In phonopy, force constants are generated based on finite displacement method. Crystal symmetry is used to reduce the calculation cost and numerical noise of the force constants. Firstly a symmetry reduced set of atomic displacements is generated. After the atomic force calculations, the set of atomic displacements are expanded using the symmetry and then all the elements of force constans between atoms in a primitive cell and the supercell are fit to the symmetry expanded forces of atoms in supercells using Moore–Penrose pseudoinverse. This procedure may considered as a variant of Parlinski-Li-Kawazoe method. Some of the details are found in the appendix of the following paper:

- L. Chaput, A. Togo, I. Tanaka, and G. Hug, Phys. Rev. B, 84, 094302 (2011)

Parlinski-Li-Kawazoe method is based on the supercell approach with the finite displacement method.

Force constants are calculated using Moore–Penrose pseudoinverse by fitting symmetry reduced elements of force constans to the linear relations between atomic forces and atomic displacements. The pseudoinverse is easy to handle arbitrary number of displacements amplitudes and directions, and can rely on the exisiting library, e.g., LAPACK.

- K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997)

In phonopy-qha, thermal properties at constant
pressure is obtained from the thermodynamic definition. To achieve
Legendre transformation, volume-energy function is generated from a
set of Helmholtz free energies and *pV* terms at volumes by fitting to
a smooth function for which equations of states are prepared in phonopy-qha.

The volume dependence of the Helmholtz free energy is included from quasi-harmonicity. When using DFT-GGA (-LDA), often we should have some amount of error in the absolute value since phonon frequencies are underestimated (overestimated). However the value of some ratio like thermal expansion coefficient is often very well estimated. An example is shown in the following paper:

- A. Togo, L. Chaput, I. Tanaka, G. Hug, Phys. Rev. B, 81, 174301-1-6 (2010)

Non-metallic crystals are polarized by atomic displacements and the generated macroscopic field changes force constants near \(\Gamma\) point. This contribution is included through non-analytical term correction.

- R. M. Pick, M. H. Cohen, and R. M. Martin, Phys. Rev. B 1, 910, (1970)

- P. Giannozzi, S. Degironcoli, P. Pavone, and S. Baroni, Phys. Rev. B 43, 7231 (1991)
- X. Gonze, J.-C. Charlier, D.C. Allan, and M.P. Teter Phys. Rev. B 50, 13035(R) (1994)
- X. Gonze, and C. Lee, Phys. Rev. B 55, 10355 (1997)

Currently phonopy employs the method by Gonze *et al.* written in the
above two papers as the default method.

This is an interpolation scheme using phonons at
\(\mathbf{q}\rightarrow \mathbf{0}\) with the correction by Pick
*et al.* and other commensurate points.

- Y. Wang , J. J. Wang , W. Y. Wang , Z. G. Mei , S. L. Shang , L. Q. Chen and Z K Liu, J. Phys.: Condens. Matter. 22, 202201 (2010)

The first derivative of this expression, which is for example used for group velocity calclation, is described in the following paper:

- Atsushi Togo, Laurent Chaput, and Isao Tanaka, Phys. Rev. B, 91, 094306-1-31 (2015)

PHONON is the original implementation of the Parlinsk-Li-Kawazoe method.

- Dario Alfè, Computer Physics Communications, 180, 2622 (2009)

PHON is based on the small displacement method.

- Xavier Gonze and Changyol Lee, Phys. Rev. B 55, 10355 (1997)

Currently there are several many implementations such as Abinit, Quantum espresso, Elk, etc. VASP can calculate force constants using DFPT however only at Gamma-point.

- Introduction to Lattice Dynamics, Martin. T. Dove, Cambridge university press
- Thermodynamics of Crystals, Duane C. Wallace, Dover Publications