Quasi harmonic approximation

Usage of phonopy-qha

Using phonopy results of thermal properties, thermal expansion and heat capacity at constant pressure can be calculated under the quasi-harmonic approximation. phonopy-qha is the script to calculate them. An example of the usage is as follows:

phonopy-qha e-v.dat thermal_properties-{1..10}.yaml

1st argument is the filename of volume-energy data (in the above expample, e-v.dat). The volume and energy of the cell (default units are in \mathrm{\AA}^3 and eV, respectively). An example of the volume-energy file is:

#   cell volume        energy of cell other than phonon
   156.7387309525      -104.5290025375
   154.4138492700      -104.6868148175
   152.2544070150      -104.8064238800
   150.2790355600      -104.8911768625
   148.4469296725      -104.9470385875
   146.7037426750      -104.9783724075
   145.1182305450      -104.9871878600
   143.5676103350      -104.9765270775
   142.1282086200      -104.9485225225
   139.4989658225      -104.8492814250

Lines starting with # are ignored. The other arguments are the filenames of thermal_properties.yaml calculated at the respective volumes given in the 1st argument. The thermal_properties.yaml at volume points have to be calculated with the same temperature ranges and same temperature steps. thermal_properties.yaml can be calculated by following Thermal properties related tags, where the physical unit of the Helmholtz free energy is kJ/mol as the default, i.e., no need to convert the physical unit in usual cases.

The example for Aluminum is found in the example directory.

If the condition under puressure is expected, PV terms may be included in the energies.

Options

-h

Show help. The available options are shown. Without any option, the results are saved into text files in simple data format.

--tmax

The maximum temperature calculated is specified. This temperature has to be lower than the maximum temperature calculated in thermal_properties.yaml to let at least two temperature points fewer. The default value is --tmax=1000.

-p

The fitting results, volume-temperature relation, and thermal expansion coefficient are plotted on the display.

--sparse

This is used with -s or -p to thin out the number of plots of the fitting results at temperatures. When --sparse=10, 1/10 is only plotted.

-s

The calculated values are written into files.

--pressure

Pressure is specified in GPa. This corresponds to the pV term described in the following section Theory of quasi-harmonic approximation. Note that bulk modulus obtained with this option than 0 GPa is incorrect.

-b

Fitting volume-energy data to an EOS, and show bulk modulus (without considering phonons). This is made by:

phonopy-qha -b e-v.dat

--eos

EOS is chosen among vinet, birch_murnaghan, and murnaghan. The default EOS is vinet.

phonopy-qha --eos='birch_murnaghan' -b e-v.dat

Output files

The physical units of V and T are \AA^3 and K, respectively. The unit of eV for Helmholtz and Gibbs energies, J/K/mol for C_V and entropy, GPa for for bulk modulus and pressure are used.

  • Bulk modulus (GPa) vs T (bulk_modulus-temperature.*)
  • Gibbs free energy (eV) vs T (gibbs-temperature.*)
  • Volume change with respect to the volume at 300 K vs T (volume_expansion.*)
  • Heat capacity at constant pressure (J/K/mol) vs T derived by -T\frac{\partial^2 G}{\partial T^2} (Cp-temperature.*)
  • Heat capacity at constant puressure (J/K/mol) vs T by polynomial fittings of Cv and S (Cp-temperature_polyfit.*)
  • Helmholtz free energy (eV) vs volume (helmholtz-volume.*). When --pressure option is specified, energy offset of pV is added. See also the following section (Theory of quasi-harmonic approximation).
  • Volume vs T (volume-temperature.*)
  • Thermal expansion coefficient vs T (thermal_expansion.*)
  • Thermodynamics Grüneisen parameter (no unit) vs T (gruneisen-temperature.dat)

Cv-volume.dat, entropy-volume.dat, and dsdv-temperature.dat (dS/dV) are the data internally used.

Theory of quasi-harmonic approximation

Here the word ‘quasi-harmonic approximation’ is used for an approximation that introduces volume dependence of phonon frequencies as a part of anharmonic effect.

A part of temperature effect can be included into total energy of electronic structure through phonon (Helmholtz) free energy at constant volume. But what we want to know is thermal properties at constant pressure. We need some transformation from function of V to function of p. Gibbs free energy is defined at a constant pressure by the transformation:

G(T, p) = \min_V \left[ U(V) + F_\mathrm{phonon}(T;\,V) + pV \right],

where

\min_V[ \text{function of } V ]

means to find unique minimum value in the brackets by changing volume. Since volume dependencies of energies in electronic and phonon structures are different, volume giving the minimum value of the energy function in the square brackets shifts from the value calculated only from electronic structure even at 0 K. By increasing temperature, the volume dependence of phonon free energy changes, then the equilibrium volume at temperatures changes. This is considered as thermal expansion under this approximation.

phonopy-qha collects the values at volumes and transforms into the thermal properties at constant pressure.