"""Module for calculating phonons of periodic systems."""
import warnings
from math import pi, sqrt
from pathlib import Path
import numpy as np
import numpy.fft as fft
import numpy.linalg as la
import ase
import ase.units as units
from ase.dft import monkhorst_pack
from ase.io.trajectory import Trajectory
from ase.parallel import world
from ase.utils import deprecated
from ase.utils.filecache import MultiFileJSONCache
class Displacement:
"""Abstract base class for phonon and el-ph supercell calculations.
Both phonons and the electron-phonon interaction in periodic systems can be
calculated with the so-called finite-displacement method where the
derivatives of the total energy and effective potential are obtained from
finite-difference approximations, i.e. by displacing the atoms. This class
provides the required functionality for carrying out the calculations for
the different displacements in its ``run`` member function.
Derived classes must overwrite the ``__call__`` member function which is
called for each atomic displacement.
"""
def __init__(self, atoms, calc=None, supercell=(1, 1, 1), name=None,
delta=0.01, center_refcell=False, comm=None):
"""Init with an instance of class ``Atoms`` and a calculator.
Parameters:
atoms: Atoms object
The atoms to work on.
calc: Calculator
Calculator for the supercell calculation.
supercell: tuple
Size of supercell given by the number of repetitions (l, m, n) of
the small unit cell in each direction.
name: str
Base name to use for files.
delta: float
Magnitude of displacement in Ang.
center_refcell: bool
Reference cell in which the atoms will be displaced. If False, then
corner cell in supercell is used. If True, then cell in the center
of the supercell is used.
comm: communicator
MPI communicator for the phonon calculation.
Default is to use world.
"""
# Store atoms and calculator
self.atoms = atoms
self.calc = calc
# Displace all atoms in the unit cell by default
self.indices = np.arange(len(atoms))
self.name = name
self.delta = delta
self.center_refcell = center_refcell
self.supercell = supercell
if comm is None:
comm = world
self.comm = comm
self.cache = MultiFileJSONCache(self.name)
def define_offset(self): # Reference cell offset
if not self.center_refcell:
# Corner cell
self.offset = 0
else:
# Center cell
N_c = self.supercell
self.offset = (N_c[0] // 2 * (N_c[1] * N_c[2]) +
N_c[1] // 2 * N_c[2] +
N_c[2] // 2)
return self.offset
@property
@ase.utils.deprecated('Please use phonons.supercell instead of .N_c')
def N_c(self):
return self._supercell
@property
def supercell(self):
return self._supercell
@supercell.setter
def supercell(self, supercell):
assert len(supercell) == 3
self._supercell = tuple(supercell)
self.define_offset()
self._lattice_vectors_array = self.compute_lattice_vectors()
@ase.utils.deprecated('Please use phonons.compute_lattice_vectors()'
' instead of .lattice_vectors()')
def lattice_vectors(self):
return self.compute_lattice_vectors()
def compute_lattice_vectors(self):
"""Return lattice vectors for cells in the supercell."""
# Lattice vectors -- ordered as illustrated in class docstring
# Lattice vectors relevative to the reference cell
R_cN = np.indices(self.supercell).reshape(3, -1)
N_c = np.array(self.supercell)[:, np.newaxis]
if self.offset == 0:
R_cN += N_c // 2
R_cN %= N_c
R_cN -= N_c // 2
return R_cN
def __call__(self, *args, **kwargs):
"""Member function called in the ``run`` function."""
raise NotImplementedError("Implement in derived classes!.")
def set_atoms(self, atoms):
"""Set the atoms to vibrate.
Parameters:
atoms: list
Can be either a list of strings, ints or ...
"""
assert isinstance(atoms, list)
assert len(atoms) <= len(self.atoms)
if isinstance(atoms[0], str):
assert np.all([isinstance(atom, str) for atom in atoms])
sym_a = self.atoms.get_chemical_symbols()
# List for atomic indices
indices = []
for type in atoms:
indices.extend([a for a, atom in enumerate(sym_a)
if atom == type])
else:
assert np.all([isinstance(atom, int) for atom in atoms])
indices = atoms
self.indices = indices
def _eq_disp(self):
return self._disp(0, 0, 0)
def _disp(self, a, i, step):
from ase.vibrations.vibrations import Displacement as VDisplacement
return VDisplacement(a, i, np.sign(step), abs(step), self)
def run(self):
"""Run the calculations for the required displacements.
