def part_int_daeg1_pseudo_ellipse_22(phi, x, y):
"""The theta-sigma partial integral of the 1st degree Frame Order matrix element 22 for the pseudo-ellipse.
@param phi: The azimuthal tilt-torsion angle.
@type phi: float
@param x: The cone opening angle along x.
@type x: float
@param y: The cone opening angle along y.
@type y: float
@return: The theta-sigma partial integral.
@rtype: float
"""
# Theta max.
tmax = tmax_pseudo_ellipse(phi, x, y)
# The theta-sigma integral.
return sin(tmax)**2
def part_int_daeg2_pseudo_ellipse_free_rotor_88(phi, x, y):
"""The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.
@param phi: The azimuthal tilt-torsion angle.
@type phi: float
@param x: The cone opening angle along x.
@type x: float
@param y: The cone opening angle along y.
@type y: float
@return: The theta-sigma partial integral.
@rtype: float
"""
# Theta max.
tmax = tmax_pseudo_ellipse(phi, x, y)
# The theta-sigma integral.
return cos(tmax)**3
def part_int_daeg2_pseudo_ellipse_torsionless_55(phi, x, y):
"""The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.
@param phi: The azimuthal tilt-torsion angle.
@type phi: float
@param x: The cone opening angle along x.
@type x: float
@param y: The cone opening angle along y.
@type y: float
@return: The theta partial integral.
@rtype: float
"""
# Theta max.
tmax = tmax_pseudo_ellipse(phi, x, y)
# The theta integral.
return 2.0*sin(phi)**2*cos(tmax)**3 + 3.0*cos(phi)**2*cos(tmax)**2
def part_int_daeg2_pseudo_ellipse_free_rotor_88(phi, x, y):
"""The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.
@param phi: The azimuthal tilt-torsion angle.
@type phi: float
@param x: The cone opening angle along x.
@type x: float
@param y: The cone opening angle along y.
@type y: float
@return: The theta-sigma partial integral.
@rtype: float
"""
# Theta max.
tmax = tmax_pseudo_ellipse(phi, x, y)
# The theta-sigma integral.
return cos(tmax)**3
def part_int_daeg2_pseudo_ellipse_torsionless_55(phi, x, y):
"""The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.
@param phi: The azimuthal tilt-torsion angle.
@type phi: float
@param x: The cone opening angle along x.
@type x: float
@param y: The cone opening angle along y.
@type y: float
@return: The theta partial integral.
@rtype: float
"""
# Theta max.
tmax = tmax_pseudo_ellipse(phi, x, y)
# The theta integral.
return 2.0 * sin(phi)**2 * cos(tmax)**3 + 3.0 * cos(phi)**2 * cos(tmax)**2
def part_int_daeg1_pseudo_ellipse_22(phi, x, y):
"""The theta-sigma partial integral of the 1st degree Frame Order matrix element 22 for the pseudo-ellipse.
@param phi: The azimuthal tilt-torsion angle.
@type phi: float
@param x: The cone opening angle along x.
@type x: float
@param y: The cone opening angle along y.
@type y: float
@return: The theta-sigma partial integral.
@rtype: float
"""
# Theta max.
tmax = tmax_pseudo_ellipse(phi, x, y)
# The theta-sigma integral.
return sin(tmax)**2
def pseudo_ellipse(phi):
"""The pseudo-ellipse wrapper formula."""
return tmax_pseudo_ellipse(phi, theta_x, theta_y)
def pseudo_ellipse(phi):
"""The pseudo-ellipse wrapper formula."""
return tmax_pseudo_ellipse(phi, theta_x, theta_y)
def pcs_numeric_int_pseudo_ellipse_torsionless_qrint(points=None, theta_x=None, theta_y=None, c=None, full_in_ref_frame=None, r_pivot_atom=None, r_pivot_atom_rev=None, r_ln_pivot=None, A=None, R_eigen=None, RT_eigen=None, Ri_prime=None, pcs_theta=None, pcs_theta_err=None, missing_pcs=None, error_flag=False):
"""Determine the averaged PCS value via numerical integration.
