def toSphere(self):
""" Create a sphere which is surely encompassing the *full* shape """
from .ellipsoid import Sphere
# Retrieve spheres
A = self.A.toSphere()
Ar = A.radius
Ac = A.center
B = self.B.toSphere()
Br = B.radius
Bc = B.center
# Calculate the distance between the spheres
dist = fnorm(Ac - Bc)
# If one is fully enclosed in the other, we can simply neglect the other
if dist + Ar <= Br:
# A is fully enclosed in B (or they are the same)
return A
elif dist + Br <= Ar:
# B is fully enclosed in A (or they are the same)
return B
elif dist <= (Ar + Br):
# We can reduce the sphere drastically because only the overlapping region is important
# i_r defines the intersection radius, search for Sphere-Sphere Intersection
dx = (dist**2 - Br**2 + Ar**2) / (2 * dist)
if dx > dist:
# the intersection is placed after the radius of B
# And in this case B is smaller (otherwise dx < 0)
return B
elif dx < 0:
return A
i_r = msqrt(4 * (dist * Ar)**2 -
(dist**2 - Br**2 + Ar**2)**2) / (2 * dist)
# Now we simply need to find the dx point along the vector Bc - Ac
# Then we can easily calculate the point from A
center = Bc - Ac
center = Ac + center / fnorm(center) * dx
A = i_r
B = i_r
else:
# In this case there is actually no overlap. So perhaps we should
# create an infinitisemal sphere such that no point will ever be found
# Or we should return a new Shape which *always* return False for indices etc.
center = (Ac + Bc) * 0.5
# Currently we simply use a very small sphere and put it in the middle between
# the spheres
# This should at least speed up comparisons
A = 0.001
B = 0.001
return Sphere(max(A, B), center)
def rotate(self, angle, v, only='abc', rad=False):
""" Rotates the supercell, in-place by the angle around the vector
One can control which cell vectors are rotated by designating them
individually with ``only='[abc]'``.
Parameters
----------
angle : float
the angle of which the geometry should be rotated
v : array_like [3]
the vector around the rotation is going to happen
v = [1,0,0] will rotate in the ``yz`` plane
rad : bool, optional
Whether the angle is in radians (True) or in degrees (False)
only : ('abc'), str, optional
only rotate the designated cell vectors.
"""
# flatte => copy
vn = np.asarray(v, dtype=np.float64).flatten()
vn /= fnorm(vn)
q = Quaternion(angle, vn, rad=rad)
q /= q.norm() # normalize the quaternion
cell = np.copy(self.cell)
if 'a' in only:
cell[0, :] = q.rotate(self.cell[0, :])
if 'b' in only:
cell[1, :] = q.rotate(self.cell[1, :])
if 'c' in only:
cell[2, :] = q.rotate(self.cell[2, :])
return self.copy(cell)
def test_sphere_and():
# For all intersections after
v = np.array([1.] * 3)
v /= fnorm(v)
D = np.linspace(0, 5, 50)
inside = np.ones(len(D), dtype=bool)
A = Sphere(2.)
is_first = True
inside[:] = True
for i, d in enumerate(D):
B = Sphere(1., center=v * d)
C = (A & B).toSphere()
if is_first and C.radius < B.radius:
is_first = False
inside[i:] = False
if inside[i]:
assert C.radius == pytest.approx(B.radius)
else:
assert C.radius < B.radius
A = Sphere(0.5)
is_first = True
inside[:] = True
for i, d in enumerate(D):
B = Sphere(1., center=v * d)
C = (A & B).toSphere()
str(A) + str(B) + str(C)
if is_first and C.radius < A.radius:
inside[i:] = False
is_first = False
if inside[i]:
assert C.radius == pytest.approx(A.radius)
else:
assert C.radius < A.radius
def find_all_bonds(geometry, tol=0.2):
"""
Finds all bonds present in a geometry.
Parameters
-----------
geometry: sisl.Geometry
the structure where the bonds should be found.
tol: float
the fraction that the distance between atoms is allowed to differ from
the "standard" in order to be considered a bond.
Return
---------
np.ndarray of shape (nbonds, 2)
each item of the array contains the 2 indices of the atoms that participate in the
bond.
"""
pt = PeriodicTable()
bonds = []
for at in geometry:
neighs = geometry.close(at, R=[0.1, 3])[-1]
for neigh in neighs[neighs > at]:
summed_radius = pt.radius([
abs(geometry.atoms[at].Z),
abs(geometry.atoms[neigh % geometry.na].Z)
]).sum()
bond_thresh = (1 + tol) * summed_radius
if bond_thresh > fnorm(geometry[neigh] - geometry[at]):
bonds.append([at, neigh])
return np.array(bonds, dtype=int)
def toSphere(self):
""" Create a sphere which is surely encompassing the *full* shape """
from .ellipsoid import Sphere
# Retrieve spheres
A = self.A.toSphere()
Ar = A.radius
Ac = A.center
B = self.B.toSphere()
Br = B.radius
Bc = B.center
center = (Ac + Bc) * 0.5
A = Ar + fnorm(center - Ac)
B = Br + fnorm(center - Bc)
return Sphere(max(A, B), center)
def parallel(self, other, axis=(0, 1, 2)):
""" Returns true if the cell vectors are parallel to `other`
Parameters
----------
other : SuperCell
the other object to check whether the axis are parallel
axis : int or array_like
only check the specified axis (default to all)
"""
axis = _a.asarrayi(axis).ravel()
# Convert to unit-vector cell
for i in axis:
a = self.cell[i, :] / fnorm(self.cell[i, :])
b = other.cell[i, :] / fnorm(other.cell[i, :])
if abs(dot3(a, b) - 1) > 0.001:
return False
return True
def _projected_1Dcoords(cls, geometry, xyz=None, axis="x", nsc=(1, 1, 1)):
"""
Moves the 3D positions of the atoms to a 2D supspace.
