""" Partly occupied Wannier functions
Find the set of partly occupied Wannier functions using the method from
Thygesen, Hansen and Jacobsen PRB v72 i12 p125119 2005.
"""
import functools
import warnings
from math import pi, sqrt
from time import time
import numpy as np
from scipy.linalg import qr
from ase.dft.bandgap import bandgap
from ase.dft.kpoints import get_monkhorst_pack_size_and_offset
from ase.io.jsonio import read_json, write_json
from ase.parallel import paropen
from ase.transport.tools import dagger, normalize
dag = dagger
def silent(*args, **kwargs):
"""Dummy logging function."""
def gram_schmidt(U):
"""Orthonormalize columns of U according to the Gram-Schmidt procedure."""
for i, col in enumerate(U.T):
for col2 in U.T[:i]:
col -= col2 * (col2.conj() @ col)
col /= np.linalg.norm(col)
def lowdin(U, S=None):
"""Orthonormalize columns of U according to the symmetric Lowdin procedure.
The implementation uses SVD, like symm. Lowdin it returns the nearest
orthonormal matrix, but is more robust.
"""
L, s, R = np.linalg.svd(U, full_matrices=False)
U[:] = L @ R
def neighbor_k_search(k_c, G_c, kpt_kc, tol=1e-4):
# search for k1 (in kpt_kc) and k0 (in alldir), such that
# k1 - k - G + k0 = 0
alldir_dc = np.array([[0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1],
[1, 1, 0], [1, 0, 1], [0, 1, 1]], dtype=int)
for k0_c in alldir_dc:
for k1, k1_c in enumerate(kpt_kc):
if np.linalg.norm(k1_c - k_c - G_c + k0_c) < tol:
return k1, k0_c
raise ValueError(f'Wannier: Did not find matching kpoint for kpt={k_c}. '
'Probably non-uniform k-point grid')
def calculate_weights(cell_cc, normalize=True):
""" Weights are used for non-cubic cells, see PRB **61**, 10040
If normalized they lose the physical dimension."""
alldirs_dc = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1],
[1, 1, 0], [1, 0, 1], [0, 1, 1]], dtype=int)
g = cell_cc @ cell_cc.T
# NOTE: Only first 3 of following 6 weights are presently used:
w = np.zeros(6)
w[0] = g[0, 0] - g[0, 1] - g[0, 2]
w[1] = g[1, 1] - g[0, 1] - g[1, 2]
w[2] = g[2, 2] - g[0, 2] - g[1, 2]
w[3] = g[0, 1]
w[4] = g[0, 2]
w[5] = g[1, 2]
# Make sure that first 3 Gdir vectors are included -
# these are used to calculate Wanniercenters.
Gdir_dc = alldirs_dc[:3]
weight_d = w[:3]
for d in range(3, 6):
if abs(w[d]) > 1e-5:
Gdir_dc = np.concatenate((Gdir_dc, alldirs_dc[d:d + 1]))
weight_d = np.concatenate((weight_d, w[d:d + 1]))
if normalize:
weight_d /= max(abs(weight_d))
return weight_d, Gdir_dc
def steepest_descent(func, step=.005, tolerance=1e-6, log=silent, **kwargs):
fvalueold = 0.
fvalue = fvalueold + 10
count = 0
while abs((fvalue - fvalueold) / fvalue) > tolerance:
fvalueold = fvalue
dF = func.get_gradients()
func.step(dF * step, **kwargs)
fvalue = func.get_functional_value()
count += 1
log(f'SteepestDescent: iter={count}, value={fvalue}')
def md_min(func, step=.25, tolerance=1e-6, max_iter=10000,
log=silent, **kwargs):
log('Localize with step =', step, 'and tolerance =', tolerance)
finit = func.get_functional_value()
t = -time()
fvalueold = 0.
fvalue = fvalueold + 10
count = 0
V = np.zeros(func.get_gradients().shape, dtype=complex)
while abs((fvalue - fvalueold) / fvalue) > tolerance:
fvalueold = fvalue
dF = func.get_gradients()
V *= (dF * V.conj()).real > 0
V += step * dF
func.step(V, **kwargs)
fvalue = func.get_functional_value()
if fvalue < fvalueold:
step *= 0.5
count += 1
log(f'MDmin: iter={count}, step={step}, value={fvalue}')
if count > max_iter:
t += time()
warnings.warn('Max iterations reached: '
'iters=%s, step=%s, seconds=%0.2f, value=%0.4f'
% (count, step, t, fvalue.real))
break
t += time()
log('%d iterations in %0.2f seconds (%0.2f ms/iter), endstep = %s' %
(count, t, t * 1000. / count, step))
log(f'Initial value={finit}, Final value={fvalue}')
def rotation_from_projection(proj_nw, fixed, ortho=True):
"""Determine rotation and coefficient matrices from projections
proj_nw = <psi_n|p_w>
psi_n: eigenstates
p_w: localized function
Nb (n) = Number of bands
Nw (w) = Number of wannier functions
M (f) = Number of fixed states
L (l) = Number of extra degrees of freedom
U (u) = Number of non-fixed states
"""
Nb, Nw = proj_nw.shape
M = fixed
L = Nw - M
U = Nb - M
U_ww = np.empty((Nw, Nw), dtype=proj_nw.dtype)
# Set the section of the rotation matrix about the 'fixed' states
U_ww[:M] = proj_nw[:M]
if L > 0:
# If there are extra degrees of freedom we have to select L of them
C_ul = np.empty((U, L), dtype=proj_nw.dtype)
# Get the projections on the 'non fixed' states
proj_uw = proj_nw[M:]
# Obtain eigenvalues and eigevectors matrix
eig_w, C_ww = np.linalg.eigh(dag(proj_uw) @ proj_uw)
