With this option the code can compute the elastic constants and elastic
compliances as a function of temperature using the quasi-harmonic
approximation. For the same geometries that are used to compute the
elastic constants with the `elastic_algorithm='energy_std'` or
`elastic_algorithm=`
`'energy'`, the code can compute the phonon
dispersions and compute the elastic constants at each temperature
as the second derivatives of the Helmholtz free energy
with respect to strain. The second derivatives are corrected so that
the stress-strain elastic constants are shown in the plots and in output.
The temperature dependent elastic constants are calculated on a regular
grid of unperturbed geometries, the same geometries chosen by the option
`what='mur_lc'`, and written on separate files, one for
each unperturbed geometry, inside the directory `anhar_files`.
In order to plot the elastic constants as a function of temperature
within the `quasi-harmonic' approximation,
it is necessary to make another calculation with `what='mur_lc_t'`
having on files the elastic constants calculated for each geometry
with the present option.
In this case `thermo_pw` will be able to calculate the anharmonic
properties using temperature dependent elastic constants and bulk moduli
obtained by interpolating the ``fixed-geometry quasi-harmonic'' elastic
constants computed by this option at the crystal parameters found at
each temperature from the minimization of the free energy.
The variables `fact_ngeo` and `ngeo_ph` are not
available with this option.
Using `start_geometry_qha` and `last_geometry_qha` it is
possible to compute the temperature dependent elastic constants for
selected or for a single unperturbed configuration.
The use of `start_geometry` and `last_geometry` is
also allowed but it refers to the global number of geometries necessary
to compute the elastic constants in all the grid.

Since the calculation of the Helmholtz free energy derivatives is
quite heavy, it has to be requested explicitly using the flag
`use_free_energy=.TRUE.`.

By default, the code computes only the elastic constants at
*T* = 0
`elastic_` `constants`. A run of `thermo_pw` using
`what='mur_lc_t'` having on files the
*T* = 0
`elastic_constants`
allows the calculation of the anharmonic properties using temperature
dependent elastic constants and bulk moduli obtained by interpolating
(within the ``quasi-static approximation") the elastic constants computed
by this option at the crystal parameters that, at each temperature,
minimize the free energy.
When both the
*T* = 0
`elastic_constants` and
`anhar_files` respectively, the latter are used.
When both `use_free_energy=.TRUE.` and
`lel_free_energy=.TRUE.` the electronic free energy is
added to the free energy before computing the elastic constants. In this case
the code expects to find on file (in `therm_files`) the electronic
thermodynamic properties for each perturbed geometry. These files are
produced with this same option and the flags
`use_free_energy=.FALSE.` and
`lel_free_energy=` `.TRUE.`. In this case the code
computes the electronic
thermodynamic properties at each perturbed geometry and writes them on
file, without calculating the elastic constants.
Note that the user must be careful to use the same value for the
`lel_free_energy` flag with this option and in the following
`mur_lc_t` calculation that interpolates the elastic constants.
After computing the elastic constants at
*T* = 0
`thermo_pw` calculation with the option
`what='mur_lc'` computing the crystal parameters for a range
of pressures (using `pmax` and `pmin` input variables). If
the elastic constants are found in the directory `elastic_constants`
they are interpolated at the pressure dependent crystal parameters and
plotted on output.
The variables that control this run are:

use_free_energy : when .TRUE. computes the elastic constants as second derivatives of the Helmholtz free energy with respect to strain. When .FALSE. the elastic constants are computed as second derivatives of the energy or using the stress-strain algorithms. Default: .FALSE. start_geometry_qha : Among the geometries considered by the option mur_lc_t the calculations of elastic constants are done starting from this geometry. Default: integer 1 last_geometry_qha : Among the geometries considered by the option mur_lc_t the calculations of elastic constants are done only up to this geometry. Default: integer total number of geometries.

An example for this option with `use_free_energy=.FALSE.`
can be found in `example22` while an example with
`use_free_energy=.TRUE.` can be found in `example23`.

Number of tasks for this option: The product of the number of tasks
needed by the `what='scf_elastic_constants'` option
and the number of geometries used with `what=mur_lc_t` when
`use_free_energy=.FALSE.`. When `use_free_`
`energy=.TRUE.`
and `all_geometries_together=.TRUE.` the number of tasks of the
previous case is further multiplied by the number of tasks needed to
compute a phonon dispersion (see above the option `what='scf_ph'`).

When `use_free_energy=.TRUE.` and
`all_geometries_together=.FALSE.` the number of tasks of this
option is equal to the number of tasks needed to compute a phonon dispersion.