from __future__ import division
import warnings
import numpy as np
[docs]def monkhorst_pack(size):
"""Construct a uniform sampling of k-space of given size."""
if np.less_equal(size, 0).any():
raise ValueError('Illegal size: %s' % list(size))
kpts = np.indices(size).transpose((1, 2, 3, 0)).reshape((-1, 3))
return (kpts + 0.5) / size - 0.5
[docs]def get_monkhorst_pack_size_and_offset(kpts):
"""Find Monkhorst-Pack size and offset.
Returns (size, offset), where::
kpts = monkhorst_pack(size) + offset.
The set of k-points must not have been symmetry reduced."""
if len(kpts) == 1:
return np.ones(3, int), np.array(kpts[0], dtype=float)
size = np.zeros(3, int)
for c in range(3):
# Determine increment between k-points along current axis
delta = max(np.diff(np.sort(kpts[:, c])))
# Determine number of k-points as inverse of distance between kpoints
if delta > 1e-8:
size[c] = int(round(1.0 / delta))
else:
size[c] = 1
if size.prod() == len(kpts):
kpts0 = monkhorst_pack(size)
offsets = kpts - kpts0
# All offsets must be identical:
if (offsets.ptp(axis=0) < 1e-9).all():
return size, offsets[0].copy()
raise ValueError('Not an ASE-style Monkhorst-Pack grid!')
def get_monkhorst_shape(kpts):
warnings.warn('Use get_monkhorst_pack_size_and_offset()[0] instead.')
return get_monkhorst_pack_size_and_offset(kpts)[0]
def kpoint_convert(cell_cv, skpts_kc=None, ckpts_kv=None):
"""Convert k-points between scaled and cartesian coordinates.
Given the atomic unit cell, and either the scaled or cartesian k-point
coordinates, the other is determined.
The k-point arrays can be either a single point, or a list of points,
i.e. the dimension k can be empty or multidimensional.
"""
if ckpts_kv is None:
icell_cv = 2 * np.pi * np.linalg.inv(cell_cv).T
return np.dot(skpts_kc, icell_cv)
elif skpts_kc is None:
return np.dot(ckpts_kv, cell_cv.T) / (2 * np.pi)
else:
raise KeyError('Either scaled or cartesian coordinates must be given.')
[docs]def get_bandpath(points, cell, npoints=50):
"""Make a list of kpoints defining the path between the given points.
points: list
List of special IBZ point pairs, e.g. ``points =
[W, L, Gamma, X, W, K]``. These should be given in
scaled coordinates.
cell: 3x3 ndarray
Unit cell of the atoms.
npoints: int
Length of the output kpts list.
Return list of k-points, list of x-coordinates and list of
x-coordinates of special points."""
points = np.asarray(points)
dists = points[1:] - points[:-1]
lengths = [np.linalg.norm(d) for d in kpoint_convert(cell, skpts_kc=dists)]
length = sum(lengths)
kpts = []
x0 = 0
x = []
X = [0]
for P, d, L in zip(points[:-1], dists, lengths):
n = int(round(L * (npoints - 1 - len(x)) / (length - x0)))
for t in np.linspace(0, 1, n, endpoint=False):
kpts.append(P + t * d)
x.append(x0 + t * L)
x0 += L
X.append(x0)
kpts.append(points[-1])
x.append(x0)
return np.array(kpts), np.array(x), np.array(X)
