**This is under development. Configurations may alter.** Requests or
suggestions are very welcome.

After setting the phonopy python path, the phonopy module is imported by:

```
from phonopy import Phonopy
```

Crystal structure is defined by the `PhonopyAtoms`

class. This
class is made to be similar to the ASE’s `Atoms`

class. The `PhonopyAtoms`

module is imported by:

```
from phonopy.structure.atoms import PhonopyAtoms
```

The work flow is schematically shown in Work flow.

The first step is to create a `Phonopy`

object with at least two
arguments, a unit cell (`PhonopyAtoms`

object, see
PhonopyAtoms class) and a supercell matrix (3x3 array, see
Supercell matrix). In the following example, a
\(2\times 2\times 2\) supercell is created. The displacements to
be introduced to the supercell are internally generated by the
`generate_displacements()`

method with the `distance`

keyword
argument. The supercells with displacements are obtained by
`get_supercells_with_displacements()`

method as a list of
`PhonopyAtoms`

objects.

```
import numpy as np
from phonopy import Phonopy
from phonopy.structure.atoms import PhonopyAtoms
a = 5.404
unitcell = PhonopyAtoms(symbols=['Si'] * 8,
cell=(np.eye(3) * a),
scaled_positions=[[0, 0, 0],
[0, 0.5, 0.5],
[0.5, 0, 0.5],
[0.5, 0.5, 0],
[0.25, 0.25, 0.25],
[0.25, 0.75, 0.75],
[0.75, 0.25, 0.75],
[0.75, 0.75, 0.25]])
phonon = Phonopy(unitcell,
[[2, 0, 0], [0, 2, 0], [0, 0, 2]],
primitive_matrix=[[0, 0.5, 0.5],
[0.5, 0, 0.5],
[0.5, 0.5, 0]])
phonon.generate_displacements(distance=0.03)
supercells = phonon.get_supercells_with_displacements()
```

In this example, the displacement distance is set to 0.03 (in Angstrom if the crystal structure uses the Angstrom unit and the default value is 0.01.)

The frequency unit conversion factor to THz has to be set by using the `factor`

keyword in `Phonopy`

class. The factors are `VaspToTHz`

for VASP,
`Wien2kToTHz`

for Wien2k, `AbinitToTHz`

for Abinit,
`PwscfToTHz`

for Pwscf, `ElkToTHz`

for Elk, `SiestaToTHz`

for Siesta, and `CrystalToTHz for CRYSTAL`

. `VaspToTHz`

is the default value.
For example:

```
from phonopy.units import AbinitToTHz
phonon = Phonopy(unitcell,
[[2, 0, 0], [0, 2, 0], [0, 0, 2]],
primitive_matrix=[[0, 0.5, 0.5],
[0.5, 0, 0.5],
[0.5, 0.5, 0]],
distance=0.03,
factor=AbinitToTHz)
```

Some more information on physical unit conversion is found at FREQUENCY_CONVERSION_FACTOR, Physical unit conversion, and Interfaces to calculators.

Forces on atoms are supposed to be obtained by running force
calculator (e.g. VASP) with each supercell with a displacement. Then
the forces in the calculation outputs have to be collected by
users. However output parsers for selected calculators are found under
`phonopy.interface`

, which may be useful. The forces have to be
stored in a specific structure: a numpy array (or nested list) as follows:

```
[ [ [ f_1x, f_1y, f_1z ], [ f_2x, f_2y, f_2z ], ... ], # first supercell
[ [ f_1x, f_1y, f_1z ], [ f_2x, f_2y, f_2z ], ... ], # second supercell
... ]
```

This array (`sets_of_forces`

) is set to the `Phonopy`

object by:

```
phonon.set_forces(sets_of_forces)
```

This is the case when the set of atomic displacements is generated
internally. The information of displacements is already stored in the
`Phonopy`

object. But if you want to input the forces together with
the corresponding custom set of displacements,
`displacement_dataset`

has to be prepared as a python dictionary as
follows:

```
displacement_dataset =
{'natom': number_of_atoms_in_supercell,
'first_atoms': [
{'number': atom index of displaced atom (starting with 0),
'displacement': displacement in Cartesian coordinates,
'forces': forces on atoms in supercell},
{...}, ...]}
```

