def unfold_points(self, k):
r""" Return a list of k-points to be evaluated for this objects unfolding
The k-point `k` is with respect to the unfolded geometry.
The return list of `k` points are the k-points required to be sampled in the
folded geometry.
Parameters
----------
k : (3,) of float
k-point evaluation corresponding to the unfolded unit-cell
Returns
-------
k_unfold
a list of ``np.prod(self.bloch)`` k-points used for the unfolding
"""
k = _a.arrayd(k)
# Create expansion points
B = self._bloch
unfold = _a.emptyd([B[2], B[1], B[0], 3])
# Use B-casting rules (much simpler)
unfold[:, :, :, 0] = (aranged(B[0]).reshape(1, 1, -1) + k[0]) / B[0]
unfold[:, :, :, 1] = (aranged(B[1]).reshape(1, -1, 1) + k[1]) / B[1]
unfold[:, :, :, 2] = (aranged(B[2]).reshape(-1, 1, 1) + k[2]) / B[2]
# Back-transform shape
return unfold.reshape(-1, 3)
def test_oplist_math(op, key1, key2):
d = {
1: oplist([ar.aranged(1, 10), ar.aranged(1, 10)]),
2: 2,
}
l1 = d[key1]
l2 = d[key2]
op(l1, l2)
def test_oplist_imath(op, key):
d = {
1: oplist([ar.aranged(1, 10), ar.aranged(1, 10)]),
2: 2,
}
l2 = d[key]
try:
l1 = oplist([data * 2 for data in l2])
except:
l1 = oplist([ar.aranged(1, 10), ar.aranged(2, 3)])
op(l1, l2)
def grid(n, displ=0., size=1., trs=False):
r""" Create a grid of `n` points with an offset of `displ` and sampling `size` around `displ`
The :math:`k`-points are :math:`\Gamma` centered.
Parameters
----------
n : int
number of points in the grid. If `trs` is ``True`` this may be smaller than `n`
displ : float, optional
the displacement of the grid
size : float, optional
the total size of the Brillouin zone to sample
trs : bool, optional
whether time-reversal-symmetry is applied
Returns
-------
k : np.ndarray
the list of k-points in the Brillouin zone to be sampled
w : np.ndarray
weights for the k-points
"""
# First ensure that displ is in the Brillouin
displ = displ % 1.
if displ > 0.5:
displ -= 1.
if trs and displ == 0.:
n_half = n // 2
if n % 2 == 1:
# Odd case, we have Gamma and remove all negative values
k = _a.aranged(n_half + 1) * size / n + displ
# Weights are all twice (except Gamma)
w = _a.onesd(len(k)) / n * size
w[1:] *= 2
else:
# Even case, we do not have Gamma, but we shift to Gamma
# All points except Gamma and edge have weights doubled
k = _a.aranged(n_half + 1) * size / n + displ
# Weights are all twice (except Gamma and band-edge)
w = _a.onesd(len(k)) / n * size
w[1:-1] *= 2
else:
# Not TRS
if n % 2 == 0:
k = (_a.aranged(n) * 2 - n) * size / (2 * n) + displ
else:
k = (_a.aranged(n) * 2 - n + 1) * size / (2 * n) + displ
w = _a.onesd(n) * size / n
# Return values
return k, w
def read_basis(self):
""" Returns data associated with the ion.xml file """
# Get the element-tree
ET = ElementTree(None, self.file)
root = ET.getroot()
# Get number of orbitals
label = root.find('label').text.strip()
Z = int(root.find('z').text) # atomic number
mass = float(root.find('mass').text)
# Read in the PAO's
paos = root.find('paos')
# Now loop over all orbitals
orbital = []
# All orbital data
Bohr2Ang = unit_convert('Bohr', 'Ang')
for orb in paos:
n = int(orb.get('n'))
l = int(orb.get('l'))
z = int(orb.get('z')) # zeta
q0 = float(orb.get('population'))
P = not int(orb.get('ispol')) == 0
# Radial components
rad = orb.find('radfunc')
npts = int(rad.find('npts').text)
# Grid spacing in Bohr (conversion is done later
# because the normalization is easier)
delta = float(rad.find('delta').text)
# Read in data to a list
dat = list(map(float, rad.find('data').text.split()))
