# Source code for ase.geometry.geometry

# Copyright (C) 2010, Jesper Friis
# (see accompanying license files for details).

"""Utility tools for atoms/geometry manipulations.
- convenient creation of slabs and interfaces of
different orientations.
- detection of duplicate atoms / atoms within cutoff radius
"""

import itertools
import numpy as np
from ase.geometry import complete_cell
from ase.geometry.minkowski_reduction import minkowski_reduce
from ase.utils import pbc2pbc
from ase.cell import Cell

def translate_pretty(fractional, pbc):
"""Translates atoms such that fractional positions are minimized."""

for i in range(3):
if not pbc[i]:
continue

indices = np.argsort(fractional[:, i])
sp = fractional[indices, i]

widths = (np.roll(sp, 1) - sp) % 1.0
fractional[:, i] -= sp[np.argmin(widths)]
fractional[:, i] %= 1.0
return fractional

[docs]def wrap_positions(positions, cell, pbc=True, center=(0.5, 0.5, 0.5),
pretty_translation=False, eps=1e-7):
"""Wrap positions to unit cell.

Returns positions changed by a multiple of the unit cell vectors to
:meth:ase.Atoms.wrap method.

Parameters:

positions: float ndarray of shape (n, 3)
Positions of the atoms
cell: float ndarray of shape (3, 3)
Unit cell vectors.
pbc: one or 3 bool
For each axis in the unit cell decides whether the positions
will be moved along this axis.
center: three float
The positons in fractional coordinates that the new positions
will be nearest possible to.
pretty_translation: bool
Translates atoms such that fractional coordinates are minimized.
eps: float
Small number to prevent slightly negative coordinates from being
wrapped.

Example:

>>> from ase.geometry import wrap_positions
>>> wrap_positions([[-0.1, 1.01, -0.5]],
...                [[1, 0, 0], [0, 1, 0], [0, 0, 4]],
...                pbc=[1, 1, 0])
array([[ 0.9 ,  0.01, -0.5 ]])
"""

if not hasattr(center, '__len__'):
center = (center,) * 3

pbc = pbc2pbc(pbc)
shift = np.asarray(center) - 0.5 - eps

# Don't change coordinates when pbc is False
shift[np.logical_not(pbc)] = 0.0

assert np.asarray(cell)[np.asarray(pbc)].any(axis=1).all(), (cell, pbc)

cell = complete_cell(cell)
fractional = np.linalg.solve(cell.T,
np.asarray(positions).T).T - shift

if pretty_translation:
fractional = translate_pretty(fractional, pbc)
shift = np.asarray(center) - 0.5
shift[np.logical_not(pbc)] = 0.0
fractional += shift
else:
for i, periodic in enumerate(pbc):
if periodic:
fractional[:, i] %= 1.0
fractional[:, i] += shift[i]

return np.dot(fractional, cell)

[docs]def get_layers(atoms, miller, tolerance=0.001):
"""Returns two arrays describing which layer each atom belongs
to and the distance between the layers and origo.

Parameters:

miller: 3 integers
The Miller indices of the planes. Actually, any direction
in reciprocal space works, so if a and b are two float
vectors spanning an atomic plane, you can get all layers
parallel to this with miller=np.cross(a,b).
tolerance: float
The maximum distance in Angstrom along the plane normal for
counting two atoms as belonging to the same plane.

Returns:

tags: array of integres
Array of layer indices for each atom.
levels: array of floats
Array of distances in Angstrom from each layer to origo.

Example:

>>> import numpy as np
>>> from ase.spacegroup import crystal
>>> atoms = crystal('Al', [(0,0,0)], spacegroup=225, cellpar=4.05)
>>> np.round(atoms.positions, decimals=5)
array([[ 0.   ,  0.   ,  0.   ],
[ 0.   ,  2.025,  2.025],
[ 2.025,  0.   ,  2.025],
[ 2.025,  2.025,  0.   ]])
>>> get_layers(atoms, (0,0,1))  # doctest: +ELLIPSIS
(array([0, 1, 1, 0]...), array([ 0.   ,  2.025]))
"""
miller = np.asarray(miller)

metric = np.dot(atoms.cell, atoms.cell.T)
c = np.linalg.solve(metric.T, miller.T).T
miller_norm = np.sqrt(np.dot(c, miller))
d = np.dot(atoms.get_scaled_positions(), miller) / miller_norm

