# 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 for example/Si-QHA is as follows.

To watch selected plots:

phonopy-qha -p e-v.dat thermal_properties.yaml-{-{5..1},{0..5}}


Without plots:

phonopy-qha e-v.dat thermal_properties.yaml-{-{5..1},{0..5}}


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

#   cell volume   energy of cell other than phonon
140.030000           -42.132246
144.500000           -42.600974
149.060000           -42.949142
153.720000           -43.188162
158.470000           -43.326751
163.320000           -43.375124
168.270000           -43.339884
173.320000           -43.230619
178.470000           -43.054343
183.720000           -42.817825
189.070000           -42.527932


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. For the numerical differentiation in phonopy-qha against temperatures, one more temperature points is needed in thermal_properties.yaml for phonopy-qha calculations.

Another example for Aluminum is found in the example/Al-QHA directory.

If the condition under puressure is expected, $$PV$$ terms may be included in the energies, or equivalent effect is applied using --pressure option.

### 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 one temperature points fewer. The default value is --tmax=1000.

#### --pressure¶

Pressure is specified in GPa. This corresponds to the $$pV$$ term described in the following section Thermal properties in (T, p) space calculated under QHA. 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


#### -p¶

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

#### -s¶

The calculated values are written into files.

#### --sparse¶

This is used with -s or -p to thin out the number of plots of the fitting results at temperatures. For example with --sparse=10, 1 in 10 temperature curves is only plotted.

### Output files¶

The physical units of V and T are $$\text{Angstrom}^3$$ 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 $$B_T$$ (GPa) vs $$T$$ (bulk_modulus-temperature.*)
• Gibbs free energy $$G$$ (eV) vs $$T$$ (gibbs-temperature.*)
• Heat capacity at constant pressure $$C_p$$ (J/K/mol) vs $$T$$ computed by $$-T\frac{\partial^2 G}{\partial T^2}$$ from three $$G(T)$$ points (Cp-temperature.*)
• Heat capacity at constant puressure $$C_p$$ (J/K/mol) vs $$T$$ computed by polynomial fittings of $$C_V(V)$$ (Cv-volume.dat) and $$S(V)$$ (entropy-volume.dat) for $$\partial S/\partial V$$ (dsdv-temperature.dat) and numerical differentiation of $$\partial V/\partial T$$, e.g., see Eq.(5) of PRB 81, 17430 by Togo et al. (Cp-temperature_polyfit.*). This may give smoother $$C_p$$ than that from $$-T\frac{\partial^2 G}{\partial T^2}$$.
• Volumetric thermal expansion coefficient $$\beta$$ vs $$T$$ computed by numerical differentiation (thermal_expansion.*)
• Volume vs $$T$$ (volume-temperature.*)
• Thermodynamics Grüneisen parameter $$\gamma = V\beta B_T/C_V$$ (no unit) vs $$T$$ (gruneisen-temperature.dat)
• Helmholtz free energy (eV) vs volume (helmholtz-volume.*). When --pressure option is specified, energy offset of $$pV$$ is added. See also the following section (Thermal properties in (T, p) space calculated under QHA).

## Thermal properties in (T, p) space calculated under QHA¶

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.