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import numpy as np
def cellvector_products(cell):
cell = _pad_nonpbc(cell)
g0 = np.empty(6, dtype=float)
g0[0] = cell[0] @ cell[0]
g0[1] = cell[1] @ cell[1]
g0[2] = cell[2] @ cell[2]
g0[3] = 2 * (cell[1] @ cell[2])
g0[4] = 2 * (cell[2] @ cell[0])
g0[5] = 2 * (cell[0] @ cell[1])
return g0
def _pad_nonpbc(cell):
# Add "infinitely long" lattice vectors for non-periodic directions,
# perpendicular to the periodic ones.
maxlen = max(cell.lengths())
mask = cell.any(1)
cell = cell.complete()
cell[~mask] *= 2 * maxlen
return cell
def niggli_reduce_cell(cell, epsfactor=None):
from ase.cell import Cell
cell = Cell.new(cell)
npbc = cell.rank
if epsfactor is None:
epsfactor = 1e-5
vol_normalization_exponent = 1 if npbc == 0 else 1 / npbc
vol_normalization = cell.complete().volume**vol_normalization_exponent
eps = epsfactor * vol_normalization
g0 = cellvector_products(cell)
g, C = _niggli_reduce(g0, eps)
abc = np.sqrt(g[:3])
# Prevent division by zero e.g. for cell==zeros((3, 3)):
abcprod = max(abc.prod(), 1e-100)
cosangles = abc * g[3:] / (2 * abcprod)
angles = 180 * np.arccos(cosangles) / np.pi
# Non-periodic directions have artificial infinitely long lattice vectors.
# We re-zero their lengths before returning:
abc[npbc:] = 0.0
newcell = Cell.fromcellpar(np.concatenate([abc, angles]))
newcell[npbc:] = 0.0
return newcell, C
def lmn_to_ijk(lmn):
if lmn.prod() == 1:
ijk = lmn.copy()
for idx in range(3):
if ijk[idx] == 0:
ijk[idx] = 1
else:
ijk = np.ones(3, dtype=int)
if np.any(lmn != -1):
r = None
for idx in range(3):
if lmn[idx] == 1:
ijk[idx] = -1
elif lmn[idx] == 0:
r = idx
if ijk.prod() == -1:
ijk[r] = -1
return ijk
def _niggli_reduce(g0, eps):
I3 = np.eye(3, dtype=int)
I6 = np.eye(6, dtype=int)
C = I3.copy()
D = I6.copy()
g = D @ g0
def lt(x, y, eps=eps):
return x < y - eps
def gt(x, y, eps=eps):
return lt(y, x, eps)
def eq(x, y, eps=eps):
return not (lt(x, y, eps) or gt(x, y, eps))
for _ in range(10000):
if (gt(g[0], g[1])
or (eq(g[0], g[1]) and gt(abs(g[3]), abs(g[4])))):
C = C @ (-I3[[1, 0, 2]])
D = I6[[1, 0, 2, 4, 3, 5]] @ D
g = D @ g0
continue
elif (gt(g[1], g[2])
or (eq(g[1], g[2]) and gt(abs(g[4]), abs(g[5])))):
C = C @ (-I3[[0, 2, 1]])
D = I6[[0, 2, 1, 3, 5, 4]] @ D
g = D @ g0
continue
lmn = np.array(gt(g[3:], 0, eps=eps / 2), dtype=int)
lmn -= np.array(lt(g[3:], 0, eps=eps / 2), dtype=int)
ijk = lmn_to_ijk(lmn)
C *= ijk[np.newaxis]
D[3] *= ijk[1] * ijk[2]
D[4] *= ijk[0] * ijk[2]
D[5] *= ijk[0] * ijk[1]
g = D @ g0
if (gt(abs(g[3]), g[1])
or (eq(g[3], g[1]) and lt(2 * g[4], g[5]))
or (eq(g[3], -g[1]) and lt(g[5], 0))):
s = int(np.sign(g[3]))
A = I3.copy()
A[1, 2] = -s
C = C @ A
B = I6.copy()
B[2, 1] = 1
B[2, 3] = -s
B[3, 1] = -2 * s
B[4, 5] = -s
D = B @ D
g = D @ g0
elif (gt(abs(g[4]), g[0])
or (eq(g[4], g[0]) and lt(2 * g[3], g[5]))
or (eq(g[4], -g[0]) and lt(g[5], 0))):
s = int(np.sign(g[4]))
A = I3.copy()
A[0, 2] = -s
C = C @ A
B = I6.copy()
B[2, 0] = 1
B[2, 4] = -s
B[3, 5] = -s
B[4, 0] = -2 * s
D = B @ D
g = D @ g0
elif (gt(abs(g[5]), g[0])
or (eq(g[5], g[0]) and lt(2 * g[3], g[4]))
or (eq(g[5], -g[0]) and lt(g[4], 0))):
s = int(np.sign(g[5]))
A = I3.copy()
A[0, 1] = -s
C = C @ A
B = I6.copy()
B[1, 0] = 1
B[1, 5] = -s
B[3, 4] = -s
B[5, 0] = -2 * s
D = B @ D
g = D @ g0
elif (lt(g[[0, 1, 3, 4, 5]].sum(), 0)
or (eq(g[[0, 1, 3, 4, 5]].sum(), 0)
and gt(2 * (g[0] + g[4]) + g[5], 0))):
A = I3.copy()
A[:, 2] = 1
C = C @ A
B = I6.copy()
B[2, :] = 1
B[3, 1] = 2
B[3, 5] = 1
B[4, 0] = 2
B[4, 5] = 1
D = B @ D
g = D @ g0
else:
break
else:
raise RuntimeError('Niggli reduction not done in 10000 steps!\n'
'g={}\n'
'operation={}'
.format(g.tolist(), C.tolist()))
return g, C
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