# Phonopy API for Python#

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

## Import modules#

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

```
from phonopy import Phonopy
```

Crystal structure is defined by the `PhonopyAtoms`

class. The `PhonopyAtoms`

module is imported by:

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

The instance of `PhonopyAtoms`

can be made by reading a crystal structure in a
variety of calculator formats found at Interfaces to calculators.

```
from phonopy.interface.calculator import read_crystal_structure
unitcell, _ = read_crystal_structure("POSCAR-unitcell", interface_mode='vasp')
```

For VASP format, the keyword argument of `interface_mode`

can be omitted. For
QE,

```
unitcell, optional_structure_info = read_crystal_structure("NaCl.in", interface_mode='qe')
```

Note that `read_crystal_structure`

returns a tuple and the first element is the
`PhonopyAtoms`

instance.

## Work flow#

The work flow is schematically shown in Work flow.

### Pre-process#

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,
supercell_matrix=[[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.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
supercells with displacements are given as a list of `PhonopyAtoms`

. See
Read and write crystal structures to write
those into files in a crystal structure format.

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, `CrystalToTHz`

for CRYSTAL, `FleurToTHz`

for
Fleur, `VaspToTHz`

, and `DftbpToTHz`

for DFTB+ is the default value. For
example:

```
from phonopy.units import AbinitToTHz
phonon = Phonopy(unitcell,
supercell_matrix=[[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]],
factor=AbinitToTHz)
```

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

### Post process#

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.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.dataset = displacement_dataset
```

From the set of displacements and forces, force constants internally with calculated supercell 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.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.

## Phonon calculation#

### Save parameters (`phonopy.save`

)#

Basic information and parameters needed for phonon calculation are saved into a
file by `phonopy.save`

.

```
phonon.save()
```

The default file name is `phonopy_params.yaml`

. Force sets, displacements, Born
effective charges, and dielectric constant are written in the default behaviour.
If force constants are needed to be written in the yaml file, the argument
`settings`

is set as follows:

```
phonon.save(settings={'force_constants': True})
```

### Shortcut to load input files (`phonopy.load`

)#

`phonopy.load`

is a convenient python method to create `Phonopy`

instance
loading forces, displacements, and parameters for non-analytical term
correction. The details are found in the docstring that can be seen by (e.g., in
ipython)

```
In [1]: import phonopy
In [2]: help(phonopy.load)
```

Examples of how to use `phonopy.load`

are listed below.

`phonopy_params.yaml`

may contain all information needed to prepare phonon
calculation:

```
phonon = phonopy.load("phonopy_params.yaml")
```

More detailed configuration can be given as follows:

```
phonon = phonopy.load(supercell_matrix=[2, 2, 2],
primitive_matrix='auto',
unitcell_filename="POSCAR",
force_constants_filename="force_constants.hdf5")
```

With `is_nac=True`

(default), `BORN`

file in the current directory is read if it
exists. If supercell is passed and `primitive matrix`

and `supercell_matrix`

are
not set, the primitive cell is automatically searched:

```
phonon = phonopy.load(supercell_filename="SPOSCAR",
force_constants_filename="force_constants.hdf5")
```

If `FORCE_SETS`

exists in the current directory, this below works to be ready
for post-process calculation with automatic choice of primitive matrix:

```
phonon = phonopy.load(supercell_filename="SPOSCAR")
```

For example, in the `example/NaCl`

directory, phonon band structure of NaCl is
easily plotted by

