Exact-Exchange Calculations

by Ute Werner for exciting nitrogen

Purpose: In this tutorial, you will learn how to perform band-structure calculations using the optimized effecitve 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: First, one needs to create good starting Kohn-Sham orbitals, e.g. by performing a LDA ground-state calculation. Using these data, a self consistent EXX calculation is performed. We assume you are already familiar with the ideas and the scripts shown in the tutorial Electronic-structure calculation. Here, we will use some ideas of this tutorial to compare the band stucture of diamond obtained from LDA and from EXX.

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2. LDA calculation

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 inputfile 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>

For the meaning of the different arguments and keywors of the properties section have a look to the tutorial Electronic-structure calculation. 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. Excact-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 50 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-compare-bands.py. Move to the parent directory (the one where you can find the LDA and EXX subdirectories) and execute the script. It will produce the following output on the screen.

$ PLOT-compare-bands.py -25 25

################################################

 Enter the names of the 2 directories to compare

------------------------------------------------

 Directory 1 ==> LDA
 Directory 2 ==> EXX

################################################

$

In this example, the first input entry is preceded by " Directory 1 ==> ". This entry value must be typed on the screen and can be the name of the directory which contains your LDA results. The second entry should then be the EXX directory. The script PLOT-compare-bands.py produces a PostScript (PLOT.eps) and a PNG (PLOT.png) output file, as well as an image on the screen. Using the examples above 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)