def tors(xyz, *inds, units='rad', absv=False):
"""Returns dihedral angle for 4 atoms in a chain based on index.
Parameters
----------
xyz : (N, 3) array_like
The atomic cartesian coordinates.
inds : list
The indices for which the dihedral angle is measured.
units : str, optional
The units of angle for the output. Default is radians.
absv : bool, optional
Specifies if the absolute value is returned.
Returns
-------
float
The dihedral angle.
"""
e1 = xyz[inds[0]] - xyz[inds[1]]
e2 = xyz[inds[2]] - xyz[inds[1]]
e3 = xyz[inds[2]] - xyz[inds[3]]
# get normals to 3-atom planes
cp1 = con.unit_vec(np.cross(e1, e2))
cp2 = con.unit_vec(np.cross(e2, e3))
if absv:
coord = con.arccos(np.dot(cp1, cp2))
else:
# get cross product of plane normals for signed dihedral angle
cp3 = np.cross(cp1, cp2)
coord = np.sign(np.dot(cp3, e2)) * con.arccos(np.dot(cp1, cp2))
return coord * con.conv('rad', units)
def edgetors(xyz, *inds, units='rad', absv=False):
"""Returns the torsional angle based on the vector difference of the
two external atoms (1-2 and 5-6) to the central 3-4 bond.
Parameters
----------
xyz : (N, 3) array_like
The atomic cartesian coordinates.
inds : list
The indices for which the edge dihedral angle is measured.
units : str, optional
The units of angle for the output. Default is radians.
absv : bool, optional
Specifies if the absolute value is returned.
Returns
-------
float
The edge dihedral angle.
"""
e1 = con.unit_vec(xyz[inds[0]] - xyz[inds[2]])
e2 = con.unit_vec(xyz[inds[1]] - xyz[inds[2]])
e3 = con.unit_vec(xyz[inds[3]] - xyz[inds[2]])
e4 = con.unit_vec(xyz[inds[4]] - xyz[inds[3]])
e5 = con.unit_vec(xyz[inds[5]] - xyz[inds[3]])
# take the difference between unit vectors of external bonds
e2 -= e1
e5 -= e4
# get cross products of difference vectors and the central bond
cp1 = con.unit_vec(np.cross(e2, e3))
cp2 = con.unit_vec(np.cross(e3, e5))
if absv:
coord = con.arccos(np.dot(cp1, cp2))
else:
# get cross product of vectors for signed dihedral angle
cp3 = np.cross(cp1, cp2)
coord = np.sign(np.dot(cp3, e3)) * con.arccos(np.dot(cp1, cp2))
return coord * con.conv('rad', units)
def planetors(xyz, *inds, units='rad', absv=False):
"""Returns the plane angle with the central bond projected out.
Parameters
----------
xyz : (N, 3) array_like
The atomic cartesian coordinates.
inds : list
The indices for which the plane dihedral angle is measured.
units : str, optional
The units of angle for the output. Default is radians.
absv : bool, optional
Specifies if the absolute value is returned.
Returns
-------
float
The plane dihedral angle.
"""
e1 = xyz[inds[0]] - xyz[inds[2]]
e2 = xyz[inds[1]] - xyz[inds[2]]
e3 = con.unit_vec(xyz[inds[3]] - xyz[inds[2]])
e4 = xyz[inds[4]] - xyz[inds[3]]
e5 = xyz[inds[5]] - xyz[inds[3]]
# get normals to 3-atom planes
cp1 = np.cross(e1, e2)
cp2 = np.cross(e4, e5)
# project out component along central bond
pj1 = con.unit_vec(cp1 - np.dot(cp1, e3) * e3)
pj2 = con.unit_vec(cp2 - np.dot(cp2, e3) * e3)
if absv:
coord = con.arccos(np.dot(pj1, pj2))
else:
# get cross product of vectors for signed dihedral angle
cp3 = np.cross(pj1, pj2)
coord = np.sign(np.dot(cp3, e3)) * con.arccos(np.dot(pj1, pj2))
return coord * con.conv('rad', units)
def bend(xyz, *inds, units='rad'):
"""Returns bending angle for 3 atoms in a chain based on index.
Parameters
----------
xyz : (N, 3) array_like
The atomic cartesian coordinates.
inds : list
The indices for which the bond angle is measured.
units : str, optional
The units of angle for the output. Default is radians.
Returns
-------
float
The bond angle.
"""
e1 = con.unit_vec(xyz[inds[0]] - xyz[inds[1]])
e2 = con.unit_vec(xyz[inds[2]] - xyz[inds[1]])
coord = con.arccos(np.dot(e1, e2))
return coord * con.conv('rad', units)
def oop(xyz, *inds, units='rad', absv=False):
"""Returns out-of-plane angle of atom 1 connected to atom 4 in the
2-3-4 plane.
Contains an additional sign convention such that rotation of the
out-of-plane atom over (under) the central plane atom gives an angle
greater than :math:`\pi/2` (less than :math:`-\pi/2`).
Parameters
----------
xyz : (N, 3) array_like
The atomic cartesian coordinates.
inds : list
The indices for which the out-of-plane angle is measured.
units : str, optional
The units of angle for the output. Default is radians.
absv : bool, optional
Specifies if the absolute value is returned.
Returns
-------
float
The out-of-plane angle.
"""
e1 = con.unit_vec(xyz[inds[0]] - xyz[inds[3]])
e2 = con.unit_vec(xyz[inds[1]] - xyz[inds[3]])
e3 = con.unit_vec(xyz[inds[2]] - xyz[inds[3]])
sintau = np.dot(np.cross(e2, e3) / np.sqrt(1 - np.dot(e2, e3)**2), e1)
coord = np.sign(np.dot(e2 + e3, e1)) * con.arccos(sintau) + np.pi / 2
# sign convention to keep |oop| < pi
if coord > np.pi:
coord -= 2 * np.pi
if absv:
return abs(coord) * con.conv('rad', units)
else:
return coord * con.conv('rad', units)
def align_axis(xyz, test_ax, ref_ax, ind=None, origin=np.zeros(3)):
"""Rotates a set of atoms such that two axes are parallel.
