def set_log(self, log=None):
"""Set output log."""
if log is None:
self.timer = nulltimer
elif log == '-':
self.timer = StepTimer(name='elph')
else:
self.timer = StepTimer(name='elph', out=open(log, 'w'))
def set_log(self, log=None):
"""Set output log."""
if log is None:
self.timer = nulltimer
elif log == '-':
self.timer = StepTimer(name='elph')
else:
self.timer = StepTimer(name='elph', out=open(log, 'w'))
def get_M(self, modes, log=None, q=0):
"""Calculate el-ph coupling matrix for given modes(s).
XXX:
kwarg "q=0" is an ugly hack for k-points.
There shuold be loops over q!
Note that modes must be given as a dictionary with mode
frequencies in eV and corresponding mode vectors in units
of 1/sqrt(amu), where amu = 1.6605402e-27 Kg is an atomic mass unit.
In short frequencies and mode vectors must be given in ase units.
::
d d ~
< w | -- v | w' > = < w | -- v | w'>
dP dP
_
\ ~a d . ~a
+ ) < w | p > -- /_\H < p | w' >
/_ i dP ij j
a,ij
_
\ d ~a . ~a
+ ) < w | -- p > /_\H < p | w' >
/_ dP i ij j
a,ij
_
\ ~a . d ~a
+ ) < w | p > /_\H < -- p | w' >
/_ i ij dP j
a,ij
"""
if log is None:
timer = nulltimer
elif log == '-':
timer = StepTimer(name='EPCM')
else:
timer = StepTimer(name='EPCM', out=open(log, 'w'))
modes1 = modes.copy()
#convert to atomic units
amu = 1.6605402e-27 # atomic unit mass [Kg]
me = 9.1093897e-31 # electron mass [Kg]
modes = {}
for k in modes1.keys():
modes[k / Hartree] = modes1[k] / np.sqrt(amu / me)
dvt_Gx, ddH_aspx = self.get_gradient()
from gpaw import restart
atoms, calc = restart('eq.gpw', txt=None)
if calc.wfs.S_qMM is None:
calc.initialize(atoms)
calc.initialize_positions(atoms)
wfs = calc.wfs
nao = wfs.setups.nao
bfs = wfs.basis_functions
dtype = wfs.dtype
spin = 0 # XXX
M_lii = {}
timer.write_now('Starting gradient of pseudo part')
for f, mode in modes.items():
mo = []
M_ii = np.zeros((nao, nao), dtype)
for a in self.indices:
mo.append(mode[a])
mode = np.asarray(mo).flatten()
dvtdP_G = np.dot(dvt_Gx, mode)
bfs.calculate_potential_matrix(dvtdP_G, M_ii, q=q)
tri2full(M_ii, 'L')
M_lii[f] = M_ii
timer.write_now('Finished gradient of pseudo part')
P_aqMi = calc.wfs.P_aqMi
# Add the term
# _
# \ ~a d . ~a
# ) < w | p > -- /_\H < p | w' >
# /_ i dP ij j
# a,ij
Ma_lii = {}
for f, mode in modes.items():
Ma_lii[f] = np.zeros_like(M_lii.values()[0])
timer.write_now('Starting gradient of dH^a part')
for f, mode in modes.items():
mo = []
for a in self.indices:
mo.append(mode[a])
mode = np.asarray(mo).flatten()
for a, ddH_spx in ddH_aspx.items():
ddHdP_sp = np.dot(ddH_spx, mode)
ddHdP_ii = unpack2(ddHdP_sp[spin])
Ma_lii[f] += dots(P_aqMi[a][q], ddHdP_ii, P_aqMi[a][q].T)
timer.write_now('Finished gradient of dH^a part')
timer.write_now('Starting gradient of projectors part')
dP_aMix = self.get_dP_aMix(calc.spos_ac, wfs, q, timer)
timer.write_now('Finished gradient of projectors part')
dH_asp = pickle.load(open('v.eq.pckl', 'rb'))[1]
Mb_lii = {}
for f, mode in modes.items():
Mb_lii[f] = np.zeros_like(M_lii.values()[0])
for f, mode in modes.items():
for a, dP_Mix in dP_aMix.items():
dPdP_Mi = np.dot(dP_Mix, mode[a])
dH_ii = unpack2(dH_asp[a][spin])
dPdP_MM = dots(dPdP_Mi, dH_ii, P_aqMi[a][q].T)
Mb_lii[f] -= dPdP_MM + dPdP_MM.T
# XXX The minus sign here is quite subtle.
# It is related to how the derivative of projector
# functions in GPAW is calculated.
# More thorough explanations, anyone...?
# Units of M_lii are Hartree/(Bohr * sqrt(m_e))
for mode in M_lii.keys():
M_lii[mode] += Ma_lii[mode] + Mb_lii[mode]
# conversion to eV. The prefactor 1 / sqrt(hb^2 / 2 * hb * f)
# has units Bohr * sqrt(me)
M_lii_1 = M_lii.copy()
M_lii = {}
for f in M_lii_1.keys():
M_lii[f * Hartree] = M_lii_1[f] * Hartree / np.sqrt(2 * f)
return M_lii
def get_M(self, modes, log=None, q=0):
"""Calculate el-ph coupling matrix for given modes(s).
XXX:
kwarg "q=0" is an ugly hack for k-points.
There shuold be loops over q!
Note that modes must be given as a dictionary with mode
frequencies in eV and corresponding mode vectors in units
of 1/sqrt(amu), where amu = 1.6605402e-27 Kg is an atomic mass unit.
In short frequencies and mode vectors must be given in ase units.
::
d d ~
< w | -- v | w' > = < w | -- v | w'>
dP dP
_
\ ~a d . ~a
+ ) < w | p > -- /_\H < p | w' >
/_ i dP ij j
a,ij
_
\ d ~a . ~a
+ ) < w | -- p > /_\H < p | w' >
/_ dP i ij j
a,ij
_
\ ~a . d ~a
+ ) < w | p > /_\H < -- p | w' >
/_ i ij dP j
a,ij
"""
if log is None:
timer = nulltimer
elif log == '-':
timer = StepTimer(name='EPCM')
else:
timer = StepTimer(name='EPCM', out=open(log, 'w'))
modes1 = modes.copy()
#convert to atomic units
amu = 1.6605402e-27 # atomic unit mass [Kg]
me = 9.1093897e-31 # electron mass [Kg]
modes = {}
for k in modes1.keys():
modes[k / Hartree] = modes1[k] / np.sqrt(amu / me)
dvt_Gx, ddH_aspx = self.get_gradient()
from gpaw import restart
atoms, calc = restart('eq.gpw', txt=None)
spos_ac = atoms.get_scaled_positions()
if calc.wfs.S_qMM is None:
calc.initialize(atoms)
calc.initialize_positions(atoms)
wfs = calc.wfs
nao = wfs.setups.nao
bfs = wfs.basis_functions
dtype = wfs.dtype
spin = 0 # XXX
M_lii = {}
timer.write_now('Starting gradient of pseudo part')
for f, mode in modes.items():
mo = []
M_ii = np.zeros((nao, nao), dtype)
for a in self.indices:
mo.append(mode[a])
mode = np.asarray(mo).flatten()
dvtdP_G = np.dot(dvt_Gx, mode)
bfs.calculate_potential_matrix(dvtdP_G, M_ii, q=q)
tri2full(M_ii, 'L')
M_lii[f] = M_ii
timer.write_now('Finished gradient of pseudo part')
P_aqMi = calc.wfs.P_aqMi
# Add the term
# _
# \ ~a d . ~a
# ) < w | p > -- /_\H < p | w' >
# /_ i dP ij j
# a,ij
Ma_lii = {}
for f, mode in modes.items():
Ma_lii[f] = np.zeros_like(M_lii.values()[0])
timer.write_now('Starting gradient of dH^a part')
for f, mode in modes.items():
mo = []
for a in self.indices:
mo.append(mode[a])
mode = np.asarray(mo).flatten()
for a, ddH_spx in ddH_aspx.items():
ddHdP_sp = np.dot(ddH_spx, mode)
ddHdP_ii = unpack2(ddHdP_sp[spin])
Ma_lii[f] += dots(P_aqMi[a][q], ddHdP_ii, P_aqMi[a][q].T)
timer.write_now('Finished gradient of dH^a part')
timer.write_now('Starting gradient of projectors part')
spos_ac = self.atoms.get_scaled_positions() % 1.0
dP_aMix = self.get_dP_aMix(spos_ac, wfs, q, timer)
timer.write_now('Finished gradient of projectors part')
dH_asp = pickle.load(open('v.eq.pckl'))[1]
Mb_lii = {}
for f, mode in modes.items():
Mb_lii[f] = np.zeros_like(M_lii.values()[0])
for f, mode in modes.items():
for a, dP_Mix in dP_aMix.items():
dPdP_Mi = np.dot(dP_Mix, mode[a])
dH_ii = unpack2(dH_asp[a][spin])
dPdP_MM = dots(dPdP_Mi, dH_ii, P_aqMi[a][q].T)
