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from __future__ import division
import re
import warnings
from math import sin, cos, pi
import numpy as np
from ase.geometry import cell_to_cellpar, crystal_structure_from_cell
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
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.')
def parse_path_string(s):
"""Parse compact string representation of BZ path.
A path string can have several non-connected sections separated by
commas. The return value is a list of sections where each section is a
list of labels.
Examples:
>>> parse_path_string('GX')
[['G', 'X']]
>>> parse_path_string('GX,M1A')
[['G', 'X'], ['M1', 'A']]
"""
paths = []
for path in s.split(','):
names = [name if name != 'Gamma' else 'G'
for name in re.split(r'([A-Z][a-z0-9]*)', path)
if name]
paths.append(names)
return paths
def bandpath(path, cell, npoints=50):
"""Make a list of kpoints defining the path between the given points.
path: list or str
Can be:
* a string that parse_path_string() understands: 'GXL'
* a list of BZ points: [(0, 0, 0), (0.5, 0, 0)]
* or several lists of BZ points if the the path is not continuous.
cell: 3x3
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."""
if isinstance(path, str):
xtal = crystal_structure_from_cell(cell)
special = get_special_points(xtal, cell)
paths = []
for names in parse_path_string(path):
paths.append([special[name] for name in names])
elif np.array(path[0]).ndim == 1:
paths = [path]
else:
paths = path
points = np.concatenate(paths)
dists = points[1:] - points[:-1]
lengths = [np.linalg.norm(d) for d in kpoint_convert(cell, skpts_kc=dists)]
i = 0
for path in paths[:-1]:
i += len(path)
lengths[i - 1] = 0
length = sum(lengths)
kpts = []
x0 = 0
x = []
X = [0]
for P, d, L in zip(points[:-1], dists, lengths):
n = max(2, int(round(L * (npoints - len(x)) / (length - x0))))
for t in np.linspace(0, 1, n)[:-1]:
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)
get_bandpath = bandpath # old name
def labels_from_kpts(kpts, cell, crystal_structure=None, eps=1e-6):
"""Get an x-axis to be used when plotting a band structure.
The first of the returned lists can be used as a x-axis when plotting
the band structure. The second list can be used as xticks, and the third
as xticklabels.
Parameters:
kpts: list
List of scaled k-points.
cell: list
Unit cell of the atomic structure.
crystal_structure: str
Crystal structure of the atoms. If None is provided the crystal
structure is determined from the cell.
Returns:
Three arrays; the first is a list of cumulative distances between kpoints,
the second is x coordinates of the special points,
the third is the special points as strings.
"""
if crystal_structure is None:
crystal_structure = crystal_structure_from_cell(cell)
points = np.asarray(kpts)
diffs = points[1:] - points[:-1]
kinks = abs(diffs[1:] - diffs[:-1]).sum(1) > eps
N = len(points)
indices = [0]
indices.extend(np.arange(1, N - 1)[kinks])
indices.append(N - 1)
special = get_special_points(crystal_structure, cell)
labels = []
for kpt in points[indices]:
for label, k in special.items():
if abs(kpt - k).sum() < eps:
break
else:
label = '?'