```
In [1]: import phonopy
In [2]: ph = phonopy.load(supercell_filename="SPOSCAR", log_level=1)
Supercell structure was read from "SPOSCAR".
NAC params were read from "BORN".
Force constants were read from "FORCE_CONSTANTS".
In [3]: print(ph.primitive)
lattice:
- [ 0.000000000000000, 2.845150738087836, 2.845150738087836 ] # a
- [ 2.845150738087836, 0.000000000000000, 2.845150738087836 ] # b
- [ 2.845150738087836, 2.845150738087836, 0.000000000000000 ] # c
points:
- symbol: Na # 1
coordinates: [ 0.000000000000000, 0.000000000000000, 0.000000000000000 ]
mass: 22.989769
- symbol: Cl # 2
coordinates: [ 0.500000000000000, 0.500000000000000, 0.500000000000000 ]
mass: 35.453000
In [4]: ph.nac_params
Out[4]:
{'born': array([[[ 1.08703000e+00, -5.17677526e-34, -1.06309751e-33],
[-5.45419984e-34, 1.08703000e+00, 1.06309751e-33],
[ 0.00000000e+00, 3.08148791e-33, 1.08703000e+00]],
[[-1.08672000e+00, -2.93244455e-35, 5.15939995e-34],
[ 5.45264441e-34, -1.08672000e+00, -5.15939995e-34],
[ 0.00000000e+00, 0.00000000e+00, -1.08672000e+00]]]),
'factor': 14.4,
'dielectric': array([[2.43533967, 0. , 0. ],
[0. , 2.43533967, 0. ],
[0. , 0. , 2.43533967]])}
In [5]: ph.auto_band_structure(plot=True).show()
```

### Band structure#

Set band paths (`run_band_structure()`

) and get the results
(`get_band_structure_dict()`

).

A dictionary with `qpoints`

, `distances`

, `frequencies`

, `eigenvectors`

,
`group_velocities`

is returned by `get_band_structure_dict()`

. Eigenvectors can
be obtained when `with_eigenvectors=True`

at `run_band_structure()`

. See the
details at docstring of `Phonopy.get_band_structure_dict`

. 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.

In `example/NaCl`

, the phonopy is executed from python script, e.g.,

```
import phonopy
from phonopy.phonon.band_structure import get_band_qpoints_and_path_connections
path = [[[0, 0, 0], [0.5, 0, 0.5], [0.625, 0.25, 0.625]],
[[0.375, 0.375, 0.75], [0, 0, 0], [0.5, 0.5, 0.5], [0.5, 0.25, 0.75]]]
labels = ["$\\Gamma$", "X", "U", "K", "$\\Gamma$", "L", "W"]
qpoints, connections = get_band_qpoints_and_path_connections(path, npoints=51)
phonon = phonopy.load("phonopy_disp.yaml")
phonon.run_band_structure(qpoints, path_connections=connections, labels=labels)
phonon.plot_band_structure().show()
# To plot DOS next to band structure
phonon.run_mesh([20, 20, 20])
phonon.run_total_dos()
phonon.plot_band_structure_and_dos().show()
# To plot PDOS next to band structure
phonon.run_mesh([20, 20, 20], with_eigenvectors=True, is_mesh_symmetry=False)
phonon.run_projected_dos()
phonon.plot_band_structure_and_dos(pdos_indices=[[0], [1]]).show()
```

`path_connections`

and `labels`

are unnecessary to set unless nice looking
plotting is needed. To obtain eigenvectors, it is necessary to inform to store
eigenvectors by:

```
phonon.run_band_structure(bands, with_eigenvectors=True)
```

To obtain group velocities:

```
phonon.run_band_structure(bands, with_group_velocities=True)
```

Automatic selection of band paths using SeeK-path is invoked by

```
phonon.auto_band_structure()
```

and to plot

```
phonon.auto_band_structure(plot=True).show()
```

To use this method, `seekpath`

python module is needed.

### Mesh sampling#

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_dict()`

. `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.run_mesh(mesh)
mesh_dict = phonon.get_mesh_dict()
qpoints = mesh_dict['qpoints']
weights = mesh_dict['weights']
frequencies = mesh_dict['frequencies']
eigenvectors = mesh_dict['eigenvectors']
group_velocities = mesh_dict['group_velocities']
```

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

```
phonon.run_mesh([20, 20, 20], with_eigenvectors=True)
```

and for group velocities:

```
phonon.run_mesh([20, 20, 20], with_group_velocities=True)
```

The first argument of `run_mesh()`

can be a float value, which is a length
measure as explained at MESH, MP, or MESH_NUMBERS, for example:

```
phonon.run_mesh(100.0)
```

### DOS and PDOS#

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

or `run_projected_dos()`

) and write the results into files
(`write_total_dos()`

and `write_projected_dos()`

). In the case of PDOS, the
eigenvectors have to be calculated in the mesh sampling. To get the results
`get_total_dos_dict()`

and `get_projected_dos_dict()`

can be used.