Mb_lii[f] -= dPdP_MM + dPdP_MM.T
# XXX The minus sign here is quite subtle.
# It is related to how the derivative of projector
# functions in GPAW is calculated.
# More thorough explanations, anyone...?
# Units of M_lii are Hartree/(Bohr * sqrt(m_e))
for mode in M_lii.keys():
M_lii[mode] += Ma_lii[mode] + Mb_lii[mode]
# conversion to eV. The prefactor 1 / sqrt(hb^2 / 2 * hb * f)
# has units Bohr * sqrt(me)
M_lii_1 = M_lii.copy()
M_lii = {}
for f in M_lii_1.keys():
M_lii[f * Hartree] = M_lii_1[f] * Hartree / np.sqrt(2 * f)
return M_lii
class ElectronPhononCoupling(Displacement):
"""Class for calculating the electron-phonon coupling in an LCAO basis.
The derivative of the effective potential wrt atomic displacements is
obtained from a finite difference approximation to the derivative by doing
a self-consistent calculation for atomic displacements in the +/-
directions. These calculations are carried out in the ``run`` member
function.
The subsequent calculation of the coupling matrix in the basis of atomic
orbitals (or Bloch-sums hereof for periodic systems) is handled by the
``calculate_matrix`` member function.
"""
def __init__(self,
atoms,
calc=None,
supercell=(1, 1, 1),
name='elph',
delta=0.01,
phonons=None):
"""Initialize with base class args and kwargs.
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
Name to use for files (default: 'elph').
delta: float
Magnitude of displacements.
phonons: Phonons object
If provided, the ``__call__`` member function of the ``Phonons``
class in ASE will be called in the ``__call__`` member function of
this class.
"""
# Init base class and make the center cell in the supercell the
# reference cell
Displacement.__init__(self,
atoms,
calc=calc,
supercell=supercell,
name=name,
delta=delta,
refcell='center')
# Store ``Phonons`` object in attribute
self.phonons = phonons
# Log
self.set_log()
# LCAO calculator
self.calc_lcao = None
# Supercell matrix
self.g_xNNMM = None
def __call__(self, atoms_N):
"""Extract effective potential and projector coefficients."""
# Do calculation
atoms_N.get_potential_energy()
# Call phonons class if provided
if self.phonons is not None:
forces = self.phonons.__call__(atoms_N)
# Get calculator
calc = atoms_N.get_calculator()
# Effective potential (in Hartree) and projector coefficients
Vt_G = calc.hamiltonian.vt_sG[0]
Vt_G = calc.wfs.gd.collect(Vt_G, broadcast=True)
dH_asp = calc.hamiltonian.dH_asp
setups = calc.wfs.setups
nspins = calc.wfs.nspins
gd_comm = calc.wfs.gd.comm
dH_all_asp = {}
for a, setup in enumerate(setups):
ni = setup.ni
nii = ni * (ni + 1) // 2
dH_tmp_sp = np.zeros((nspins, nii))
if a in dH_asp:
dH_tmp_sp[:] = dH_asp[a]
gd_comm.sum(dH_tmp_sp)
dH_all_asp[a] = dH_tmp_sp
return Vt_G, dH_all_asp
def set_lcao_calculator(self, calc):
"""Set LCAO calculator for the calculation of the supercell matrix."""
# Add parameter checks here
# - check that gamma
# - check that no symmetries are used
# - ...
assert calc.input_parameters['mode'] == 'lcao', "LCAO mode required."
symmetry = calc.input_parameters['symmetry']
assert symmetry['point_group'] != True, "Symmetries not supported."
self.calc_lcao = calc
def set_basis_info(self, *args):
"""Store lcao basis info for atoms in reference cell in attribute.
Parameters
----------
args: tuple
If the LCAO calculator is not available (e.g. if the supercell is
loaded from file), the ``load_supercell_matrix`` member function
provides the required info as arguments.
"""
if len(args) == 0:
calc = self.calc_lcao
setups = calc.wfs.setups
bfs = calc.wfs.basis_functions
nao_a = [setups[a].nao for a in range(len(self.atoms))]
M_a = [bfs.M_a[a] for a in range(len(self.atoms))]
else:
M_a = args[0]
nao_a = args[1]
self.basis_info = {'M_a': M_a, 'nao_a': nao_a}
def set_log(self, log=None):
"""Set output log."""
if log is None:
self.timer = nulltimer
elif log == '-':
self.timer = StepTimer(name='elph')
else:
self.timer = StepTimer(name='elph', out=open(log, 'w'))
def calculate_supercell_matrix(self,
dump=0,
name=None,
filter=None,
include_pseudo=True,
atoms=None):
"""Calculate matrix elements of the el-ph coupling in the LCAO basis.
This function calculates the matrix elements between LCAOs and local
atomic gradients of the effective potential. The matrix elements are
calculated for the supercell used to obtain finite-difference
approximations to the derivatives of the effective potential wrt to
atomic displacements.
Parameters
----------
dump: int
Dump supercell matrix to pickle file (default: 0).
0: Supercell matrix not saved
1: Supercell matrix saved in a single pickle file.
2: Dump matrix for different gradients in separate files. Useful
for large systems where the total array gets too large for a
single pickle file.
name: string
User specified name of the generated pickle file(s). If not
provided, the string in the ``name`` attribute is used.
filter: str
Fourier filter atomic gradients of the effective potential. The
specified components (``normal`` or ``umklapp``) are removed
(default: None).
include_pseudo: bool
Include the contribution from the psedupotential in the atomic
gradients. If ``False``, only the gradient of the effective
potential is included (default: True).
atoms: Atoms object
Calculate supercell for an ``Atoms`` object different from the one
provided in the ``__init__`` method (WARNING, NOT working!).
"""
assert self.calc_lcao is not None, "Set LCAO calculator"
# Supercell atoms
if atoms is None:
atoms_N = self.atoms * self.N_c
else:
atoms_N = atoms
# Initialize calculator if required and extract useful quantities
calc = self.calc_lcao
if not hasattr(calc.wfs, 'S_qMM'):
calc.initialize(atoms_N)
calc.initialize_positions(atoms_N)
self.set_basis_info()
basis = calc.input_parameters['basis']
# Extract useful objects from the calculator
wfs = calc.wfs
gd = calc.wfs.gd
kd = calc.wfs.kd
kpt_u = wfs.kpt_u
setups = wfs.setups
nao = setups.nao
bfs = wfs.basis_functions
dtype = wfs.dtype
spin = 0 # XXX
# If gamma calculation, overlap with neighboring cell cannot be removed
if kd.gamma:
print("WARNING: Gamma-point calculation.")
else:
# Bloch to real-space converter
tb = TightBinding(atoms_N, calc)
self.timer.write_now("Calculating supercell matrix")
self.timer.write_now("Calculating real-space gradients")
# Calculate finite-difference gradients (in Hartree / Bohr)
V1t_xG, dH1_xasp = self.calculate_gradient()
self.timer.write_now("Finished real-space gradients")
# Fourier filter the atomic gradients of the effective potential
if filter is not None:
self.timer.write_now("Fourier filtering gradients")
V1_xG = V1t_xG.copy()
self.fourier_filter(V1t_xG, components=filter)
self.timer.write_now("Finished Fourier filtering")
# For the contribution from the derivative of the projectors
dP_aqvMi = self.calculate_dP_aqvMi(wfs)
# Equilibrium atomic Hamiltonian matrix (projector coefficients)
dH_asp = pickle.load(open(self.name + '.eq.pckl'))[1]
# Check that the grid is the same as in the calculator
assert np.all(V1t_xG.shape[-3:] == (gd.N_c + gd.pbc_c - 1)), \
"Mismatch in grids."