# Since the readed data has fewer significant digits we
# might as well re-create the table of the radial component.
r = aranged(npts) * delta
# To get it per Ang**3
# TODO, check that this is correct.
# The fact that we have to have it normalized means that we need
# to convert psi /sqrt(Bohr**3) -> /sqrt(Ang**3)
# \int psi^\dagger psi == 1
psi = arrayd(dat[1::2]) * r**l / Bohr2Ang**(3. / 2.)
# Create the sphericalorbital and then the atomicorbital
sorb = SphericalOrbital(l, (r * Bohr2Ang, psi), q0)
# This will be -l:l (this is the way siesta does it)
orbital.extend(sorb.toAtomicOrbital(n=n, Z=z, P=P))
# Now create the atom and return
return Atom(Z, orbital, mass=mass, tag=label)
def _index_shape_cuboid(self, cuboid):
""" Internal routine for cuboid shape-indices """
# Construct all points on the outer rim of the cuboids
min_d = fnorm(self.dcell).min()
# Retrieve cuboids edge-lengths
v = cuboid.edge_length
# Create normalized cuboid vectors (because we expan via the lengths below
vn = cuboid._v / fnorm(cuboid._v).reshape(-1, 1)
LL = (cuboid.center - cuboid._v.sum(0) / 2).reshape(1, 3)
UR = (cuboid.center + cuboid._v.sum(0) / 2).reshape(1, 3)
# Create coordinates
a = vn[0, :].reshape(1, -1) * _a.aranged(0, v[0] + min_d, min_d).reshape(-1, 1)
b = vn[1, :].reshape(1, -1) * _a.aranged(0, v[1] + min_d, min_d).reshape(-1, 1)
c = vn[2, :].reshape(1, -1) * _a.aranged(0, v[2] + min_d, min_d).reshape(-1, 1)
# Now create all sides
sa = a.shape[0]
sb = b.shape[0]
sc = c.shape[0]
def plane(v1, v2):
return (v1.reshape(-1, 1, 3) + v2.reshape(1, -1, 3)).reshape(1, -1, 3)
# Allocate for the 6 faces of the cuboid
rxyz = _a.emptyd([2, sa * sb + sa * sc + sb * sc, 3])
# Define the LL and UR
rxyz[0, :, :] = LL
rxyz[1, :, :] = UR
i = 0
rxyz[:, i:i + sa * sb, :] += plane(a, b)
i += sa * sb
rxyz[:, i:i + sa * sc, :] += plane(a, c)
i += sa * sc
rxyz[:, i:i + sb * sc, :] += plane(b, c)
del a, b, c, sa, sb, sc
rxyz.shape = (-1, 3)
# Get all indices of the cuboid planes
return self.index(rxyz)
def _index_shape(self, shape):
""" Internal routine for shape-indices """
# First grab the sphere, subsequent indices will be reduced
# by the actual shape
cuboid = shape.toCuboid()
ellipsoid = shape.toEllipsoid()
if ellipsoid.volume() > cuboid.volume():
idx = self._index_shape_cuboid(cuboid)
else:
idx = self._index_shape_ellipsoid(ellipsoid)
# Get min/max
imin = idx.min(0)
imax = idx.max(0)