keys = np.argsort(d)
ikeys = np.argsort(keys)
mask = np.concatenate(([True], np.diff(d[keys]) > tolerance))
if tags.min() == 1:
tags -= 1

return tags, levels

def naive_find_mic(v, cell):
"""Finds the minimum-image representation of vector(s) v.
Safe to use for (pbc.all() and (norm(v_mic) < 0.5 * min(cell.lengths()))).
Can otherwise fail for non-orthorhombic cells.
Described in:
W. Smith, "The Minimum Image Convention in Non-Cubic MD Cells", 1989,
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.1696."""
f = Cell(cell).scaled_positions(v)
f -= np.floor(f + 0.5)
vmin = f @ cell
vlen = np.linalg.norm(vmin, axis=1)
return vmin, vlen

def general_find_mic(v, cell, pbc=True):
"""Finds the minimum-image representation of vector(s) v. Using the
Minkowski reduction the algorithm is relatively slow but safe for any cell.
"""

cell = complete_cell(cell)
rcell, _ = minkowski_reduce(cell, pbc=pbc)
positions = wrap_positions(v, rcell, pbc=pbc, eps=0)

# In a Minkowski-reduced cell we only need to test nearest neighbors,
# or "Voronoi-relevant" vectors. These are a subset of combinations of
# [-1, 0, 1] of the reduced cell vectors.

# Define ranges [-1, 0, 1] for periodic directions and [0] for aperiodic
# directions.
ranges = [np.arange(-1 * p, p + 1) for p in pbc]

# Get Voronoi-relevant vectors.
# Pre-pend (0, 0, 0) to resolve issue #772
hkls = np.array([(0, 0, 0)] + list(itertools.product(*ranges)))
vrvecs = hkls @ rcell

# Map positions into neighbouring cells.
x = positions + vrvecs[:, None]

# Find minimum images
lengths = np.linalg.norm(x, axis=2)
indices = np.argmin(lengths, axis=0)
vmin = x[indices, np.arange(len(positions)), :]
vlen = lengths[indices, np.arange(len(positions))]
return vmin, vlen

[docs]def find_mic(v, cell, pbc=True):
"""Finds the minimum-image representation of vector(s) v using either one
of two find mic algorithms depending on the given cell, v and pbc."""

cell = Cell(cell)
pbc = cell.any(1) & pbc2pbc(pbc)
dim = np.sum(pbc)
v = np.asarray(v)
single = v.ndim == 1
v = np.atleast_2d(v)

if dim > 0:
naive_find_mic_is_safe = False
if dim == 3:
vmin, vlen = naive_find_mic(v, cell)
# naive find mic is safe only for the following condition
if (vlen < 0.5 * min(cell.lengths())).all():
naive_find_mic_is_safe = True  # hence skip Minkowski reduction

if not naive_find_mic_is_safe:
vmin, vlen = general_find_mic(v, cell, pbc=pbc)
else:
vmin = v.copy()
vlen = np.linalg.norm(vmin, axis=1)

if single:
return vmin[0], vlen[0]
else:
return vmin, vlen

[docs]def conditional_find_mic(vectors, cell, pbc):
"""Return list of vector arrays and corresponding list of vector lengths
for a given list of vector arrays. The minimum image convention is applied
if cell and pbc are set. Can be used like a simple version of get_distances.
"""
if (cell is None) != (pbc is None):
raise ValueError("cell or pbc must be both set or both be None")
if cell is not None:
mics = [find_mic(v, cell, pbc) for v in vectors]
vectors, vector_lengths = zip(*mics)
else:
vector_lengths = np.linalg.norm(vectors, axis=2)
return [np.asarray(v) for v in vectors], vector_lengths

[docs]def get_angles(v0, v1, cell=None, pbc=None):
"""Get angles formed by two lists of vectors.

Calculate angle in degrees between vectors v0 and v1

Set a cell and pbc to enable minimum image
convention, otherwise angles are taken as-is.
"""
(v0, v1), (nv0, nv1) = conditional_find_mic([v0, v1], cell, pbc)

if (nv0 <= 0).any() or (nv1 <= 0).any():
raise ZeroDivisionError('Undefined angle')
v0n = v0 / nv0[:, np.newaxis]
v1n = v1 / nv1[:, np.newaxis]
# We just normalized the vectors, but in some cases we can get
# bad things like 1+2e-16.  These we clip away:
angles = np.arccos(np.einsum('ij,ij->i', v0n, v1n).clip(-1.0, 1.0))
return np.degrees(angles)

[docs]def get_angles_derivatives(v0, v1, cell=None, pbc=None):
"""Get derivatives of angles formed by two lists of vectors (v0, v1) w.r.t.
Cartesian coordinates in degrees.