To plot total DOS,

```
phonon.run_mesh([20, 20, 20]) phonon.run_total_dos()
phonon.plot_total_dos().show()
```

and projected DOS

```
phonon.run_mesh([20, 20, 20], with_eigenvectors=True, is_mesh_symmetry=False)
phonon.run_projected_dos()
phonon.plot_projected_dos().show()
```

Convenient shortcuts exist as follows:

```
phonon.auto_total_dos(plot=True).show()
```

and

```
phonon.auto_projected_dos(plot=True).show()
```

### Thermal properties#

Before starting the thermal property calculation, the mesh sampling calculation
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
`run_thermal_properties`

. Helmholtz free energy, entropy, heat capacity at
constant volume at temperatures are obtained by `get_thermal_properties_dict`

,
where the results are given as a dictionary of temperatures, Helmholtz free
energy, entropy, and heat capacity with keys `temperatures`

, `free_energy`

,
`entropy`

, and `heat_capacity`

, respectively.

```
phonon.run_mesh([20, 20, 20])
phonon.run_thermal_properties(t_step=10,
t_max=1000,
t_min=0)
tp_dict = phonon.get_thermal_properties_dict()
temperatures = tp_dict['temperatures']
free_energy = tp_dict['free_energy']
entropy = tp_dict['entropy']
heat_capacity = tp_dict['heat_capacity']
for t, F, S, cv in zip(temperatures, free_energy, entropy, heat_capacity):
print(("%12.3f " + "%15.7f" * 3) % ( t, F, S, cv ))
phonon.plot_thermal_properties().show()
```

### Non-analytical term correction#

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 = [[[1.08878299, 0, 0],
[0, 1.08878299, 0],
[0, 0, 1.08878299]],
[[-1.08878299, 0, 0],
[0, -1.08878299, 0],
[0, 0, -1.08878299]]]
epsilon = [[2.56544559, 0, 0],
[0, 2.56544559, 0],
[0, 0, 2.56544559]]
factors = 14.400
phonon.nac_params = {'born': born,
'factor': factors,
'dielectric': epsilon}
```

## Data structure#

### Eigenvectors#

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.

#### Band structure#

```
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)
```

#### Mesh sampling#

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

`PhonopyAtoms`

class#

### Initialization#

The usable keywords in the initialization are:

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

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

`cell`

`positions`

or`scaled_positions`

`symbols`

or`numbers`

### Variables#

The following variables are implemented in the `PhonopyAtoms`

class in
`phonopy/structure/atoms.py`

.

`cell`

#

Basis 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, cell)
```

where `np`

means the numpy module (`import numpy as np`

).

`symbols`

#

Chemical symbols, e.g.,

```
['Zn', 'Zn', 'O', 'O']
```

for the ZnO unit cell.

`numbers`

#

Atomic numbers, e.g.,

```
[30, 30, 8, 8]
```

for the ZnO unit cell.

`masses`

#

Atomic masses, e.g.,

```
[65.38, 65.38, 15.9994, 15.9994]
```

for the ZnO unit cell.

### Attributes#

```
cell
positions
scaled_positions
masses
magnetic_moments
symbols
numbers
volume
```

where `volume`

is the getter only.

### Methods#

`unitcell.get_number_of_atoms()`

is equivalent to `len(unitcell)`

. An instance
can be deep-copied by `unitcell.copy()`

. Human-readable crystal structure in
Yaml format is shown by `print(unitcell)`

. `unitcell.to_tuple`

converts to
spglib crystal structure
(https://spglib.github.io/spglib/python-spglib.html#crystal-structure-cell).

## Definitions of variables#

### Primitive matrix#

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

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 (cell). 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#

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

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 (cell). 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#

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.