# Sort columns of eigenvectors matrix according to the eigenvalues
# magnitude, select only the L largest ones. Then use them to obtain
# the parameter C matrix.
C_ul[:] = proj_uw @ C_ww[:, np.argsort(- eig_w.real)[:L]]
# Compute the section of the rotation matrix about 'non fixed' states
U_ww[M:] = dag(C_ul) @ proj_uw
normalize(C_ul)
else:
# If there are no extra degrees of freedom we do not need any parameter
# matrix C
C_ul = np.empty((U, 0), dtype=proj_nw.dtype)
if ortho:
# Orthogonalize with Lowdin to take the closest orthogonal set
lowdin(U_ww)
else:
normalize(U_ww)
return U_ww, C_ul
def search_for_gamma_point(kpts):
"""Returns index of Gamma point in a list of k-points."""
gamma_idx = np.argmin([np.linalg.norm(kpt) for kpt in kpts])
if np.linalg.norm(kpts[gamma_idx]) >= 1e-14:
gamma_idx = None
return gamma_idx
def scdm(pseudo_nkG, kpts, fixed_k, Nw):
"""Compute localized orbitals with SCDM method
This method was published by Anil Damle and Lin Lin in Multiscale
Modeling & Simulation 16, 1392–1410 (2018).
For now only the isolated bands algorithm is implemented, because it is
intended as a drop-in replacement for other initial guess methods for
the ASE Wannier class.
pseudo_nkG = pseudo wave-functions on a real grid
Ng (G) = number of real grid points
kpts = List of k-points in the BZ
Nk (k) = Number of k-points
Nb (n) = Number of bands
Nw (w) = Number of wannier functions
fixed_k = Number of fixed states for each k-point
L (l) = Number of extra degrees of freedom
U (u) = Number of non-fixed states
"""
gamma_idx = search_for_gamma_point(kpts)
Nk = len(kpts)
U_kww = []
C_kul = []
# compute factorization only at Gamma point
_, _, P = qr(pseudo_nkG[:, gamma_idx, :], mode='full',
pivoting=True, check_finite=True)
for k in range(Nk):
A_nw = pseudo_nkG[:, k, P[:Nw]]
U_ww, C_ul = rotation_from_projection(proj_nw=A_nw,
fixed=fixed_k[k],
ortho=True)
U_kww.append(U_ww)
C_kul.append(C_ul)
U_kww = np.asarray(U_kww)
return C_kul, U_kww
def arbitrary_s_orbitals(atoms, Ns, rng=np.random):
"""
Generate a list of Ns randomly placed s-orbitals close to at least
one atom (< 1.5Å).
The format of the list is the one required by GPAW in initial_wannier().
"""
# Create dummy copy of the Atoms object and dummy H atom
tmp_atoms = atoms.copy()
tmp_atoms.append('H')
s_pos = tmp_atoms.get_scaled_positions()
orbs = []
for _ in range(Ns):
fine = False
while not fine:
# Random position
x, y, z = rng.rand(3)
s_pos[-1] = [x, y, z]
tmp_atoms.set_scaled_positions(s_pos)
# Use dummy H atom to measure distance from any other atom
dists = tmp_atoms.get_distances(
a=-1,
indices=range(len(atoms)))
# Check if it is close to at least one atom
if (dists < 1.5).any():
fine = True
orbs.append([[x, y, z], 0, 1])
return orbs
def init_orbitals(atoms, ntot, rng=np.random):
"""
Place d-orbitals for every atom that has some in the valence states
and then random s-orbitals close to at least one atom (< 1.5Å).
The orbitals list format is compatible with GPAW.get_initial_wannier().
'atoms': ASE Atoms object
'ntot': total number of needed orbitals
'rng': generator random numbers
"""
# List all the elements that should have occupied d-orbitals
# in the valence states (according to GPAW setups)
d_metals = set(list(range(21, 31)) + list(range(39, 52)) +
list(range(57, 84)) + list(range(89, 113)))
orbs = []
# Start with zero orbitals
No = 0
# Add d orbitals to each d-metal
for i, z in enumerate(atoms.get_atomic_numbers()):
if z in d_metals:
No_new = No + 5
if No_new <= ntot:
orbs.append([i, 2, 1])
No = No_new
if No < ntot:
# Add random s-like orbitals if there are not enough yet
Ns = ntot - No
orbs += arbitrary_s_orbitals(atoms, Ns, rng)
assert sum(orb[1] * 2 + 1 for orb in orbs) == ntot
return orbs
def square_modulus_of_Z_diagonal(Z_dww):
"""
Square modulus of the Z matrix diagonal, the diagonal is taken
for the indexes running on the WFs.