In this way, we can plot the structure from the "point of view" that we want.
NOTE: If axis is one of {"a", "b", "c", "1", "2", "3"} the function doesn't
project the coordinates in the direction of the lattice vector. The fractional
coordinates, taking in consideration the three lattice vectors, are returned
instead.
Parameters
------------
geometry: sisl.Geometry
the geometry for which you want the projected coords
xyz: array-like of shape (natoms, 3), optional
the 3D coordinates that we want to project.
otherwise they are taken from the geometry.
axis: {"x", "y", "z", "a", "b", "c", "1", "2", "3"} or array-like of shape 3, optional
the direction to be displayed along the X axis.
nsc: array-like of shape (3, ), optional
only used if `axis` is a lattice vector. It is used to rescale everything to the unit
cell lattice vectors, otherwise `GeometryPlot` doesn't play well with `GridPlot`.
Returns
----------
np.ndarray of shape (natoms, )
the 1D coordinates of the geometry, with all positions projected into the line
defined by axis.
"""
if xyz is None:
xyz = geometry.xyz
if isinstance(axis, str) and axis in ("a", "b", "c", "0", "1", "2"):
return cls._projected_2Dcoords(geometry,
xyz,
xaxis=axis,
yaxis="a" if axis == "c" else "c",
nsc=nsc)[..., 0]
# Get the direction that the axis represents
axis = cls._direction(axis, geometry.cell)
return xyz.dot(axis / fnorm(axis)) / fnorm(axis)
def move(self, v):
""" Appends additional space to the object """
# check which cell vector resembles v the most,
# use that
cell = np.copy(self.cell)
p = np.empty([3], np.float64)
cl = fnorm(cell)
for i in range(3):
p[i] = abs(np.sum(cell[i, :] * v)) / cl[i]
cell[np.argmax(p), :] += v
return self.copy(cell)
def is_orthogonal(self):
""" Returns true if the cell vectors are orthogonal """
# Convert to unit-vector cell
cell = np.copy(self.cell)
cl = fnorm(cell)
cell[0, :] = cell[0, :] / cl[0]
cell[1, :] = cell[1, :] / cl[1]
cell[2, :] = cell[2, :] / cl[2]
i_s = dot3(cell[0, :], cell[1, :]) < 0.001
i_s = dot3(cell[0, :], cell[2, :]) < 0.001 and i_s
i_s = dot3(cell[1, :], cell[2, :]) < 0.001 and i_s
return i_s
def _bond_length(geom, bond):
"""
Returns the length of a bond between two atoms.
Parameters
------------
geom: Geometry
the structure where the atoms are
bond: array-like of two int
the indices of the atoms that form the bond
"""
return fnorm(geom[bond[1]] - geom[bond[0]])
def add_vacuum(self, vacuum, axis):
""" Add vacuum along the `axis` lattice vector
Parameters
----------
vacuum : float
amount of vacuum added, in Ang
axis : int
the lattice vector to add vacuum along
"""
cell = np.copy(self.cell)
d = cell[axis, :].copy()
# normalize to get direction vector
cell[axis, :] += d * (vacuum / fnorm(d))
return self.copy(cell)
def angle(self, i, j, rad=False):
""" The angle between two of the cell vectors
Parameters
----------
i : int
the first cell vector
j : int
the second cell vector
rad : bool, optional
whether the returned value is in radians
"""
n = fnorm(self.cell[[i, j], :])
ang = math.acos(dot3(self.cell[i, :], self.cell[j, :]) / (n[0] * n[1]))
if rad:
return ang
return math.degrees(ang)
def parameters(self, rad=False):
r""" Cell parameters of this cell in 3 lengths and 3 angles
Notes
-----
Since we return the length and angles between vectors it may not be possible to
recreate the same cell. Only in the case where the first lattice vector *only*
has a Cartesian :math:`x` component will this be the case
Parameters
----------
rad : bool, optional
whether the angles are returned in radians (otherwise in degree)
Returns
-------
float
length of first lattice vector
float
length of second lattice vector
float
length of third lattice vector
float
angle between b and c vectors
float
angle between a and c vectors
float
angle between a and b vectors
"""
if rad:
f = 1.
else:
f = 180 / np.pi
# Calculate length of each lattice vector
cell = self.cell.copy()
abc = fnorm(cell)
from math import acos
cell = cell / abc.reshape(-1, 1)
alpha = acos(dot3(cell[1, :], cell[2, :])) * f
beta = acos(dot3(cell[0, :], cell[2, :])) * f
gamma = acos(dot3(cell[0, :], cell[1, :])) * f
return abc[0], abc[1], abc[2], alpha, beta, gamma
def radius(self):
""" Return the radius of the Ellipsoid """
return fnorm(self._v)
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 initialize(self):
""" Initialize the internal data-arrays used for efficient calculation of the real-space quantities
This method should first be called *after* all options has been specified.