This will do a calculation for 6 displacements per atom, +-x, +-y, and
+-z. Only those calculations that are not already done will be
started. Be aware that an interrupted calculation may produce an empty
file (ending with .json), which must be deleted before restarting the
job. Otherwise the calculation for that displacement will not be done.
"""
# Atoms in the supercell -- repeated in the lattice vector directions
# beginning with the last
atoms_N = self.atoms * self.supercell
# Set calculator if provided
assert self.calc is not None, "Provide calculator in __init__ method"
atoms_N.calc = self.calc
# Do calculation on equilibrium structure
eq_disp = self._eq_disp()
with self.cache.lock(eq_disp.name) as handle:
if handle is not None:
output = self.calculate(atoms_N, eq_disp)
handle.save(output)
# Positions of atoms to be displaced in the reference cell
natoms = len(self.atoms)
offset = natoms * self.offset
pos = atoms_N.positions[offset: offset + natoms].copy()
# Loop over all displacements
for a in self.indices:
for i in range(3):
for sign in [-1, 1]:
disp = self._disp(a, i, sign)
with self.cache.lock(disp.name) as handle:
if handle is None:
continue
try:
atoms_N.positions[offset + a, i] = \
pos[a, i] + sign * self.delta
result = self.calculate(atoms_N, disp)
handle.save(result)
finally:
# Return to initial positions
atoms_N.positions[offset + a, i] = pos[a, i]
self.comm.barrier()
def clean(self):
"""Delete generated files."""
if self.comm.rank == 0:
nfiles = self._clean()
else:
nfiles = 0
self.comm.barrier()
return nfiles
def _clean(self):
name = Path(self.name)
nfiles = 0
if name.is_dir():
for fname in name.iterdir():
fname.unlink()
nfiles += 1
name.rmdir()
return nfiles
[docs]class Phonons(Displacement):
r"""Class for calculating phonon modes using the finite displacement method.
The matrix of force constants is calculated from the finite difference
approximation to the first-order derivative of the atomic forces as::
2 nbj nbj
nbj d E F- - F+
C = ------------ ~ ------------- ,
mai dR dR 2 * delta
mai nbj
where F+/F- denotes the force in direction j on atom nb when atom ma is
displaced in direction +i/-i. The force constants are related by various
symmetry relations. From the definition of the force constants it must
be symmetric in the three indices mai::
nbj mai bj ai
C = C -> C (R ) = C (-R ) .
mai nbj ai n bj n
As the force constants can only depend on the difference between the m and
n indices, this symmetry is more conveniently expressed as shown on the
right hand-side.
The acoustic sum-rule::
_ _
aj \ bj
C (R ) = - ) C (R )
ai 0 /__ ai m
(m, b)
!=
(0, a)
Ordering of the unit cells illustrated here for a 1-dimensional system (in
case ``refcell=None`` in constructor!):
::
m = 0 m = 1 m = -2 m = -1
-----------------------------------------------------
| | | | |
| * b | * | * | * |
| | | | |
| * a | * | * | * |
| | | | |
-----------------------------------------------------
Example:
>>> from ase.build import bulk
>>> from ase.phonons import Phonons
>>> from gpaw import GPAW, FermiDirac
>>> atoms = bulk('Si', 'diamond', a=5.4)
>>> calc = GPAW(mode='fd',
... kpts=(5, 5, 5),
... h=0.2,
... occupations=FermiDirac(0.))
>>> ph = Phonons(atoms, calc, supercell=(5, 5, 5))
>>> ph.run()
>>> ph.read(method='frederiksen', acoustic=True)
"""
def __init__(self, *args, **kwargs):
"""Initialize with base class args and kwargs."""
if 'name' not in kwargs:
kwargs['name'] = "phonon"
self.deprecate_refcell(kwargs)
Displacement.__init__(self, *args, **kwargs)
# Attributes for force constants and dynamical matrix in real space
self.C_N = None # in units of eV / Ang**2
self.D_N = None # in units of eV / Ang**2 / amu
# Attributes for born charges and static dielectric tensor
self.Z_avv = None
self.eps_vv = None
@staticmethod
def deprecate_refcell(kwargs: dict):
if 'refcell' in kwargs:
warnings.warn('Keyword refcell of Phonons is deprecated.'
