class PhononPerturbation(Perturbation):
"""Implementation of a phonon perturbation.
This class implements the change in the effective potential due to a
displacement of an atom ``a`` in direction ``v`` with wave-vector ``q``.
The action of the perturbing potential on a state vector is implemented in
the ``apply`` member function.
"""
def __init__(self, calc, kd, poisson_solver, dtype=float, **kwargs):
"""Store useful objects, e.g. lfc's for the various atomic functions.
Depending on whether the system is periodic or finite, Poisson's equation
is solved with FFT or multigrid techniques, respectively.
Parameters
----------
calc: Calculator
Ground-state calculation.
kd: KPointDescriptor
Descriptor for the q-vectors of the dynamical matrix.
"""
self.kd = kd
self.dtype = dtype
self.poisson = poisson_solver
# Gamma wrt q-vector
if self.kd.gamma:
self.phase_cd = None
else:
assert self.kd.mynks == len(self.kd.ibzk_qc)
self.phase_qcd = []
sdisp_cd = calc.wfs.gd.sdisp_cd
for q in range(self.kd.mynks):
phase_cd = np.exp(2j * np.pi * \
sdisp_cd * self.kd.ibzk_qc[q, :, np.newaxis])
self.phase_qcd.append(phase_cd)
# Store grid-descriptors
self.gd = calc.density.gd
self.finegd = calc.density.finegd
# Steal setups for the lfc's
setups = calc.wfs.setups
# Store projector coefficients
self.dH_asp = calc.hamiltonian.dH_asp.copy()
# Localized functions:
# core corections
self.nct = LFC(self.gd, [[setup.nct] for setup in setups],
integral=[setup.Nct for setup in setups], dtype=self.dtype)
# compensation charges
#XXX what is the consequence of numerical errors in the integral ??
self.ghat = LFC(self.finegd, [setup.ghat_l for setup in setups],
dtype=self.dtype)
## self.ghat = LFC(self.finegd, [setup.ghat_l for setup in setups],
## integral=sqrt(4 * pi), dtype=self.dtype)
# vbar potential
self.vbar = LFC(self.finegd, [[setup.vbar] for setup in setups],
dtype=self.dtype)
# Expansion coefficients for the compensation charges
self.Q_aL = calc.density.Q_aL.copy()
# Grid transformer -- convert array from fine to coarse grid
self.restrictor = Transformer(self.finegd, self.gd, nn=3,
dtype=self.dtype, allocate=False)
# Atom, cartesian coordinate and q-vector of the perturbation
self.a = None
self.v = None
# Local q-vector index of the perturbation
if self.kd.gamma:
self.q = -1
else:
self.q = None
def initialize(self, spos_ac):
"""Prepare the various attributes for a calculation."""
# Set positions on LFC's
self.nct.set_positions(spos_ac)
self.ghat.set_positions(spos_ac)
self.vbar.set_positions(spos_ac)
if not self.kd.gamma:
# Set q-vectors and update
self.ghat.set_k_points(self.kd.ibzk_qc)
self.ghat._update(spos_ac)
# Set q-vectors and update
self.vbar.set_k_points(self.kd.ibzk_qc)
self.vbar._update(spos_ac)
# Phase factor exp(iq.r) needed to obtian the periodic part of lfcs
coor_vg = self.finegd.get_grid_point_coordinates()
cell_cv = self.finegd.cell_cv
# Convert to scaled coordinates
scoor_cg = np.dot(la.inv(cell_cv), coor_vg.swapaxes(0, -2))
scoor_cg = scoor_cg.swapaxes(1,-2)
# Phase factor
phase_qg = np.exp(2j * pi *
np.dot(self.kd.ibzk_qc, scoor_cg.swapaxes(0,-2)))
self.phase_qg = phase_qg.swapaxes(1, -2)
#XXX To be removed from this class !!