labels.append(label)
xcoords = [0]
for i1, i2 in zip(indices[:-1], indices[1:]):
if i1 + 1 == i2:
length = 0
else:
diff = points[i2] - points[i1]
length = np.linalg.norm(kpoint_convert(cell, skpts_kc=diff))
xcoords.extend(np.linspace(0, length, i2 - i1 + 1)[1:] + xcoords[-1])
xcoords = np.array(xcoords)
return xcoords, xcoords[indices], labels
special_points = {
'cubic': {'G': [0, 0, 0],
'M': [1 / 2, 1 / 2, 0],
'R': [1 / 2, 1 / 2, 1 / 2],
'X': [0, 1 / 2, 0]},
'fcc': {'G': [0, 0, 0],
'K': [3 / 8, 3 / 8, 3 / 4],
'L': [1 / 2, 1 / 2, 1 / 2],
'U': [5 / 8, 1 / 4, 5 / 8],
'W': [1 / 2, 1 / 4, 3 / 4],
'X': [1 / 2, 0, 1 / 2]},
'bcc': {'G': [0, 0, 0],
'H': [1 / 2, -1 / 2, 1 / 2],
'P': [1 / 4, 1 / 4, 1 / 4],
'N': [0, 0, 1 / 2]},
'tetragonal': {'G': [0, 0, 0],
'A': [1 / 2, 1 / 2, 1 / 2],
'M': [1 / 2, 1 / 2, 0],
'R': [0, 1 / 2, 1 / 2],
'X': [0, 1 / 2, 0],
'Z': [0, 0, 1 / 2]},
'orthorhombic': {'G': [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]},
'hexagonal': {'G': [0, 0, 0],
'A': [0, 0, 1 / 2],
'H': [1 / 3, 1 / 3, 1 / 2],
'K': [1 / 3, 1 / 3, 0],
'L': [1 / 2, 0, 1 / 2],
'M': [1 / 2, 0, 0]}}
special_paths = {
'cubic': 'GXMGRX,MR',
'fcc': 'GXWKGLUWLK,UX',
'bcc': 'GHNGPH,PN',
'tetragonal': 'GXMGZRAZXR,MA',
'orthorhombic': 'GXSYGZURTZ,YT,UX,SR',
'hexagonal': 'GMKGALHA,LM,KH',
'monoclinic': 'GYHCEM1AXH1,MDZ,YD'}
def get_special_points(lattice, cell, eps=1e-4):
"""Return dict of special points.
The definitions are from a paper by Wahyu Setyawana and Stefano
Curtarolo::
http://dx.doi.org/10.1016/j.commatsci.2010.05.010
lattice: str
One of the following: cubic, fcc, bcc, orthorhombic, tetragonal,
hexagonal or monoclinic.
cell: 3x3 ndarray
Unit cell.
eps: float
Tolerance for cell-check.
"""
lattice = lattice.lower()
cellpar = cell_to_cellpar(cell=cell)
abc = cellpar[:3]
angles = cellpar[3:] / 180 * pi
a, b, c = abc
alpha, beta, gamma = angles
# Check that the unit-cells are as in the Setyawana-Curtarolo paper:
if lattice == 'cubic':
assert abc.ptp() < eps and abs(angles - pi / 2).max() < eps
elif lattice == 'fcc':
assert abc.ptp() < eps and abs(angles - pi / 3).max() < eps
elif lattice == 'bcc':
angle = np.arccos(-1 / 3)
assert abc.ptp() < eps and abs(angles - angle).max() < eps
elif lattice == 'tetragonal':
assert abs(a - b) < eps and abs(angles - pi / 2).max() < eps
elif lattice == 'orthorhombic':
assert abs(angles - pi / 2).max() < eps
elif lattice == 'hexagonal':
assert abs(a - b) < eps
assert abs(gamma - pi / 3 * 2) < eps
assert abs(angles[:2] - pi / 2).max() < eps
elif lattice == 'monoclinic':
assert c >= a and c >= b
assert alpha < pi / 2 and abs(angles[1:] - pi / 2).max() < eps
if lattice != 'monoclinic':
return special_points[lattice]
# Here, we need the cell:
eta = (1 - b * cos(alpha) / c) / (2 * sin(alpha)**2)
nu = 1 / 2 - eta * c * cos(alpha) / b
return {'G': [0, 0, 0],
'A': [1 / 2, 1 / 2, 0],
'C': [0, 1 / 2, 1 / 2],
'D': [1 / 2, 0, 1 / 2],
'D1': [1 / 2, 0, -1 / 2],
'E': [1 / 2, 1 / 2, 1 / 2],
'H': [0, eta, 1 - nu],
'H1': [0, 1 - eta, nu],
'H2': [0, eta, -nu],
'M': [1 / 2, eta, 1 - nu],
'M1': [1 / 2, 1 - eta, nu],
'M2': [1 / 2, eta, -nu],
'X': [0, 1 / 2, 0],
'Y': [0, 0, 1 / 2],
'Y1': [0, 0, -1 / 2],
'Z': [1 / 2, 0, 0]}
# 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
# 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]}}
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