del idx
dc = self.dcell
# Now to find the actual points inside the shape
# First create all points in the square and then retrieve all indices
# within.
ix = _a.aranged(imin[0], imax[0] + 0.5)
iy = _a.aranged(imin[1], imax[1] + 0.5)
iz = _a.aranged(imin[2], imax[2] + 0.5)
output_shape = (ix.size, iy.size, iz.size, 3)
rxyz = _a.emptyd(output_shape)
ao = add.outer
ao(ao(ix * dc[0, 0], iy * dc[1, 0]), iz * dc[2, 0], out=rxyz[:, :, :, 0])
ao(ao(ix * dc[0, 1], iy * dc[1, 1]), iz * dc[2, 1], out=rxyz[:, :, :, 1])
ao(ao(ix * dc[0, 2], iy * dc[1, 2]), iz * dc[2, 2], out=rxyz[:, :, :, 2])
idx = shape.within_index(rxyz.reshape(-1, 3))
del rxyz
i = _a.emptyi(output_shape)
i[:, :, :, 0] = ix.reshape(-1, 1, 1)
i[:, :, :, 1] = iy.reshape(1, -1, 1)
i[:, :, :, 2] = iz.reshape(1, 1, -1)
del ix, iy, iz
i.shape = (-1, 3)
i = take(i, idx, axis=0)
del idx
return i
def read_basis(self):
""" Returns data associated with the ion.xml file """
no = len(self._dimension('norbs'))
# Get number of orbitals
label = self.Label.strip()
Z = int(self.Atomic_number)
mass = float(self.Mass)
# Retrieve values
orb_l = self._variable('orbnl_l')[:] # angular quantum number
orb_n = self._variable('orbnl_n')[:] # principal quantum number
orb_z = self._variable('orbnl_z')[:] # zeta
orb_P = self._variable(
'orbnl_ispol')[:] > 0 # polarization shell, or not
orb_q0 = self._variable('orbnl_pop')[:] # q0 for the orbitals
orb_delta = self._variable('delta')[:] # delta for the functions
orb_psi = self._variable('orb')[:, :]
# Now loop over all orbitals
orbital = []
# All orbital data
Bohr2Ang = unit_convert('Bohr', 'Ang')
for io in range(no):
n = orb_n[io]
l = orb_l[io]
z = orb_z[io]
P = orb_P[io]
# Grid spacing in Bohr (conversion is done later
# because the normalization is easier)
delta = orb_delta[io]
# Since the readed data has fewer significant digits we
# might as well re-create the table of the radial component.
r = aranged(orb_psi.shape[1]) * delta
# To get it per Ang**3
# TODO, check that this is correct.
# The fact that we have to have it normalized means that we need
# to convert psi /sqrt(Bohr**3) -> /sqrt(Ang**3)
# \int psi^\dagger psi == 1
psi = orb_psi[io, :] * r**l / Bohr2Ang**(3. / 2.)
# Create the sphericalorbital and then the atomicorbital
sorb = SphericalOrbital(l, (r * Bohr2Ang, psi), orb_q0[io])
# This will be -l:l (this is the way siesta does it)
orbital.extend(sorb.toAtomicOrbital(n=n, Z=z, P=P))
# Now create the atom and return
return Atom(Z, orbital, mass=mass, tag=label)
def read_basis(self):
""" Returns a set of atoms corresponding to the basis-sets in the nc file """
if 'BASIS' not in self.groups:
return None
basis = self.groups['BASIS']
atom = [None] * len(basis.groups)
for a_str in basis.groups:
a = basis.groups[a_str]
if 'orbnl_l' not in a.variables:
# Do the easy thing.
# Get number of orbitals
label = a.Label.strip()
Z = int(a.Atomic_number)
mass = float(a.Mass)
i = int(a.ID) - 1
atom[i] = Atom(Z, [-1] * a.Number_of_orbitals,
mass=mass,
tag=label)
continue
# Retrieve values
orb_l = a.variables['orbnl_l'][:] # angular quantum number
orb_n = a.variables['orbnl_n'][:] # principal quantum number
orb_z = a.variables['orbnl_z'][:] # zeta
orb_P = a.variables[
'orbnl_ispol'][:] > 0 # polarization shell, or not
orb_q0 = a.variables['orbnl_pop'][:] # q0 for the orbitals
orb_delta = a.variables['delta'][:] # delta for the functions
orb_psi = a.variables['orb'][:, :]
# Now loop over all orbitals
orbital = []
# Number of basis-orbitals (before m-expansion)
no = len(a.dimensions['norbs'])
# All orbital data
for io in range(no):
n = orb_n[io]
l = orb_l[io]
z = orb_z[io]
P = orb_P[io]
# Grid spacing in Bohr (conversion is done later
# because the normalization is easier)
delta = orb_delta[io]
