"""Helper functions for creating supercells."""
import numpy as np
from ase import Atoms
class SupercellError(Exception):
"""Use if construction of supercell fails"""
[docs]def get_deviation_from_optimal_cell_shape(cell, target_shape="sc", norm=None):
r"""
Calculates the deviation of the given cell metric from the ideal
cell metric defining a certain shape. Specifically, the function
evaluates the expression `\Delta = || Q \mathbf{h} -
\mathbf{h}_{target}||_2`, where `\mathbf{h}` is the input
metric (*cell*) and `Q` is a normalization factor (*norm*)
while the target metric `\mathbf{h}_{target}` (via
*target_shape*) represent simple cubic ('sc') or face-centered
cubic ('fcc') cell shapes.
Parameters:
cell: 2D array of floats
Metric given as a (3x3 matrix) of the input structure.
target_shape: str
Desired supercell shape. Can be 'sc' for simple cubic or
'fcc' for face-centered cubic.
norm: float
Specify the normalization factor. This is useful to avoid
recomputing the normalization factor when computing the
deviation for a series of P matrices.
"""
if target_shape in ["sc", "simple-cubic"]:
target_metric = np.eye(3)
elif target_shape in ["fcc", "face-centered cubic"]:
target_metric = 0.5 * np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]])
if not norm:
norm = (np.linalg.det(cell) / np.linalg.det(target_metric))**(-1.0 / 3)
return np.linalg.norm(norm * cell - target_metric)
[docs]def find_optimal_cell_shape(
cell,
target_size,
target_shape,
lower_limit=-2,
upper_limit=2,
verbose=False,
):
"""Returns the transformation matrix that produces a supercell
corresponding to *target_size* unit cells with metric *cell* that
most closely approximates the shape defined by *target_shape*.
Parameters:
cell: 2D array of floats
Metric given as a (3x3 matrix) of the input structure.
target_size: integer
Size of desired super cell in number of unit cells.
target_shape: str
Desired supercell shape. Can be 'sc' for simple cubic or
'fcc' for face-centered cubic.
lower_limit: int
Lower limit of search range.
upper_limit: int
Upper limit of search range.
verbose: bool
Set to True to obtain additional information regarding
construction of transformation matrix.
"""
# Set up target metric
if target_shape in ["sc", "simple-cubic"]:
target_metric = np.eye(3)
elif target_shape in ["fcc", "face-centered cubic"]:
target_metric = 0.5 * np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]],
dtype=float)
if verbose:
print("target metric (h_target):")
print(target_metric)
# Normalize cell metric to reduce computation time during looping
norm = (target_size * np.linalg.det(cell) /
np.linalg.det(target_metric))**(-1.0 / 3)
norm_cell = norm * cell
if verbose:
print("normalization factor (Q): %g" % norm)
# Approximate initial P matrix
ideal_P = np.dot(target_metric, np.linalg.inv(norm_cell))
if verbose:
print("idealized transformation matrix:")
print(ideal_P)
starting_P = np.array(np.around(ideal_P, 0), dtype=int)
if verbose:
print("closest integer transformation matrix (P_0):")
print(starting_P)
# Prepare run.
from itertools import product
best_score = 1e6
optimal_P = None
for dP in product(range(lower_limit, upper_limit + 1), repeat=9):
dP = np.array(dP, dtype=int).reshape(3, 3)
P = starting_P + dP
if int(np.around(np.linalg.det(P), 0)) != target_size:
continue
score = get_deviation_from_optimal_cell_shape(
np.dot(P, norm_cell), target_shape=target_shape, norm=1.0)
if score < best_score:
best_score = score
optimal_P = P
if optimal_P is None:
print("Failed to find a transformation matrix.")
return None
# Finalize.
if verbose:
print("smallest score (|Q P h_p - h_target|_2): %f" % best_score)
print("optimal transformation matrix (P_opt):")
print(optimal_P)
print("supercell metric:")
print(np.round(np.dot(optimal_P, cell), 4))
print("determinant of optimal transformation matrix: %g" %
np.linalg.det(optimal_P))
return optimal_P
[docs]def make_supercell(prim, P, *, wrap=True, order="cell-major", tol=1e-5):
r"""Generate a supercell by applying a general transformation (*P*) to
the input configuration (*prim*).