Set a cell and pbc to enable minimum image
convention, otherwise derivatives of angles are taken as-is.

There is a singularity in the derivatives for sin(angle) -> 0 for which
a ZeroDivisionError is raised.

Derivative output format: [[dx_a0, dy_a0, dz_a0], [...], [..., dz_a2].
"""
(v0, v1), (nv0, nv1) = conditional_find_mic([v0, v1], cell, pbc)

angles = np.radians(get_angles(v0, v1, cell=cell, pbc=pbc))
sin_angles = np.sin(angles)
cos_angles = np.cos(angles)
if (sin_angles == 0.).any():  # identify singularities
raise ZeroDivisionError('Singularity for angle derivative')

product = nv0 * nv1
deriv_d0 = (-(v1 / product[:, np.newaxis]  # derivatives by atom 0
- np.einsum('ij,i->ij', v0, cos_angles / nv0**2))
/ sin_angles[:, np.newaxis])
deriv_d2 = (-(v0 / product[:, np.newaxis]  # derivatives by atom 2
- np.einsum('ij,i->ij', v1, cos_angles / nv1**2))
/ sin_angles[:, np.newaxis])
deriv_d1 = -(deriv_d0 + deriv_d2)  # derivatives by atom 1
derivs = np.stack((deriv_d0, deriv_d1, deriv_d2), axis=1)
return np.degrees(derivs)

[docs]def get_dihedrals(v0, v1, v2, cell=None, pbc=None):
"""Get dihedral angles formed by three lists of vectors.

Calculate dihedral angle (in degrees) between the vectors a0->a1,
a1->a2 and a2->a3, written as v0, v1 and v2.

Set a cell and pbc to enable minimum image
convention, otherwise angles are taken as-is.
"""
(v0, v1, v2), (_, nv1, _) = conditional_find_mic([v0, v1, v2], cell, pbc)

v1n = v1 / nv1[:, np.newaxis]
# v, w: projection of v0, v2 onto plane perpendicular to v1
v = -v0 - np.einsum('ij,ij,ik->ik', -v0, v1n, v1n)
w = v2 - np.einsum('ij,ij,ik->ik', v2, v1n, v1n)

# formula returns 0 for undefined dihedrals; prefer ZeroDivisionError
undefined_v = np.all(v == 0.0, axis=1)
undefined_w = np.all(w == 0.0, axis=1)
if np.any([undefined_v, undefined_w]):
raise ZeroDivisionError('Undefined dihedral')

x = np.einsum('ij,ij->i', v, w)
y = np.einsum('ij,ij->i', np.cross(v1n, v, axis=1), w)
dihedrals = np.arctan2(y, x)            # dihedral angle in [-pi, pi]
dihedrals[dihedrals < 0.] += 2 * np.pi  # dihedral angle in [0, 2*pi]
return np.degrees(dihedrals)

[docs]def get_dihedrals_derivatives(v0, v1, v2, cell=None, pbc=None):
"""Get derivatives of dihedrals formed by three lists of vectors
(v0, v1, v2) w.r.t Cartesian coordinates in degrees.

Set a cell and pbc to enable minimum image
convention, otherwise dihedrals are taken as-is.

Derivative output format: [[dx_a0, dy_a0, dz_a0], ..., [..., dz_a3]].
"""
(v0, v1, v2), (nv0, nv1, nv2) = conditional_find_mic([v0, v1, v2], cell,
pbc)

v0 /= nv0[:, np.newaxis]
v1 /= nv1[:, np.newaxis]
v2 /= nv2[:, np.newaxis]
normal_v01 = np.cross(v0, v1, axis=1)
normal_v12 = np.cross(v1, v2, axis=1)
cos_psi01 = np.einsum('ij,ij->i', v0, v1)  # == np.sum(v0 * v1, axis=1)
sin_psi01 = np.sin(np.arccos(cos_psi01))
cos_psi12 = np.einsum('ij,ij->i', v1, v2)
sin_psi12 = np.sin(np.arccos(cos_psi12))
if (sin_psi01 == 0.).any() or (sin_psi12 == 0.).any():
raise ZeroDivisionError('Undefined derivative for undefined dihedral')