# Calculate < i k | grad H | j k >, i.e. matrix elements in Bloch basis
# List for supercell matrices;
g_xNNMM = []
self.timer.write_now("Calculating gradient of PAW Hamiltonian")
# Do each cartesian component separately
for i, a in enumerate(self.indices):
for v in range(3):
# Corresponding array index
x = 3 * i + v
V1t_G = V1t_xG[x]
self.timer.write_now("%s-gradient of atom %u" %
(['x', 'y', 'z'][v], a))
# Array for different k-point components
g_qMM = np.zeros((len(kpt_u), nao, nao), dtype)
# 1) Gradient of effective potential
self.timer.write_now(
"Starting gradient of effective potential")
for kpt in kpt_u:
# Matrix elements
geff_MM = np.zeros((nao, nao), dtype)
bfs.calculate_potential_matrix(V1t_G, geff_MM, q=kpt.q)
tri2full(geff_MM, 'L')
# Insert in array
g_qMM[kpt.q] += geff_MM
self.timer.write_now(
"Finished gradient of effective potential")
if include_pseudo:
self.timer.write_now("Starting gradient of pseudo part")
# 2) Gradient of non-local part (projectors)
self.timer.write_now("Starting gradient of dH^a")
P_aqMi = calc.wfs.P_aqMi
# 2a) dH^a part has contributions from all other atoms
for kpt in kpt_u:
# Matrix elements
gp_MM = np.zeros((nao, nao), dtype)
dH1_asp = dH1_xasp[x]
for a_, dH1_sp in dH1_asp.items():
dH1_ii = unpack2(dH1_sp[spin])
gp_MM += np.dot(
P_aqMi[a_][kpt.q],
np.dot(dH1_ii,
P_aqMi[a_][kpt.q].T.conjugate()))
g_qMM[kpt.q] += gp_MM
self.timer.write_now("Finished gradient of dH^a")
self.timer.write_now("Starting gradient of projectors")
# 2b) dP^a part has only contributions from the same atoms
dP_qvMi = dP_aqvMi[a]
dH_ii = unpack2(dH_asp[a][spin])
for kpt in kpt_u:
#XXX Sort out the sign here; conclusion -> sign = +1 !
P1HP_MM = +1 * np.dot(
dP_qvMi[kpt.q][v],
np.dot(dH_ii, P_aqMi[a][kpt.q].T.conjugate()))
# Matrix elements
gp_MM = P1HP_MM + P1HP_MM.T.conjugate()
g_qMM[kpt.q] += gp_MM
self.timer.write_now("Finished gradient of projectors")
self.timer.write_now("Finished gradient of pseudo part")
# Extract R_c=(0, 0, 0) block by Fourier transforming
if kd.gamma or kd.N_c is None:
g_MM = g_qMM[0]
else:
# Convert to array
g_MM = tb.bloch_to_real_space(g_qMM, R_c=(0, 0, 0))[0]
# Reshape to global unit cell indices
N = np.prod(self.N_c)
# Number of basis function in the primitive cell
assert (nao % N) == 0, "Alarm ...!"
nao_cell = nao / N
g_NMNM = g_MM.reshape((N, nao_cell, N, nao_cell))
g_NNMM = g_NMNM.swapaxes(1, 2).copy()
self.timer.write_now("Finished supercell matrix")
if dump != 2:
g_xNNMM.append(g_NNMM)
else:
if name is not None:
fname = '%s.supercell_matrix_x_%2.2u.%s.pckl' % (
name, x, basis)
else:
fname = self.name + \
'.supercell_matrix_x_%2.2u.%s.pckl' % (x, basis)
if kd.comm.rank == 0:
fd = open(fname, 'w')
M_a = self.basis_info['M_a']
nao_a = self.basis_info['nao_a']
pickle.dump((g_NNMM, M_a, nao_a), fd, 2)
fd.close()
self.timer.write_now("Finished gradient of PAW Hamiltonian")
if dump != 2:
# Collect gradients in one array
self.g_xNNMM = np.array(g_xNNMM)
# Dump to pickle file using binary mode together with basis info
if dump and kd.comm.rank == 0:
if name is not None:
fname = '%s.supercell_matrix.%s.pckl' % (name, basis)
else:
fname = self.name + '.supercell_matrix.%s.pckl' % basis
fd = open(fname, 'w')
M_a = self.basis_info['M_a']
nao_a = self.basis_info['nao_a']
pickle.dump((self.g_xNNMM, M_a, nao_a), fd, 2)
fd.close()
def load_supercell_matrix(self, basis=None, name=None, multiple=False):
"""Load supercell matrix from pickle file.
Parameters
----------
basis: string
String specifying the LCAO basis used to calculate the supercell
matrix, e.g. 'dz(dzp)'.
name: string
User specified name of the pickle file.
multiple: bool
Load each derivative from individual files.
"""
assert (basis is not None) or (name is not None), \
"Provide basis or name."
if self.g_xNNMM is not None:
self.g_xNNMM = None
if not multiple:
# File name
if name is not None:
fname = name
else:
fname = self.name + '.supercell_matrix.%s.pckl' % basis
fd = open(fname)
self.g_xNNMM, M_a, nao_a = pickle.load(fd)
fd.close()
else:
g_xNNMM = []
for x in range(len(self.indices) * 3):
if name is not None:
fname = name
else:
fname = self.name + \
'.supercell_matrix_x_%2.2u.%s.pckl' % (x, basis)
fd = open(fname, 'r')
g_NNMM, M_a, nao_a = pickle.load(fd)
fd.close()
g_xNNMM.append(g_NNMM)
self.g_xNNMM = np.array(g_xNNMM)
self.set_basis_info(M_a, nao_a)
def apply_cutoff(self, cutmax=None, cutmin=None):
"""Zero matrix element inside/beyond the specified cutoffs.
Parameters
----------
cutmax: float
Zero matrix elements for basis functions with a distance to the
atomic gradient that is larger than the cutoff.
cutmin: float
Zero matrix elements where both basis functions have distances to
the atomic gradient that is smaller than the cutoff.
"""
if cutmax is not None:
cutmax = float(cutmax)
if cutmin is not None:
cutmin = float(cutmin)
# Reference to supercell matrix attribute
g_xNNMM = self.g_xNNMM
# Number of atoms and primitive cells
N_atoms = len(self.indices)
N = np.prod(self.N_c)
nao = g_xNNMM.shape[-1]
# Reshape array
g_avNNMM = g_xNNMM.reshape(N_atoms, 3, N, N, nao, nao)
# Make slices for orbitals on atoms
M_a = self.basis_info['M_a']
nao_a = self.basis_info['nao_a']
slice_a = []
for a in range(len(self.atoms)):
start = M_a[a]
stop = start + nao_a[a]
s = slice(start, stop)
slice_a.append(s)
# Lattice vectors
R_cN = self.lattice_vectors()
# Unit cell vectors
cell_vc = self.atoms.cell.transpose()
# Atomic positions in reference cell
pos_av = self.atoms.get_positions()
# Create a mask array to zero the relevant matrix elements
if cutmin is not None:
mask_avNNMM = np.zeros(g_avNNMM.shape, dtype=bool)
# Zero elements where one of the basis orbitals has a distance to atoms
# (atomic gradients) 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
for i, a in enumerate(self.indices):
# Atomic distances wrt to the position of the gradient
dist_a = np.sqrt(np.sum((pos_av[a] - posn_av)**2, axis=-1))
if cutmax is not None:
# Atoms indices where the distance is larger than the max
# cufoff
j_a = np.where(dist_a > cutmax)[0]
# Zero elements
for j in j_a:
g_avNNMM[a, :, n, :, slice_a[j], :] = 0.0
g_avNNMM[a, :, :, n, :, slice_a[j]] = 0.0
if cutmin is not None:
# Atoms indices where the distance is larger than the min
# cufoff
j_a = np.where(dist_a > cutmin)[0]
# Update mask to keep elements where one LCAO is outside
# the min cutoff
for j in j_a:
mask_avNNMM[a, :, n, :, slice_a[j], :] = True
mask_avNNMM[a, :, :, n, :, slice_a[j]] = True
# Zero elements where both LCAOs are located within the min cutoff
if cutmin is not None:
g_avNNMM[~mask_avNNMM] = 0.0
def lcao_matrix(self, u_l, omega_l):
"""Calculate the el-ph coupling in the electronic LCAO basis.