The transformation is described by a 3x3 integer matrix
`\mathbf{P}`. Specifically, the new cell metric
`\mathbf{h}` is given in terms of the metric of the input
configuration `\mathbf{h}_p` by `\mathbf{P h}_p =
\mathbf{h}`.
Parameters:
prim: ASE Atoms object
Input configuration.
P: 3x3 integer matrix
Transformation matrix `\mathbf{P}`.
wrap: bool
wrap in the end
order: str (default: "cell-major")
how to order the atoms in the supercell
"cell-major":
[atom1_shift1, atom2_shift1, ..., atom1_shift2, atom2_shift2, ...]
i.e. run first over all the atoms in cell1 and then move to cell2.
"atom-major":
[atom1_shift1, atom1_shift2, ..., atom2_shift1, atom2_shift2, ...]
i.e. run first over atom1 in all the cells and then move to atom2.
This may be the order preferred by most VASP users.
tol: float
tolerance for wrapping
"""
supercell_matrix = P
supercell = clean_matrix(supercell_matrix @ prim.cell)
# cartesian lattice points
lattice_points_frac = lattice_points_in_supercell(supercell_matrix)
lattice_points = np.dot(lattice_points_frac, supercell)
N = len(lattice_points)
if order == "cell-major":
shifted = prim.positions[None, :, :] + lattice_points[:, None, :]
elif order == "atom-major":
shifted = prim.positions[:, None, :] + lattice_points[None, :, :]
else:
raise ValueError(f"invalid order: {order}")
shifted_reshaped = shifted.reshape(-1, 3)
superatoms = Atoms(positions=shifted_reshaped,
cell=supercell,
pbc=prim.pbc)
# Copy over any other possible arrays, inspired by atoms.__imul__
for name, arr in prim.arrays.items():
if name == "positions":
# This was added during construction of the super cell
continue
shape = (N * arr.shape[0], *arr.shape[1:])
if order == "cell-major":
new_arr = np.repeat(arr[None, :], N, axis=0).reshape(shape)
elif order == "atom-major":
new_arr = np.repeat(arr[:, None], N, axis=1).reshape(shape)
superatoms.set_array(name, new_arr)
# check number of atoms is correct
n_target = abs(int(np.round(np.linalg.det(supercell_matrix) * len(prim))))
if n_target != len(superatoms):
msg = "Number of atoms in supercell: {}, expected: {}".format(
n_target, len(superatoms))
raise SupercellError(msg)
if wrap:
superatoms.wrap(eps=tol)
return superatoms
def lattice_points_in_supercell(supercell_matrix):
"""Find all lattice points contained in a supercell.
Adapted from pymatgen, which is available under MIT license:
The MIT License (MIT) Copyright (c) 2011-2012 MIT & The Regents of the
University of California, through Lawrence Berkeley National Laboratory
"""
diagonals = np.array([
[0, 0, 0],
[0, 0, 1],
[0, 1, 0],
[0, 1, 1],
[1, 0, 0],
[1, 0, 1],
[1, 1, 0],
[1, 1, 1],
])
d_points = np.dot(diagonals, supercell_matrix)
mins = np.min(d_points, axis=0)
maxes = np.max(d_points, axis=0) + 1
ar = np.arange(mins[0], maxes[0])[:, None] * np.array([1, 0, 0])[None, :]
br = np.arange(mins[1], maxes[1])[:, None] * np.array([0, 1, 0])[None, :]
cr = np.arange(mins[2], maxes[2])[:, None] * np.array([0, 0, 1])[None, :]
all_points = ar[:, None, None] + br[None, :, None] + cr[None, None, :]
all_points = all_points.reshape((-1, 3))
frac_points = np.dot(all_points, np.linalg.inv(supercell_matrix))
tvects = frac_points[np.all(frac_points < 1 - 1e-10, axis=1)
& np.all(frac_points >= -1e-10, axis=1)]
assert len(tvects) == round(abs(np.linalg.det(supercell_matrix)))
return tvects
def clean_matrix(matrix, eps=1e-12):
""" clean from small values"""
matrix = np.array(matrix)
for ij in np.ndindex(matrix.shape):
if abs(matrix[ij]) < eps:
matrix[ij] = 0
return matrix