If the user hasn't specified the ``bz`` value as an option this method will update the internal
integration Brillouin zone based on the ``dk`` option.
"""
# Try and guess the directions
unfold = self._unfold
nsc = self.parent.nsc.copy()
axes = self._options['axes']
if axes is None:
if nsc[2] == 1:
axes = _a.arrayi([0, 1])
elif nsc[1] == 1:
axes = _a.arrayi([0, 2])
elif nsc[0] == 1:
axes = _a.arrayi([1, 2])
else:
raise ValueError(self.__class__.__name__ + '.initialize currently only supports a 2D real-space self-energy, hence the MonkhorstPack grid should reflect only 2 periodic directions.')
self._options['axes'] = axes
# Check that we have periodicity along the chosen axes
nsc_sum = nsc[axes].sum()
if nsc_sum == 1:
raise ValueError(self.__class__.__name__ + '.initialize found no periodic directions for the chosen integration axes: {} and {}.'.format(*nsc[axes]))
elif nsc_sum < 6:
raise ValueError((self.__class__.__name__ + '.initialize found one periodic direction '
'out of two for the chosen integration axes: {} and {}. '
'For 1D systems the regular surface self-energy method is appropriate.').format(*nsc[axes]))
if self._options['semi_axis'] is None and self._options['k_axis'] is None:
# None of the axis has been described
if nsc[axes[0]] > 3:
k_ax = axes[0]
s_ax = axes[1]
elif nsc[axes[1]] > 3:
k_ax = axes[1]
s_ax = axes[0]
else:
# Choose the direction of k to be the smallest shortest
sc = self.parent.sc.tile(unfold[axes[0]], axes[0]).tile(unfold[axes[1]], axes[1])
rcell = fnorm(sc.rcell)[axes]
k_ax = axes[np.argmax(rcell)]
if k_ax == axes[0]:
s_ax = axes[1]
else:
s_ax = axes[0]
self._options['semi_axis'] = s_ax
self._options['k_axis'] = k_ax
s_ax = 'ABC'[s_ax]
k_ax = 'ABC'[k_ax]
info(self.__class__.__name__ + '.initialize determined the semi-inf- and k-directions to be: {} and {}'.format(s_ax, k_ax))
elif self._options['k_axis'] is None:
s_ax = self._options['semi_axis']
if s_ax == axes[0]:
k_ax = axes[1]
else:
k_ax = axes[0]
self._options['k_axis'] = k_ax
k_ax = 'ABC'[k_ax]
info(self.__class__.__name__ + '.initialize determined the k direction to be: {}'.format(k_ax))
elif self._options['semi_axis'] is None:
k_ax = self._options['k_axis']
if k_ax == axes[0]:
s_ax = axes[1]
else:
s_ax = axes[0]
self._options['semi_axis'] = s_ax
s_ax = 'ABC'[s_ax]
info(self.__class__.__name__ + '.initialize determined the semi-infinite direction to be: {}'.format(s_ax))
k_ax = self._options['k_axis']
s_ax = self._options['semi_axis']
if nsc[s_ax] != 3:
raise ValueError(self.__class__.__name__ + '.initialize found the self-energy direction to be '
'incompatible with the parent object. It *must* have 3 supercells along the '
'semi-infinite direction.')
P0 = self.real_space_parent()
V_atoms = self.real_space_coupling(True)[1]
self._calc = {
# The below algorithm requires the direction to be negative
# if changed, B, C should be reversed below
'SE': RecursiveSI(self.parent, '-' + 'ABC'[s_ax], eta=self._options['eta']),
# Used to calculate the real-space self-energy
'P0': P0.Pk(), # in sparse format
'S0': P0.Sk(), # in sparse format
# Orbitals in the coupling atoms
'orbs': P0.a2o(V_atoms, True).reshape(-1, 1),
}
# Update the BrillouinZone integration grid in case it isn't specified
if self._options['bz'] is None:
# Update the integration grid
# Note this integration grid is based on the big system.
sc = self.parent.sc.tile(unfold[axes[0]], axes[0]).tile(unfold[axes[1]], axes[1])
rcell = fnorm(sc.rcell)[k_ax]
nk = [1] * 3
nk[k_ax] = int(self._options['dk'] * rcell)
self._options['bz'] = MonkhorstPack(sc, nk, trs=self._options['trs'])
info(self.__class__.__name__ + '.initialize determined the number of k-points: {}'.format(nk[k_ax]))
def edge_length(self):
""" The lengths of each of the vector that defines the cuboid """
return fnorm(self._v)
def length(self):
""" Length of each lattice vector """
return fnorm(self.cell)