# Setup the Poisson solver -- to be used on the fine grid
self.poisson.set_grid_descriptor(self.finegd)
self.poisson.initialize()
# Grid transformer
self.restrictor.allocate()
def set_q(self, q):
"""Set the index of the q-vector of the perturbation."""
assert not self.kd.gamma, "Gamma-point calculation"
self.q = q
# Update phases and Poisson solver
self.phase_cd = self.phase_qcd[q]
self.poisson.set_q(self.kd.ibzk_qc[q])
# Invalidate calculated quantities
# - local part of perturbing potential
self.v1_G = None
def set_av(self, a, v):
"""Set atom and cartesian component of the perturbation.
Parameters
----------
a: int
Index of the atom.
v: int
Cartesian component (0, 1 or 2) of the atomic displacement.
"""
assert self.q is not None
self.a = a
self.v = v
# Update derivative of local potential
self.calculate_local_potential()
def get_phase_cd(self):
"""Overwrite base class member function."""
return self.phase_cd
def has_q(self):
"""Overwrite base class member function."""
return (not self.kd.gamma)
def get_q(self):
"""Return q-vector."""
assert not self.kd.gamma, "Gamma-point calculation."
return self.kd.ibzk_qc[self.q]
def solve_poisson(self, phi_g, rho_g):
"""Solve Poisson's equation for a Bloch-type charge distribution.
More to come here ...
Parameters
----------
phi_g: GridDescriptor
Grid for the solution of Poissons's equation.
rho_g: GridDescriptor
Grid with the charge distribution.
"""
#assert phi_g.shape == rho_g.shape == self.phase_qg.shape[-3:], \
# ("Arrays have incompatible shapes.")
assert self.q is not None, ("q-vector not set")
# Gamma point calculation wrt the q-vector -> rho_g periodic
if self.kd.gamma:
#XXX NOTICE: solve_neutral
self.poisson.solve_neutral(phi_g, rho_g)
else:
# Divide out the phase factor to get the periodic part
rhot_g = rho_g/self.phase_qg[self.q]
# Solve Poisson's equation for the periodic part of the potential
#XXX NOTICE: solve_neutral
self.poisson.solve_neutral(phi_g, rhot_g)
# Return to Bloch form
phi_g *= self.phase_qg[self.q]
def calculate_local_potential(self):
"""Derivate of the local potential wrt an atomic displacements.
The local part of the PAW potential has contributions from the
compensation charges (``ghat``) and a spherical symmetric atomic
potential (``vbar``).
"""
assert self.a is not None
assert self.v is not None
assert self.q is not None
a = self.a
v = self.v
# Expansion coefficients for the ghat functions
Q_aL = self.ghat.dict(zero=True)
# Remember sign convention for add_derivative method
# And be sure not to change the dtype of the arrays by assigning values
# to array elements.
Q_aL[a][:] = -1 * self.Q_aL[a]
# Grid for derivative of compensation charges
ghat1_g = self.finegd.zeros(dtype=self.dtype)
self.ghat.add_derivative(a, v, ghat1_g, c_axi=Q_aL, q=self.q)
# Solve Poisson's eq. for the potential from the periodic part of the
# compensation charge derivative
v1_g = self.finegd.zeros(dtype=self.dtype)
self.solve_poisson(v1_g, ghat1_g)
# Store potential from the compensation charge
self.vghat1_g = v1_g.copy()
# Add derivative of vbar - sign convention in add_derivative method
c_ai = self.vbar.dict(zero=True)
c_ai[a][0] = -1.
self.vbar.add_derivative(a, v, v1_g, c_axi=c_ai, q=self.q)
# Store potential for the evaluation of the energy derivative
self.v1_g = v1_g.copy()
# Transfer to coarse grid
v1_G = self.gd.zeros(dtype=self.dtype)
self.restrictor.apply(v1_g, v1_G, phases=self.phase_cd)
self.v1_G = v1_G
def apply(self, psi_nG, y_nG, wfs, k, kplusq):
"""Apply perturbation to unperturbed wave-functions.