# Since the readed data has fewer significant digits we
# might as well re-create the table of the radial component.
r = aranged(orb_psi.shape[1]) * delta
# To get it per Ang**3
# TODO, check that this is correct.
# The fact that we have to have it normalized means that we need
# to convert psi /sqrt(Bohr**3) -> /sqrt(Ang**3)
# \int psi^\dagger psi == 1
psi = orb_psi[io, :] * r**l / Bohr2Ang**(3. / 2.)
# Create the sphericalorbital and then the atomicorbital
sorb = SphericalOrbital(l, (r * Bohr2Ang, psi), orb_q0[io])
# This will be -l:l (this is the way siesta does it)
orbital.extend(sorb.toAtomicOrbital(n=n, Z=z, P=P))
# Get number of orbitals
label = a.Label.strip()
Z = int(a.Atomic_number)
mass = float(a.Mass)
i = int(a.ID) - 1
atom[i] = Atom(Z, orbital, mass=mass, tag=label)
return atom
def tile(self, reps, axis, normalize=False, offset=0):
r"""Tile the state vectors for a new supercell
Tiling a state vector makes use of the Bloch factors for a state by utilizing
.. math::
\psi_{\mathbf k}(\mathbf r + \mathbf T) \propto e^{i\mathbf k\cdot \mathbf T}
where :math:`\mathbf T = i\mathbf a_0 + j\mathbf a_1 + l\mathbf a_2`. Note that `axis`
selects which of the :math:`\mathbf a_i` vectors that are translated and `reps` corresponds
to the :math:`i`, :math:`j` and :math:`l` variables. The `offset` moves the individual states
by said amount, i.e. :math:`i\to i+\mathrm{offset}`.
Parameters
----------
reps : int
number of repetitions along a specific lattice vector
axis : int
lattice vector to tile along
normalize: bool, optional
whether the states are normalized upon return, may be useful for
eigenstates
offset: float, optional
the offset for the phase factors
See Also
--------
Geometry.tile
"""
# the parent gets tiled
parent = self.parent.tile(reps, axis)
# the k-point gets reduced
k = _a.asarrayd(self.info.get("k", [0] * 3))
# now tile the state vectors
state = np.tile(self.state, (1, reps)).astype(np.complex128,
copy=False)
# re-shape to apply phase-factors
state.shape = (len(self), reps, -1)
# Tiling stuff is trivial since we simply
# translate the bloch coefficients with:
# exp(i k.T)
# with T being
# i * a_0 + j * a_1 + k * a_2
# We can leave out the lattice vectors entirely
phase = exp(2j * _pi * k[axis] * (_a.aranged(reps) + offset))
state *= phase.reshape(1, -1, 1)
state.shape = (len(self), -1)
# update new k; when we double the system, we halve the periodicity
# and hence we need to account for this
k[axis] = (k[axis] * reps % 1)
while k[axis] > 0.5:
k[axis] -= 1
while k[axis] <= -0.5:
k[axis] += 1
# this allows us to make the same usable for StateC classes
s = self.copy()
s.parent = parent
s.state = state
# update the k-point
s.info = dict(**self.info)
s.info.update({'k': k})
if normalize:
return s.normalize()
return s
def unfold(self, M, k_unfold):
r""" Unfold the matrix list of matrices `M` into a corresponding k-point (unfolding k-points are `k_unfold`)
Parameters
----------
