How to Calculate the Stress Tensor

by Rostam Golesorkhtabar, Pasquale Pavone, & Konstantin Lion for exciting nitrogen

Purpose: In this tutorial you will learn how to set up and execute exciting calculations using the STRESS-exciting interface, which allows to obtain full stress tensors of crystal systems for any crystal structure. In addition, the application of STRESS-exciting to the determination of stress tensor for the hexagonal Be is explicitly presented.

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0. Define relevant environment variables

Read the following paragraphs before starting with the rest of this tutorial!

Before starting, be sure that relevant environment variables are already defined as specified in How to set environment variables for tutorials scripts. Here is a list of the scripts which are relevant for this tutorial with a short description.

• STRESS-exciting-setup.py: Python script for generating structures at different strains.
• STRESS-exciting-submit.sh: (Bash) shell script for running a series of exciting calculations.
• STRESS-exciting-analyze.py: Python script for fitting the energy-vs-strain and CVe-vs-strain curves.
• STRESS-exciting-clean.sh: (Bash) shell script for cleaning unnecessary files.
• STRESS-exciting-result.py: Python script for calculating the stress components.
• exciting2sgroup.py: Python script for converting the exciting input to an sgroup input file.
• Grace.par: File containing xmgrace parameters needed by visualization tools.

Requirements: Bash shell. Python numpy, lxml, math, sys and os libraries. xmgrace (visualization tool).

From now on the symbol $ will indicate the shell prompt.

Extra requirement: Tool for space-group determination

The scripts in this tutorial use the sgroup tool. If you have not done before, this tool should be downloaded and installed.

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[How to check if "sgroup" is already installed]

In order to check if sgroup is already installed in the computer you are working, write on the command line:

$ sgroup

Two possible messages may appear now on the screen.

• First case: sgroup is already installed:

No input file specified.

• Second case: sgroup must be installed:

bash: sgroup: command not found

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[How to install "sgroup"]

The code sgroup is a utility which allows to determine the space group and symmetry operations of a crystal structure.

• Tool name: sgroup
• Authors: B.Z. Yanchitsky, A.N. Timoshevskii
• Website: http://cpc.cs.qub.ac.uk/summaries/adon
• Download sgroup here (open the link, push the Download botton, and save the file containg the sgroup package)

After the download, you will get a tar.gz file, go to the directory where you saved this file and execute the following commands.

$ tar xfvp DownloadedFile.tar.gz
$ cd SpaceGroups
$ make
$ cp sgroup $EXCITINGTOOLS

Now, you have everything you need for starting and performing general lattice optimization.

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1. Set up and perform a reference calculation

In order to test the calculation of the stress tensor presented in the next sections, we set up and perform a calculation for the system we have chosen in a reference configuration. The example which is shown here is the hexagona structure of Be under an applied pressure. First, create a working directory Be-reference and move inside it.

$ mkdir Be-reference
$ cd Be-reference

Here, copy the following lines inside a file input.xml.

<input>
 
   <title>Be: Stress calculation</title>
 
   <structure speciespath="$EXCITINGROOT/species">
 
      <crystal scale="4.300">
         <basevect>  1.00000000  0.00000000  0.00000000 </basevect>
         <basevect> -0.50000000  0.86602540  0.00000000 </basevect>
         <basevect>  0.00000000  0.00000000  1.50000000 </basevect>
      </crystal>
 
      <species speciesfile="Be.xml" rmt="1.95">
         <atom coord="0.66666667 0.33333333 0.75000000"/>
         <atom coord="0.33333333 0.66666667 0.25000000"/>
      </species>
 
   </structure>
 
   <groundstate 
      ngridk="6 6 4"
      xctype="GGA_PBE_SOL">
   </groundstate>
 
   <relax/>
 
</input>

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

Now, we procede as alredy seen in General lattice optimization in order to generate new structures and determine the corressponding pressure.

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[Click here for the complete procedure for generating structures and determining pressures]

First, we generate input files for a series of volumes using the script OPTIMIZE-lattice.py.

$ OPTIMIZE-lattice.py 

>>>> Please enter the exciting input file name: input.xml

     Number and name of space group: 194 (P 63/m m c)       
     Hexagonal I structure in the Laue classification.

     Which parameter would you like to optimize?
     1 ... volume                               
     2 ... c/a ratio with constant volume       
>>>> Please choose '1' or '2': 1

>>>> Please enter the maximum physical strain value  
     The suggested value is between 0.001 and 0.050: 0.03
     The maximum physical strain is 0.03

>>>> Please enter the number of the distorted structures [odd number > 4]: 5
     The number of the distorted structures is 5

$

Entry values must be typed on the screen when requested. In this case, entries are the following.