@keyword points: The Sobol points in the torsion-tilt angle space.
@type points: numpy rank-2, 3D array
@keyword theta_x: The x-axis half cone angle.
@type theta_x: float
@keyword theta_y: The y-axis half cone angle.
@type theta_y: float
@keyword c: The PCS constant (without the interatomic distance and in Angstrom units).
@type c: numpy rank-1 array
@keyword full_in_ref_frame: An array of flags specifying if the tensor in the reference frame is the full or reduced tensor.
@type full_in_ref_frame: numpy rank-1 array
@keyword r_pivot_atom: The pivot point to atom vector.
@type r_pivot_atom: numpy rank-2, 3D array
@keyword r_pivot_atom_rev: The reversed pivot point to atom vector.
@type r_pivot_atom_rev: numpy rank-2, 3D array
@keyword r_ln_pivot: The lanthanide position to pivot point vector.
@type r_ln_pivot: numpy rank-2, 3D array
@keyword A: The full alignment tensor of the non-moving domain.
@type A: numpy rank-2, 3D array
@keyword R_eigen: The eigenframe rotation matrix.
@type R_eigen: numpy rank-2, 3D array
@keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations).
@type RT_eigen: numpy rank-2, 3D array
@keyword Ri_prime: The empty rotation matrix for the in-frame isotropic cone motion, used to calculate the PCS for each state i in the numerical integration.
@type Ri_prime: numpy rank-2, 3D array
@keyword pcs_theta: The storage structure for the back-calculated PCS values.
@type pcs_theta: numpy rank-2 array
@keyword pcs_theta_err: The storage structure for the back-calculated PCS errors.
@type pcs_theta_err: numpy rank-2 array
@keyword missing_pcs: A structure used to indicate which PCS values are missing.
@type missing_pcs: numpy rank-2 array
@keyword error_flag: A flag which if True will cause the PCS errors to be estimated and stored in pcs_theta_err.
@type error_flag: bool
"""
# Clear the data structures.
for i in range(len(pcs_theta)):
for j in range(len(pcs_theta[i])):
pcs_theta[i, j] = 0.0
pcs_theta_err[i, j] = 0.0
# Loop over the samples.
num = 0
for i in range(len(points)):
# Unpack the point.
theta, phi = points[i]
# Calculate theta_max.
theta_max = tmax_pseudo_ellipse(phi, theta_x, theta_y)
# Outside of the distribution, so skip the point.
if theta > theta_max:
continue
# Calculate the PCSs for this state.
pcs_pivot_motion_torsionless_qrint(theta_i=theta, phi_i=phi, full_in_ref_frame=full_in_ref_frame, r_pivot_atom=r_pivot_atom, r_pivot_atom_rev=r_pivot_atom_rev, r_ln_pivot=r_ln_pivot, A=A, R_eigen=R_eigen, RT_eigen=RT_eigen, Ri_prime=Ri_prime, pcs_theta=pcs_theta, pcs_theta_err=pcs_theta_err, missing_pcs=missing_pcs)
# Increment the number of points.
num += 1
# Calculate the PCS and error.
for i in range(len(pcs_theta)):
for j in range(len(pcs_theta[i])):
# The average PCS.
pcs_theta[i, j] = c[i] * pcs_theta[i, j] / float(num)
# The error.
if error_flag:
pcs_theta_err[i, j] = abs(pcs_theta_err[i, j] / float(num) - pcs_theta[i, j]**2) / float(num)
pcs_theta_err[i, j] = c[i] * sqrt(pcs_theta_err[i, j])
print("%8.3f +/- %-8.3f" % (pcs_theta[i, j]*1e6, pcs_theta_err[i, j]*1e6))