Additional Material: Calculating Useful Properties from Total Energies

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Once you can calculate a converged total energy for a given structure, you already know enough to calculate many useful materials properties.

What can we do with total energies?

The DFT total energy on its own is not a very useful number, but changes and differences in the total energy with respect to some input parameter can be used to calculate many materials properties.

Predicting volume

If we calculate the total energy of say diamond as a function of volume, the volume at which the total energy is minimized will be the theoretically predicted volume.

Let’s do this in the directory 01_diamond_volume. We could set up a series of calculations manually, where we generate a set of input files each with a slightly different value of the input lattice length. Then we would run each of these calculations, and finally gather the total energy vs cell volume in a data file. Or, we could write a script that would do all this for us.

We already have a template file set up as C_diamond_base.in. Take a look at this now. It’s set up such that the value of the A variable will be replaced in the automatically generated input files. Note, we’re using a fairly high energy cut-off here also. In principle once the calculation is set up, we will need to explicitly test the convergence of the predicted volume versus energy cut-off.

The script will need to be slightly more complex than previously, as we’d managed to get by with just producing integer values in our scripts previously. Now we’ll need to produce floating point values for A to get the resolution we would like. To do this, we can use the bc command to perform the calculation of the lattice length for a given input file.

Also Quantum Espresso outputs the unit cell volume (in Bohr cubed) so we can read this rather than calculating it ourselves.

This gives us the following script:

#!/bin/bash

template="C_diamond_base.in"
repstr="xxxx"

# The "for" construction we've used previously only handles integers.
# So we set an initial value for A and its delta value as variables.
a1=3.40
da=0.02

for i in {00..10..1}
do
  inp="C_diamond_${i}.in"
  # bc is a calculator that can calculate expressions in the shell. We
  # pass it our calculation with echo and save the result as a variable.
  a=$(echo "$a1 + $i * $da" | bc)
  sed "s/$repstr/$a/" $template > $inp
  pw.x < $inp &> ${inp%.*}.out
done

awk '/unit-cell volume/{vol=$4}
     /^!.*total/{print vol, $5}' *out > etot_v_vol.dat

Save this script in the calculation directory and use it to obtain a file with the total energy versus volume of carbon diamond.

Let’s generate a plot of this with gnuplot and make sure it looks sensible. A simple gnuplot script etot_v_vol.gpl is already in the calculation directory. The contents are as follows:

set title "Carbon Diamond, ecutwfc=60 Ry"
set xlabel "Unit Cell Volume (Bohr^3)"
set ylabel "Total Energy (Ry)"
set term pngcairo
set output "etot_v_vol.png"
plot "etot_v_vol.dat"

This sets a title and axes labels, sets the output type to png, and sets and output filename, then plots the data as we’ve seen before. All these commands could be entered directly in gnuplot, but it can be easier to save them as a script if you want to come back in the future and make minor modifications. If your run this script with gnuplot etot_v_vol.gpl you’ll see a png file of the plot has been produced in the directory. You can quickly view this with display etot_v_vol.png.

Task

  • What is the calculated volume with the lowest total energy?
  • What lattice length does this correspond to?
  • Modify the script to calculate 21 points from A=3.40 to A=3.60 inclusive.
  • Repeat this last calculation for energy cut-offs of 40 Ry and 80 Ry. Note, you should set a different filename for the data file each time or rename the previous data file before running this so you can compare them all directly..
  • Plot all the data for all three energy cut-offs together in gnuplot. Note you can plot several data files in gnuplot by separating them with a comma - e.g. plot "data1.dat", "data2.dat". Save the output as a png file.

Bulk Modulus of diamond

You may have realised that we could use the results of our previous calculations to predict the bulk modulus of diamond. The bulk modulus is proportional to the second derivative of the energy with respect to volume at the equilibrium volume. Say we approximate our crystal as a harmonic solid, then we could fit a harmonic expression to it to obtain a value for the bulk modulus, K in gnuplot. We can do that, and plot the fit together with our data with the following gnuplot script.:

# Define a function for a simple harmonic equation of state
E(x) = E0 + K*(x - V0)**2/(2*V0)
# It's better to give initial guesses for a good fit. You can make a good
# guess at E0 and V0 from the data plot you generated earlier.
E0=-22.782
V0=74

# The actual fit command. The parameters and errors will be output to your
# terminal and to the file 'fit.log'.
fit E(x) "etot_v_vol.dat" via E0, V0, K

# Let's also produce a plot of the fit along with our data. We can use the
# same settings as before.
set title "Carbon Diamond, ecutwfc=60 Ry"
set xlabel "Unit Cell Volume (Bohr^3)"
set ylabel "Total Energy (Ry)"
set term pngcairo
set output "etot_v_vol_fit.png"

