mirror of https://github.com/phonopy/phonopy.git
182 lines
5.8 KiB
ReStructuredText
182 lines
5.8 KiB
ReStructuredText
.. _phonopy_qha:
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Quasi harmonic approximation
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=============================================
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.. contents::
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:depth: 2
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:local:
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Usage of ``phonopy-qha``
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------------------------
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Using phonopy results of thermal properties, thermal expansion and
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heat capacity at constant pressure can be calculated under the
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quasi-harmonic approximation. ``phonopy-qha`` is the script to
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calculate them. An example of the usage is as follows:
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::
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phonopy-qha e-v.dat thermal_properties-{1..10}.yaml
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1st argument is the filename of volume-energy data (in the above
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expample, ``e-v.dat``). The volume and energy of the cell (default
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units are in :math:`\mathrm{\AA}^3` and eV, respectively). An example of the
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volume-energy file is::
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# cell volume energy of cell other than phonon
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156.7387309525 -104.5290025375
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154.4138492700 -104.6868148175
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152.2544070150 -104.8064238800
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150.2790355600 -104.8911768625
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148.4469296725 -104.9470385875
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146.7037426750 -104.9783724075
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145.1182305450 -104.9871878600
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143.5676103350 -104.9765270775
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142.1282086200 -104.9485225225
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139.4989658225 -104.8492814250
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Lines starting with ``#`` are ignored. The other arguments are the
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filenames of ``thermal_properties.yaml`` calculated at the respective
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volumes given in the 1st argument. The ``thermal_properties.yaml`` at
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volume points have to be calculated with the same temperature ranges
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and same temperature steps. ``thermal_properties.yaml`` can be
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calculated by following :ref:`thermal_properties_tag`, where the
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physical unit of the Helmholtz free energy is kJ/mol as the default,
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i.e., no need to convert the physical unit in usual cases.
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The example for Aluminum is found in the ``example`` directory.
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If the condition under puressure is expected, :math:`PV` terms may be
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included in the energies.
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.. _phonopy_qha_options:
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Options
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^^^^^^^
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``-h``
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~~~~~~~
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Show help. The available options are shown. Without any option, the
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results are saved into text files in simple data format.
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``--tmax``
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~~~~~~~~~~~~
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The maximum temperature calculated is specified. This temperature has
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to be lower than the maximum temperature calculated in
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``thermal_properties.yaml`` to let at least two temperature points
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fewer. The default value is ``--tmax=1000``.
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``-p``
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~~~~~~~
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The fitting results, volume-temperature relation, and thermal expansion
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coefficient are plotted on the display.
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``--sparse``
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~~~~~~~~~~~~~~
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This is used with ``-s`` or ``-p`` to thin out the number of plots of
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the fitting results at temperatures. When ``--sparse=10``, 1/10 is
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only plotted.
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``-s``
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~~~~~~~
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The calculated values are written into files.
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``--pressure``
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~~~~~~~~~~~~~~~~
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Pressure is specified in GPa. This corresponds to the :math:`pV` term
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described in the following section :ref:`theory_of_qha`. Note that
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bulk modulus obtained with this option than 0 GPa is incorrect.
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``-b``
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~~~~~~~
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Fitting volume-energy data to an EOS, and show bulk
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modulus (without considering phonons). This is made by::
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phonopy-qha -b e-v.dat
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``--eos``
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~~~~~~~~~~~
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EOS is chosen among ``vinet``, ``birch_murnaghan``, and
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``murnaghan``. The default EOS is ``vinet``.
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::
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phonopy-qha --eos='birch_murnaghan' -b e-v.dat
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.. _phonopy_qha_output_files:
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Output files
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^^^^^^^^^^^^^
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The physical units of V and T are :math:`\AA^3` and K,
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respectively. The unit of eV for Helmholtz and Gibbs energies, J/K/mol
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for :math:`C_V` and entropy, GPa for for bulk modulus and pressure
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are used.
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- Bulk modulus (GPa) vs T (``bulk_modulus-temperature.*``)
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- Gibbs free energy (eV) vs T (``gibbs-temperature.*``)
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- Volume change with respect to the volume at 300 K vs T (``volume_expansion.*``)
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- Heat capacity at constant pressure (J/K/mol) vs T derived by
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:math:`-T\frac{\partial^2 G}{\partial T^2}` (``Cp-temperature.*``)
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- Heat capacity at constant puressure (J/K/mol) vs T by polynomial
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fittings of Cv and S (``Cp-temperature_polyfit.*``)
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- Helmholtz free energy (eV) vs volume
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(``helmholtz-volume.*``). When ``--pressure`` option is specified,
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energy offset of :math:`pV` is added. See also the following section
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(:ref:`theory_of_qha`).
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- Volume vs T (``volume-temperature.*``)
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- Thermal expansion coefficient vs T (``thermal_expansion.*``)
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- Thermodynamics Grüneisen parameter (no unit) vs T (``gruneisen-temperature.dat``)
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``Cv-volume.dat``, ``entropy-volume.dat``,
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and ``dsdv-temperature.dat`` (:math:`dS/dV`) are the data internally
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used.
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.. _theory_of_qha:
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Theory of quasi-harmonic approximation
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--------------------------------------
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Here the word 'quasi-harmonic approximation' is used for an
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approximation that introduces volume dependence of phonon frequencies
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as a part of anharmonic effect.
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A part of temperature effect can be included into total energy of
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electronic structure through phonon (Helmholtz) free energy at
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constant volume. But what we want to know is thermal properties at
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constant pressure. We need some transformation from function of *V* to
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function of *p*. Gibbs free energy is defined at a constant pressure by
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the transformation:
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.. math::
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G(T, p) = \min_V \left[ U(V) + F_\mathrm{phonon}(T;\,V) + pV \right],
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where
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.. math::
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\min_V[ \text{function of } V ]
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means to find unique minimum value in the brackets by changing
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volume. Since volume dependencies of energies in electronic and phonon
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structures are different, volume giving the minimum value of the
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energy function in the square brackets shifts from the value
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calculated only from electronic structure even at 0 K. By increasing
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temperature, the volume dependence of phonon free energy changes, then
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the equilibrium volume at temperatures changes. This is considered as
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thermal expansion under this approximation.
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``phonopy-qha`` collects the values at volumes and transforms into the
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thermal properties at constant pressure.
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