'Please use center_refcell (bool)', FutureWarning)
kwargs['center_refcell'] = bool(kwargs['refcell'])
kwargs.pop('refcell')
return kwargs
def __call__(self, atoms_N):
"""Calculate forces on atoms in supercell."""
return atoms_N.get_forces()
def calculate(self, atoms_N, disp):
forces = self(atoms_N)
return {'forces': forces}
[docs] def check_eq_forces(self):
"""Check maximum size of forces in the equilibrium structure."""
eq_disp = self._eq_disp()
feq_av = self.cache[eq_disp.name]['forces']
fmin = feq_av.min()
fmax = feq_av.max()
i_min = np.where(feq_av == fmin)
i_max = np.where(feq_av == fmax)
return fmin, fmax, i_min, i_max
[docs] @deprecated('Current implementation of non-analytical correction is '
'likely incorrect, see '
'https://gitlab.com/ase/ase/-/issues/941')
def read_born_charges(self, name='born', neutrality=True):
r"""Read Born charges and dieletric tensor from JSON file.
The charge neutrality sum-rule::
_ _
\ a
) Z = 0
/__ ij
a
Parameters:
neutrality: bool
Restore charge neutrality condition on calculated Born effective
charges.
name: str
Key used to identify the file with Born charges for the unit cell
in the JSON cache.
.. deprecated:: 3.22.1
Current implementation of non-analytical correction is likely
incorrect, see :issue:`941`
"""
# Load file with Born charges and dielectric tensor for atoms in the
# unit cell
Z_avv, eps_vv = self.cache[name]
# Neutrality sum-rule
if neutrality:
Z_mean = Z_avv.sum(0) / len(Z_avv)
Z_avv -= Z_mean
self.Z_avv = Z_avv[self.indices]
self.eps_vv = eps_vv
[docs] def read(self, method='Frederiksen', symmetrize=3, acoustic=True,
cutoff=None, born=False, **kwargs):
"""Read forces from json files and calculate force constants.
Extra keyword arguments will be passed to ``read_born_charges``.
Parameters:
method: str
Specify method for evaluating the atomic forces.
symmetrize: int
Symmetrize force constants (see doc string at top) when
``symmetrize != 0`` (default: 3). Since restoring the acoustic sum
rule breaks the symmetry, the symmetrization must be repeated a few
times until the changes a insignificant. The integer gives the
number of iterations that will be carried out.
acoustic: bool
Restore the acoustic sum rule on the force constants.
cutoff: None or float
Zero elements in the dynamical matrix between atoms with an
interatomic distance larger than the cutoff.
born: bool
Read in Born effective charge tensor and high-frequency static
dielelctric tensor from file.
"""
method = method.lower()
assert method in ['standard', 'frederiksen']
if cutoff is not None:
cutoff = float(cutoff)
# Read Born effective charges and optical dielectric tensor
if born:
self.read_born_charges(**kwargs)
# Number of atoms
natoms = len(self.indices)
# Number of unit cells
N = np.prod(self.supercell)
# Matrix of force constants as a function of unit cell index in units
# of eV / Ang**2
C_xNav = np.empty((natoms * 3, N, natoms, 3), dtype=float)
# Loop over all atomic displacements and calculate force constants
for i, a in enumerate(self.indices):
for j, v in enumerate('xyz'):
# Atomic forces for a displacement of atom a in direction v
# basename = '%s.%d%s' % (self.name, a, v)
basename = '%d%s' % (a, v)
fminus_av = self.cache[basename + '-']['forces']
fplus_av = self.cache[basename + '+']['forces']
if method == 'frederiksen':
fminus_av[a] -= fminus_av.sum(0)
fplus_av[a] -= fplus_av.sum(0)
# Finite difference derivative
C_av = fminus_av - fplus_av
C_av /= 2 * self.delta
# Slice out included atoms
C_Nav = C_av.reshape((N, len(self.atoms), 3))[:, self.indices]
index = 3 * i + j
C_xNav[index] = C_Nav
# Make unitcell index the first and reshape
C_N = C_xNav.swapaxes(0, 1).reshape((N,) + (3 * natoms, 3 * natoms))
# Cut off before symmetry and acoustic sum rule are imposed
if cutoff is not None:
self.apply_cutoff(C_N, cutoff)
# Symmetrize force constants
if symmetrize:
for _ in range(symmetrize):
# Symmetrize
C_N = self.symmetrize(C_N)
# Restore acoustic sum-rule
if acoustic:
self.acoustic(C_N)
else:
break
# Store force constants and dynamical matrix
self.C_N = C_N
self.D_N = C_N.copy()
# Add mass prefactor
m_a = self.atoms.get_masses()
self.m_inv_x = np.repeat(m_a[self.indices]**-0.5, 3)
M_inv = np.outer(self.m_inv_x, self.m_inv_x)
for D in self.D_N:
D *= M_inv
[docs] def symmetrize(self, C_N):
"""Symmetrize force constant matrix."""