This is set to the `Phonopy`

object by:

```
phonopy.set_displacement_dataset(displacement_dataset)
```

From the set of displacements and forces, force constants internally with calculated suuprcell sets of forces by

```
phonon.produce_force_constants()
```

If you have force constants and don’t need to create force constants from forces and displacements, simply set your force constants by

```
phonon.set_force_constants(force_constants)
```

The force constants matrix is given in 4 dimensional array (better to
be a numpy array of `dtype='double', order='C'`

). The shape of
force constants matrix is `(N, N, 3, 3)`

where `N`

is the number
of atoms in the supercell and 3 gives Cartesian axes. The compact
force constants matrix with `(Np, N, 3, 3)`

where `Np`

is the
number of atoms in the primitive cell is also supported. See the
details at FORCE_CONSTANTS and force_constants.hdf5.

Set band paths (`set_band_structure`

) and get the results
(`get_band_structure`

).

A tuple of (q-points, distances, frequencies, eigenvectors) is
obtained by `get_band_structure()`

. Eigenvectors can be obtained
when `is_eigenvectors=True`

at `set_band_structure()`

. Eigenvalues
are stored in a numpy array with the shape of (number_of_bands,
len(distances)). Phonon frequency is sqrt(eigenvalue). A negative
eigenvalue has to correspond to the imaginary frequency, but for the
plotting, it is set as the negative value in the above example. In
addition, you need to multiply by your unit conversion factor. In the
case of VASP to transform to THz, the factor is 15.633302.

```
bands = []
q_start = np.array([0.5, 0.5, 0.0])
q_end = np.array([0.0, 0.0, 0.0])
band = []
for i in range(51):
band.append(q_start + (q_end - q_start) / 50 * i)
bands.append(band)
q_start = np.array([0.0, 0.0, 0.0])
q_end = np.array([0.5, 0.0, 0.0])
band = []
for i in range(51):
band.append(q_start + (q_end - q_start) / 50 * i)
bands.append(band)
phonon.set_band_structure(bands)
phonon.plot_band_structure().show()
q_points, distances, frequencies, eigvecs = phonon.get_band_structure()
```

To obtain eigenvectors, it is necessary to inform to store eigenvectors by:

```
phonon.set_band_structure(bands, is_eigenvectors=True)
```

Set sampling mesh (`set_mesh`

) in reciprocal space. The irreducible
*q*-points and corresponding *q*-point weights, eigenvalues, and
eigenvectors are obtained by `get_mesh`

. `mesh`

gives the
sampling mesh with Monkhorst-Pack scheme. The keyword `shift`

gives
the fractional mesh shift with respect to the neighboring grid points.

```
mesh = [20, 20, 20]
phonon.set_mesh(mesh)
qpoints, weights, frequencies, eigvecs = phonon.get_mesh()
```

To obtain eigenvectors, it is necessary to inform to store eigenvectors by:

```
phonon.set_mesh([20, 20, 20], is_eigenvectors=True)
```

Before starting mesh sampling has to be finished. Then set parameters
(`set_total_DOS`

or `set_partial_DOS`

) and write the results into
files (`write_total_DOS`

and `write_partial_DOS`

). In the case of
PDOS, the eigenvectors have to be calculated in the mesh
sampling. `get_total_DOS`

and `get_partial_DOS`

are under preparation.

```
phonon.set_total_DOS()
phonon.plot_total_DOS().show()
```

Before starting the thermal property calculation, the mesh sampling
calclation has to be done in the **THz unit**. The unit conversion
factor for phonon frequency is set in the pre-process of Phonopy with
the `factor`

keyword. Calculation range of temperature is set by the
parameters `set_thermal_properties`

. Helmholtz free energy, entropy,
heat capacity at contant volume at temperaturs are obtained by
`get_thermal_properties`