M : list of numpy arrays
matrices used for unfolding
k_unfold : (*, 3) of float
unfolding k-points as returned by `Bloch.unfold_points`
Returns
-------
M_unfold : unfolded matrix of size ``M[0].shape * k_unfold.shape[0] ** 2``
"""
Bi, Bj, Bk = self.bloch
# Retrieve shapes
M0, M1 = M[0].shape
shape = (Bk, Bj, Bi, M0, Bk, Bj, Bi, M1)
Mshape = (M0, 1, 1, 1, M1)
# Allocate the unfolded matrix
Mu = zeros(shape, dtype=dtype_real_to_complex(M[0].dtype))
# Use B-casting rules (much simpler)
# Perform unfolding
N = len(self)
w = 1 / N
if Bi == 1:
if Bj == 1:
K = aranged(Bk).reshape(1, Bk, 1, 1, 1)
for T in range(N):
m = M[T].reshape(Mshape) * w
kjpi = 2j * pi * k_unfold[T, 2]
for k in range(Bk):
add(Mu[k, 0, 0], m * exp(kjpi * (K - k)), out=Mu[k, 0, 0])
elif Bk == 1:
J = aranged(Bj).reshape(1, 1, Bj, 1, 1)
for T in range(N):
m = M[T].reshape(Mshape) * w
kjpi = 2j * pi * k_unfold[T, 1]
for j in range(Bj):
add(Mu[0, j, 0], m * exp(kjpi * (J - j)), out=Mu[0, j, 0])
else:
J = aranged(Bj).reshape(1, 1, Bj, 1, 1)
K = aranged(Bk).reshape(1, Bk, 1, 1, 1)
for T in range(N):
m = M[T].reshape(Mshape) * w
kpi = pi * k_unfold[T, :]
for k in range(Bk):
Kk = (K - k) * kpi[2]
for j in range(Bj):
add(Mu[k, j, 0], m * exp(2j * (Kk + kpi[1] * (J - j))), out=Mu[k, j, 0])
elif Bj == 1:
if Bk == 1:
I = aranged(Bi).reshape(1, 1, 1, Bi, 1)
for T in range(N):
m = M[T].reshape(Mshape) * w
kjpi = 2j * pi * k_unfold[T, 0]
for i in range(Bi):
add(Mu[0, 0, i], m * exp(kjpi * (I - i)), out=Mu[0, 0, i])
else:
I = aranged(Bi).reshape(1, 1, 1, Bi, 1)
K = aranged(Bk).reshape(1, Bk, 1, 1, 1)
for T in range(N):
m = M[T].reshape(Mshape) * w
kpi = pi * k_unfold[T, :]
for k in range(Bk):
Kk = (K - k) * kpi[2]
for i in range(Bi):
add(Mu[k, 0, i], m * exp(2j * (Kk + kpi[0] * (I - i))), out=Mu[k, 0, i])
elif Bk == 1:
I = aranged(Bi).reshape(1, 1, 1, Bi, 1)
J = aranged(Bj).reshape(1, 1, Bj, 1, 1)
for T in range(N):
m = M[T].reshape(Mshape) * w
kpi = pi * k_unfold[T, :]
for j in range(Bj):
Jj = (J - j) * kpi[1]
for i in range(Bi):
add(Mu[0, j, i], m * exp(2j * (Jj + kpi[0] * (I - i))), out=Mu[0, j, i])
else:
I = aranged(Bi).reshape(1, 1, 1, Bi, 1)
J = aranged(Bj).reshape(1, 1, Bj, 1, 1)
K = aranged(Bk).reshape(1, Bk, 1, 1, 1)
for T in range(N):
m = M[T].reshape(Mshape) * w
kpi = pi * k_unfold[T, :]
for k in range(Bk):
Kk = (K - k) * kpi[2]
for j in range(Bj):
KkJj = Kk + (J - j) * kpi[1]
for i in range(Bi):
# Calculate phases and add for all expansions
add(Mu[k, j, i], m * exp(2j * (KkJj + kpi[0] * (I - i))), out=Mu[k, j, i])
return Mu.reshape(N * M0, N * M1)
def ArgumentParser(self, p=None, *args, **kwargs):
""" Returns the arguments that is available for this Sile """
#limit_args = kwargs.get('limit_arguments', True)
short = kwargs.get('short', False)
def opts(*args):
if short:
return args
return [args[0]]
# We limit the import to occur here
import argparse
Bohr2Ang = unit_convert('Bohr', 'Ang')
Ry2eV = unit_convert('Bohr', 'Ang')