Parameters
----------
xyz : (N, 3) array_like
The atomic cartesian coordinates.
test_crd : (3,) array_like
Cartesian coordinates of the original axis.
test_crd : (3,) array_like
Cartesian coordinates of the final axis.
ind : array_like, optional
List of atomic indices to specify which atoms are displaced. If
`ind == None` (default) then all atoms are displaced.
origin : (3,) array_like, optional
The origin of rotation. Default is the cartesian origin.
Returns
-------
(N, 3) ndarray
The atomic cartesian coordinates of the displaced molecule.
"""
vp = VectorParser(xyz)
test = vp(test_ax, unit=True)
ref = vp(ref_ax, unit=True)
if np.allclose(test, ref):
return xyz
elif np.allclose(test, -ref):
rotax = np.array([0., 0., 1.])
if np.allclose(test, rotax) or np.allclose(test, -rotax):
rotax = np.array([0., 1., 0.])
rotax -= np.dot(rotax, test) * test
return rotate(xyz, np.pi, rotax, ind=ind, origin=origin)
else:
angle = con.arccos(np.dot(test, ref))
rotax = np.cross(test, ref)
return rotate(xyz, angle, rotax, ind=ind, origin=origin)
def test_arccos_minusone():
ang = con.arccos(-1 - 1e-10)
assert np.isclose(ang, np.pi)
def test_arccos_plusone():
ang = con.arccos(1 + 1e-10)
assert np.isclose(ang, 0)
def angax(rotmat, units='rad'):
r"""Returns the angle, axis of rotation and determinant of a
rotational matrix.
Based on the form of **R**, it can be separated into symmetric
and antisymmetric components with :math:`(r_{ij} + r_{ji})/2` and
:math:`(r_{ij} - r_{ji})/2`, respectively. Then,
.. math::
r_{ii} = \cos(a) + u_i^2 (\det(\mathbf{R}) - \cos(a)),
\cos(a) = (-\det(\mathbf{R}) + \sum_j r_{jj}) / 2 =
(\mathrm{tr}(\mathbf{R}) - \det(\mathbf{R})) / 2.
From the expression for :math:`r_{ii}`, the magnitude of :math:`u_i`
can be found
.. math::
|u_i| = \sqrt{\frac{1 + \det(\mathbf{R}) [2 r_{ii} -
\mathrm{tr}(\mathbf{R})])}{3 - \det(\mathbf{R}) \mathrm{tr}(\mathbf{R})}},
which satisfies :math:`u \cdot u = 1`. Note that if
:math:`\det(\mathbf{R}) \mathrm{tr}(\mathbf{R}) = 3`, the axis is arbitrary
(identity or inversion). Otherwise, the sign can be found from the
antisymmetric component of **R**.
.. math::
u_i \sin(a) = (r_{jk} - r_{kj}) / 2, \quad i \neq j \neq k,
\mathrm{sign}(u_i) = \mathrm{sign}(r_{jk} - r_{kj}),
since :math:`\sin(a)` is positive in the range 0 to :math:`\pi`. :math:`i`,
:math:`j` and :math:`k` obey the cyclic relation 3 -> 2 -> 1 -> 3 -> ...
This fails when :math:`det(\mathbf{R}) \mathrm{tr}(\mathbf{R}) = -1`, in
which case the symmetric component of **R** is used
.. math::
u_i u_j (\det(\mathbf{R}) - \cos(a)) = (r_{ij} + r_{ji}) / 2,
\mathrm{sign}(u_i) \mathrm{sign}(u_j) = \det(\mathbf{R}) \mathrm{sign}(r_{ij} + r_{ji}).
The signs can then be found by letting :math:`\mathrm{sign}(u_3) = +1`,
since a rotation of :math:`pi` or a reflection are equivalent for
antiparallel axes. See
http://scipp.ucsc.edu/~haber/ph251/rotreflect_17.pdf
Parameters
----------
rotmat : (3, 3) array_like
The rotational matrix.
units : str, optional
The output units for the angle. Default is radians.
Returns
-------
ang : float
The angle of rotation.
u : (3,) ndarray
The axis of rotation as a 3D vector.
det : int
The determinant of the rotation matrix.
Raises
------
ValueError
When the absolute value of the determinant is not equal to 1.
"""
det = np.linalg.det(rotmat)
if not np.isclose(np.abs(det), 1):
raise ValueError('Determinant of a rotational matrix must be +/- 1')
tr = np.trace(rotmat)
ang = con.arccos((tr - det) / 2) * con.conv('rad', units)
if np.isclose(det * tr, 3):
u = np.array([0, 0, 1])
else:
u = np.sqrt((1 + det * (2 * np.diag(rotmat) - tr)) / (3 - det * tr))
if np.isclose(det * tr, -1):
sgn = np.ones(3)
sgn[1] = det * _nonzero_sign(rotmat[1, 2] + rotmat[2, 1])
sgn[0] = det * sgn[1] * _nonzero_sign(rotmat[0, 1] + rotmat[1, 0])
u *= sgn
else:
u[0] *= _nonzero_sign(rotmat[1, 2] - rotmat[2, 1])
u[1] *= _nonzero_sign(rotmat[2, 0] - rotmat[0, 2])
u[2] *= _nonzero_sign(rotmat[0, 1] - rotmat[1, 0])
return ang, u, det