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1import numpy as np

4def rotation_matrix_from_points(m0, m1):

5 """Returns a rigid transformation/rotation matrix that minimizes the

6 RMSD between two set of points.

8 m0 and m1 should be (3, npoints) numpy arrays with

9 coordinates as columns::

11 (x1 x2 x3 ... xN

12 y1 y2 y3 ... yN

13 z1 z2 z3 ... zN)

15 The centeroids should be set to origin prior to

16 computing the rotation matrix.

18 The rotation matrix is computed using quaternion

19 algebra as detailed in::

21 Melander et al. J. Chem. Theory Comput., 2015, 11,1055

22 """

24 v0 = np.copy(m0)

25 v1 = np.copy(m1)

27 # compute the rotation quaternion

29 R11, R22, R33 = np.sum(v0 * v1, axis=1)

30 R12, R23, R31 = np.sum(v0 * np.roll(v1, -1, axis=0), axis=1)

31 R13, R21, R32 = np.sum(v0 * np.roll(v1, -2, axis=0), axis=1)

33 f = [[R11 + R22 + R33, R23 - R32, R31 - R13, R12 - R21],

34 [R23 - R32, R11 - R22 - R33, R12 + R21, R13 + R31],

35 [R31 - R13, R12 + R21, -R11 + R22 - R33, R23 + R32],

36 [R12 - R21, R13 + R31, R23 + R32, -R11 - R22 + R33]]

38 F = np.array(f)

40 w, V = np.linalg.eigh(F)

41 # eigenvector corresponding to the most

42 # positive eigenvalue

43 q = V[:, np.argmax(w)]

45 # Rotation matrix from the quaternion q

47 R = quaternion_to_matrix(q)

49 return R

52def quaternion_to_matrix(q):

53 """Returns a rotation matrix.

55 Computed from a unit quaternion Input as (4,) numpy array.

56 """

58 q0, q1, q2, q3 = q

59 R_q = [[q0**2 + q1**2 - q2**2 - q3**2,

60 2 * (q1 * q2 - q0 * q3),

61 2 * (q1 * q3 + q0 * q2)],

62 [2 * (q1 * q2 + q0 * q3),

63 q0**2 - q1**2 + q2**2 - q3**2,

64 2 * (q2 * q3 - q0 * q1)],

65 [2 * (q1 * q3 - q0 * q2),

66 2 * (q2 * q3 + q0 * q1),

67 q0**2 - q1**2 - q2**2 + q3**2]]

68 return np.array(R_q)

71def minimize_rotation_and_translation(target, atoms):

72 """Minimize RMSD between atoms and target.

74 Rotate and translate atoms to best match target. For more details, see::

76 Melander et al. J. Chem. Theory Comput., 2015, 11,1055

77 """

79 p = atoms.get_positions()

80 p0 = target.get_positions()

82 # centeroids to origin

83 c = np.mean(p, axis=0)

84 p -= c

85 c0 = np.mean(p0, axis=0)

86 p0 -= c0

88 # Compute rotation matrix

89 R = rotation_matrix_from_points(p.T, p0.T)

91 atoms.set_positions(np.dot(p, R.T) + c0)