For now, only works for Gamma-point phonons.
Parameters
----------
u_l: ndarray
Mass-scaled polarization vectors (in units of 1 / sqrt(amu)) of the
phonons.
omega_l: ndarray
Vibrational frequencies in eV.
"""
# Supercell matrix (Hartree / Bohr)
assert self.g_xNNMM is not None, "Load supercell matrix."
assert self.g_xNNMM.shape[1:3] == (1, 1)
g_xMM = self.g_xNNMM[:, 0, 0, :, :]
# Number of atomic orbitals
nao = g_xMM.shape[-1]
# Number of phonon modes
nmodes = u_l.shape[0]
#
u_lx = u_l.reshape(nmodes, 3 * len(self.atoms))
g_lMM = np.dot(u_lx, g_xMM.transpose(1, 0, 2))
# Multiply prefactor sqrt(hbar / 2 * M * omega) in units of Bohr
amu = units._amu # atomic mass unit
me = units._me # electron mass
g_lMM /= np.sqrt(2 * amu / me / units.Hartree * \
omega_l[:, np.newaxis, np.newaxis])
# Convert to eV
g_lMM *= units.Hartree
return g_lMM
def bloch_matrix(self,
kpts,
qpts,
c_kn,
u_ql,
omega_ql=None,
kpts_from=None):
"""Calculate el-ph coupling in the Bloch basis for the electrons.
This function calculates the electron-phonon coupling between the
specified Bloch states, i.e.::
______
mnl / hbar ^
g = /------- < m k + q | e . grad V | n k >
kq \/ 2 M w ql q
ql
In case the ``omega_ql`` keyword argument is not given, the bare matrix
element (in units of eV / Ang) without the sqrt prefactor is returned.
Parameters
----------
kpts: ndarray or tuple.
k-vectors of the Bloch states. When a tuple of integers is given, a
Monkhorst-Pack grid with the specified number of k-points along the
directions of the reciprocal lattice vectors is generated.
qpts: ndarray or tuple.
q-vectors of the phonons.
c_kn: ndarray
Expansion coefficients for the Bloch states. The ordering must be
the same as in the ``kpts`` argument.
u_ql: ndarray
Mass-scaled polarization vectors (in units of 1 / sqrt(amu)) of the
phonons. Again, the ordering must be the same as in the
corresponding ``qpts`` argument.
omega_ql: ndarray
Vibrational frequencies in eV.
kpts_from: list of ints or int
Calculate only the matrix element for the k-vectors specified by
their index in the ``kpts`` argument (default: all).
In short, phonon frequencies and mode vectors must be given in ase units.
"""
assert self.g_xNNMM is not None, "Load supercell matrix."
assert len(c_kn.shape) == 3
assert len(u_ql.shape) == 4
if omega_ql is not None:
assert np.all(u_ql.shape[:2] == omega_ql.shape[:2])
# Translate k-points into 1. BZ (required by ``find_k_plus_q``` member
# function of the ```KPointDescriptor``).
if isinstance(kpts, np.ndarray):
assert kpts.shape[1] == 3, "kpts_kc array must be given"
# XXX This does not seem to cause problems!
kpts -= kpts.round()
# Use the KPointDescriptor to keep track of the k and q-vectors
kd_kpts = KPointDescriptor(kpts)
kd_qpts = KPointDescriptor(qpts)
# Check that number of k- and q-points agree with the number of Bloch
# functions and polarization vectors
assert kd_kpts.nbzkpts == len(c_kn)
assert kd_qpts.nbzkpts == len(u_ql)
# Include all k-point per default
if kpts_from is None:
kpts_kc = kd_kpts.bzk_kc
kpts_k = range(kd_kpts.nbzkpts)
else:
kpts_kc = kd_kpts.bzk_kc[kpts_from]
if isinstance(kpts_from, int):
kpts_k = list([kpts_from])
else:
kpts_k = list(kpts_from)
# Supercell matrix (real matrix in Hartree / Bohr)
g_xNNMM = self.g_xNNMM
# Number of phonon modes and electronic bands
nmodes = u_ql.shape[1]
nbands = c_kn.shape[1]
# Number of atoms displacements and basis functions
ndisp = np.prod(u_ql.shape[2:])
assert ndisp == (3 * len(self.indices))
nao = c_kn.shape[2]
assert ndisp == g_xNNMM.shape[0]
assert nao == g_xNNMM.shape[-1]
# Lattice vectors
R_cN = self.lattice_vectors()
# Number of unit cell in supercell
N = np.prod(self.N_c)
# Allocate array for couplings
g_qklnn = np.zeros(
(kd_qpts.nbzkpts, len(kpts_kc), nmodes, nbands, nbands),
dtype=complex)
self.timer.write_now("Calculating coupling matrix elements")
for q, q_c in enumerate(kd_qpts.bzk_kc):
# Find indices of k+q for the k-points
kplusq_k = kd_kpts.find_k_plus_q(q_c, kpts_k=kpts_k)
# Here, ``i`` is counting from 0 and ``k`` is the global index of
# the k-point
for i, (k, k_c) in enumerate(zip(kpts_k, kpts_kc)):
# Check the wave vectors (adapted to the ``KPointDescriptor`` class)
kplusq_c = k_c + q_c
kplusq_c -= kplusq_c.round()
assert np.allclose(kplusq_c, kd_kpts.bzk_kc[kplusq_k[i]] ), \
(i, k, k_c, q_c, kd_kpts.bzk_kc[kplusq_k[i]])
# Allocate array
g_xMM = np.zeros((ndisp, nao, nao), dtype=complex)
# Multiply phase factors
for m in range(N):
for n in range(N):
Rm_c = R_cN[:, m]
Rn_c = R_cN[:, n]
phase = np.exp(
2.j * pi *
(np.dot(k_c, Rm_c - Rn_c) + np.dot(q_c, Rm_c)))
# Sum contributions from different cells
g_xMM += g_xNNMM[:, m, n, :, :] * phase
# LCAO coefficient for Bloch states
ck_nM = c_kn[k]
ckplusq_nM = c_kn[kplusq_k[i]]
# Mass scaled polarization vectors
u_lx = u_ql[q].reshape(nmodes, 3 * len(self.atoms))
g_nxn = np.dot(ckplusq_nM.conj(), np.dot(g_xMM, ck_nM.T))
g_lnn = np.dot(u_lx, g_nxn)
# Insert value
g_qklnn[q, i] = g_lnn
# XXX Temp
if np.all(q_c == 0.0):
# These should be real
print(g_qklnn[q].imag.min(), g_qklnn[q].imag.max())
self.timer.write_now(
"Finished calculation of coupling matrix elements")
# Return the bare matrix element if frequencies are not given
if omega_ql is None:
# Convert to eV / Ang
g_qklnn *= units.Hartree / units.Bohr
else:
# Multiply prefactor sqrt(hbar / 2 * M * omega) in units of Bohr
amu = units._amu # atomic mass unit
me = units._me # electron mass
g_qklnn /= np.sqrt(2 * amu / me / units.Hartree * \
omega_ql[:, np.newaxis, :, np.newaxis, np.newaxis])
# Convert to eV
g_qklnn *= units.Hartree
# Return couplings in eV (or eV / Ang)
return g_qklnn
def fourier_filter(self, V1t_xG, components='normal', criteria=1):
"""Fourier filter atomic gradients of the effective potential.
Parameters
----------
V1t_xG: ndarray
Array representation of atomic gradients of the effective potential
in the supercell grid.
components: str
Fourier components to filter out (``normal`` or ``umklapp``).