# The first thing we do is adding the geometry to the NameSpace of the
# parser.
# This will enable custom actions to interact with the geometry in a
# straight forward manner.
# convert netcdf file to a dictionary
ion_nc = PropertyDict()
ion_nc.n = self._variable('orbnl_n')[:]
ion_nc.l = self._variable('orbnl_l')[:]
ion_nc.zeta = self._variable('orbnl_z')[:]
ion_nc.pol = self._variable('orbnl_ispol')[:]
ion_nc.orbital = self._variable('orb')[:]
# this gets converted later
delta = self._variable('delta')[:]
r = aranged(ion_nc.orbital.shape[1]).reshape(1, -1) * delta.reshape(-1, 1)
ion_nc.orbital *= r ** ion_nc.l.reshape(-1, 1) / Bohr2Ang * (3./2.)
ion_nc.r = r * Bohr2Ang
ion_nc.kb = PropertyDict()
ion_nc.kb.n = self._variable('pjnl_n')[:]
ion_nc.kb.l = self._variable('pjnl_l')[:]
ion_nc.kb.e = self._variable('pjnl_ekb')[:] * Ry2eV
ion_nc.kb.proj = self._variable('proj')[:]
delta = self._variable('kbdelta')[:]
r = aranged(ion_nc.kb.proj.shape[1]).reshape(1, -1) * delta.reshape(-1, 1)
ion_nc.kb.proj *= r ** ion_nc.kb.l.reshape(-1, 1) / Bohr2Ang * (3./2.)
ion_nc.kb.r = r * Bohr2Ang
vna = self._variable('vna')
r = aranged(vna[:].size) * vna.Vna_delta
ion_nc.vna = PropertyDict()
ion_nc.vna.v = vna[:] * Ry2eV * r / Bohr2Ang ** 3
ion_nc.vna.r = r * Bohr2Ang
# this is charge (not 1/sqrt(charge))
chlocal = self._variable('chlocal')
r = aranged(chlocal[:].size) * chlocal.Chlocal_delta
ion_nc.chlocal = PropertyDict()
ion_nc.chlocal.v = chlocal[:] * r / Bohr2Ang ** 3
ion_nc.chlocal.r = r * Bohr2Ang
vlocal = self._variable('reduced_vlocal')
r = aranged(vlocal[:].size) * vlocal.Reduced_vlocal_delta
ion_nc.vlocal = PropertyDict()
ion_nc.vlocal.v = vlocal[:] * r / Bohr2Ang ** 3
ion_nc.vlocal.r = r * Bohr2Ang
if "core" in self.variables:
# this is charge (not 1/sqrt(charge))
core = self._variable('core')
r = aranged(core[:].size) * core.Core_delta
ion_nc.core = PropertyDict()
ion_nc.core.v = core[:] * r / Bohr2Ang ** 3
ion_nc.core.r = r * Bohr2Ang
d = {
"_data": ion_nc,
"_kb_proj": False,
"_l": True,
"_n": True,
}
namespace = default_namespace(**d)
# l-quantum number
class lRange(argparse.Action):
def __call__(self, parser, ns, value, option_string=None):
value = (value
.replace("s", 0)
.replace("p", 1)
.replace("d", 2)
.replace("f", 3)
.replace("g", 4)
)
ns._l = strmap(int, value)[0]
p.add_argument('-l',
action=lRange,
help='Denote the sub-section of l-shells that are plotted: "s,f"')
# n quantum number
class nRange(argparse.Action):
def __call__(self, parser, ns, value, option_string=None):
ns._n = strmap(int, value)[0]
p.add_argument('-n',
action=nRange,
help='Denote the sub-section of n quantum numbers that are plotted: "2-4,6"')
class Plot(argparse.Action):
def __call__(self, parser, ns, value, option_string=None):
import matplotlib.pyplot as plt
# Retrieve values
data = ns._data
# We have these plots:
# - orbitals
# - projectors
# - chlocal
# - vna
# - vlocal
# - core (optional)
# We'll plot them like this:
# orbitals | projectors
# vna + vlocal | chlocal + core
#
# Determine different n, l
fig, axs = plt.subplots(2, 2)
# Now plot different orbitals
for n, l, zeta, pol, r, orb in zip(data.n, data.l, data.zeta,
data.pol, data.r, data.orbital):
if pol == 1:
pol = 'P'
else:
pol = ''
axs[0][0].plot(r, orb, label=f"n{n}l{l}Z{zeta}{pol}")
axs[0][0].set_title("Orbitals")
axs[0][0].set_xlabel("Distance [Ang]")
axs[0][0].set_ylabel("Value [a.u.]")
