import numpy as np
from ase.geometry import find_mic
def rotation_matrix_from_points(m0, m1):
"""Returns a rigid transformation/rotation matrix that minimizes the
RMSD between two set of points.
m0 and m1 should be (3, npoints) numpy arrays with
coordinates as columns::
(x1 x2 x3 ... xN
y1 y2 y3 ... yN
z1 z2 z3 ... zN)
The centeroids should be set to origin prior to
computing the rotation matrix.
The rotation matrix is computed using quaternion
algebra as detailed in::
Melander et al. J. Chem. Theory Comput., 2015, 11,1055
"""
v0 = np.copy(m0)
v1 = np.copy(m1)
# compute the rotation quaternion
R11, R22, R33 = np.sum(v0 * v1, axis=1)
R12, R23, R31 = np.sum(v0 * np.roll(v1, -1, axis=0), axis=1)
R13, R21, R32 = np.sum(v0 * np.roll(v1, -2, axis=0), axis=1)
f = [[R11 + R22 + R33, R23 - R32, R31 - R13, R12 - R21],
[R23 - R32, R11 - R22 - R33, R12 + R21, R13 + R31],
[R31 - R13, R12 + R21, -R11 + R22 - R33, R23 + R32],
[R12 - R21, R13 + R31, R23 + R32, -R11 - R22 + R33]]
F = np.array(f)
w, V = np.linalg.eigh(F)
# eigenvector corresponding to the most
# positive eigenvalue
q = V[:, np.argmax(w)]
# Rotation matrix from the quaternion q
R = quaternion_to_matrix(q)
return R
def quaternion_to_matrix(q):
"""Returns a rotation matrix.
Computed from a unit quaternion Input as (4,) numpy array.
"""
q0, q1, q2, q3 = q
R_q = [[q0**2 + q1**2 - q2**2 - q3**2,
2 * (q1 * q2 - q0 * q3),
2 * (q1 * q3 + q0 * q2)],
[2 * (q1 * q2 + q0 * q3),
q0**2 - q1**2 + q2**2 - q3**2,
2 * (q2 * q3 - q0 * q1)],
[2 * (q1 * q3 - q0 * q2),
2 * (q2 * q3 + q0 * q1),
q0**2 - q1**2 - q2**2 + q3**2]]
return np.array(R_q)
[docs]def minimize_rotation_and_translation(target, atoms):
"""Minimize RMSD between atoms and target.
Rotate and translate atoms to best match target. Disregards rotation if PBC
are found. Does not accound for changes in the cell. For more details, see::
Melander et al. J. Chem. Theory Comput., 2015, 11,1055
"""
p = atoms.get_positions()
p0 = target.get_positions()
if sum(atoms.pbc) != 0:
# maybe we can raise a warning about cell changes here since we don't
# account for them?
# is this the best form of *find_mic version to use?
dp_min, dp_len = find_mic(p - p0, cell=target.cell, pbc=target.pbc)
# add displacement without net translation
p = p0 + dp_min - np.mean(dp_min, axis=0)
R = np.eye(3) # null rotation
# centeroids to origin
c = np.mean(p, axis=0)
p -= c
c0 = np.mean(p0, axis=0)
p0 -= c0
if sum(atoms.pbc) == 0:
# Compute rotation matrix
R = rotation_matrix_from_points(p.T, p0.T)
atoms.set_positions(np.dot(p, R.T) + c0)