"""
assert components in ['normal', 'umklapp']
# Grid shape
shape = V1t_xG.shape[-3:]
# Primitive unit cells in Bohr/Bohr^-1
cell_cv = self.atoms.get_cell() / units.Bohr
reci_vc = 2 * pi * la.inv(cell_cv)
norm_c = np.sqrt(np.sum(reci_vc**2, axis=0))
# Periodic BC array
pbc_c = np.array(self.atoms.get_pbc(), dtype=bool)
# Supercell atoms and cell
atoms_N = self.atoms * self.N_c
supercell_cv = atoms_N.get_cell() / units.Bohr
# q-grid in units of the grid spacing (FFT ordering)
q_cG = np.indices(shape).reshape(3, -1)
q_c = np.array(shape)[:, np.newaxis]
q_cG += q_c // 2
q_cG %= q_c
q_cG -= q_c // 2
# Locate q-points inside the Brillouin zone
if criteria == 0:
# Works for all cases
# Grid spacing in direction of reciprocal lattice vectors
h_c = np.sqrt(np.sum((2 * pi * la.inv(supercell_cv))**2, axis=0))
# XXX Why does a "*=" operation on q_cG not work here ??
q1_cG = q_cG * h_c[:, np.newaxis] / (norm_c[:, np.newaxis] / 2)
mask_G = np.ones(np.prod(shape), dtype=bool)
for i, pbc in enumerate(pbc_c):
if not pbc:
continue
mask_G &= (-1. < q1_cG[i]) & (q1_cG[i] <= 1.)
else:
# 2D hexagonal lattice
# Projection of q points onto the periodic directions. Only in
# these directions do normal and umklapp processees make sense.
q_vG = np.dot(q_cG[pbc_c].T,
2 * pi * la.inv(supercell_cv).T[pbc_c]).T.copy()
# Parametrize the BZ boundary in terms of the angle theta
theta_G = np.arctan2(q_vG[1], q_vG[0]) % (pi / 3)
phi_G = pi / 6 - np.abs(theta_G)
qmax_G = norm_c[0] / 2 / np.cos(phi_G)
norm_G = np.sqrt(np.sum(q_vG**2, axis=0))
# Includes point on BZ boundary with +1e-2
mask_G = (norm_G <= qmax_G + 1e-2
) # & (q_vG[1] < (norm_c[0] / 2 - 1e-3))
if components != 'normal':
mask_G = ~mask_G
# Reshape to grid shape
mask_G.shape = shape
for V1t_G in V1t_xG:
# Fourier transform atomic gradient
V1tq_G = fft.fftn(V1t_G)
# Zero normal/umklapp components
V1tq_G[mask_G] = 0.0
# Fourier transform back
V1t_G[:] = fft.ifftn(V1tq_G).real
def calculate_gradient(self):
"""Calculate gradient of effective potential and projector coefs.
This function loads the generated pickle files and calculates
finite-difference derivatives.
"""
# Array and dict for finite difference derivatives
V1t_xG = []
dH1_xasp = []
x = 0
for a in self.indices:
for v in range(3):
name = '%s.%d%s' % (self.name, a, 'xyz'[v])
# Potential and atomic density matrix for atomic displacement
try:
Vtm_G, dHm_asp = pickle.load(open(name + '-.pckl'))
Vtp_G, dHp_asp = pickle.load(open(name + '+.pckl'))
except (IOError, EOFError):
raise IOError, "%s(-/+).pckl" % name
# FD derivatives in Hartree / Bohr
V1t_G = (Vtp_G - Vtm_G) / (2 * self.delta / units.Bohr)
V1t_xG.append(V1t_G)
dH1_asp = {}
for atom in dHm_asp.keys():
dH1_asp[atom] = (dHp_asp[atom] - dHm_asp[atom]) / \
(2 * self.delta / units.Bohr)
dH1_xasp.append(dH1_asp)
x += 1
return np.array(V1t_xG), dH1_xasp
def calculate_dP_aqvMi(self, wfs):
"""Overlap between LCAO basis functions and gradient of projectors.
Only the gradient wrt the atomic positions in the reference cell is
computed.
"""
nao = wfs.setups.nao
nq = len(wfs.ibzk_qc)
atoms = [self.atoms[i] for i in self.indices]
# Derivatives in reference cell
dP_aqvMi = {}
for atom, setup in zip(atoms, wfs.setups):
a = atom.index
dP_aqvMi[a] = np.zeros((nq, 3, nao, setup.ni), wfs.dtype)
# Calculate overlap between basis function and gradient of projectors
# NOTE: the derivative is calculated wrt the atomic position and not
# the real-space coordinate
calc = TwoCenterIntegralCalculator(wfs.ibzk_qc, derivative=True)
expansions = ManySiteDictionaryWrapper(wfs.tci.P_expansions, dP_aqvMi)
calc.calculate(wfs.tci.atompairs, [expansions], [dP_aqvMi])
# Extract derivatives in the reference unit cell
# dP_aqvMi = {}
# for atom in self.atoms:
# dP_aqvMi[atom.index] = dPall_aqvMi[atom.index]
return dP_aqvMi
class ElectronPhononCoupling(Displacement):
"""Class for calculating the electron-phonon coupling in an LCAO basis.
The derivative of the effective potential wrt atomic displacements is
obtained from a finite difference approximation to the derivative by doing
a self-consistent calculation for atomic displacements in the +/-
directions. These calculations are carried out in the ``run`` member
function.
The subsequent calculation of the coupling matrix in the basis of atomic
orbitals (or Bloch-sums hereof for periodic systems) is handled by the
``calculate_matrix`` member function.
"""
def __init__(self, atoms, calc=None, supercell=(1, 1, 1), name='elph',
delta=0.01, phonons=None):
"""Initialize with base class args and kwargs.
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
Name to use for files (default: 'elph').
delta: float
Magnitude of displacements.
phonons: Phonons object
If provided, the ``__call__`` member function of the ``Phonons``
class in ASE will be called in the ``__call__`` member function of
this class.
"""
# Init base class and make the center cell in the supercell the
# reference cell
Displacement.__init__(self, atoms, calc=calc, supercell=supercell,
name=name, delta=delta, refcell='center')
# Store ``Phonons`` object in attribute
self.phonons = phonons
# Log
self.set_log()
# LCAO calculator
self.calc_lcao = None
# Supercell matrix
self.g_xNNMM = None
def __call__(self, atoms_N):
"""Extract effective potential and projector coefficients."""
# Do calculation
atoms_N.get_potential_energy()
# Call phonons class if provided
if self.phonons is not None:
forces = self.phonons.__call__(atoms_N)
# Get calculator
calc = atoms_N.get_calculator()
# Effective potential (in Hartree) and projector coefficients
Vt_G = calc.hamiltonian.vt_sG[0]
Vt_G = calc.wfs.gd.collect(Vt_G, broadcast=True)
dH_asp = calc.hamiltonian.dH_asp
setups = calc.wfs.setups
nspins = calc.wfs.nspins
gd_comm = calc.wfs.gd.comm
dH_all_asp = {}
for a, setup in enumerate(setups):
ni = setup.ni
nii = ni * (ni + 1) // 2
dH_tmp_sp = np.zeros((nspins, nii))
if a in dH_asp:
dH_tmp_sp[:] = dH_asp[a]
gd_comm.sum(dH_tmp_sp)
dH_all_asp[a] = dH_tmp_sp
return Vt_G, dH_all_asp
def set_lcao_calculator(self, calc):
"""Set LCAO calculator for the calculation of the supercell matrix."""
# Add parameter checks here
# - check that gamma
# - check that no symmetries are used
# - ...
parameters = calc.input_parameters
assert parameters['mode'] == 'lcao', "LCAO mode required."
assert parameters['usesymm'] != True, "Symmetries not supported."
self.calc_lcao = calc
def set_basis_info(self, *args):
"""Store lcao basis info for atoms in reference cell in attribute.
Parameters
----------
args: tuple
If the LCAO calculator is not available (e.g. if the supercell is
loaded from file), the ``load_supercell_matrix`` member function
provides the required info as arguments.
