# 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 run
fitting and calculation to perform it. Mind that at leave 5 volume
points are needed to run `phonopy-qha`

for fitting.

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}}
```

The first argument is the filename of volume-energy data (in the above
expample, `e-v.dat`

). The volumes and energies are given in
\(\text{Angstrom}^3\) and eV, respectively. Theses energies are
only dependent on volume but not on temperature unless using `--efe`

option. Therefore in the simplest case, these are taken as the
electronic total energies at 0K. 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 following arguments of `phonopy-qha`

are the filenames of
`thermal_properties.yaml`

’s calculated at the volumes given in the
volume-energy file. These filenames have to be ordered in the same
order as the volumes written in the volume-energy file. Since the
volume v.s. free energy fitting is done at each temperature given in
`thermal_properties.yaml`

, all `thermal_properties.yaml`

’s have to
be calculated in the same temperature ranges and with the same
temperature step. `phonopy-qha`

can calculate thermal properties at
constant pressure up to the temperature point that is one point less
than that in `thermal_properties.yaml`

because of the numerical
differentiation with respect to temperature points. Therefore
`thermal_properties.yaml`

has to be calculated up to higher
temperatures than that expected by `phonopy-qha`

.

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.

Experimentally, temperature dependent energies are supported by
`--efe`

option. The usage is written at
https://github.com/phonopy/phonopy/blob/develop/example/Cu-QHA/README.

### 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.

`--efe`

¶

**Experimental**

Temperature dependent energies other than phonon free energy are included with this option. This is used such as:

```
% phonopy-qha -p --tmax=1300 --efe fe-v.dat e-v.dat thermal_properties.yaml-{00..10}
```

The temperature dependent energies are stored in `fe-v.dat`

. The
file format is:

```
# volume: 43.08047896 43.97798894 44.87549882 45.77300889 46.67051887 47.56802885 48.46553883 49.36304881 50.26055878 51.15806876 52.05557874
# T(K) Free energies
0.0000 -17.27885993 -17.32227490 -17.34336569 -17.34479760 -17.32843604 -17.29673896 -17.25081954 -17.19263337 -17.12356816 -17.04467997 -16.95752155
10.0000 -17.27886659 -17.32228126 -17.34337279 -17.34481060 -17.32844885 -17.29675204 -17.25083261 -17.19264615 -17.12358094 -17.04469309 -16.95753464
20.0000 -17.27887453 -17.32228804 -17.34338499 -17.34482383 -17.32846353 -17.29676491 -17.25084547 -17.19265900 -17.12359399 -17.04470709 -16.95754774
...
```

The first column gives temperatures in K and the following columns
give electronic free energies in eV at temperatures and at unit
(primitive) cell volumes. The lines starting with `#`

are
ignored. This file doesn’t contain the information about cell
volumes. Instead, the volumes are obtained from `e-v.dat`

file. The
energies in `e-v.dat`

are not used when `--efe`

option is
used. The temperature points are expected to be the same as those in
`thermal_properties.yaml`

at least up to the maximum temperature
specified for `phonopy-qha`

.

An example is given in `example/Cu-QHA`

. The `fe-v.dat`

contains
electronic free energy calculated following, e.g., Eqs. (11) and (12)
in the paper by Wolverton and Zunger, Phys. Rev. B, **52**, 8813
(1994) (of course this paper is not the first one that showed these
equations):

with

and

where \(g\) is 1 or 2 for collinear spin polarized and non-spin
polarized systems, respectively. For VASP, a script to create
`fe-v.dat`

and `e-v.dat`

by these equations is prepared as
`phonopy-vasp-efe`

, which is used as:

```
% phonopy-vasp-efe --tmax=1500 vasprun.xml-{00..10}
```

where `vasprun.xml-{00..10}`

have to be computed for the same unit
cells as those used for `thermal_properties.yaml`

. When `phonopy`

was run with `PRIMITIVE_AXES`

or `--pa`

option, the unit cells for
computing electronic eigenvalues have to be carefully chosen to agree
with those after applying `PRIMITIVE_AXES`

, or energies are scaled a
posteriori.

### 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**, 174301 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:

where

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.