#
Direct method

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**PHONON Software**

An *ab initio* program optimizes the structure of the crystalline
supercell within constraints imposed by a crystallographic space group.
In optimized configuration the forces acting on all atoms of the
supercell vanish.
Displacing one atom from its equilibium position,
one generates non-zero Hellmann-Feynman forces acting on all atoms of
the supercell.

The lattice dynamics requires to know the force constants,
which are the second derivatives of the potential energy.
In the direct method one calculates the force constants from the
Hellmann-Feynman forces.
The Hellmann-Feynman forces, in turn, are found when a single atom
is displaced.
In this way all force constants of the supercell
can be found.

To obtain the reliable phonon dispersion relations the supercell
diameter should be about 8 - 10 angsterms. The magnitude of the force
constants
beyond that distnce is usually negligible, and in such a case the
phonon dispersion curve can be calculated exactly.
But even if the supercell size is small, exact phonon frequencies can
be obtained for the wave vectors
commensurate with the supercell size. Then, the phonon dispersion
curves are the
symmetry controlled interpolations between the exact points.

For high-symmetry supercells treated by an *ab initio* program,
**Phonon** calculates the phonon dispersion curves
within minutes.

Within the direct method approach and according to the symmetry
imposed by the crystal space group
**Phonon** calculates the following items:
- Required atomic displacements for generating the
Hellmann-Feynman forces
- Wave vectors at which exact phonon frequencies can be calculated
- Neighbor list of atoms in the supercell
- Symmetry of force constant's matrices
- Number of independent parameters for each force constant matrix
- Modification of the force constants due to
translational-rotational invariances
- Phonon dispersion relations along any line of reciprocal space
- Form factors of phonon dispersion relations
- LO/TO splitting from known effective charges and dielectric
constant
- Polarization vectors for any mode
- Irreducible representations of all phonon modes at the Brillouin
zone center
- Total density of phonon states
- Partial density of phonon states for each degree of freedom and
each atom
- Thermodynamic functions: internal energy, free energy, entropy,
heat capacity
- Thermal displacement tensors (Debey-Waller fator)
- Dynamical structure factor giving intensity of coherently
scattering neutrons i different Brillouin zones.
- Doubly differential incoherent inelastic scattering cross
section on monocrystals and polycrystals, including multi-phonon
processes

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