"""
return np.abs(Z_dww.diagonal(0, 1, 2))**2
def get_kklst(kpt_kc, Gdir_dc):
# Set the list of neighboring k-points k1, and the "wrapping" k0,
# such that k1 - k - G + k0 = 0
#
# Example: kpoints = (-0.375,-0.125,0.125,0.375), dir=0
# G = [0.25,0,0]
# k=0.375, k1= -0.375 : -0.375-0.375-0.25 => k0=[1,0,0]
#
# For a gamma point calculation k1 = k = 0, k0 = [1,0,0] for dir=0
Nk = len(kpt_kc)
Ndir = len(Gdir_dc)
if Nk == 1:
kklst_dk = np.zeros((Ndir, 1), int)
k0_dkc = Gdir_dc.reshape(-1, 1, 3)
else:
kklst_dk = np.empty((Ndir, Nk), int)
k0_dkc = np.empty((Ndir, Nk, 3), int)
# Distance between kpoints
kdist_c = np.empty(3)
for c in range(3):
# make a sorted list of the kpoint values in this direction
slist = np.argsort(kpt_kc[:, c], kind='mergesort')
skpoints_kc = np.take(kpt_kc, slist, axis=0)
kdist_c[c] = max(
skpoints_kc[n + 1, c] - skpoints_kc[n, c]
for n in range(Nk - 1)
)
for d, Gdir_c in enumerate(Gdir_dc):
for k, k_c in enumerate(kpt_kc):
# setup dist vector to next kpoint
G_c = np.where(Gdir_c > 0, kdist_c, 0)
if max(G_c) < 1e-4:
kklst_dk[d, k] = k
k0_dkc[d, k] = Gdir_c
else:
kklst_dk[d, k], k0_dkc[d, k] = \
neighbor_k_search(k_c, G_c, kpt_kc)
return kklst_dk, k0_dkc
def get_invkklst(kklst_dk):
Ndir, Nk = kklst_dk.shape
invkklst_dk = np.empty(kklst_dk.shape, int)
for d in range(Ndir):
for k1 in range(Nk):
invkklst_dk[d, k1] = kklst_dk[d].tolist().index(k1)
return invkklst_dk
def choose_states(calcdata, fixedenergy, fixedstates, Nk, nwannier, log, spin):
if fixedenergy is None and fixedstates is not None:
if isinstance(fixedstates, int):
fixedstates = [fixedstates] * Nk
fixedstates_k = np.array(fixedstates, int)
elif fixedenergy is not None and fixedstates is None:
# Setting number of fixed states and EDF from given energy cutoff.
# All states below this energy cutoff are fixed.
# The reference energy is Ef for metals and CBM for insulators.
if calcdata.gap < 0.01 or fixedenergy < 0.01:
cutoff = fixedenergy + calcdata.fermi_level
else:
cutoff = fixedenergy + calcdata.lumo
# Find the states below the energy cutoff at each k-point
tmp_fixedstates_k = []
for k in range(Nk):
eps_n = calcdata.eps_skn[spin, k]
kindex = eps_n.searchsorted(cutoff)
tmp_fixedstates_k.append(kindex)
fixedstates_k = np.array(tmp_fixedstates_k, int)
elif fixedenergy is not None and fixedstates is not None:
raise RuntimeError(
'You can not set both fixedenergy and fixedstates')
if nwannier == 'auto':
if fixedenergy is None and fixedstates is None:
# Assume the fixedexergy parameter equal to 0 and
# find the states below the Fermi level at each k-point.
log("nwannier=auto but no 'fixedenergy' or 'fixedstates'",
"parameter was provided, using Fermi level as",
"energy cutoff.")
tmp_fixedstates_k = []
for k in range(Nk):
eps_n = calcdata.eps_skn[spin, k]
kindex = eps_n.searchsorted(calcdata.fermi_level)
tmp_fixedstates_k.append(kindex)
fixedstates_k = np.array(tmp_fixedstates_k, int)
nwannier = np.max(fixedstates_k)
# Without user choice just set nwannier fixed states without EDF
if fixedstates is None and fixedenergy is None:
fixedstates_k = np.array([nwannier] * Nk, int)
return fixedstates_k, nwannier
def get_eigenvalues(calc):
nspins = calc.get_number_of_spins()
nkpts = len(calc.get_ibz_k_points())
nbands = calc.get_number_of_bands()
eps_skn = np.empty((nspins, nkpts, nbands))
for ispin in range(nspins):
for ikpt in range(nkpts):
eps_skn[ispin, ikpt] = calc.get_eigenvalues(kpt=ikpt, spin=ispin)
return eps_skn
class CalcData:
def __init__(self, kpt_kc, atoms, fermi_level, lumo, eps_skn,
gap):
self.kpt_kc = kpt_kc
self.atoms = atoms
self.fermi_level = fermi_level
self.lumo = lumo
self.eps_skn = eps_skn
self.gap = gap
@property
def nbands(self):
return self.eps_skn.shape[2]
def get_calcdata(calc):
kpt_kc = calc.get_bz_k_points()
# Make sure there is no symmetry reduction
assert len(calc.get_ibz_k_points()) == len(kpt_kc)
lumo = calc.get_homo_lumo()[1]
gap = bandgap(calc=calc)[0]
return CalcData(
kpt_kc=kpt_kc,
atoms=calc.get_atoms(),
fermi_level=calc.get_fermi_level(),
lumo=lumo,
eps_skn=get_eigenvalues(calc),
gap=gap)
[docs]class Wannier:
"""Partly occupied Wannier functions
Find the set of partly occupied Wannier functions according to
Thygesen, Hansen and Jacobsen PRB v72 i12 p125119 2005.