axs[0][0].legend()
# plot projectors
for n, l, e, r, proj in zip(
data.kb.n, data.kb.l, data.kb.e, data.kb.r, data.kb.proj):
axs[0][1].plot(r, proj, label=f"n{n}l{l} e={e:.5f}")
axs[0][1].set_title("KB projectors")
axs[0][1].set_xlabel("Distance [Ang]")
axs[0][1].set_ylabel("Value [a.u.]")
axs[0][1].legend()
axs[1][0].plot(data.vna.r, data.vna.v, label='Vna')
axs[1][0].plot(data.vlocal.r, data.vlocal.v, label='Vlocal')
axs[1][0].set_title("Potentials")
axs[1][0].set_xlabel("Distance [Ang]")
axs[1][0].set_ylabel("Potential [eV]")
axs[1][0].legend()
axs[1][1].plot(data.chlocal.r, data.chlocal.v, label='Chlocal')
if "core" in data:
axs[1][1].plot(data.core.r, data.core.v, label='core')
axs[1][1].set_title("Charge")
axs[1][1].set_xlabel("Distance [Ang]")
axs[1][1].set_ylabel("Charge [Ang^3]")
axs[1][1].legend()
if value is None:
plt.show()
else:
plt.savefig(value)
p.add_argument(*opts('--plot', '-p'), action=Plot, nargs='?', metavar='FILE',
help='Plot the content basis set file, possibly saving plot to a file.')
return p, namespace
def test_oplist_single(op):
d = oplist([ar.aranged(1, 10), ar.aranged(1, 10)])
op(d)
def param_circle(self, sc, N_or_dk, kR, normal, origo, loop=False):
r""" Create a parameterized k-point list where the k-points are generated on a circle around an origo
The generated circle is a perfect circle in the reciprocal space (Cartesian coordinates).
To generate a perfect circle in units of the reciprocal lattice vectors one can
generate the circle for a diagonal supercell with side-length :math:`2\pi`, see
example below.
Parameters
----------
sc : SuperCell, or SuperCellChild
the supercell used to construct the k-points
N_or_dk : int
number of k-points generated using the parameterization (if an integer),
otherwise it specifies the discretization length on the circle (in 1/Ang),
If the latter case will use less than 4 points a warning will be raised and
the number of points increased to 4.
kR : float
radius of the k-point. In 1/Ang
normal : array_like of float
normal vector to determine the circle plane
origo : array_like of float
origo of the circle used to generate the circular parameterization
loop : bool, optional
whether the first and last point are equal
Examples
--------
>>> sc = SuperCell([1, 1, 10, 90, 90, 60])
>>> bz = BrillouinZone.param_circle(sc, 10, 0.05, [0, 0, 1], [1./3, 2./3, 0])
To generate a circular set of k-points in reduced coordinates (reciprocal
>>> sc = SuperCell([1, 1, 10, 90, 90, 60])
>>> bz = BrillouinZone.param_circle(sc, 10, 0.05, [0, 0, 1], [1./3, 2./3, 0])
>>> bz_rec = BrillouinZone.param_circle(2*np.pi, 10, 0.05, [0, 0, 1], [1./3, 2./3, 0])
>>> bz.k[:, :] = bz_rec.k[:, :]
Returns
-------
BrillouinZone : with the parameterized k-points.