1. input.xml, the name of the input file you have created.
2. 1, in order to change only the volume.
3. 0.03, the absolute value of the maximum strain for which we want to perform the calculation. Notice that in this case the physical strain ϵ is defined by the relationship V=V_initial·(1+ϵ)³, where V_initial is the volume of the unit cell defined in input.xml.
4. 5, the number of deformed structures equally spaced in strain, which are generated between the maximum negative strain and the maximum positive one.

Now, you must move to the directory VOL and execute the calculation there using the script OPTIMIZE-submit.sh.

$ cd VOL
$ OPTIMIZE-submit.sh 

SCF calculation of "vol_01" starts --------------------------------
   Elapsed time = 0m3s

SCF calculation of "vol_02" starts --------------------------------
   Elapsed time = 0m3s

...

SCF calculation of "vol_05" starts --------------------------------
   Elapsed time = 0m3s

$

When calculations finished, run again the OPTIMIZE-lattice.py script in the current directory choosing the Birch-Murnaghan EoS option.

$ OPTIMIZE-lattice.py

>>>> Murnaghan or Birch-Murnaghan EOS: [M/B] B

 ===================================================================== 
 Fit accuracy: 
     Log(Final residue in [Ha]): -6.04 

 Final parameters:
     E_min = -29.36096 [Ha]
     V_min = 106.5871 [Bohr^3]
     B_0   = 121.948 [GPa]
     B'    = 3.554

 Optimized lattice parameter saved into the file: "BM-optimized.xml".
 =====================================================================

$

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Now, you should have a file which called BM_eos.out inside the current directory. It contains all information which is related to the Birch-Murnaghan fit including the hydrostatic pressure corresponding to the calculated structures. The content of this file should be similar to the following:

 === Birch-Murnaghan eos =========================
 Fit accuracy: 
     Log(Final residue in [Ha]): -6.04 

 Final parameters:
     E_min = -29.36096 [Ha]
     V_min = 106.5871 [Bohr^3]
     B_0   = 121.948 [GPa]
     B'    = 3.554
 =================================================

 Volume     E_dft-E_eos     Pressure [GPa]
94.2633     +0.00000009     +18.616
98.7043     -0.00000041     +10.735
103.2857    +0.00000065     +4.057
108.0004    -0.00000047     -1.569
112.8597    +0.00000012     -6.298

As can be seen, the last 5 lines of this file show the volumes of the calculated structures (first column) and the corresponding hydrostatic pressure (third column). The first structure, (with a volume of 94.2633 Bohr³) corresponds to a hydrostatic pressure of +18.616 GPa. In the following sections, this structure will be denoted as reference configuration.

We notice here that the pressure is related to the components of the stress tensor, $\sigma_{\alpha \beta}$, as:

(1)

\begin{align} \nonumber P = -\frac{1}{3}\sum_{\alpha = 1}^{3} \sigma_{\alpha \alpha}\,. \end{align}

Next, we will calculate directly the stress tensor of this reference configuration and will use the equation above to (re)calculate the corresponding hydrostatic pressure.

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2. Set up and perform a calculation of the stress for the reference configuration

Before continuing, we remind here that all strains considered in this tutorial are physical strains. To start the preparation of the calculation of the stress we move to the parent directory, create a new directory Be-stress and move inside it.

$ cd ../
$ mkdir Be-stress
$ cd Be-stress

Inside this directory, we copy the input file corresponding to the reference configuration:

$ cp ../Be-reference/VOL/vol_01/input.xml ./

i) Generation of input files for distorted structures

In order to generate input files for a series of distorted structures, you have to run the script STRESS-exciting-setup.py, which will produce the following output on the screen.

$ STRESS-exciting-setup.py 

>>>> Please enter the exciting input file name: input.xml

     Number and name of space group: 194 (P 63/m m c)
     Hexagonal I structure in the Laue classification.       
     This structure has 2 independent stress components.

>>>> Please enter the maximum amount of strain 
     The suggested value is between 0.0010 and 0.0100 : 0.002
     The maximum amount of strain is 0.002

>>>> Please enter the number of the distorted structures [odd number > 4]: 21
     The number of the distorted structures is 21

$

Entry values must be typed on the screen when requested. In this case, the entries are the following.

1. input.xml, the name of the input file.
2. 0.002, the absolute value of the maximum strain for which we want to perform the calculation.
3. 21, the number of deformed structures equally spaced in strain, which are generated between the maximum negative strain and the maximum positive one.

ii) Execute the calculations

To execute the series of calculations, run the script STRESS-exciting-submit.sh, which will produce an output on the screen similar to the following.

