Potential energy of phonon system is represented as functions of atomic positions:
where is the point of the -th atom in the -th unit cell and and are the number of atoms in a unit cell and the number of unit cells, respectively. A force and a second-order force constant are given by
respectively, where , , ..., are the Cartesian indices, , , ..., are the indices of atoms in a unit cell, and , , ..., are the indices of unit cells. In the finite displacement method, the equation for the force constants is approximated as
where are the forces on atoms with a finite displacement and usually .
The following is a modified and simplified version of the Parlinski-Li-Kawazoe method, which is just a numerical fitting approach to obtain force constants from forces and displacements.
The last equation above is represented by matrices as
where , , and for a pair of atoms, e.g. , are given by
The matrix equation is expanded for number of forces and displacements as follows:
With sufficient number of atomic displacements, this may be solved by pseudo inverse such as
Required number of atomic displacements to solve the simultaneous equations may be reduced using site-point symmetries. The matrix equation can be written using a symmetry operation as
where is the site symmetry operation centring at . and are defined as and , respectively, where is the matrix representation of the rotation operation. The combined simultaneous equations are built such as
where the superscript with parenthesis gives the index of site-symmetry operations. This is solved by pseudo inverse.
In phonopy, a phase convention of dynamical matrix is used as follows:
where is the atomic mass and is the wave vector. An equation of motion is written as
where the eigenvector of the band index at is obtained by the diagonalization of :
The atomic displacements are given as
where and are the creation and annihilation operators of phonon, is the reduced Planck constant, and is the time.
To correct long range interaction of macroscopic electric field induced by polarization of collective ionic motions near the -point, non-analytical term is added to dynamical matrix (Non-analytical term correction). At , the dynamical matrix with non-analytical term is given by,
Phonon frequencies at general q-points are interpolated by the method of Wang et al. (Interpolation scheme at general q-points with non-analytical term correction).
Phonopy calculates phonon frequencies based on input values from users. In the default case, the physical units of distance, atomic mass, force, and force constants are supposed to be , , , and , respectively, and the physical unit of the phonon frequency is converted to THz. This conversion is made as follows:
Internally phonon frequency has the physical unit of
frequency. To convert this unit to THz (not angular frequency), the
sqrt(EV/AMU)/Angstrom/(2*pi)/1e12 is made.
Angstrom are the values to convert them to those in the
SI base unit, i.e., to Joule, kg, and metre, respectively. These values
implemented in phonopy are found at a phonopy github page. This
unit conversion factor can be manually specified. See
The unit conversion factor in the
BORN file is multiplied with the second
term of the right hand side of the equation in
Non-analytical term correction where this equation is written
with atomic units (Gonze and Lee, 1997).
The physical unit of the part of the equation corresponding to force
is . In the default case for the VASP interface, internally is given in . In total, the necessary unit conversion is . In the default case of the Wien2k interface, the conversion factor is . For the other interfaces, the conversion factors are similarly calculated following the unit systems employed in phonopy (Interfaces to calculators).
In phonopy, so-called commensurate points mean the q-points whose waves are confined in the supercell used in the phonon calculation.
To explain about the commensurate points, let basis vectors of a primitive cell in direct space cell be the column vectors and those of the supercell be . The transformation of the basis vectors from the primitive cell to the supercell is written as
is given as a matrix and its elements are all integers, which is a constraint we have. The resolution for q-points being the commensurate points is determined by since one period of a wave has to be bound by any of lattice points inside the supercell. Therefore the number of commensurate points becomes the same as the number of the primitive cell that can be contained in the supercell, i.e., .
Then let the basis vectors in reciprocal space be the column vectors . Note that often reciprocal vectors are deifned by row vectors, but column vectors are chosen here to formulate. Formally we see the set of besis vectors are matrices, we have the following relation:
Similarly for the supercell, we define a relation
To multiply an arbitrary q-point on both sides
we find the constraint of a q-point being one of the commensurate points is the elements of to be integers.