# Number of atoms
natoms = len(self.indices)
# Number of unit cells
N = np.prod(self.supercell)
# Reshape force constants to (l, m, n) cell indices
C_lmn = C_N.reshape(self.supercell + (3 * natoms, 3 * natoms))
# Shift reference cell to center index
if self.offset == 0:
C_lmn = fft.fftshift(C_lmn, axes=(0, 1, 2)).copy()
# Make force constants symmetric in indices -- in case of an even
# number of unit cells don't include the first cell
i, j, k = 1 - np.asarray(self.supercell) % 2
C_lmn[i:, j:, k:] *= 0.5
C_lmn[i:, j:, k:] += \
C_lmn[i:, j:, k:][::-1, ::-1, ::-1].transpose(0, 1, 2, 4, 3).copy()
if self.offset == 0:
C_lmn = fft.ifftshift(C_lmn, axes=(0, 1, 2)).copy()
# Change to single unit cell index shape
C_N = C_lmn.reshape((N, 3 * natoms, 3 * natoms))
return C_N
[docs] def acoustic(self, C_N):
"""Restore acoustic sumrule on force constants."""
# Number of atoms
natoms = len(self.indices)
# Copy force constants
C_N_temp = C_N.copy()
# Correct atomic diagonals of R_m = (0, 0, 0) matrix
for C in C_N_temp:
for a in range(natoms):
for a_ in range(natoms):
C_N[self.offset,
3 * a: 3 * a + 3,
3 * a: 3 * a + 3] -= C[3 * a: 3 * a + 3,
3 * a_: 3 * a_ + 3]
[docs] def apply_cutoff(self, D_N, r_c):
"""Zero elements for interatomic distances larger than the cutoff.
Parameters:
D_N: ndarray
Dynamical/force constant matrix.
r_c: float
Cutoff in Angstrom.
"""
# Number of atoms and primitive cells
natoms = len(self.indices)
N = np.prod(self.supercell)
# Lattice vectors
R_cN = self._lattice_vectors_array
# Reshape matrix to individual atomic and cartesian dimensions
D_Navav = D_N.reshape((N, natoms, 3, natoms, 3))
# Cell vectors
cell_vc = self.atoms.cell.transpose()
# Atomic positions in reference cell
pos_av = self.atoms.get_positions()
# Zero elements with a distance to atoms in the reference cell
# larger than the cutoff
for n in range(N):
# Lattice vector to cell
R_v = np.dot(cell_vc, R_cN[:, n])
# Atomic positions in cell
posn_av = pos_av + R_v
# Loop over atoms and zero elements
for i, a in enumerate(self.indices):
dist_a = np.sqrt(np.sum((pos_av[a] - posn_av)**2, axis=-1))
# Atoms where the distance is larger than the cufoff
i_a = dist_a > r_c # np.where(dist_a > r_c)
# Zero elements
D_Navav[n, i, :, i_a, :] = 0.0
[docs] def get_force_constant(self):
"""Return matrix of force constants."""
assert self.C_N is not None
return self.C_N
[docs] def get_band_structure(self, path, modes=False, born=False, verbose=True):
"""Calculate and return the phonon band structure.
This method computes the phonon band structure for a given path
in reciprocal space. It is a wrapper around the internal
`band_structure` method of the `Phonons` class. The method can
optionally calculate and return phonon modes.
Parameters:
path : BandPath object
The BandPath object defining the path in the reciprocal
space over which the phonon band structure is calculated.
modes : bool, optional
If True, phonon modes will also be calculated and returned.
Defaults to False.
born : bool, optional
If True, includes the effect of Born effective charges in
the phonon calculations.
Defaults to False.
verbose : bool, optional
If True, enables verbose output during the calculation.
Defaults to True.
Returns:
BandStructure or tuple of (BandStructure, ndarray)
If `modes` is False, returns a `BandStructure` object
containing the phonon band structure. If `modes` is True,
returns a tuple, where the first element is the
`BandStructure` object and the second element is an ndarray
of phonon modes.
Example:
>>> from ase.dft.kpoints import BandPath
>>> path = BandPath(...) # Define the band path
>>> phonons = Phonons(...)