# The following is a list of the critical points in the 1. Brillouin zone
# for some typical crystal structures.
# (In units of the reciprocal basis vectors)
# See http://en.wikipedia.org/wiki/Brillouin_zone
ibz_points = {'cubic': {'Gamma': [0, 0, 0 ],
'X': [0, 0 / 2, 1 / 2],
'R': [1 / 2, 1 / 2, 1 / 2],
'M': [0 / 2, 1 / 2, 1 / 2]},
'fcc': {'Gamma': [0, 0, 0 ],
'X': [1 / 2, 0, 1 / 2],
'W': [1 / 2, 1 / 4, 3 / 4],
'K': [3 / 8, 3 / 8, 3 / 4],
'U': [5 / 8, 1 / 4, 5 / 8],
'L': [1 / 2, 1 / 2, 1 / 2]},
'bcc': {'Gamma': [0, 0, 0 ],
'H': [1 / 2, -1 / 2, 1 / 2],
'N': [0, 0, 1 / 2],
'P': [1 / 4, 1 / 4, 1 / 4]},
'hexagonal':
{'Gamma': [0, 0, 0 ],
'M': [0, 1 / 2, 0 ],
'K': [-1 / 3, 1 / 3, 0 ],
'A': [0, 0, 1 / 2 ],
'L': [0, 1 / 2, 1 / 2 ],
'H': [-1 / 3, 1 / 3, 1 / 2 ]},
'tetragonal':
{'Gamma': [0, 0, 0 ],
'X': [1 / 2, 0, 0 ],
'M': [1 / 2, 1 / 2, 0 ],
'Z': [0, 0, 1 / 2 ],
'R': [1 / 2, 0, 1 / 2 ],
'A': [1 / 2, 1 / 2, 1 / 2 ]},
'orthorhombic':
{'Gamma': [0, 0, 0 ],
'R': [1 / 2, 1 / 2, 1 / 2 ],
'S': [1 / 2, 1 / 2, 0 ],
'T': [0, 1 / 2, 1 / 2 ],
'U': [1 / 2, 0, 1 / 2 ],
'X': [1 / 2, 0, 0 ],
'Y': [0, 1 / 2, 0 ],
'Z': [0, 0, 1 / 2 ]},
}
# ChadiCohen k point grids. The k point grids are given in units of the
# reciprocal unit cell. The variables are named after the following
# convention: cc+'<Nkpoints>'+_+'shape'. For example an 18 k point
# sq(3)xsq(3) is named 'cc18_sq3xsq3'.
cc6_1x1 = np.array([
1, 1, 0, 1, 0, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0,
0, 1, 0]).reshape((6, 3)) / 3.0
cc12_2x3 = np.array([
3, 4, 0, 3, 10, 0, 6, 8, 0, 3, -2, 0, 6, -4, 0,
6, 2, 0, -3, 8, 0, -3, 2, 0, -3, -4, 0, -6, 4, 0, -6, -2, 0, -6,
-8, 0]).reshape((12, 3)) / 18.0
cc18_sq3xsq3 = np.array([
2, 2, 0, 4, 4, 0, 8, 2, 0, 4, -2, 0, 8, -4,
0, 10, -2, 0, 10, -8, 0, 8, -10, 0, 2, -10, 0, 4, -8, 0, -2, -8,
0, 2, -4, 0, -4, -4, 0, -2, -2, 0, -4, 2, 0, -2, 4, 0, -8, 4, 0,
-4, 8, 0]).reshape((18, 3)) / 18.0
cc18_1x1 = np.array([
2, 4, 0, 2, 10, 0, 4, 8, 0, 8, 4, 0, 8, 10, 0,
10, 8, 0, 2, -2, 0, 4, -4, 0, 4, 2, 0, -2, 8, 0, -2, 2, 0, -2, -4,
0, -4, 4, 0, -4, -2, 0, -4, -8, 0, -8, 2, 0, -8, -4, 0, -10, -2,
0]).reshape((18, 3)) / 18.0
cc54_sq3xsq3 = np.array([
4, -10, 0, 6, -10, 0, 0, -8, 0, 2, -8, 0, 6,
-8, 0, 8, -8, 0, -4, -6, 0, -2, -6, 0, 2, -6, 0, 4, -6, 0, 8, -6,