deriv_d0 = -normal_v01 / (nv0 * sin_psi01**2)[:, np.newaxis]  # by atom 0
deriv_d3 = normal_v12 / (nv2 * sin_psi12**2)[:, np.newaxis]  # by atom 3
deriv_d1 = (((nv1 + nv0 * cos_psi01) / nv1)[:, np.newaxis] * -deriv_d0
+ (cos_psi12 * nv2 / nv1)[:, np.newaxis] * deriv_d3)  # by a1
deriv_d2 = (-((nv1 + nv2 * cos_psi12) / nv1)[:, np.newaxis] * deriv_d3
- (cos_psi01 * nv0 / nv1)[:, np.newaxis] * -deriv_d0)  # by a2
derivs = np.stack((deriv_d0, deriv_d1, deriv_d2, deriv_d3), axis=1)
return np.degrees(derivs)

[docs]def get_distances(p1, p2=None, cell=None, pbc=None):
"""Return distance matrix of every position in p1 with every position in p2

If p2 is not set, it is assumed that distances between all positions in p1
are desired. p2 will be set to p1 in this case.

Use set cell and pbc to use the minimum image convention.
"""
p1 = np.atleast_2d(p1)
if p2 is None:
np1 = len(p1)
ind1, ind2 = np.triu_indices(np1, k=1)
D = p1[ind2] - p1[ind1]
else:
p2 = np.atleast_2d(p2)
D = (p2[np.newaxis, :, :] - p1[:, np.newaxis, :]).reshape((-1, 3))

(D, ), (D_len, ) = conditional_find_mic([D], cell=cell, pbc=pbc)

if p2 is None:
Dout = np.zeros((np1, np1, 3))
Dout[(ind1, ind2)] = D
Dout -= np.transpose(Dout, axes=(1, 0, 2))

Dout_len = np.zeros((np1, np1))
Dout_len[(ind1, ind2)] = D_len
Dout_len += Dout_len.T
return Dout, Dout_len

# Expand back to matrix indexing
D.shape = (-1, len(p2), 3)
D_len.shape = (-1, len(p2))

return D, D_len

[docs]def get_distances_derivatives(v0, cell=None, pbc=None):
"""Get derivatives of distances for all vectors in v0 w.r.t. Cartesian
coordinates in Angstrom.

Set cell and pbc to use the minimum image convention.

There is a singularity for distances -> 0 for which a ZeroDivisionError is
raised.
Derivative output format: [[dx_a0, dy_a0, dz_a0], [dx_a1, dy_a1, dz_a1]].
"""
(v0, ), (dists, ) = conditional_find_mic([v0], cell, pbc)

if (dists <= 0.).any():  # identify singularities
raise ZeroDivisionError('Singularity for distance derivative')

derivs_d0 = np.einsum('i,ij->ij', -1. / dists, v0)  # derivatives by atom 0
derivs_d1 = -derivs_d0                              # derivatives by atom 1
derivs = np.stack((derivs_d0, derivs_d1), axis=1)
return derivs

[docs]def get_duplicate_atoms(atoms, cutoff=0.1, delete=False):
"""Get list of duplicate atoms and delete them if requested.

Identify all atoms which lie within the cutoff radius of each other.
Delete one set of them if delete == True.
"""
from scipy.spatial.distance import pdist
dists = pdist(atoms.get_positions(), 'sqeuclidean')
dup = np.nonzero(dists < cutoff**2)
rem = np.array(_row_col_from_pdist(len(atoms), dup[0]))
if delete:
if rem.size != 0:
del atoms[rem[:, 0]]
else:
return rem

def _row_col_from_pdist(dim, i):
"""Calculate the i,j index in the square matrix for an index in a
condensed (triangular) matrix.
"""
i = np.array(i)
b = 1 - 2 * dim
x = (np.floor((-b - np.sqrt(b**2 - 8 * i)) / 2)).astype(int)
y = (i + x * (b + x + 2) / 2 + 1).astype(int)
if i.shape:
return list(zip(x, y))
else:
return [(x, y)]

[docs]def permute_axes(atoms, permutation):
"""Permute axes of unit cell and atom positions. Considers only cell and
atomic positions. Other vector quantities such as momenta are not
modified."""
assert (np.sort(permutation) == np.arange(3)).all()

permuted = atoms.copy()
scaled = permuted.get_scaled_positions()
permuted.set_cell(permuted.cell.permute_axes(permutation),
scale_atoms=False)
permuted.set_scaled_positions(scaled[:, permutation])
permuted.set_pbc(permuted.pbc[permutation])
return permuted