$ STRESS-exciting-submit.sh 

        +--------------------------------------+
        | SCF calculation of "Dst01_01" starts |
        +--------------------------------------+
   Elapsed time = 0m5s
Mon 16 Jul 16:53:43 CEST 2018

        +--------------------------------------+
        | SCF calculation of "Dst01_02" starts |
        +--------------------------------------+
   Elapsed time = 0m5s
Mon 16 Jul 16:53:48 CEST 2018

        +--------------------------------------+
        | SCF calculation of "Dst01_03" starts |
        +--------------------------------------+

...

        +--------------------------------------+
        | SCF calculation of "Dst02_21" starts |
        +--------------------------------------+
   Elapsed time = 0m5s
Mon 16 Jul 16:57:29 CEST 2018
$

After the complete run, stress components can be calculated.

iii) Analyzing the calculations

Stress components are obtained from energy calculations using a similar method to the one illustrated in Energy vs. strain calculations. Within this approach, first derivatives of the energy curves are evaluated with the help of a curve fitting. Calculations performed above were producing 21 points for each energy curve. The quality of the fitting procedure can be improved by increasing the number of data points per energy curve.

Now, we analyze our calculations. The script STRESS-exciting-analyze.py allows the analysis of the dependence of the calculated derivatives of the energy-vs-strain curve on

1. the range of distortions included in the fitting procedure (the x axis in xmgrace plots),
2. the degree of the polynomial fit used in the fitting procedure (different color curves in xmgrace plots).

The STRESS-exciting-analyze.py script is executed as follows.

$ STRESS-exciting-analyze.py

At this point, two xmgrace plots will appear on your screen automatically (for more information on how to deal with xmgrace plots, see Xmgrace: A Quickstart).

• Results for the distortion Dst01

• Results for the distortion Dst02

The previous plots can be used to determine the best range of deformations and order of polynomial fit for each distortion.

As an example, we analyze the first plot, corresponding to the distortion Dst01. Distortion types are listed in the file Distorted_Parameters. By examining this file, we can see that the Dst01 distortion corresponds to an applied physical strain in the Voigt notation with the form (η,η,η,0,0,0), where η is a strain parameter. This deformation type is directly connected with the hydrostatic pressure. For each distorsion type, a plot appear. This plot contains the first derivative of the energy with respect to the strain parameter η as a function of the maximum strain and of the order of polynomial fit.

Analyzing the plot for the Dst01 distortion, we notice that curves corresponding to the lower order of the polynomial used in the fit show a horizontal plateau at about -555 kbar. This can be assumed to be the converged value for the first derivative, from the point of view of the fit (further information on this topic can be found here). In a similar way, the results for the Dst02 distortion can be analyzed, too.

The script STRESS-exciting-analyze.py generates the file STRESS.IN, which will be discussed and used in the next section.

iv)Numerical values of the stress components

In order to obtain the numerical values of the stress tensor, you should use the following procedure. The first step is to edit the file STRESS.IN, which should have the form

Dst01    eta_max    Fit_order
Dst02    eta_max    Fit_order

In each row of this file, you should insert a value for eta_max and Fit_order. According to the result of the visual analysis of the previous figures, these two values are extracted by considering the plateau regions of the corresponding plot: eta_max is a value of maximum distortion located in the plateau region of the curve representing the polynomial fit of order Fit_order. In our case, we can choose for each distortion 0.002 and 1 as the best distortion range and polynomial fit, respectively. Thus, the STRESS.IN file will now contain

Dst01     0.002     1
Dst02     0.002     1

Now, you run the STRESS-exciting-result.py script.

$ STRESS-exciting-result.py

Finally, numerical values of the stress components are written in the output file STRESS.OUT. In this case the content of STRESS.OUT should be similar to

 Tue Jul 17 12:38:53 2018

 The hydrostatic pressure and stress tensor in kilobar (kbar) 
 P =  184.841019 [kbar]
        -150.863125           0.000000           0.000000
           0.000000        -150.863125           0.000000
           0.000000           0.000000        -252.796807

 The hydrostatic pressure and stress tensor in gigapascal (GPa) 
 P =  18.484102 [GPa]
         -15.086313           0.000000           0.000000
           0.000000         -15.086313           0.000000
           0.000000           0.000000         -25.279681

From this output, we can extract a value of +18.484 GPa for the hydrostatic pressure of the reference configuration, which is in very good agreement with the value obtained in Section 1.

Exercise

• Repeat all calculations using a denser k-point grid and a larger number of strain points.