/usr/lib/python2.7/dist-packages/ase/phonons.py is in python-ase 3.9.1.4567-3.
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"""Module for calculating phonons of periodic systems."""
import sys
import pickle
from math import sin, pi, sqrt
from os import remove
from os.path import isfile
import numpy as np
import numpy.linalg as la
import numpy.fft as fft
has_spglib = False
try:
from pyspglib import spglib
has_spglib = True
except ImportError:
pass
import ase.units as units
from ase.parallel import rank, barrier
from ase.dft import monkhorst_pack
from ase.io.trajectory import Trajectory
from ase.utils import opencew
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, refcell=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.
refcell: str
Reference cell in which the atoms will be displaced. If ``None``,
corner cell in supercell is used. If ``str``, cell in the center of
the supercell is used.
"""
# Store atoms and calculator
self.atoms = atoms
self.calc = calc
# Displace all atoms in the unit cell by default
self.indices = range(len(atoms))
self.name = name
self.delta = delta
self.N_c = supercell
# Reference cell offset
if refcell is None:
# Corner cell
self.offset = 0
else:
# Center cell
N_c = self.N_c
self.offset = N_c[0] // 2 * (N_c[1] * N_c[2]) + N_c[1] // \
2 * N_c[2] + N_c[2] // 2
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 lattice_vectors(self):
"""Return lattice vectors for cells in the supercell."""
# Lattice vectors relevative to the reference cell
R_cN = np.indices(self.N_c).reshape(3, -1)
N_c = np.array(self.N_c)[:, np.newaxis]
if self.offset == 0:
R_cN += N_c // 2
R_cN %= N_c
R_cN -= N_c // 2
return R_cN
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 .pckl), 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.N_c
# Set calculator if provided
assert self.calc is not None, "Provide calculator in __init__ method"
atoms_N.set_calculator(self.calc)
# Do calculation on equilibrium structure
filename = self.name + '.eq.pckl'
fd = opencew(filename)
if fd is not None:
# Call derived class implementation of __call__
output = self.__call__(atoms_N)
# Write output to file
if rank == 0:
pickle.dump(output, fd)
sys.stdout.write('Writing %s\n' % filename)
fd.close()
sys.stdout.flush()
# 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]:
# Filename for atomic displacement
filename = '%s.%d%s%s.pckl' % \
(self.name, a, 'xyz'[i], ' +-'[sign])
# Wait for ranks before checking for file
# barrier()
fd = opencew(filename)
if fd is None:
# Skip if already done
continue
# Update atomic positions
atoms_N.positions[offset + a, i] = \
pos[a, i] + sign * self.delta
# Call derived class implementation of __call__
output = self.__call__(atoms_N)
# Write output to file
if rank == 0:
pickle.dump(output, fd)
sys.stdout.write('Writing %s\n' % filename)
fd.close()
sys.stdout.flush()
# Return to initial positions
atoms_N.positions[offset + a, i] = pos[a, i]
def clean(self):
"""Delete generated pickle files."""
if isfile(self.name + '.eq.pckl'):
remove(self.name + '.eq.pckl')
for a in self.indices:
for i in 'xyz':
for sign in '-+':
name = '%s.%d%s%s.pckl' % (self.name, a, i, sign)
if isfile(name):
remove(name)
class Phonons(Displacement):
"""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.lattice import bulk
>>> from ase.phonons import Phonons
>>> from gpaw import GPAW, FermiDirac
>>> atoms = bulk('Si', 'diamond', a=5.4)
>>> calc = GPAW(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.keys():
kwargs['name'] = "phonon"
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
def __call__(self, atoms_N):
"""Calculate forces on atoms in supercell."""
# Calculate forces
forces = atoms_N.get_forces()
return forces
def check_eq_forces(self):
"""Check maximum size of forces in the equilibrium structure."""
fname = '%s.eq.pckl' % self.name
feq_av = pickle.load(open(fname))
fmin = feq_av.max()
fmax = feq_av.min()
i_min = np.where(feq_av == fmin)
i_max = np.where(feq_av == fmax)
return fmin, fmax, i_min, i_max
def read_born_charges(self, name=None, neutrality=True):
"""Read Born charges and dieletric tensor from pickle file.
The charge neutrality sum-rule::
_ _
\ a
) Z = 0
/__ ij
a
Parameters
----------
neutrality: bool
Restore charge neutrality condition on calculated Born effective
charges.
"""
# Load file with Born charges and dielectric tensor for atoms in the
# unit cell
if name is None:
filename = '%s.born.pckl' % self.name
else:
filename = name
fd = open(filename)
Z_avv, eps_vv = pickle.load(fd)
fd.close()
# 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
def read(self, method='Frederiksen', symmetrize=3, acoustic=True,
cutoff=None, born=False, **kwargs):
"""Read forces from pickle 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.N_c)
# 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)
fminus_av = pickle.load(open(basename + '-.pckl'))
fplus_av = pickle.load(open(basename + '+.pckl'))
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 i 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
def symmetrize(self, C_N):
"""Symmetrize force constant matrix."""
# Number of atoms
natoms = len(self.indices)
# Number of unit cells
N = np.prod(self.N_c)
# Reshape force constants to (l, m, n) cell indices
C_lmn = C_N.reshape(self.N_c + (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
i, j, k = np.asarray(self.N_c) % 2 - 1
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
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]
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.N_c)
# Lattice vectors
R_cN = self.lattice_vectors()
# 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
# print ""
def get_force_constant(self):
"""Return matrix of force constants."""
assert self.C_N is not None
return self.C_N
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.
Eigenvalues 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
# Lattice vectors -- ordered as illustrated in class docstring
R_cN = self.lattice_vectors()
# 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.N_c)
## 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
phase_N = np.exp(-2.j * pi * np.dot(q_c, R_cN))
D_q = np.sum(phase_N[:, np.newaxis, np.newaxis] * D_N, axis=0)
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 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
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 l in branch_l:
omega = omega_l[0, l]
u_av = u_l[0, l]
# 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]
traj = Trajectory('%s.mode.%d.traj' % (self.name, l), 'w')
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)
traj.close()
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