>>> bs, modes = phonons.get_band_structure(path, modes=True)
"""
result = self.band_structure(path.kpts,
modes=modes,
born=born,
verbose=verbose)
if modes:
omega_kl, omega_modes = result
else:
omega_kl = result
from ase.spectrum.band_structure import BandStructure
bs = BandStructure(path, energies=omega_kl[None])
# Return based on the modes flag
return (bs, omega_modes) if modes else bs
[docs] def compute_dynamical_matrix(self, q_scaled: np.ndarray, D_N: np.ndarray):
""" Computation of the dynamical matrix in momentum space D_ab(q).
This is a Fourier transform from real-space dynamical matrix D_N
for a given momentum vector q.
q_scaled: q vector in scaled coordinates.
D_N: the dynamical matrix in real-space. It is necessary, at least
currently, to provide this matrix explicitly (rather than use
self.D_N) because this matrix is modified by the Born charges
contributions and these modifications are momentum (q) dependent.
Result:
D(q): two-dimensional, complex-valued array of
shape=(3 * natoms, 3 * natoms).
"""
# Evaluate fourier sum
R_cN = self._lattice_vectors_array
phase_N = np.exp(-2.j * pi * np.dot(q_scaled, R_cN))
D_q = np.sum(phase_N[:, np.newaxis, np.newaxis] * D_N, axis=0)
return D_q
[docs] def band_structure(self, path_kc, modes=False, born=False, verbose=True):
"""Calculate phonon dispersion along a path in the Brillouin zone.
The dynamical matrix at arbitrary q-vectors is obtained by Fourier
transforming the real-space force constants. In case of negative
eigenvalues (squared frequency), the corresponding negative frequency
is returned.
Frequencies and modes are in units of eV and Ang/sqrt(amu),
respectively.
Parameters:
path_kc: ndarray
List of k-point coordinates (in units of the reciprocal lattice
vectors) specifying the path in the Brillouin zone for which the
dynamical matrix will be calculated.
modes: bool
Returns both frequencies and modes when True.
born: bool
Include non-analytic part given by the Born effective charges and
the static part of the high-frequency dielectric tensor. This
contribution to the force constant accounts for the splitting
between the LO and TO branches for q -> 0.
verbose: bool
Print warnings when imaginary frequncies are detected.
"""
assert self.D_N is not None
if born:
assert self.Z_avv is not None
assert self.eps_vv is not None
# Dynamical matrix in real-space
D_N = self.D_N
# Lists for frequencies and modes along path
omega_kl = []
u_kl = []
# Reciprocal basis vectors for use in non-analytic contribution
reci_vc = 2 * pi * la.inv(self.atoms.cell)
# Unit cell volume in Bohr^3
vol = abs(la.det(self.atoms.cell)) / units.Bohr**3
for q_c in path_kc:
# Add non-analytic part
if born:
# q-vector in cartesian coordinates
q_v = np.dot(reci_vc, q_c)
# Non-analytic contribution to force constants in atomic units
qdotZ_av = np.dot(q_v, self.Z_avv).ravel()
C_na = (4 * pi * np.outer(qdotZ_av, qdotZ_av) /
np.dot(q_v, np.dot(self.eps_vv, q_v)) / vol)
self.C_na = C_na / units.Bohr**2 * units.Hartree
# Add mass prefactor and convert to eV / (Ang^2 * amu)
M_inv = np.outer(self.m_inv_x, self.m_inv_x)
D_na = C_na * M_inv / units.Bohr**2 * units.Hartree
self.D_na = D_na
D_N = self.D_N + D_na / np.prod(self.supercell)
# if np.prod(self.N_c) == 1:
#
# q_av = np.tile(q_v, len(self.indices))
# q_xx = np.vstack([q_av]*len(self.indices)*3)
# D_m += q_xx
# Evaluate fourier sum
D_q = self.compute_dynamical_matrix(q_c, D_N)
if modes:
omega2_l, u_xl = la.eigh(D_q, UPLO='U')
# Sort eigenmodes according to eigenvalues (see below) and
# multiply with mass prefactor
u_lx = (self.m_inv_x[:, np.newaxis] *
u_xl[:, omega2_l.argsort()]).T.copy()
u_kl.append(u_lx.reshape((-1, len(self.indices), 3)))
else:
omega2_l = la.eigvalsh(D_q, UPLO='U')
# Sort eigenvalues in increasing order
omega2_l.sort()
# Use dtype=complex to handle negative eigenvalues
omega_l = np.sqrt(omega2_l.astype(complex))
# Take care of imaginary frequencies
if not np.all(omega2_l >= 0.):
indices = np.where(omega2_l < 0)[0]
if verbose:
print('WARNING, %i imaginary frequencies at '
'q = (% 5.2f, % 5.2f, % 5.2f) ; (omega_q =% 5.3e*i)'
% (len(indices), q_c[0], q_c[1], q_c[2],
omega_l[indices][0].imag))
omega_l[indices] = -1 * np.sqrt(np.abs(omega2_l[indices].real))
omega_kl.append(omega_l.real)
# Conversion factor: sqrt(eV / Ang^2 / amu) -> eV
s = units._hbar * 1e10 / sqrt(units._e * units._amu)
omega_kl = s * np.asarray(omega_kl)
if modes:
return omega_kl, np.asarray(u_kl)
return omega_kl
def get_dos(self, kpts=(10, 10, 10), npts=1000, delta=1e-3, indices=None):
from ase.spectrum.dosdata import RawDOSData
# dos = self.dos(kpts, npts, delta, indices)
kpts_kc = monkhorst_pack(kpts)
omega_w = self.band_structure(kpts_kc).ravel()
dos = RawDOSData(omega_w, np.ones_like(omega_w))
return dos
[docs] def dos(self, kpts=(10, 10, 10), npts=1000, delta=1e-3, indices=None):
"""Calculate phonon dos as a function of energy.