Parameters
----------
psi_nG: ndarray
Set of grid vectors to which the perturbation is applied.
y_nG: ndarray
Output vectors.
wfs: WaveFunctions
Instance of class ``WaveFunctions``.
k: int
Index of the k-point for the vectors.
kplusq: int
Index of the k+q vector.
"""
assert self.a is not None
assert self.v is not None
assert self.q is not None
assert psi_nG.ndim in (3, 4)
assert tuple(self.gd.n_c) == psi_nG.shape[-3:]
if psi_nG.ndim == 3:
y_nG += self.v1_G * psi_nG
else:
y_nG += self.v1_G[np.newaxis, :] * psi_nG
self.apply_nonlocal_potential(psi_nG, y_nG, wfs, k, kplusq)
def apply_nonlocal_potential(self, psi_nG, y_nG, wfs, k, kplusq):
"""Derivate of the non-local PAW potential wrt an atomic displacement.
Parameters
----------
k: int
Index of the k-point being operated on.
kplusq: int
Index of the k+q vector.
"""
assert self.a is not None
assert self.v is not None
assert psi_nG.ndim in (3, 4)
assert tuple(self.gd.n_c) == psi_nG.shape[-3:]
if psi_nG.ndim == 3:
n = 1
else:
n = psi_nG.shape[0]
a = self.a
v = self.v
P_ani = wfs.kpt_u[k].P_ani
dP_aniv = wfs.kpt_u[k].dP_aniv
pt = wfs.pt
# < p_a^i | Psi_nk >
P_ni = P_ani[a]
# < dp_av^i | Psi_nk > - remember the sign convention of the derivative
dP_ni = -1 * dP_aniv[a][...,v]
# Expansion coefficients for the projectors on atom a
dH_ii = unpack(self.dH_asp[a][0])
# The derivative of the non-local PAW potential has two contributions
# 1) Sum over projectors
c_ni = np.dot(dP_ni, dH_ii)
c_ani = pt.dict(shape=n, zero=True)
c_ani[a] = c_ni
# k+q !!
pt.add(y_nG, c_ani, q=kplusq)
# 2) Sum over derivatives of the projectors
dc_ni = np.dot(P_ni, dH_ii)
dc_ani = pt.dict(shape=n, zero=True)
# Take care of sign of derivative in the coefficients
dc_ani[a] = -1 * dc_ni
# k+q !!
pt.add_derivative(a, v, y_nG, dc_ani, q=kplusq)
class WaveFunctions:
"""Class for wave-function related stuff (e.g. projectors)."""
def __init__(self, nbands, kpt_u, setups, kd, gd, dtype=float):
"""Store and initialize required attributes.
Parameters
----------
nbands: int
Number of occupied bands.
kpt_u: list of KPoints
List of KPoint instances from a ground-state calculation (i.e. the
attribute ``calc.wfs.kpt_u``).
setups: Setups
LocalizedFunctionsCollection setups.
kd: KPointDescriptor
K-point and symmetry related stuff.
gd: GridDescriptor
Descriptor for the coarse grid.
dtype: dtype
This is the ``dtype`` for the wave-function derivatives (same as
the ``dtype`` for the ground-state wave-functions).
"""
self.dtype = dtype
# K-point related attributes
self.kd = kd
# Number of occupied bands
self.nbands = nbands
# Projectors
self.pt = LFC(gd, [setup.pt_j for setup in setups], dtype=self.dtype)
# Store grid
self.gd = gd
# Unfold the irreducible BZ to the full BZ
# List of KPointContainers for the k-points in the full BZ
self.kpt_u = []
# No symmetries or only time-reversal symmetry used
if kd.symmetry is None:
# For now, time-reversal symmetry not allowed
assert len(kpt_u) == kd.nbzkpts
for k in range(kd.nbzkpts):
kpt_ = kpt_u[k]
psit_nG = gd.empty(nbands, dtype=self.dtype)
for n, psit_G in enumerate(psit_nG):
psit_G[:] = kpt_.psit_nG[n]
