Source code for ase.phonons

from __future__ import print_function
"""Module for calculating phonons of periodic systems."""

import sys
import pickle
from math import 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

import ase.units as units
from ase.parallel import rank
from ase.dft import monkhorst_pack
from ase.io.trajectory import Trajectory
from ase.utils import opencew, pickleload, basestring


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 = np.arange(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], basestring):
            assert np.all([isinstance(atom, basestring) 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
        self.state = 'eq.pckl'
        filename = self.name + '.' + self.state

        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, protocol=2)
                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
                    self.state = '%d%s%s.pckl' % (a, 'xyz'[i], ' +-'[sign])
                    filename = self.name + '.' + self.state
                    # 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, protocol=2)
                        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)


[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(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
[docs] def check_eq_forces(self): """Check maximum size of forces in the equilibrium structure.""" fname = '%s.eq.pckl' % self.name feq_av = pickleload(open(fname, 'rb')) 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
[docs] def read_born_charges(self, name=None, neutrality=True): r"""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 with open(filename, 'rb') as fd: Z_avv, eps_vv = pickleload(fd) # 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 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 = pickleload(open(basename + '-.pckl', 'rb')) fplus_av = pickleload(open(basename + '+.pckl', 'rb')) 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
[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.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 cell i, j, k = 1 - np.asarray(self.N_c) % 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.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 ""
[docs] def get_force_constant(self): """Return matrix of force constants.""" assert self.C_N is not None return self.C_N
def get_band_structure(self, path, modes=False, born=False, verbose=True): omega_kl = self.band_structure(path.kpts, modes, born, verbose) if modes: assert 0 omega_kl, modes = omega_kl from ase.dft.band_structure import BandStructure bs = BandStructure(path, energies=omega_kl[None]) return bs
[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 # 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 get_dos(self, kpts=(10, 10, 10), npts=1000, delta=1e-3, indices=None): #dos = self.dos(kpts, npts, delta, indices) kpts_kc = monkhorst_pack(kpts) omega_w = self.band_structure(kpts_kc).ravel() from ase.dft.pdos import DOS dos = DOS(omega_w, np.ones_like(omega_w)[None]) 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 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()