Parameters:
qpts: tuple
Shape of Monkhorst-Pack grid for sampling the Brillouin zone.
npts: int
Number of energy points.
delta: float
Broadening of Lorentzian line-shape in eV.
indices: list
If indices is not None, the atomic-partial dos for the specified
atoms will be calculated.
"""
# Monkhorst-Pack grid
kpts_kc = monkhorst_pack(kpts)
N = np.prod(kpts)
# Get frequencies
omega_kl = self.band_structure(kpts_kc)
# Energy axis and dos
omega_e = np.linspace(0., np.amax(omega_kl) + 5e-3, num=npts)
dos_e = np.zeros_like(omega_e)
# Sum up contribution from all q-points and branches
for omega_l in omega_kl:
diff_el = (omega_e[:, np.newaxis] - omega_l[np.newaxis, :])**2
dos_el = 1. / (diff_el + (0.5 * delta)**2)
dos_e += dos_el.sum(axis=1)
dos_e *= 1. / (N * pi) * 0.5 * delta
return omega_e, dos_e
[docs] def write_modes(self, q_c, branches=0, kT=units.kB * 300, born=False,
repeat=(1, 1, 1), nimages=30, center=False):
"""Write modes to trajectory file.
Parameters:
q_c: ndarray
q-vector of the modes.
branches: int or list
Branch index of modes.
kT: float
Temperature in units of eV. Determines the amplitude of the atomic
displacements in the modes.
born: bool
Include non-analytic contribution to the force constants at q -> 0.
repeat: tuple
Repeat atoms (l, m, n) times in the directions of the lattice
vectors. Displacements of atoms in repeated cells carry a Bloch
phase factor given by the q-vector and the cell lattice vector R_m.
nimages: int
Number of images in an oscillation.
center: bool
Center atoms in unit cell if True (default: False).
"""
if isinstance(branches, int):
branch_l = [branches]
else:
branch_l = list(branches)
# Calculate modes
omega_l, u_l = self.band_structure([q_c], modes=True, born=born)
# Repeat atoms
atoms = self.atoms * repeat
# Center
if center:
atoms.center()
# Here ``Na`` refers to a composite unit cell/atom dimension
pos_Nav = atoms.get_positions()
# Total number of unit cells
N = np.prod(repeat)
# Corresponding lattice vectors R_m
R_cN = np.indices(repeat).reshape(3, -1)
# Bloch phase
phase_N = np.exp(2.j * pi * np.dot(q_c, R_cN))
phase_Na = phase_N.repeat(len(self.atoms))
for lval in branch_l:
omega = omega_l[0, lval]
u_av = u_l[0, lval]
# Mean displacement of a classical oscillator at temperature T
u_av *= sqrt(kT) / abs(omega)
mode_av = np.zeros((len(self.atoms), 3), dtype=complex)
# Insert slice with atomic displacements for the included atoms
mode_av[self.indices] = u_av
# Repeat and multiply by Bloch phase factor
mode_Nav = np.vstack(N * [mode_av]) * phase_Na[:, np.newaxis]
with Trajectory('%s.mode.%d.traj'
% (self.name, lval), 'w') as traj:
for x in np.linspace(0, 2 * pi, nimages, endpoint=False):
atoms.set_positions((pos_Nav + np.exp(1.j * x) *
mode_Nav).real)
traj.write(atoms)