"""
if isinstance(N_or_dk, Integral):
N = N_or_dk
else:
# Calculate the required number of points
N = int(kR ** 2 * np.pi / N_or_dk + 0.5)
if N < 4:
N = 4
info('BrillouinZone.param_circle increased the number of circle points to 4.')
# Conversion object
bz = BrillouinZone(sc)
normal = _a.arrayd(normal)
origo = _a.arrayd(origo)
k_n = bz.tocartesian(normal)
k_o = bz.tocartesian(origo)
# Generate a preset list of k-points on the unit-circle
if loop:
radians = _a.aranged(N) / (N-1) * 2 * np.pi
else:
radians = _a.aranged(N) / N * 2 * np.pi
k = _a.emptyd([N, 3])
k[:, 0] = np.cos(radians)
k[:, 1] = np.sin(radians)
k[:, 2] = 0.
# Now generate the rotation
_, theta, phi = cart2spher(k_n)
if theta != 0:
pv = _a.arrayd([k_n[0], k_n[1], 0])
pv /= fnorm(pv)
q = Quaternion(phi, pv, rad=True) * Quaternion(theta, [0, 0, 1], rad=True)
else:
q = Quaternion(0., [0, 0, k_n[2] / abs(k_n[2])], rad=True)
# Calculate k-points
k = q.rotate(k)
k *= kR / fnorm(k).reshape(-1, 1)
k = bz.toreduced(k + k_o)
# The sum of weights is equal to the BZ area
W = np.pi * kR ** 2
w = np.repeat([W / N], N)
return BrillouinZone(sc, k, w)
def grid(cls, n, displ=0., size=1., centered=True, trs=False):
r""" Create a grid of `n` points with an offset of `displ` and sampling `size` around `displ`
The :math:`k`-points are :math:`\Gamma` centered.
Parameters
----------
n : int
number of points in the grid. If `trs` is ``True`` this may be smaller than `n`
displ : float, optional
the displacement of the grid
size : float, optional
the total size of the Brillouin zone to sample
centered : bool, optional
if the points are centered
trs : bool, optional
whether time-reversal-symmetry is applied
Returns
-------
k : np.ndarray
the list of k-points in the Brillouin zone to be sampled
w : np.ndarray
weights for the k-points
"""
# First ensure that displ is in the Brillouin
displ = displ % 1.
if displ > 0.5:
displ -= 1.
if displ < -0.5:
displ += 1.
# Centered _only_ has effect IFF
# displ == 0. and size == 1
# Otherwise we resort to other schemes
if displ != 0. or size != 1.:
centered = False
# We create the full grid, then afterwards we figure out TRS
n_half = n // 2
if n % 2 == 1:
k = _a.aranged(-n_half, n_half + 1) * size / n + displ
else:
k = _a.aranged(-n_half, n_half) * size / n + displ
if not centered:
# Shift everything by halve the size each occupies
k += size / (2 * n)
# Move k to the primitive cell and generate weights
k = cls.in_primitive(k)
w = _a.onesd(n) * size / n
# Check for TRS points
if trs and np.any(k < 0.):
# Make all positive to remove the double conting terms
k_pos = np.abs(k)
# Sort k-points and weights
idx = np.argsort(k_pos)
# Re-arange according to k value
k_pos = k_pos[idx]
w = w[idx]
# Find indices of all equivalent k-points (tolerance of 1e-10 in reciprocal units)
# 1e-10 ~ 1e10 k-points (no body will do this!)
idx_same = (np.diff(k_pos) < 1e-10).nonzero()[0]
# The above algorithm should never create more than two duplicates.
# Hence we can simply remove all idx_same and double the weight for all
# idx_same + 1.
w[idx_same + 1] *= 2
# Delete the duplicated k-points (they are already sorted)
k = np.delete(k_pos, idx_same, axis=0)
w = np.delete(w, idx_same)
else:
# Sort them, because it makes more visual sense
idx = np.argsort(k)
k = k[idx]
w = w[idx]
# Return values
return k, w