"""
def __init__(self, nwannier, calc,
file=None,
nbands=None,
fixedenergy=None,
fixedstates=None,
spin=0,
initialwannier='orbitals',
functional='std',
rng=np.random,
log=silent):
"""
Required arguments:
``nwannier``: The number of Wannier functions you wish to construct.
This must be at least half the number of electrons in the system
and at most equal to the number of bands in the calculation.
It can also be set to 'auto' in order to automatically choose the
minimum number of needed Wannier function based on the
``fixedenergy`` / ``fixedstates`` parameter.
``calc``: A converged DFT calculator class.
If ``file`` arg. is not provided, the calculator *must* provide the
method ``get_wannier_localization_matrix``, and contain the
wavefunctions (save files with only the density is not enough).
If the localization matrix is read from file, this is not needed,
unless ``get_function`` or ``write_cube`` is called.
Optional arguments:
``nbands``: Bands to include in localization.
The number of bands considered by Wannier can be smaller than the
number of bands in the calculator. This is useful if the highest
bands of the DFT calculation are not well converged.
``spin``: The spin channel to be considered.
The Wannier code treats each spin channel independently.
``fixedenergy`` / ``fixedstates``: Fixed part of Hilbert space.
Determine the fixed part of Hilbert space by either a maximal
energy *or* a number of bands (possibly a list for multiple
k-points).
Default is None meaning that the number of fixed states is equated
to ``nwannier``.
The maximal energy is relative to the CBM if there is a finite
bandgap or to the Fermi level if there is none.
``file``: Read localization and rotation matrices from this file.
``initialwannier``: Initial guess for Wannier rotation matrix.
Can be 'bloch' to start from the Bloch states, 'random' to be
randomized, 'orbitals' to start from atom-centered d-orbitals and
randomly placed gaussian centers (see init_orbitals()),
'scdm' to start from localized state selected with SCDM
or a list passed to calc.get_initial_wannier.
``functional``: The functional used to measure the localization.
Can be 'std' for the standard quadratic functional from the PRB
paper, 'var' to add a variance minimizing term.
``rng``: Random number generator for ``initialwannier``.
``log``: Function which logs, such as print().
"""
# Bloch phase sign convention.
# May require special cases depending on which code is used.
sign = -1
self.log = log
self.calc = calc
self.spin = spin
self.functional = functional
self.initialwannier = initialwannier
self.log('Using functional:', functional)
self.calcdata = get_calcdata(calc)
self.kptgrid = get_monkhorst_pack_size_and_offset(self.kpt_kc)[0]
self.calcdata.kpt_kc *= sign
self.largeunitcell_cc = (self.unitcell_cc.T * self.kptgrid).T
self.weight_d, self.Gdir_dc = calculate_weights(self.largeunitcell_cc)
assert len(self.weight_d) == len(self.Gdir_dc)
if nbands is None:
# XXX Can work with other number of bands than calculator.
# Is this case properly tested, lest we confuse them?
nbands = self.calcdata.nbands
self.nbands = nbands
self.fixedstates_k, self.nwannier = choose_states(
self.calcdata, fixedenergy, fixedstates, self.Nk, nwannier,
log, spin)
# Compute the number of extra degrees of freedom (EDF)
self.edf_k = self.nwannier - self.fixedstates_k
self.log(f'Wannier: Fixed states : {self.fixedstates_k}')
self.log(f'Wannier: Extra degrees of freedom: {self.edf_k}')
self.kklst_dk, k0_dkc = get_kklst(self.kpt_kc, self.Gdir_dc)
# Set the inverse list of neighboring k-points
self.invkklst_dk = get_invkklst(self.kklst_dk)
Nw = self.nwannier
Nb = self.nbands
self.Z_dkww = np.empty((self.Ndir, self.Nk, Nw, Nw), complex)
self.V_knw = np.zeros((self.Nk, Nb, Nw), complex)
if file is None:
self.Z_dknn = self.new_Z(calc, k0_dkc)
self.initialize(file=file, initialwannier=initialwannier, rng=rng)
@property
def atoms(self):
return self.calcdata.atoms
@property
def kpt_kc(self):
return self.calcdata.kpt_kc
@property
def Ndir(self):
return len(self.weight_d) # Number of directions
@property
def Nk(self):
return len(self.kpt_kc)
def new_Z(self, calc, k0_dkc):
Nb = self.nbands
Z_dknn = np.empty((self.Ndir, self.Nk, Nb, Nb), complex)
for d, dirG in enumerate(self.Gdir_dc):
for k in range(self.Nk):
k1 = self.kklst_dk[d, k]
k0_c = k0_dkc[d, k]
Z_dknn[d, k] = calc.get_wannier_localization_matrix(
nbands=Nb, dirG=dirG, kpoint=k, nextkpoint=k1,
G_I=k0_c, spin=self.spin)
return Z_dknn
@property
def unitcell_cc(self):
return self.atoms.cell
@property
def U_kww(self):
return self.wannier_state.U_kww
@property
def C_kul(self):
return self.wannier_state.C_kul
[docs] def initialize(self, file=None, initialwannier='random', rng=np.random):
"""Re-initialize current rotation matrix.