"""
if len(args) == 0:
calc = self.calc_lcao
setups = calc.wfs.setups
bfs = calc.wfs.basis_functions
nao_a = [setups[a].nao for a in range(len(self.atoms))]
M_a = [bfs.M_a[a] for a in range(len(self.atoms))]
else:
M_a = args[0]
nao_a = args[1]
self.basis_info = {'M_a': M_a,
'niAO_a': nao_a} # XXX niAO -> nao
def set_log(self, log=None):
"""Set output log."""
if log is None:
self.timer = nulltimer
elif log == '-':
self.timer = StepTimer(name='elph')
else:
self.timer = StepTimer(name='elph', out=open(log, 'w'))
def calculate_supercell_matrix(self, dump=0, name=None, filter=None,
include_pseudo=True, atoms=None):
"""Calculate matrix elements of the el-ph coupling in the LCAO basis.
This function calculates the matrix elements between LCAOs and local
atomic gradients of the effective potential. The matrix elements are
calculated for the supercell used to obtain finite-difference
approximations to the derivatives of the effective potential wrt to
atomic displacements.
Parameters
----------
dump: int
Dump supercell matrix to pickle file (default: 0).
0: Supercell matrix not saved
1: Supercell matrix saved in a single pickle file.
2: Dump matrix for different gradients in separate files. Useful
for large systems where the total array gets too large for a
single pickle file.
name: string
User specified name of the generated pickle file(s). If not
provided, the string in the ``name`` attribute is used.
filter: str
Fourier filter atomic gradients of the effective potential. The
specified components (``normal`` or ``umklapp``) are removed
(default: None).
include_pseudo: bool
Include the contribution from the psedupotential in the atomic
gradients. If ``False``, only the gradient of the effective
potential is included (default: True).
atoms: Atoms object
Calculate supercell for an ``Atoms`` object different from the one
provided in the ``__init__`` method (WARNING, NOT working!).
"""
assert self.calc_lcao is not None, "Set LCAO calculator"
# Supercell atoms
if atoms is None:
atoms_N = self.atoms * self.N_c
else:
atoms_N = atoms
# Initialize calculator if required and extract useful quantities
calc = self.calc_lcao
if not hasattr(calc.wfs, 'S_qMM'):
calc.initialize(atoms_N)
calc.initialize_positions(atoms_N)
self.set_basis_info()
basis = calc.input_parameters['basis']
# Extract useful objects from the calculator
wfs = calc.wfs
gd = calc.wfs.gd
kd = calc.wfs.kd
kpt_u = wfs.kpt_u
setups = wfs.setups
nao = setups.nao
bfs = wfs.basis_functions
dtype = wfs.dtype
spin = 0 # XXX
# If gamma calculation, overlap with neighboring cell cannot be removed
if kd.gamma:
print "WARNING: Gamma-point calculation."
else:
# Bloch to real-space converter
tb = TightBinding(atoms_N, calc)
self.timer.write_now("Calculating supercell matrix")
self.timer.write_now("Calculating real-space gradients")
# Calculate finite-difference gradients (in Hartree / Bohr)
V1t_xG, dH1_xasp = self.calculate_gradient()
self.timer.write_now("Finished real-space gradients")
# Fourier filter the atomic gradients of the effective potential
if filter is not None:
self.timer.write_now("Fourier filtering gradients")
V1_xG = V1t_xG.copy()
self.fourier_filter(V1t_xG, components=filter)
self.timer.write_now("Finished Fourier filtering")
# For the contribution from the derivative of the projectors
dP_aqvMi = self.calculate_dP_aqvMi(wfs)
# Equilibrium atomic Hamiltonian matrix (projector coefficients)
dH_asp = pickle.load(open(self.name + '.eq.pckl'))[1]
# Check that the grid is the same as in the calculator
assert np.all(V1t_xG.shape[-3:] == (gd.N_c + gd.pbc_c - 1)), \
"Mismatch in grids."
# Calculate < i k | grad H | j k >, i.e. matrix elements in Bloch basis
# List for supercell matrices;
g_xNNMM = []
self.timer.write_now("Calculating gradient of PAW Hamiltonian")
# Do each cartesian component separately
for i, a in enumerate(self.indices):
for v in range(3):
# Corresponding array index
x = 3 * i + v
V1t_G = V1t_xG[x]
self.timer.write_now("%s-gradient of atom %u" %
(['x','y','z'][v], a))
# Array for different k-point components
g_qMM = np.zeros((len(kpt_u), nao, nao), dtype)
# 1) Gradient of effective potential
self.timer.write_now("Starting gradient of effective potential")
for kpt in kpt_u:
# Matrix elements
geff_MM = np.zeros((nao, nao), dtype)
bfs.calculate_potential_matrix(V1t_G, geff_MM, q=kpt.q)
tri2full(geff_MM, 'L')
# Insert in array
g_qMM[kpt.q] += geff_MM
self.timer.write_now("Finished gradient of effective potential")
if include_pseudo:
self.timer.write_now("Starting gradient of pseudo part")
# 2) Gradient of non-local part (projectors)
self.timer.write_now("Starting gradient of dH^a")
P_aqMi = calc.wfs.P_aqMi
# 2a) dH^a part has contributions from all other atoms
for kpt in kpt_u:
# Matrix elements
gp_MM = np.zeros((nao, nao), dtype)
dH1_asp = dH1_xasp[x]
for a_, dH1_sp in dH1_asp.items():
dH1_ii = unpack2(dH1_sp[spin])
gp_MM += np.dot(P_aqMi[a_][kpt.q], np.dot(dH1_ii,
P_aqMi[a_][kpt.q].T.conjugate()))
g_qMM[kpt.q] += gp_MM
self.timer.write_now("Finished gradient of dH^a")
self.timer.write_now("Starting gradient of projectors")
# 2b) dP^a part has only contributions from the same atoms
dP_qvMi = dP_aqvMi[a]
dH_ii = unpack2(dH_asp[a][spin])
for kpt in kpt_u:
#XXX Sort out the sign here; conclusion -> sign = +1 !
P1HP_MM = +1 * np.dot(dP_qvMi[kpt.q][v], np.dot(dH_ii,
P_aqMi[a][kpt.q].T.conjugate()))
# Matrix elements
gp_MM = P1HP_MM + P1HP_MM.T.conjugate()
g_qMM[kpt.q] += gp_MM
self.timer.write_now("Finished gradient of projectors")
self.timer.write_now("Finished gradient of pseudo part")
# Extract R_c=(0, 0, 0) block by Fourier transforming
if kd.gamma or kd.N_c is None:
g_MM = g_qMM[0]
else:
# Convert to array
g_MM = tb.bloch_to_real_space(g_qMM, R_c=(0, 0, 0))[0]
# Reshape to global unit cell indices
N = np.prod(self.N_c)
# Number of basis function in the primitive cell
assert (nao % N) == 0, "Alarm ...!"
nao_cell = nao / N
g_NMNM = g_MM.reshape((N, nao_cell, N, nao_cell))
g_NNMM = g_NMNM.swapaxes(1, 2).copy()
self.timer.write_now("Finished supercell matrix")
if dump != 2:
g_xNNMM.append(g_NNMM)
else:
if name is not None:
fname = '%s.supercell_matrix_x_%2.2u.%s.pckl' % (name, x, basis)
else:
fname = self.name + \
'.supercell_matrix_x_%2.2u.%s.pckl' % (x, basis)
if kd.comm.rank == 0:
fd = open(fname, 'w')
M_a = self.basis_info['M_a']
nao_a = self.basis_info['niAO_a']
pickle.dump((g_NNMM, M_a, nao_a), fd, 2)
fd.close()
self.timer.write_now("Finished gradient of PAW Hamiltonian")
if dump != 2:
# Collect gradients in one array
self.g_xNNMM = np.array(g_xNNMM)
# Dump to pickle file using binary mode together with basis info
if dump and kd.comm.rank == 0:
if name is not None:
fname = '%s.supercell_matrix.%s.pckl' % (name, basis)
else:
fname = self.name + '.supercell_matrix.%s.pckl' % basis
fd = open(fname, 'w')
M_a = self.basis_info['M_a']
nao_a = self.basis_info['nao_a']
pickle.dump((self.g_xNNMM, M_a, nao_a), fd, 2)
fd.close()
def load_supercell_matrix(self, basis=None, name=None, multiple=False):
"""Load supercell matrix from pickle file.