## Read and write crystal structures#

There is a function to write the `PhonopyAtoms`

instance into crystal structure
formats of different force calculators, `write_crystal_structure`

. This works as
a partner of `read_crystal_structure`

. Taking an example of QE interface, how to
use these functions is shown below.

```
In [1]: from phonopy.interface.calculator import read_crystal_structure, write_crystal_structure
In [2]: !cat "NaCl.in"
&control
calculation = 'scf'
tprnfor = .true.
tstress = .true.
pseudo_dir = '/home/togo/espresso/pseudo/'
/
&system
ibrav = 0
nat = 8
ntyp = 2
ecutwfc = 70.0
/
&electrons
diagonalization = 'david'
conv_thr = 1.0d-9
/
ATOMIC_SPECIES
Na 22.98976928 Na.pbe-spn-kjpaw_psl.0.2.UPF
Cl 35.453 Cl.pbe-n-kjpaw_psl.0.1.UPF
ATOMIC_POSITIONS crystal
Na 0.0000000000000000 0.0000000000000000 0.0000000000000000
Na 0.0000000000000000 0.5000000000000000 0.5000000000000000
Na 0.5000000000000000 0.0000000000000000 0.5000000000000000
Na 0.5000000000000000 0.5000000000000000 0.0000000000000000
Cl 0.5000000000000000 0.5000000000000000 0.5000000000000000
Cl 0.5000000000000000 0.0000000000000000 0.0000000000000000
Cl 0.0000000000000000 0.5000000000000000 0.0000000000000000
Cl 0.0000000000000000 0.0000000000000000 0.5000000000000000
CELL_PARAMETERS angstrom
5.6903014761756712 0 0
0 5.6903014761756712 0
0 0 5.6903014761756712
K_POINTS automatic
8 8 8 1 1 1
In [3]: cell, optional_structure_info = read_crystal_structure("NaCl.in", interface_mode='qe')
In [4]: optional_structure_info
Out[4]:
('NaCl.in',
{'Na': 'Na.pbe-spn-kjpaw_psl.0.2.UPF', 'Cl': 'Cl.pbe-n-kjpaw_psl.0.1.UPF'})
In [5]: write_crystal_structure("NaCl-out.in", cell, interface_mode='qe', optional_structure_info=optional_structure_info)
In [6]: !cat "NaCl-out.in"
! ibrav = 0, nat = 8, ntyp = 2
CELL_PARAMETERS bohr
10.7531114272216008 0.0000000000000000 0.0000000000000000
0.0000000000000000 10.7531114272216008 0.0000000000000000
0.0000000000000000 0.0000000000000000 10.7531114272216008
ATOMIC_SPECIES
Na 22.98977 Na.pbe-spn-kjpaw_psl.0.2.UPF
Cl 35.45300 Cl.pbe-n-kjpaw_psl.0.1.UPF
ATOMIC_POSITIONS crystal
Na 0.0000000000000000 0.0000000000000000 0.0000000000000000
Na 0.0000000000000000 0.5000000000000000 0.5000000000000000
Na 0.5000000000000000 0.0000000000000000 0.5000000000000000
Na 0.5000000000000000 0.5000000000000000 0.0000000000000000
Cl 0.5000000000000000 0.5000000000000000 0.5000000000000000
Cl 0.5000000000000000 0.0000000000000000 0.0000000000000000
Cl 0.0000000000000000 0.5000000000000000 0.0000000000000000
Cl 0.0000000000000000 0.0000000000000000 0.5000000000000000
```

Depending on calculator interfaces, all the information can not be recovered
from the information obtained from `read_crystal_structure`

. More details about
how `write_crystal_structure`

works may need to read directly the
code.

## Getting parameters for non-analytical term correction#

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 by the unit cell.

```
import io
import numpy as np
from phonopy.units import Hartree, Bohr
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)
vasprun.parse():
epsilon = vasprun.epsilon
borns = vasprun.born
unitcell = vasprun.cell
borns_, epsilon_ = symmetrize_borns_and_epsilon(
borns,
epsilon,
unitcell,
primitive_matrix=[[0, 0.5, 0.5],
[0.5, 0, 0.5],
[0.5, 0.5, 0]],
supercell_matrix=np.diag([2, 2, 2]),
symprec=1e-5)
nac_params = {'born': borns_,
'factor': Hartree * Bohr,
'dielectric': epsilon_}
```