, where the results are given as a tuple of
temperaturs, Helmholtz free energy, entropy, and heat capacity.

```
phonon.set_thermal_properties(t_step=10,
t_max=1000,
t_min=0)
for t, free_energy, entropy, cv in np.array(phonon.get_thermal_properties()).T:
print ("%12.3f " + "%15.7f" * 3) % ( t, free_energy, entropy, cv )
phonon.plot_thermal_properties().show()
```

To apply non-analytical term correction, Born effective charge tensors
for all atoms in **primitive** cell, dielectric constant tensor, and
the unit conversion factor have to be correctly set. The tensors are
given in Cartesian coordinates.

```
born = [[[1.08703, 0, 0],
[0, 1.08703, 0],
[0, 0, 1.08703]],
[[-1.08672, 0, 0],
[0, -1.08672, 0],
[0, 0, -1.08672]]]
epsilon = [[2.43533967, 0, 0],
[0, 2.43533967, 0],
[0, 0, 2.43533967]]
factors = 14.400
phonon.set_nac_params({'born': born,
'factor': factors,
'dielectric': epsilon})
```

A group velocity at a q-point is obtained by:

```
phonon.get_group_velocity_at_q(q_point)
```

Group velocities with mesh sampling, band structure, or q-points calculations are given as follows.

First inform phonopy object to calculate group velocity:

```
phonon.set_group_velocity()
```

Then the respective group velocities are obtained by:

```
phonon.get_group_velocity()
```

The shape of group velocity array is to follow those array shapes of calculation modes.

Eigenvectors are given as the column vectors. Internally phonopy uses numpy.linalg.eigh and eigh is a wrapper of LAPACK. So eigenvectors follow the convention of LAPACK, which can be shown at http://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.eigh.html

Eigenvectors corresponding to phonopy yaml output are obtained as follows.

```
if eigvecs is not None:
for eigvecs_on_path in eigvecs:
for eigvecs_at_q in eigvecs_on_path:
for vec in eigvecs_at_q.T:
print vec
```

```
if eigvecs is not None:
for eigvecs_at_q in eigvecs:
for vec in eigvecs_at_q.T:
print vec
```

`PhonopyAtoms`

class¶The usable keywords in the initialization are:

```
symbols=None,
positions=None,
numbers=None,
masses=None,
scaled_positions=None,
cell=None
```

At least three arguments have to be given at the initialization, which are

`cell`

`positions`

or`scaled_positions`

`symbols`

or`numbers`

The following variables are implemented in the `PhonopyAtoms`

class
in `atoms.py`

.

`lattice_vectors`

¶Lattice vectors are given in the matrix form in Cartesian coordinates.

```
[ [ a_x, a_y, a_z ],
[ b_x, b_y, b_z ],
[ c_x, c_y, c_z ] ]
```

`scaled_positions`

¶Atomic positions in fractional coordinates.

```
[ [ x1_a, x1_b, x1_c ],
[ x2_a, x2_b, x2_c ],
[ x3_a, x3_b, x3_c ],
... ]
```

`positions`

¶Cartesian positions of atoms.

```
positions = np.dot(scaled_positions, lattice_vectors)
```

where `np`

means the numpy module (`import numpy as np`

).

```
set_cell(lattice_vectors)
get_cell()
set_positions(positions)
get_positions()
set_scaled_positions(scaled_positions)
get_scaled_positions()
set_masses(masses)
get_masses()
set_chemical_symbols(symbols)
get_chemical_symbols()
get_number_of_atoms()
get_atomic_numbers()
get_volume()
```

These methods are designed to be compatible to the ASE’s `Atoms`

class. The arguments have to be set in the structures shown in
Variables.

