def external_basis(self):
"""The basis for small displacements in the external degrees of freedom.
The basis is expressed in mass-weighted Cartesian coordinates.
The three translations are along the x, y, and z-axis. The three
rotations are about an axis parallel to the x, y, and z-axis
through the center of mass.
The returned result depends on the periodicity of the system:
* When the system is periodic, only the translation external degrees
are included. The result is an array with shape (3,3N)
* When the system is not periodic, the rotational external
degrees are also included. The result is an array with shape (6,3N). The
first three rows correspond to translation, the latter three rows
correspond to rotation.
"""
center = (self.coordinates*self.masses3.reshape((-1,3))).sum(0)/self.mass # center of mass
result = transrot_basis(self.coordinates - center, not self.periodic)
result *= np.sqrt(self.masses3) # transform basis to mass weighted coordinates
return result
def external_basis(self):
"""The basis for small displacements in the external degrees of freedom.
The basis is expressed in mass-weighted Cartesian coordinates.
The three translations are along the x, y, and z-axis. The three
rotations are about an axis parallel to the x, y, and z-axis
through the center of mass.
The returned result depends on the periodicity of the system:
* When the system is periodic, only the translation external degrees
are included. The result is an array with shape (3,3N)
* When the system is not periodic, the rotational external
degrees are also included. The result is an array with shape (6,3N). The
first three rows correspond to translation, the latter three rows
correspond to rotation.
"""
center = (self.coordinates*self.masses3.reshape((-1,3))).sum(0)/self.mass # center of mass
result = transrot_basis(self.coordinates - center, not self.periodic)
result *= np.sqrt(self.masses3) # transform basis to mass weighted coordinates
return result
def compute_moments(coordinates, masses3, center, axis, indexes):
"""Computes the absolute and the relative moment of an internal rotor
Arguments:
| ``coordinates`` -- The coordinates of all atoms, float numpy array
with shape (N,3).
| ``masses3`` -- The diagonal of the mass matrix, each mass is repeated
three tines, float numpy array with shape 3N.
| ``center`` -- A point on the rotation axis. Float numpy array with
shape 3.
| ``axis`` -- A unit vector with the direction of the rotation axis.
Float numpy array with shape 3.
| ``indexes`` -- The indexes of the atoms that belong to the rotor.
The implementation is based on the transformation of the mass-matrix to
the internal rotation coordinate. The derivative of the internal
coordinate towards cartesian coordinates represents a small displacement
of the atoms. This displacement is constrained to show no external
linear or rotational moment.
"""
# the derivative of the cartesian coordinates towards the rotation
# angle of the top:
rot_tangent = numpy.zeros((len(coordinates)*3), float)
for i in indexes:
rot_tangent[3*i:3*i+3] = numpy.cross(coordinates[i]-center, axis)
# transform this to a derivative without global linear or angular
# momentum. This is done by projecting on the basis of external degrees
# of freedom in mass-weighted coordinates, and subsequently subtracting
# that projection from the original tangent
basis = transrot_basis(coordinates)
A = numpy.dot(basis*masses3, basis.transpose())
B = numpy.dot(basis*masses3, rot_tangent)
alphas = numpy.linalg.solve(A,B)
rot_tangent_relative = rot_tangent - numpy.dot(alphas, basis)
return (
(rot_tangent**2*masses3).sum(),
(rot_tangent_relative**2*masses3).sum()
)
def compute_moments(coordinates, masses3, center, axis, indexes):
"""Computes the absolute and the relative moment of an internal rotor
Arguments:
| ``coordinates`` -- The coordinates of all atoms, float numpy array
with shape (N,3).
| ``masses3`` -- The diagonal of the mass matrix, each mass is repeated
three tines, float numpy array with shape 3N.
| ``center`` -- A point on the rotation axis. Float numpy array with
shape 3.
| ``axis`` -- A unit vector with the direction of the rotation axis.
Float numpy array with shape 3.
| ``indexes`` -- The indexes of the atoms that belong to the rotor.
The implementation is based on the transformation of the mass-matrix to
the internal rotation coordinate. The derivative of the internal
coordinate towards cartesian coordinates represents a small displacement
of the atoms. This displacement is constrained to show no external
linear or rotational moment.
"""
# the derivative of the cartesian coordinates towards the rotation
# angle of the top:
rot_tangent = np.zeros((len(coordinates) * 3), float)
for i in indexes:
rot_tangent[3 * i:3 * i + 3] = np.cross(coordinates[i] - center, axis)
# transform this to a derivative without global linear or angular
# momentum. This is done by projecting on the basis of external degrees
# of freedom in mass-weighted coordinates, and subsequently subtracting
# that projection from the original tangent
basis = transrot_basis(coordinates)
A = np.dot(basis * masses3, basis.transpose())
B = np.dot(basis * masses3, rot_tangent)
alphas = np.linalg.solve(A, B)
rot_tangent_relative = rot_tangent - np.dot(alphas, basis)
return ((rot_tangent**2 * masses3).sum(),
(rot_tangent_relative**2 * masses3).sum())