Keywords are identical to those of the constructor.
"""
from ase.dft.wannierstate import WannierSpec, WannierState
spec = WannierSpec(self.Nk, self.nwannier, self.nbands,
self.fixedstates_k)
if file is not None:
with paropen(file, 'r') as fd:
Z_dknn, U_kww, C_kul = read_json(fd, always_array=False)
self.Z_dknn = Z_dknn
wannier_state = WannierState(C_kul, U_kww)
elif initialwannier == 'bloch':
# Set U and C to pick the lowest Bloch states
wannier_state = spec.bloch(self.edf_k)
elif initialwannier == 'random':
wannier_state = spec.random(rng, self.edf_k)
elif initialwannier == 'orbitals':
orbitals = init_orbitals(self.atoms, self.nwannier, rng)
wannier_state = spec.initial_orbitals(
self.calc, orbitals, self.kptgrid, self.edf_k, self.spin)
elif initialwannier == 'scdm':
wannier_state = spec.scdm(self.calc, self.kpt_kc, self.spin)
else:
wannier_state = spec.initial_wannier(
self.calc, initialwannier, self.kptgrid,
self.edf_k, self.spin)
self.wannier_state = wannier_state
self.update()
[docs] def save(self, file):
"""Save information on localization and rotation matrices to file."""
with paropen(file, 'w') as fd:
write_json(fd, (self.Z_dknn, self.U_kww, self.C_kul))
def update(self):
# Update large rotation matrix V (from rotation U and coeff C)
for k, M in enumerate(self.fixedstates_k):
self.V_knw[k, :M] = self.U_kww[k, :M]
if M < self.nwannier:
self.V_knw[k, M:] = self.C_kul[k] @ self.U_kww[k, M:]
# else: self.V_knw[k, M:] = 0.0
# Calculate the Zk matrix from the large rotation matrix:
# Zk = V^d[k] Zbloch V[k1]
for d in range(self.Ndir):
for k in range(self.Nk):
k1 = self.kklst_dk[d, k]
self.Z_dkww[d, k] = dag(self.V_knw[k]) \
@ (self.Z_dknn[d, k] @ self.V_knw[k1])
# Update the new Z matrix
self.Z_dww = self.Z_dkww.sum(axis=1) / self.Nk
[docs] def get_optimal_nwannier(self, nwrange=5, random_reps=5, tolerance=1e-6):
"""
The optimal value for 'nwannier', maybe.
The optimal value is the one that gives the lowest average value for
the spread of the most delocalized Wannier function in the set.
``nwrange``: number of different values to try for 'nwannier', the
values will span a symmetric range around ``nwannier`` if possible.
``random_reps``: number of repetitions with random seed, the value is
then an average over these repetitions.
``tolerance``: tolerance for the gradient descent algorithm, can be
useful to increase the speed, with a cost in accuracy.
"""
# Define the range of values to try based on the maximum number of fixed
# states (that is the minimum number of WFs we need) and the number of
# available bands we have.
max_number_fixedstates = np.max(self.fixedstates_k)
min_range_value = max(self.nwannier - int(np.floor(nwrange / 2)),
max_number_fixedstates)
max_range_value = min(min_range_value + nwrange, self.nbands + 1)
Nws = np.arange(min_range_value, max_range_value)
# If there is no randomness, there is no need to repeat
random_initials = ['random', 'orbitals']
if self.initialwannier not in random_initials:
random_reps = 1
t = -time()
avg_max_spreads = np.zeros(len(Nws))
for j, Nw in enumerate(Nws):
self.log('Trying with Nw =', Nw)
# Define once with the fastest 'initialwannier',
# then initialize with random seeds in the for loop
wan = Wannier(nwannier=int(Nw),
calc=self.calc,
nbands=self.nbands,
spin=self.spin,
functional=self.functional,
initialwannier='bloch',
log=self.log,
rng=self.rng)
wan.fixedstates_k = self.fixedstates_k
wan.edf_k = wan.nwannier - wan.fixedstates_k
max_spreads = np.zeros(random_reps)
for i in range(random_reps):
wan.initialize(initialwannier=self.initialwannier)
wan.localize(tolerance=tolerance)
max_spreads[i] = np.max(wan.get_spreads())
avg_max_spreads[j] = max_spreads.mean()
self.log('Average spreads: ', avg_max_spreads)
t += time()
self.log(f'Execution time: {t:.1f}s')
return Nws[np.argmin(avg_max_spreads)]
[docs] def get_centers(self, scaled=False):
"""Calculate the Wannier centers
::
pos = L / 2pi * phase(diag(Z))
"""
coord_wc = np.angle(self.Z_dww[:3].diagonal(0, 1, 2)).T / (2 * pi) % 1
if not scaled:
coord_wc = coord_wc @ self.largeunitcell_cc
return coord_wc
[docs] def get_radii(self):
r"""Calculate the spread of the Wannier functions.
::
-- / L \ 2 2
radius**2 = - > | --- | ln |Z|
--d \ 2pi /
Note that this function can fail with some Bravais lattices,
see `get_spreads()` for a more robust alternative.