Parameters
----------
basis: string
String specifying the LCAO basis used to calculate the supercell
matrix, e.g. 'dz(dzp)'.
name: string
User specified name of the pickle file.
multiple: bool
Load each derivative from individual files.
"""
assert (basis is not None) or (name is not None), \
"Provide basis or name."
if self.g_xNNMM is not None:
self.g_xNNMM = None
if not multiple:
# File name
if name is not None:
fname = name
else:
fname = self.name + '.supercell_matrix.%s.pckl' % basis
fd = open(fname)
self.g_xNNMM, M_a, nao_a = pickle.load(fd)
fd.close()
else:
g_xNNMM = []
for x in range(len(self.indices)*3):
if name is not None:
fname = name
else:
fname = self.name + \
'.supercell_matrix_x_%2.2u.%s.pckl' % (x, basis)
fd = open(fname, 'r')
g_NNMM, M_a, nao_a = pickle.load(fd)
fd.close()
g_xNNMM.append(g_NNMM)
self.g_xNNMM = np.array(g_xNNMM)
self.set_basis_info(M_a, nao_a)
def apply_cutoff(self, cutmax=None, cutmin=None):
"""Zero matrix element inside/beyond the specified cutoffs.
Parameters
----------
cutmax: float
Zero matrix elements for basis functions with a distance to the
atomic gradient that is larger than the cutoff.
cutmin: float
Zero matrix elements where both basis functions have distances to
the atomic gradient that is smaller than the cutoff.
"""
if cutmax is not None:
cutmax = float(cutmax)
if cutmin is not None:
cutmin = float(cutmin)
# Reference to supercell matrix attribute
g_xNNMM = self.g_xNNMM
# Number of atoms and primitive cells
N_atoms = len(self.indices)
N = np.prod(self.N_c)
nao = g_xNNMM.shape[-1]
# Reshape array
g_avNNMM = g_xNNMM.reshape(N_atoms, 3, N, N, nao, nao)
# Make slices for orbitals on atoms
M_a = self.basis_info['M_a']
nao_a = self.basis_info['niAO_a']
slice_a = []
for a in range(len(self.atoms)):
start = M_a[a] ;
stop = start + nao_a[a]
s = slice(start, stop)
slice_a.append(s)
# Lattice vectors
R_cN = self.lattice_vectors()
# Unit cell vectors
cell_vc = self.atoms.cell.transpose()
# Atomic positions in reference cell
pos_av = self.atoms.get_positions()
# Create a mask array to zero the relevant matrix elements
if cutmin is not None:
mask_avNNMM = np.zeros(g_avNNMM.shape, dtype=bool)
# Zero elements where one of the basis orbitals has a distance to atoms
# (atomic gradients) 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
for i, a in enumerate(self.indices):
# Atomic distances wrt to the position of the gradient
dist_a = np.sqrt(np.sum((pos_av[a] - posn_av)**2, axis=-1))
if cutmax is not None:
# Atoms indices where the distance is larger than the max
# cufoff
j_a = np.where(dist_a > cutmax)[0]
# Zero elements
for j in j_a:
g_avNNMM[a, :, n, :, slice_a[j], :] = 0.0
g_avNNMM[a, :, :, n, :, slice_a[j]] = 0.0
if cutmin is not None:
# Atoms indices where the distance is larger than the min
# cufoff
j_a = np.where(dist_a > cutmin)[0]
# Update mask to keep elements where one LCAO is outside
# the min cutoff
for j in j_a:
mask_avNNMM[a, :, n, :, slice_a[j], :] = True
mask_avNNMM[a, :, :, n, :, slice_a[j]] = True
# Zero elements where both LCAOs are located within the min cutoff
if cutmin is not None:
g_avNNMM[~mask_avNNMM] = 0.0
def lcao_matrix(self, u_l, omega_l):
"""Calculate the el-ph coupling in the electronic LCAO basis.
For now, only works for Gamma-point phonons.
Parameters
----------
u_l: ndarray
Mass-scaled polarization vectors (in units of 1 / sqrt(amu)) of the
phonons.
omega_l: ndarray
Vibrational frequencies in eV.
"""
# Supercell matrix (Hartree / Bohr)
assert self.g_xNNMM is not None, "Load supercell matrix."
assert self.g_xNNMM.shape[1:3] == (1, 1)
g_xMM = self.g_xNNMM[:, 0, 0, :, :]
# Number of atomic orbitals
nao = g_xMM.shape[-1]
# Number of phonon modes
nmodes = u_l.shape[0]
#
u_lx = u_l.reshape(nmodes, 3 * len(self.atoms))
g_lMM = np.dot(u_lx, g_xMM.transpose(1, 0, 2))
# Multiply prefactor sqrt(hbar / 2 * M * omega) in units of Bohr
amu = units._amu # atomic mass unit
me = units._me # electron mass
g_lMM /= np.sqrt(2 * amu / me / units.Hartree * \
omega_l[:, np.newaxis, np.newaxis])
# Convert to eV
g_lMM *= units.Hartree
return g_lMM
def bloch_matrix(self, kpts, qpts, c_kn, u_ql, omega_ql=None, kpts_from=None):
"""Calculate el-ph coupling in the Bloch basis for the electrons.
This function calculates the electron-phonon coupling between the
specified Bloch states, i.e.::
______
mnl / hbar ^
g = /------- < m k + q | e . grad V | n k >
kq \/ 2 M w ql q
ql
In case the ``omega_ql`` keyword argument is not given, the bare matrix
element (in units of eV / Ang) without the sqrt prefactor is returned.
Parameters
----------
kpts: ndarray or tuple.
k-vectors of the Bloch states. When a tuple of integers is given, a
Monkhorst-Pack grid with the specified number of k-points along the
directions of the reciprocal lattice vectors is generated.
qpts: ndarray or tuple.
q-vectors of the phonons.
c_kn: ndarray
Expansion coefficients for the Bloch states. The ordering must be
the same as in the ``kpts`` argument.
u_ql: ndarray
Mass-scaled polarization vectors (in units of 1 / sqrt(amu)) of the
phonons. Again, the ordering must be the same as in the
corresponding ``qpts`` argument.
omega_ql: ndarray
Vibrational frequencies in eV.
kpts_from: list of ints or int
Calculate only the matrix element for the k-vectors specified by
their index in the ``kpts`` argument (default: all).
In short, phonon frequencies and mode vectors must be given in ase units.
"""
assert self.g_xNNMM is not None, "Load supercell matrix."
assert len(c_kn.shape) == 3
assert len(u_ql.shape) == 4
if omega_ql is not None:
assert np.all(u_ql.shape[:2] == omega_ql.shape[:2])
# Translate k-points into 1. BZ (required by ``find_k_plus_q``` member
# function of the ```KPointDescriptor``).
if isinstance(kpts, np.ndarray):
assert kpts.shape[1] == 3, "kpts_kc array must be given"