# The fit parameters are set a their optimized values at the end of the fit
# command above, so we can simply plot E(x).
plot "etot_v_vol.dat", E(x)

Looking at the fit parameters, we can see that we obtain a value for K and the equilibrium volume V0 directly. The value of V0 will be in whatever units were used for input volumes, so Bohr^3 in our case. The value of K will be in units of Energy/Volume. If we want to compare to the experimental value we’ll need to convert to a more standard unit. 1 Ry is 2.179872325E-18 J, and 1 Bohr^3 is (5.2917721067E-11)^3 m^3. If you don’t want to be copy and pasting numbers to a spreadsheet or similar, another useful way to do calculations in the terminal is using python. Type python to start an interactive session. Type whatever expressions you want, and type ctrl+d or quit() to exit python once you’re done.

python
>>> K * 2.179872325E-18 / (5.2917721067E-11)**3 # Enter your K value here.
>>> _ * 1e-9 # Convert the last result ('_') to GPa

The usual measured value for the bulk modulus of diamond is around 540 GPa. How close to this are we?

Task

  • Convert the fit error to GPa also so we have an estimate of our accuracy.
  • Repeat the above fit for the data produced with different energy cut-off earlier. How do the values and errors compare?
  • A more advanced expression for the equation of state in a solid is the Birch Murnaghan Equation of State. See if you can fit this expression to your data. The E(V) expression could be written in gnuplot as E(x) = E0 + 9.0/16.0*V0*K*( Kp*((V0/x)**(2.0/3.0) - 1)**3 + ((V0/x)**(2.0/3.0) - 1)**2 * (6.0 - 4.0*(V0/x)**(2.0/3.0)))
    • Note the expression also depends on the parameter Kp so you’ll need to add this to the list of fit parameters in the gnuplot fit command.
    • How does the error in the fit value of K change with this new fit?
    • Generate a plot showing both fits for your most well converged set of data along with the data points.

H2 Bond Length

For simple molecules we can calculate the total energy versus bond length in the same way. This calculation is set up in the directory 02_H2_bond. Again, we’re going to use a short script to modify a single value in a template input file and run a series of calculations. Take a look at the template input file first. We’re not using any new inputs here. We’ve placed one atom at the origin, and the second will be moved along the x-axis.

We’ll make a few modifications to the script this time though so that we parse the data from the output files as they are generated and use the bond-length value from the script. Save the following to a script in the calculation directory:

#!/bin/bash

template="H2_base.in"
repstr="xxxx"

# Again we set an initial value for the x-coordinate of on H-atom and a delta
# value as variables.
hx1=0.700
dhx=0.005

# Empty the file, since we'll be appending to it in the calculation loop.
> etot_v_bl.dat

for i in {00..40..1}
do
  inp="H2_${i}.in"
  # We save the output filename to a variable also.
  out="${inp%.*}.out"

  # Again we use bc to get the atomic position for each input.
  hx=$(echo "$hx1 + $i * $dhx" | bc)
  sed "s/$repstr/$hx/" $template > $inp
  pw.x < $inp &> $out

  # awk is inside the loop this time, and we are appending to the data file
  # after each calculation completes.
  awk -v bl=$hx '/^!.*total/{print bl, $5}' $out >> etot_v_bl.dat
done

Take a look at the data file. What bond length minimizes the energy?

Plot this data in gnuplot. You may want to copy and modify the script we used earlier for diamond.

H2 Vibration Frequency

As you may have realised, we can use what we already calculated here to find the H2 vibration frequency. If we assume the variation in energy with bond length is approximately quadratic (how good an assumption do you think this is?) we can fit a function to the data in gnuplot and find the vibrational frequency.

We can modify the gnuplot script we used earlier to fit and plot this function with our data as follows:

E(x) = E0 + k*(x-x0)**2
fit E(x) "etot_v_bl.dat" via E0, x0, k

set title "Hydrogen Molecule"
set xlabel "Bond Length (Angstrom)"
set ylabel "Total Energy (Ry)"
set term pngcairo
set output "etot_v_bl_fit.png"

plot "etot_v_bl.dat", E(x)

Use this script to make a png plot and inspect it. How good a fit do you think we have?

Task

  • Modify the calculation script to calculate 40 values over a range of 0.1 Angstrom near the minimum. Also repeat the fit.
    • How does the fit compare to the previous one?
  • The fit has produced a value for an effective spring-constant k for the H2 vibration. k is in units of Rydberg per Angstrom^2. Find the frequency of vibration of the molecule in THz. Note, since we calculated k by fixing one atom and moving the other, you will need to use the reduced mass of the pair of atoms.
    • What would the frequency be for a molecule made of two deuterium atoms?