"""
r2 = - (self.largeunitcell_cc.diagonal()**2 / (2 * pi)**2) \
@ np.log(abs(self.Z_dww[:3].diagonal(0, 1, 2))**2)
return np.sqrt(r2)
[docs] def get_spreads(self):
r"""Calculate the spread of the Wannier functions in Ų.
The definition is based on eq. 13 in PRB61-15 by Berghold and Mundy.
::
/ 1 \ 2 -- 2
spread = - |----| > W_d ln |Z_dw|
\2 pi/ --d
"""
# compute weights without normalization, to keep physical dimension
weight_d, _ = calculate_weights(self.largeunitcell_cc, normalize=False)
Z2_dw = square_modulus_of_Z_diagonal(self.Z_dww)
spread_w = - (np.log(Z2_dw).T @ weight_d).real / (2 * np.pi)**2
return spread_w
def get_spectral_weight(self, w):
return abs(self.V_knw[:, :, w])**2 / self.Nk
[docs] def get_pdos(self, w, energies, width):
"""Projected density of states (PDOS).
Returns the (PDOS) for Wannier function ``w``. The calculation
is performed over the energy grid specified in energies. The
PDOS is produced as a sum of Gaussians centered at the points
of the energy grid and with the specified width.
"""
spec_kn = self.get_spectral_weight(w)
dos = np.zeros(len(energies))
for k, spec_n in enumerate(spec_kn):
eig_n = self.calcdata.eps_skn[self.spin, k]
for weight, eig in zip(spec_n, eig_n):
# Add gaussian centered at the eigenvalue
x = ((energies - eig) / width)**2
dos += weight * np.exp(-x.clip(0., 40.)) / (sqrt(pi) * width)
return dos
[docs] def translate(self, w, R):
"""Translate the w'th Wannier function
The distance vector R = [n1, n2, n3], is in units of the basis
vectors of the small cell.
"""
for kpt_c, U_ww in zip(self.kpt_kc, self.U_kww):
U_ww[:, w] *= np.exp(2.j * pi * (np.array(R) @ kpt_c))
self.update()
[docs] def translate_to_cell(self, w, cell):
"""Translate the w'th Wannier function to specified cell"""
scaled_c = np.angle(self.Z_dww[:3, w, w]) * self.kptgrid / (2 * pi)
trans = np.array(cell) - np.floor(scaled_c)
self.translate(w, trans)
[docs] def translate_all_to_cell(self, cell=(0, 0, 0)):
r"""Translate all Wannier functions to specified cell.
Move all Wannier orbitals to a specific unit cell. There
exists an arbitrariness in the positions of the Wannier
orbitals relative to the unit cell. This method can move all
orbitals to the unit cell specified by ``cell``. For a
`\Gamma`-point calculation, this has no effect. For a
**k**-point calculation the periodicity of the orbitals are
given by the large unit cell defined by repeating the original
unitcell by the number of **k**-points in each direction. In
this case it is useful to move the orbitals away from the
boundaries of the large cell before plotting them. For a bulk
calculation with, say 10x10x10 **k** points, one could move
the orbitals to the cell [2,2,2]. In this way the pbc
boundary conditions will not be noticed.
"""
scaled_wc = (np.angle(self.Z_dww[:3].diagonal(0, 1, 2)).T *
self.kptgrid / (2 * pi))
trans_wc = np.array(cell)[None] - np.floor(scaled_wc)
for kpt_c, U_ww in zip(self.kpt_kc, self.U_kww):
U_ww *= np.exp(2.j * pi * (trans_wc @ kpt_c))
self.update()
[docs] def distances(self, R):
"""Relative distances between centers.
Returns a matrix with the distances between different Wannier centers.
R = [n1, n2, n3] is in units of the basis vectors of the small cell
and allows one to measure the distance with centers moved to a
different small cell.
The dimension of the matrix is [Nw, Nw].
"""
Nw = self.nwannier
cen = self.get_centers()
r1 = cen.repeat(Nw, axis=0).reshape(Nw, Nw, 3)
r2 = cen.copy()
for i in range(3):
r2 += self.unitcell_cc[i] * R[i]
r2 = np.swapaxes(r2.repeat(Nw, axis=0).reshape(Nw, Nw, 3), 0, 1)
return np.sqrt(np.sum((r1 - r2)**2, axis=-1))
@functools.lru_cache(maxsize=10000)
def _get_hopping(self, n1, n2, n3):
"""Returns the matrix H(R)_nm=<0,n|H|R,m>.
::
1 _ -ik.R
H(R) = <0,n|H|R,m> = --- >_ e H(k)
Nk k
where R = (n1, n2, n3) is the cell-distance (in units of the basis
vectors of the small cell) and n,m are indices of the Wannier functions.
This function caches up to 'maxsize' results.
"""
R = np.array([n1, n2, n3], float)
H_ww = np.zeros([self.nwannier, self.nwannier], complex)
for k, kpt_c in enumerate(self.kpt_kc):
phase = np.exp(-2.j * pi * (np.array(R) @ kpt_c))
H_ww += self.get_hamiltonian(k) * phase
return H_ww / self.Nk
[docs] def get_hopping(self, R):
"""Returns the matrix H(R)_nm=<0,n|H|R,m>.
::
1 _ -ik.R
H(R) = <0,n|H|R,m> = --- >_ e H(k)
Nk k
where R is the cell-distance (in units of the basis vectors of
the small cell) and n,m are indices of the Wannier functions.