0, 10, -6, 0, -6, -4, 0, -2, -4, 0, 0, -4, 0, 4, -4, 0, 6, -4, 0,
10, -4, 0, -6, -2, 0, -4, -2, 0, 0, -2, 0, 2, -2, 0, 6, -2, 0, 8,
-2, 0, -8, 0, 0, -4, 0, 0, -2, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0,
-8, 2, 0, -6, 2, 0, -2, 2, 0, 0, 2, 0, 4, 2, 0, 6, 2, 0, -10, 4,
0, -6, 4, 0, -4, 4, 0, 0, 4, 0, 2, 4, 0, 6, 4, 0, -10, 6, 0, -8,
6, 0, -4, 6, 0, -2, 6, 0, 2, 6, 0, 4, 6, 0, -8, 8, 0, -6, 8, 0,
-2, 8, 0, 0, 8, 0, -6, 10, 0, -4, 10, 0]).reshape((54, 3)) / 18.0
cc54_1x1 = np.array([
2, 2, 0, 4, 4, 0, 8, 8, 0, 6, 8, 0, 4, 6, 0, 6,
10, 0, 4, 10, 0, 2, 6, 0, 2, 8, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, -2,
6, 0, -2, 4, 0, -4, 6, 0, -6, 4, 0, -4, 2, 0, -6, 2, 0, -2, 0, 0,
-4, 0, 0, -8, 0, 0, -8, -2, 0, -6, -2, 0, -10, -4, 0, -10, -6, 0,
-6, -4, 0, -8, -6, 0, -2, -2, 0, -4, -4, 0, -8, -8, 0, 4, -2, 0,
6, -2, 0, 6, -4, 0, 2, 0, 0, 4, 0, 0, 6, 2, 0, 6, 4, 0, 8, 6, 0,
8, 0, 0, 8, 2, 0, 10, 4, 0, 10, 6, 0, 2, -4, 0, 2, -6, 0, 4, -6,
0, 0, -2, 0, 0, -4, 0, -2, -6, 0, -4, -6, 0, -6, -8, 0, 0, -8, 0,
-2, -8, 0, -4, -10, 0, -6, -10, 0]).reshape((54, 3)) / 18.0
cc162_sq3xsq3 = np.array([
-8, 16, 0, -10, 14, 0, -7, 14, 0, -4, 14,
0, -11, 13, 0, -8, 13, 0, -5, 13, 0, -2, 13, 0, -13, 11, 0, -10,
11, 0, -7, 11, 0, -4, 11, 0, -1, 11, 0, 2, 11, 0, -14, 10, 0, -11,
10, 0, -8, 10, 0, -5, 10, 0, -2, 10, 0, 1, 10, 0, 4, 10, 0, -16,
8, 0, -13, 8, 0, -10, 8, 0, -7, 8, 0, -4, 8, 0, -1, 8, 0, 2, 8, 0,
5, 8, 0, 8, 8, 0, -14, 7, 0, -11, 7, 0, -8, 7, 0, -5, 7, 0, -2, 7,
0, 1, 7, 0, 4, 7, 0, 7, 7, 0, 10, 7, 0, -13, 5, 0, -10, 5, 0, -7,
5, 0, -4, 5, 0, -1, 5, 0, 2, 5, 0, 5, 5, 0, 8, 5, 0, 11, 5, 0,
-14, 4, 0, -11, 4, 0, -8, 4, 0, -5, 4, 0, -2, 4, 0, 1, 4, 0, 4, 4,
0, 7, 4, 0, 10, 4, 0, -13, 2, 0, -10, 2, 0, -7, 2, 0, -4, 2, 0,
-1, 2, 0, 2, 2, 0, 5, 2, 0, 8, 2, 0, 11, 2, 0, -11, 1, 0, -8, 1,
0, -5, 1, 0, -2, 1, 0, 1, 1, 0, 4, 1, 0, 7, 1, 0, 10, 1, 0, 13, 1,
0, -10, -1, 0, -7, -1, 0, -4, -1, 0, -1, -1, 0, 2, -1, 0, 5, -1,
0, 8, -1, 0, 11, -1, 0, 14, -1, 0, -11, -2, 0, -8, -2, 0, -5, -2,
0, -2, -2, 0, 1, -2, 0, 4, -2, 0, 7, -2, 0, 10, -2, 0, 13, -2, 0,
-10, -4, 0, -7, -4, 0, -4, -4, 0, -1, -4, 0, 2, -4, 0, 5, -4, 0,
8, -4, 0, 11, -4, 0, 14, -4, 0, -8, -5, 0, -5, -5, 0, -2, -5, 0,
1, -5, 0, 4, -5, 0, 7, -5, 0, 10, -5, 0, 13, -5, 0, 16, -5, 0, -7,
-7, 0, -4, -7, 0, -1, -7, 0, 2, -7, 0, 5, -7, 0, 8, -7, 0, 11, -7,