# psit_0 = psit_G[0, 0, 0]
# psit_G *= psit_0.conj() / (abs(psit_0))
# Strip off KPoint attributes and store in the KPointContainer
# Note, only the occupied GS wave-functions are retained here !
kpt = KPointContainer(weight=kpt_.weight,
k=kpt_.k,
s=kpt_.s,
phase_cd=kpt_.phase_cd,
eps_n=kpt_.eps_n[:nbands],
psit_nG=psit_nG,
psit1_nG=None,
P_ani=None,
dP_aniv=None)
# q=kpt.q,
# f_n=kpt.f_n[:nbands])
self.kpt_u.append(kpt)
else:
assert len(kpt_u) == kd.nibzkpts
for k, k_c in enumerate(kd.bzk_kc):
# Index of symmetry related point in the irreducible BZ
ik = kd.kibz_k[k]
# Index of point group operation
s = kd.sym_k[k]
# Time-reversal symmetry used
time_reversal = kd.time_reversal_k[k]
# Coordinates of symmetry related point in the irreducible BZ
ik_c = kd.ibzk_kc[ik]
# Point group operation
op_cc = kd.symmetry.op_scc[s]
# KPoint from ground-state calculation
kpt_ = kpt_u[ik]
weight = 1. / kd.nbzkpts * (2 - kpt_.s)
phase_cd = np.exp(2j * pi * gd.sdisp_cd * k_c[:, np.newaxis])
psit_nG = gd.empty(nbands, dtype=self.dtype)
for n, psit_G in enumerate(psit_nG):
#XXX Seems to corrupt my memory somehow ???
psit_G[:] = kd.symmetry.symmetrize_wavefunction(
kpt_.psit_nG[n], ik_c, k_c, op_cc, time_reversal)
# Choose gauge
# psit_0 = psit_G[0, 0, 0]
# psit_G *= psit_0.conj() / (abs(psit_0))
kpt = KPointContainer(weight=weight,
k=k,
s=kpt_.s,
phase_cd=phase_cd,
eps_n=kpt_.eps_n[:nbands],
psit_nG=psit_nG,
psit1_nG=None,
P_ani=None,
dP_aniv=None)
self.kpt_u.append(kpt)
def initialize(self, spos_ac):
"""Initialize projectors according to the ``gamma`` attribute."""
# Set positions on LFC's
self.pt.set_positions(spos_ac)
if not self.kd.gamma:
# Set k-vectors and update
self.pt.set_k_points(self.kd.ibzk_kc)
self.pt._update(spos_ac)
# Calculate projector coefficients for the GS wave-functions
self.calculate_projector_coef()
def reset(self):
"""Make fresh arrays for wave-function derivatives."""
for kpt in self.kpt_u:
kpt.psit1_nG = self.gd.zeros(n=self.nbands, dtype=self.dtype)
def calculate_projector_coef(self):
"""Coefficients for the derivative of the non-local part of the PP.
Parameters
----------
k: int
Index of the k-point of the Bloch state on which the non-local
potential operates on.
The calculated coefficients are the following (except for an overall
sign of -1; see ``derivative`` member function of class ``LFC``):
1. Coefficients from the projector functions::
/ a
P_ani = | dG p (G) Psi (G) ,
/ i n
2. Coefficients from the derivative of the projector functions::
/ a
dP_aniv = | dG dp (G) Psi (G) ,
/ iv n
where::
a d a
dp (G) = --- Phi (G) .
iv a i
dR
"""
n = self.nbands
for kpt in self.kpt_u:
# K-point index and wave-functions
k = kpt.k
psit_nG = kpt.psit_nG
# Integration dicts
P_ani = self.pt.dict(shape=n)
dP_aniv = self.pt.dict(shape=n, derivative=True)
# 1) Integrate with projectors
self.pt.integrate(psit_nG, P_ani, q=k)
kpt.P_ani = P_ani
# 2) Integrate with derivative of projectors
self.pt.derivative(psit_nG, dP_aniv, q=k)
kpt.dP_aniv = dP_aniv