Primitive matrix \(M_\mathrm{p}\) is a tranformation matrix from lattice vectors to those of a primitive cell if there exists the primitive cell in the lattice vectors. Following a crystallography convention, the transformation is given by

\[( \mathbf{a}_\mathrm{p} \; \mathbf{b}_\mathrm{p} \; \mathbf{c}_\mathrm{p} )
= ( \mathbf{a}_\mathrm{u} \; \mathbf{b}_\mathrm{u} \;
\mathbf{c}_\mathrm{u} ) M_\mathrm{p}\]

where \(\mathbf{a}_\mathrm{u}\), \(\mathbf{b}_\mathrm{u}\),
and \(\mathbf{c}_\mathrm{u}\) are the column vectors of the
original lattice vectors, and \(\mathbf{a}_\mathrm{p}\),
\(\mathbf{b}_\mathrm{p}\), and \(\mathbf{c}_\mathrm{p}\) are
the column vectors of the primitive lattice vectors. Be careful that
the lattice vectors of the `PhonopyAtoms`

class are the row vectors
(lattice_vectors). Therefore the phonopy code, which relies
on the `PhonopyAtoms`

class, is usually written such as

```
primitive_lattice = np.dot(original_lattice.T, primitive_matrix).T,
```

or equivalently,

```
primitive_lattice = np.dot(primitive_matrix.T, original_lattice)
```

Supercell matrix \(M_\mathrm{s}\) is a tranformation matrix from lattice vectors to those of a super cell. Following a crystallography convention, the transformation is given by

\[( \mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \; \mathbf{c}_\mathrm{s} )
= ( \mathbf{a}_\mathrm{u} \; \mathbf{b}_\mathrm{u} \;
\mathbf{c}_\mathrm{u} ) M_\mathrm{s}\]

where \(\mathbf{a}_\mathrm{u}\), \(\mathbf{b}_\mathrm{u}\),
and \(\mathbf{c}_\mathrm{u}\) are the column vectors of the
original lattice vectors, and \(\mathbf{a}_\mathrm{s}\),
\(\mathbf{b}_\mathrm{s}\), and \(\mathbf{c}_\mathrm{s}\) are
the column vectors of the supercell lattice vectors. Be careful that
the lattice vectors of the `PhonopyAtoms`

class are the row vectors
(lattice_vectors). Therefore the phonopy code, which relies
on the `PhonopyAtoms`

class, is usually written such as

```
supercell_lattice = np.dot(original_lattice.T, supercell_matrix).T,
```

or equivalently,

```
supercell_lattice = np.dot(supercell_matrix.T, original_lattice)
```

Symmetry search tolerance (often the name `symprec`

is used in
phonopy) is used to determine symmetry operations of the crystal
structures. The physical unit follows that of input crystal structure.

Parameters for non-analytical term correction may be made as follows. This example assumes that the user knows what are the unit cell and primitive cell and that the Born effective charge and dielectric constant were calculated using VASP code for the unit cell.

```
import io
import numpy as np
from phonopy import Phonopy
from phonopy.units import Hartree, Bohr
from phonopy.structure.atoms import PhonopyAtoms
from phonopy.structure.symmetry import symmetrize_borns_and_epsilon
from phonopy.interface.vasp import VasprunxmlExpat
with io.open("vasprun.xml", "rb") as f:
vasprun = VasprunxmlExpat(f)
if vasprun.parse():
epsilon = vasprun.get_epsilon()
borns = vasprun.get_born()
lattice = vasprun.get_lattice()[-1]
points = vasprun.get_points()[-1]
symbols = vasprun.get_symbols()
unit_cell = PhonopyAtoms(symbols=symbols,
scaled_positions=points,
cell=lattice)
symprec = 1e-5
phonon = Phonopy(unit_cell,
np.diag([1, 1, 1]),
primitive_matrix=[[0, 0.5, 0.5],
[0.5, 0, 0.5],
[0.5, 0.5, 0]],
symprec=symprec)
borns_, epsilon_ = symmetrize_borns_and_epsilon(
borns, epsilon, unit_cell, symprec=symprec)
np.testing.assert_allclose(borns, borns_, atol=1e-1)
np.testing.assert_allclose(epsilon, epsilon_, atol=1e-1)
pcell = phonon.get_primitive()
p2s_map = pcell.get_primitive_to_supercell_map()
nac_params = {'born': np.array([borns_[i] for i in p2s_map]),
'factor': Hartree * Bohr,
'dielectric': epsilon_}
```