"""
return self._get_hopping(R[0], R[1], R[2])
[docs] def get_hamiltonian(self, k):
"""Get Hamiltonian at existing k-vector of index k
::
dag
H(k) = V diag(eps ) V
k k k
"""
eps_n = self.calcdata.eps_skn[self.spin, k, :self.nbands]
V_nw = self.V_knw[k]
return (dag(V_nw) * eps_n) @ V_nw
[docs] def get_hamiltonian_kpoint(self, kpt_c):
"""Get Hamiltonian at some new arbitrary k-vector
::
_ ik.R
H(k) = >_ e H(R)
R
Warning: This method moves all Wannier functions to cell (0, 0, 0)
"""
self.log('Translating all Wannier functions to cell (0, 0, 0)')
self.translate_all_to_cell()
max = (self.kptgrid - 1) // 2
N1, N2, N3 = max
Hk = np.zeros([self.nwannier, self.nwannier], complex)
for n1 in range(-N1, N1 + 1):
for n2 in range(-N2, N2 + 1):
for n3 in range(-N3, N3 + 1):
R = np.array([n1, n2, n3], float)
hop_ww = self.get_hopping(R)
phase = np.exp(+2.j * pi * (R @ kpt_c))
Hk += hop_ww * phase
return Hk
[docs] def get_function(self, index, repeat=None):
r"""Get Wannier function on grid.
Returns an array with the funcion values of the indicated Wannier
function on a grid with the size of the *repeated* unit cell.
For a calculation using **k**-points the relevant unit cell for
eg. visualization of the Wannier orbitals is not the original unit
cell, but rather a larger unit cell defined by repeating the
original unit cell by the number of **k**-points in each direction.
Note that for a `\Gamma`-point calculation the large unit cell
coinsides with the original unit cell.
The large unitcell also defines the periodicity of the Wannier
orbitals.
``index`` can be either a single WF or a coordinate vector in terms
of the WFs.
"""
# Default size of plotting cell is the one corresponding to k-points.
if repeat is None:
repeat = self.kptgrid
N1, N2, N3 = repeat
dim = self.calc.get_number_of_grid_points()
largedim = dim * [N1, N2, N3]
wanniergrid = np.zeros(largedim, dtype=complex)
for k, kpt_c in enumerate(self.kpt_kc):
# The coordinate vector of wannier functions
if isinstance(index, int):
vec_n = self.V_knw[k, :, index]
else:
vec_n = self.V_knw[k] @ index
wan_G = np.zeros(dim, complex)
for n, coeff in enumerate(vec_n):
wan_G += coeff * self.calc.get_pseudo_wave_function(
n, k, self.spin, pad=True)
# Distribute the small wavefunction over large cell:
for n1 in range(N1):
for n2 in range(N2):
for n3 in range(N3): # sign?
e = np.exp(-2.j * pi * np.array([n1, n2, n3]) @ kpt_c)
wanniergrid[n1 * dim[0]:(n1 + 1) * dim[0],
n2 * dim[1]:(n2 + 1) * dim[1],
n3 * dim[2]:(n3 + 1) * dim[2]] += e * wan_G
# Normalization
wanniergrid /= np.sqrt(self.Nk)
return wanniergrid
[docs] def write_cube(self, index, fname, repeat=None, angle=False):
"""
Dump specified Wannier function to a cube file.
Arguments:
``index``: Integer, index of the Wannier function to save.
``repeat``: Array of integer, repeat supercell and Wannier function.
``fname``: Name of the cube file.
``angle``: If False, save the absolute value. If True, save
the complex phase of the Wannier function.
"""
from ase.io import write
# Default size of plotting cell is the one corresponding to k-points.
if repeat is None:
repeat = self.kptgrid
# Remove constraints, some are not compatible with repeat()
atoms = self.atoms.copy()
atoms.set_constraint()
atoms = atoms * repeat
func = self.get_function(index, repeat)
# Compute absolute value or complex angle
if angle:
data = np.angle(func)
else:
if self.Nk == 1:
func *= np.exp(-1.j * np.angle(func.max()))
func = abs(func)
data = func
write(fname, atoms, data=data, format='cube')
[docs] def localize(self, step=0.25, tolerance=1e-08,
updaterot=True, updatecoeff=True):
"""Optimize rotation to give maximal localization"""
md_min(self, step=step, tolerance=tolerance, log=self.log,
updaterot=updaterot, updatecoeff=updatecoeff)
[docs] def get_functional_value(self):
"""Calculate the value of the spread functional.
::
Tr[|ZI|^2]=sum(I)sum(n) w_i|Z_(i)_nn|^2,
where w_i are weights.
If the functional is set to 'var' it subtracts a variance term
::
Nw * var(sum(n) w_i|Z_(i)_nn|^2),
where Nw is the number of WFs ``nwannier``.