0, 14, -7, 0, 17, -7, 0, -8, -8, 0, -5, -8, 0, -2, -8, 0, 1, -8,
0, 4, -8, 0, 7, -8, 0, 10, -8, 0, 13, -8, 0, 16, -8, 0, -7, -10,
0, -4, -10, 0, -1, -10, 0, 2, -10, 0, 5, -10, 0, 8, -10, 0, 11,
-10, 0, 14, -10, 0, 17, -10, 0, -5, -11, 0, -2, -11, 0, 1, -11, 0,
4, -11, 0, 7, -11, 0, 10, -11, 0, 13, -11, 0, 16, -11, 0, -1, -13,
0, 2, -13, 0, 5, -13, 0, 8, -13, 0, 11, -13, 0, 14, -13, 0, 1,
-14, 0, 4, -14, 0, 7, -14, 0, 10, -14, 0, 13, -14, 0, 5, -16, 0,
8, -16, 0, 11, -16, 0, 7, -17, 0, 10, -17, 0]).reshape((162, 3)) / 27.0
cc162_1x1 = np.array([
-8, -16, 0, -10, -14, 0, -7, -14, 0, -4, -14,
0, -11, -13, 0, -8, -13, 0, -5, -13, 0, -2, -13, 0, -13, -11, 0,
-10, -11, 0, -7, -11, 0, -4, -11, 0, -1, -11, 0, 2, -11, 0, -14,
-10, 0, -11, -10, 0, -8, -10, 0, -5, -10, 0, -2, -10, 0, 1, -10,
0, 4, -10, 0, -16, -8, 0, -13, -8, 0, -10, -8, 0, -7, -8, 0, -4,
-8, 0, -1, -8, 0, 2, -8, 0, 5, -8, 0, 8, -8, 0, -14, -7, 0, -11,
-7, 0, -8, -7, 0, -5, -7, 0, -2, -7, 0, 1, -7, 0, 4, -7, 0, 7, -7,
0, 10, -7, 0, -13, -5, 0, -10, -5, 0, -7, -5, 0, -4, -5, 0, -1,
-5, 0, 2, -5, 0, 5, -5, 0, 8, -5, 0, 11, -5, 0, -14, -4, 0, -11,
-4, 0, -8, -4, 0, -5, -4, 0, -2, -4, 0, 1, -4, 0, 4, -4, 0, 7, -4,
0, 10, -4, 0, -13, -2, 0, -10, -2, 0, -7, -2, 0, -4, -2, 0, -1,
-2, 0, 2, -2, 0, 5, -2, 0, 8, -2, 0, 11, -2, 0, -11, -1, 0, -8,
-1, 0, -5, -1, 0, -2, -1, 0, 1, -1, 0, 4, -1, 0, 7, -1, 0, 10, -1,
0, 13, -1, 0, -10, 1, 0, -7, 1, 0, -4, 1, 0, -1, 1, 0, 2, 1, 0, 5,
1, 0, 8, 1, 0, 11, 1, 0, 14, 1, 0, -11, 2, 0, -8, 2, 0, -5, 2, 0,
-2, 2, 0, 1, 2, 0, 4, 2, 0, 7, 2, 0, 10, 2, 0, 13, 2, 0, -10, 4,
0, -7, 4, 0, -4, 4, 0, -1, 4, 0, 2, 4, 0, 5, 4, 0, 8, 4, 0, 11, 4,
0, 14, 4, 0, -8, 5, 0, -5, 5, 0, -2, 5, 0, 1, 5, 0, 4, 5, 0, 7, 5,
0, 10, 5, 0, 13, 5, 0, 16, 5, 0, -7, 7, 0, -4, 7, 0, -1, 7, 0, 2,
7, 0, 5, 7, 0, 8, 7, 0, 11, 7, 0, 14, 7, 0, 17, 7, 0, -8, 8, 0,
-5, 8, 0, -2, 8, 0, 1, 8, 0, 4, 8, 0, 7, 8, 0, 10, 8, 0, 13, 8, 0,
16, 8, 0, -7, 10, 0, -4, 10, 0, -1, 10, 0, 2, 10, 0, 5, 10, 0, 8,
10, 0, 11, 10, 0, 14, 10, 0, 17, 10, 0, -5, 11, 0, -2, 11, 0, 1,
11, 0, 4, 11, 0, 7, 11, 0, 10, 11, 0, 13, 11, 0, 16, 11, 0, -1,
13, 0, 2, 13, 0, 5, 13, 0, 8, 13, 0, 11, 13, 0, 14, 13, 0, 1, 14,
0, 4, 14, 0, 7, 14, 0, 10, 14, 0, 13, 14, 0, 5, 16, 0, 8, 16, 0,
11, 16, 0, 7, 17, 0, 10, 17, 0]).reshape((162, 3)) / 27.0