# XXX This does not seem to cause problems!
kpts -= kpts.round()
# Use the KPointDescriptor to keep track of the k and q-vectors
kd_kpts = KPointDescriptor(kpts)
kd_qpts = KPointDescriptor(qpts)
# Check that number of k- and q-points agree with the number of Bloch
# functions and polarization vectors
assert kd_kpts.nbzkpts == len(c_kn)
assert kd_qpts.nbzkpts == len(u_ql)
# Include all k-point per default
if kpts_from is None:
kpts_kc = kd_kpts.bzk_kc
kpts_k = range(kd_kpts.nbzkpts)
else:
kpts_kc = kd_kpts.bzk_kc[kpts_from]
if isinstance(kpts_from, int):
kpts_k = list([kpts_from])
else:
kpts_k = list(kpts_from)
# Supercell matrix (real matrix in Hartree / Bohr)
g_xNNMM = self.g_xNNMM
# Number of phonon modes and electronic bands
nmodes = u_ql.shape[1]
nbands = c_kn.shape[1]
# Number of atoms displacements and basis functions
ndisp = np.prod(u_ql.shape[2:])
assert ndisp == (3 * len(self.indices))
nao = c_kn.shape[2]
assert ndisp == g_xNNMM.shape[0]
assert nao == g_xNNMM.shape[-1]
# Lattice vectors
R_cN = self.lattice_vectors()
# Number of unit cell in supercell
N = np.prod(self.N_c)
# Allocate array for couplings
g_qklnn = np.zeros((kd_qpts.nbzkpts, len(kpts_kc), nmodes,
nbands, nbands), dtype=complex)
self.timer.write_now("Calculating coupling matrix elements")
for q, q_c in enumerate(kd_qpts.bzk_kc):
# Find indices of k+q for the k-points
kplusq_k = kd_kpts.find_k_plus_q(q_c, kpts_k=kpts_k)
# Here, ``i`` is counting from 0 and ``k`` is the global index of
# the k-point
for i, (k, k_c) in enumerate(zip(kpts_k, kpts_kc)):
# Check the wave vectors (adapted to the ``KPointDescriptor`` class)
kplusq_c = k_c + q_c
kplusq_c -= kplusq_c.round()
assert np.allclose(kplusq_c, kd_kpts.bzk_kc[kplusq_k[i]] ), \
(i, k, k_c, q_c, kd_kpts.bzk_kc[kplusq_k[i]])
# Allocate array
g_xMM = np.zeros((ndisp, nao, nao), dtype=complex)
# Multiply phase factors
for m in range(N):
for n in range(N):
Rm_c = R_cN[:, m]
Rn_c = R_cN[:, n]
phase = np.exp(2.j * pi * (np.dot(k_c, Rm_c - Rn_c)
+ np.dot(q_c, Rm_c)))
# Sum contributions from different cells
g_xMM += g_xNNMM[:, m, n, :, :] * phase
# LCAO coefficient for Bloch states
ck_nM = c_kn[k]
ckplusq_nM = c_kn[kplusq_k[i]]
# Mass scaled polarization vectors
u_lx = u_ql[q].reshape(nmodes, 3 * len(self.atoms))
g_nxn = np.dot(ckplusq_nM.conj(), np.dot(g_xMM, ck_nM.T))
g_lnn = np.dot(u_lx, g_nxn)
# Insert value
g_qklnn[q, i] = g_lnn
# XXX Temp
if np.all(q_c == 0.0):
# These should be real
print g_qklnn[q].imag.min(), g_qklnn[q].imag.max()
self.timer.write_now("Finished calculation of coupling matrix elements")
# Return the bare matrix element if frequencies are not given
if omega_ql is None:
# Convert to eV / Ang
g_qklnn *= units.Hartree / units.Bohr
else:
# Multiply prefactor sqrt(hbar / 2 * M * omega) in units of Bohr
amu = units._amu # atomic mass unit
me = units._me # electron mass
g_qklnn /= np.sqrt(2 * amu / me / units.Hartree * \
omega_ql[:, np.newaxis, :, np.newaxis, np.newaxis])
# Convert to eV
g_qklnn *= units.Hartree
# Return couplings in eV (or eV / Ang)
return g_qklnn
def fourier_filter(self, V1t_xG, components='normal', criteria=1):
"""Fourier filter atomic gradients of the effective potential.
Parameters
----------
V1t_xG: ndarray
Array representation of atomic gradients of the effective potential
in the supercell grid.
components: str
Fourier components to filter out (``normal`` or ``umklapp``).
"""
assert components in ['normal', 'umklapp']
# Grid shape
shape = V1t_xG.shape[-3:]
# Primitive unit cells in Bohr/Bohr^-1
cell_cv = self.atoms.get_cell() / units.Bohr
reci_vc = 2 * pi * la.inv(cell_cv)
norm_c = np.sqrt(np.sum(reci_vc**2, axis=0))
# Periodic BC array
pbc_c = np.array(self.atoms.get_pbc(), dtype=bool)
# Supercell atoms and cell
atoms_N = self.atoms * self.N_c
supercell_cv = atoms_N.get_cell() / units.Bohr
# q-grid in units of the grid spacing (FFT ordering)
q_cG = np.indices(shape).reshape(3, -1)
q_c = np.array(shape)[:, np.newaxis]
q_cG += q_c // 2
q_cG %= q_c
q_cG -= q_c // 2
# Locate q-points inside the Brillouin zone
if criteria == 0:
# Works for all cases
# Grid spacing in direction of reciprocal lattice vectors
h_c = np.sqrt(np.sum((2 * pi * la.inv(supercell_cv))**2, axis=0))
# XXX Why does a "*=" operation on q_cG not work here ??
q1_cG = q_cG * h_c[:, np.newaxis] / (norm_c[:, np.newaxis] / 2)
mask_G = np.ones(np.prod(shape), dtype=bool)
for i, pbc in enumerate(pbc_c):
if not pbc:
continue
mask_G &= (-1. < q1_cG[i]) & (q1_cG[i] <= 1.)
else:
# 2D hexagonal lattice
# Projection of q points onto the periodic directions. Only in
# these directions do normal and umklapp processees make sense.
q_vG = np.dot(q_cG[pbc_c].T,
2 * pi * la.inv(supercell_cv).T[pbc_c]).T.copy()
# Parametrize the BZ boundary in terms of the angle theta
theta_G = np.arctan2(q_vG[1], q_vG[0]) % (pi / 3)
phi_G = pi / 6 - np.abs(theta_G)
qmax_G = norm_c[0] / 2 / np.cos(phi_G)
norm_G = np.sqrt(np.sum(q_vG**2, axis=0))
# Includes point on BZ boundary with +1e-2
mask_G = (norm_G <= qmax_G + 1e-2) # & (q_vG[1] < (norm_c[0] / 2 - 1e-3))
if components != 'normal':
mask_G = ~mask_G
# Reshape to grid shape
mask_G.shape = shape
for V1t_G in V1t_xG:
# Fourier transform atomic gradient
V1tq_G = fft.fftn(V1t_G)
# Zero normal/umklapp components
V1tq_G[mask_G] = 0.0
# Fourier transform back
V1t_G[:] = fft.ifftn(V1tq_G).real
def calculate_gradient(self):
"""Calculate gradient of effective potential and projector coefs.
This function loads the generated pickle files and calculates
finite-difference derivatives.
"""
# Array and dict for finite difference derivatives
V1t_xG = []
dH1_xasp = []
x = 0
for a in self.indices:
for v in range(3):
name = '%s.%d%s' % (self.name, a, 'xyz'[v])
# Potential and atomic density matrix for atomic displacement
try:
Vtm_G, dHm_asp = pickle.load(open(name + '-.pckl'))
Vtp_G, dHp_asp = pickle.load(open(name + '+.pckl'))
except (IOError, EOFError):
raise IOError, "%s(-/+).pckl" % name
# FD derivatives in Hartree / Bohr
V1t_G = (Vtp_G - Vtm_G) / (2 * self.delta / units.Bohr)
V1t_xG.append(V1t_G)
dH1_asp = {}
for atom in dHm_asp.keys():
dH1_asp[atom] = (dHp_asp[atom] - dHm_asp[atom]) / \
(2 * self.delta / units.Bohr)
dH1_xasp.append(dH1_asp)
x += 1
return np.array(V1t_xG), dH1_xasp
def calculate_dP_aqvMi(self, wfs):
"""Overlap between LCAO basis functions and gradient of projectors.
Only the gradient wrt the atomic positions in the reference cell is
computed.
"""
nao = wfs.setups.nao
nq = len(wfs.ibzk_qc)
atoms = [self.atoms[i] for i in self.indices]
# Derivatives in reference cell
dP_aqvMi = {}
for atom, setup in zip(atoms, wfs.setups):
a = atom.index
dP_aqvMi[a] = np.zeros((nq, 3, nao, setup.ni), wfs.dtype)
# Calculate overlap between basis function and gradient of projectors
# NOTE: the derivative is calculated wrt the atomic position and not
# the real-space coordinate
calc = TwoCenterIntegralCalculator(wfs.ibzk_qc, derivative=True)
expansions = ManySiteDictionaryWrapper(wfs.tci.P_expansions, dP_aqvMi)
calc.calculate(wfs.tci.atompairs, [expansions], [dP_aqvMi])
# Extract derivatives in the reference unit cell
# dP_aqvMi = {}
# for atom in self.atoms:
# dP_aqvMi[atom.index] = dPall_aqvMi[atom.index]
return dP_aqvMi