"""
a_w = self._spread_contributions()
if self.functional == 'std':
fun = np.sum(a_w)
elif self.functional == 'var':
fun = np.sum(a_w) - self.nwannier * np.var(a_w)
self.log(f'std: {np.sum(a_w):.4f}',
f'\tvar: {self.nwannier * np.var(a_w):.4f}')
return fun
def get_gradients(self):
# Determine gradient of the spread functional.
#
# The gradient for a rotation A_kij is::
#
# dU = dRho/dA_{k,i,j} = sum(I) sum(k')
# + Z_jj Z_kk',ij^* - Z_ii Z_k'k,ij^*
# - Z_ii^* Z_kk',ji + Z_jj^* Z_k'k,ji
#
# The gradient for a change of coefficients is::
#
# dRho/da^*_{k,i,j} = sum(I) [[(Z_0)_{k} V_{k'} diag(Z^*) +
# (Z_0_{k''})^d V_{k''} diag(Z)] *
# U_k^d]_{N+i,N+j}
#
# where diag(Z) is a square,diagonal matrix with Z_nn in the diagonal,
# k' = k + dk and k = k'' + dk.
#
# The extra degrees of freedom chould be kept orthonormal to the fixed
# space, thus we introduce lagrange multipliers, and minimize instead::
#
# Rho_L = Rho - sum_{k,n,m} lambda_{k,nm} <c_{kn}|c_{km}>
#
# for this reason the coefficient gradients should be multiplied
# by (1 - c c^dag).
Nb = self.nbands
Nw = self.nwannier
if self.functional == 'var':
O_w = self._spread_contributions()
O_sum = np.sum(O_w)
dU = []
dC = []
for k in range(self.Nk):
M = self.fixedstates_k[k]
L = self.edf_k[k]
U_ww = self.U_kww[k]
C_ul = self.C_kul[k]
Utemp_ww = np.zeros((Nw, Nw), complex)
Ctemp_nw = np.zeros((Nb, Nw), complex)
for d, weight in enumerate(self.weight_d):
if abs(weight) < 1.0e-6:
continue
Z_knn = self.Z_dknn[d]
diagZ_w = self.Z_dww[d].diagonal()
Zii_ww = np.repeat(diagZ_w, Nw).reshape(Nw, Nw)
if self.functional == 'var':
diagOZ_w = O_w * diagZ_w
OZii_ww = np.repeat(diagOZ_w, Nw).reshape(Nw, Nw)
k1 = self.kklst_dk[d, k]
k2 = self.invkklst_dk[d, k]
V_knw = self.V_knw
Z_kww = self.Z_dkww[d]
if L > 0:
Ctemp_nw += weight * (
((Z_knn[k] @ V_knw[k1]) * diagZ_w.conj() +
(dag(Z_knn[k2]) @ V_knw[k2]) * diagZ_w) @ dag(U_ww))
if self.functional == 'var':
# Gradient of the variance term, split in two terms
def variance_term_computer(factor):
result = (
self.nwannier * 2 * weight * (
((Z_knn[k] @ V_knw[k1]) * factor.conj() +
(dag(Z_knn[k2]) @ V_knw[k2]) * factor) @
dag(U_ww)) / Nw**2
)
return result
first_term = \
O_sum * variance_term_computer(diagZ_w) / Nw**2
second_term = \
- variance_term_computer(diagOZ_w) / Nw
Ctemp_nw += first_term + second_term
temp = Zii_ww.T * Z_kww[k].conj() - Zii_ww * Z_kww[k2].conj()
Utemp_ww += weight * (temp - dag(temp))
if self.functional == 'var':
Utemp_ww += (self.nwannier * 2 * O_sum * weight *
(temp - dag(temp)) / Nw**2)
temp = (OZii_ww.T * Z_kww[k].conj()
- OZii_ww * Z_kww[k2].conj())
Utemp_ww -= (self.nwannier * 2 * weight *
(temp - dag(temp)) / Nw)
dU.append(Utemp_ww.ravel())
if L > 0:
# Ctemp now has same dimension as V, the gradient is in the
# lower-right (Nb-M) x L block
Ctemp_ul = Ctemp_nw[M:, M:]
G_ul = Ctemp_ul - ((C_ul @ dag(C_ul)) @ Ctemp_ul)
dC.append(G_ul.ravel())
return np.concatenate(dU + dC)
def _spread_contributions(self):
"""
Compute the contribution of each WF to the spread functional.
"""
return (square_modulus_of_Z_diagonal(self.Z_dww).T
@ self.weight_d).real
def step(self, dX, updaterot=True, updatecoeff=True):
# dX is (A, dC) where U->Uexp(-A) and C->C+dC
Nw = self.nwannier
Nk = self.Nk
M_k = self.fixedstates_k
L_k = self.edf_k
if updaterot:
A_kww = dX[:Nk * Nw**2].reshape(Nk, Nw, Nw)
for U, A in zip(self.U_kww, A_kww):
H = -1.j * A.conj()
epsilon, Z = np.linalg.eigh(H)
# Z contains the eigenvectors as COLUMNS.
# Since H = iA, dU = exp(-A) = exp(iH) = ZDZ^d
dU = Z * np.exp(1.j * epsilon) @ dag(Z)
if U.dtype == float:
U[:] = (U @ dU).real
else:
U[:] = U @ dU
if updatecoeff:
start = 0
for C, unocc, L in zip(self.C_kul, self.nbands - M_k, L_k):
if L == 0 or unocc == 0:
continue
Ncoeff = L * unocc
deltaC = dX[Nk * Nw**2 + start: Nk * Nw**2 + start + Ncoeff]
C += deltaC.reshape(unocc, L)
gram_schmidt(C)
start += Ncoeff
self.update()