground state self-consistent cycle, for example with LDA, is performed to create good Kohn-Sham orbitals as initial guess for the EXX calculation. Afterwards, starting from these data, a self consistent EXX calculation is executed.

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2. LDA calculation

We start by performing an LDA calculation for comparing with EXX. This initial LDA calculation is not related to the initial self-consistent cycle performed at the beginning of the EXX calculation, since the latter is done internally by exciting.

As usual, the first thing to do is to create the working directory for this tutorial and move inside it.

$ mkdir exact-exchange
$ cd exact-exchange

i) Preparation of the calculation

Create an input file input.xml for the ground-state and band-structure calculation of diamond using LDA which could look like this

<input>
 
   <title>Diamond LDA</title>
 
   <structure speciespath="$EXCITINGROOT/species" >
      <crystal scale="6.7425">
         <basevect> 0.0     0.5     0.5 </basevect>
         <basevect> 0.5     0.0     0.5 </basevect>
         <basevect> 0.5     0.5     0.0 </basevect>
      </crystal>
      <species speciesfile="C.xml">
         <atom coord="0.00 0.00 0.00" />
         <atom coord="0.25 0.25 0.25" />
      </species>
   </structure>
 
   <groundstate 
      ngridk="8 8 8"
      xctype="LDA_PW">
   </groundstate>
 
   <properties>
      <bandstructure>
         <plot1d>
            <path steps="200">
               <point coord="1.0     0.0     0.0" label="Gamma"/>
               <point coord="0.625   0.375   0.0" label="K"/>
               <point coord="0.5     0.5     0.0" label="X"/>
               <point coord="0.0     0.0     0.0" label="Gamma"/>
               <point coord="0.5     0.0     0.0" label="L"/> 
            </path>
         </plot1d>
      </bandstructure>
   </properties>
 
</input>

We assume you are already familiar with band-structure calculations. However, if you need information on the meaning of the different arguments and keywords inside the element properties, you can have a look to the tutorial Electronic band-structure and density of states.

Be sure to set the correct path for the exciting root directory (indicated in this example by $EXCITINGROOT) to the one pointing to the place where the exciting directory is placed. In order to do this, use the command

$ SETUP-excitingroot.sh

ii) Execute the calculation

Perform the LDA calculation using the script EXECUTE-single.sh.

$ EXECUTE-single.sh LDA

Be sure that your calculation is completed before proceeding with the next steps!

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3. Exact-exchange calculation

i) Preparation of the calculation

In the next step, we will calculate the ground state and band structure using EXX. Therefore, replace the groundstate section in the file input.xml with the following lines.

...
   <groundstate 
      ngridk="3 3 3"
      xctype="EXX"
      vkloff="0.5 0.5 0.5"
      nempty="29">
 
      <OEP 
         maxitoep="100"> 
      </OEP>
 
   </groundstate>
...

Let's have a closer look to the parameters in this file.

Parameter Description xctype = "EXX" The calculation is performed using the exact-exchange functional nempty = "29" The number of empty bands, crucial for the convergence vkloff = "0.5 0.5 0.5" The origin of the k-point grid, crucial for the convergence maxitoep = "100" The number of iterations for the evaluation of integrals in EXX

Since the EXX calculations are much more time consuming than corresponding LDA ones, computational parameters have been chosen in a way to speed up the calculation. However, the calculation will still last approximately 10-20 min. For further details on the parameters see Input Reference.

ii) Execute the calculation

The calculation can be started by the following command.

$ EXECUTE-single.sh EXX

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4. Post processing

For a graphical comparison of the band structures calculated with the LDA and EXX functionals you can use the script PLOT-band-structure.py, which is described in details in The python script "PLOT-band-structure.py". Move to the parent directory (the one where you can find the LDA and EXX subdirectories) and execute the script as follows.

$ PLOT-band-structure.py -d LDA EXX  -e -25 25  -z vbM  -l 'lower right'

The options we used for the script PLOT-band-structure.py are also fully described in The python script "PLOT-band-structure.py". This script produces an encapsulated PostScript (PLOT.eps) and a PNG (PLOT.png) output file. In this examples you will obtain the following plot.

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5. Excercise

Try to modify some of the parameters in the groundstate element of the EXX input file and investigate the effect on the band structure and on the computational time.

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6. Literature

• S. Sharma, J. K. Dewhurst, and C. Ambrosch-Draxl, Phys. Rev. Lett. 95, 136402 (2005); S. Sharma, J. K. Dewhurst, C. Ambrosch-Draxl, S. Kurth, N. Helbig, S. Pittalis, S. Shallcross, L. Nordström, and E. K. U. Gross, Phys. Rev. Lett. 98, 196405 (2007)