Package lib :: Package frame_order :: Module iso_cone
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Source Code for Module lib.frame_order.iso_cone

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  2  #                                                                             # 
  3  # Copyright (C) 2009-2014 Edward d'Auvergne                                   # 
  4  #                                                                             # 
  5  # This file is part of the program relax (http://www.nmr-relax.com).          # 
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 21   
 22  # Module docstring. 
 23  """Module for the handling of Frame Order.""" 
 24   
 25  # Python module imports. 
 26  from math import cos, pi, sqrt 
 27  from numpy import sinc 
 28  try: 
 29      from scipy.integrate import tplquad 
 30  except ImportError: 
 31      pass 
 32   
 33  # relax module imports. 
 34  from lib.frame_order.matrix_ops import pcs_pivot_motion_full, pcs_pivot_motion_full_qrint, rotate_daeg 
 35   
 36   
37 -def compile_2nd_matrix_iso_cone(matrix, Rx2_eigen, cone_theta, sigma_max):
38 """Generate the rotated 2nd degree Frame Order matrix for the isotropic cone. 39 40 The cone axis is assumed to be parallel to the z-axis in the eigenframe. 41 42 @param matrix: The Frame Order matrix, 2nd degree to be populated. 43 @type matrix: numpy 9D, rank-2 array 44 @param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself. 45 @type Rx2_eigen: numpy 9D, rank-2 array 46 @param cone_theta: The cone opening angle. 47 @type cone_theta: float 48 @param sigma_max: The maximum torsion angle. 49 @type sigma_max: float 50 """ 51 52 # Populate the Frame Order matrix in the eigenframe. 53 populate_2nd_eigenframe_iso_cone(matrix, cone_theta, sigma_max) 54 55 # Rotate and return the frame order matrix. 56 return rotate_daeg(matrix, Rx2_eigen)
57 58
59 -def pcs_numeric_int_iso_cone(theta_max=None, sigma_max=None, c=None, r_pivot_atom=None, r_ln_pivot=None, A=None, R_eigen=None, RT_eigen=None, Ri_prime=None):
60 """Determine the averaged PCS value via numerical integration. 61 62 @keyword theta_max: The half cone angle. 63 @type theta_max: float 64 @keyword sigma_max: The maximum torsion angle. 65 @type sigma_max: float 66 @keyword c: The PCS constant (without the interatomic distance and in Angstrom units). 67 @type c: float 68 @keyword r_pivot_atom: The pivot point to atom vector. 69 @type r_pivot_atom: numpy rank-1, 3D array 70 @keyword r_ln_pivot: The lanthanide position to pivot point vector. 71 @type r_ln_pivot: numpy rank-1, 3D array 72 @keyword A: The full alignment tensor of the non-moving domain. 73 @type A: numpy rank-2, 3D array 74 @keyword R_eigen: The eigenframe rotation matrix. 75 @type R_eigen: numpy rank-2, 3D array 76 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 77 @type RT_eigen: numpy rank-2, 3D array 78 @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. 79 @type Ri_prime: numpy rank-2, 3D array 80 @return: The averaged PCS value. 81 @rtype: float 82 """ 83 84 # Perform numerical integration. 85 result = tplquad(pcs_pivot_motion_full, -sigma_max, sigma_max, lambda phi: -pi, lambda phi: pi, lambda theta, phi: 0.0, lambda theta, phi: theta_max, args=(r_pivot_atom, r_ln_pivot, A, R_eigen, RT_eigen, Ri_prime)) 86 87 # The surface area normalisation factor. 88 SA = 4.0 * pi * sigma_max * (1.0 - cos(theta_max)) 89 90 # Return the value. 91 return c * result[0] / SA
92 93
94 -def pcs_numeric_int_iso_cone_qrint(points=None, theta_max=None, sigma_max=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):
95 """Determine the averaged PCS value via numerical integration. 96 97 @keyword points: The Sobol points in the torsion-tilt angle space. 98 @type points: numpy rank-2, 3D array 99 @keyword theta_max: The half cone angle. 100 @type theta_max: float 101 @keyword sigma_max: The maximum torsion angle. 102 @type sigma_max: float 103 @keyword c: The PCS constant (without the interatomic distance and in Angstrom units). 104 @type c: numpy rank-1 array 105 @keyword full_in_ref_frame: An array of flags specifying if the tensor in the reference frame is the full or reduced tensor. 106 @type full_in_ref_frame: numpy rank-1 array 107 @keyword r_pivot_atom: The pivot point to atom vector. 108 @type r_pivot_atom: numpy rank-2, 3D array 109 @keyword r_pivot_atom_rev: The reversed pivot point to atom vector. 110 @type r_pivot_atom_rev: numpy rank-2, 3D array 111 @keyword r_ln_pivot: The lanthanide position to pivot point vector. 112 @type r_ln_pivot: numpy rank-2, 3D array 113 @keyword A: The full alignment tensor of the non-moving domain. 114 @type A: numpy rank-2, 3D array 115 @keyword R_eigen: The eigenframe rotation matrix. 116 @type R_eigen: numpy rank-2, 3D array 117 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 118 @type RT_eigen: numpy rank-2, 3D array 119 @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. 120 @type Ri_prime: numpy rank-2, 3D array 121 @keyword pcs_theta: The storage structure for the back-calculated PCS values. 122 @type pcs_theta: numpy rank-2 array 123 @keyword pcs_theta_err: The storage structure for the back-calculated PCS errors. 124 @type pcs_theta_err: numpy rank-2 array 125 @keyword missing_pcs: A structure used to indicate which PCS values are missing. 126 @type missing_pcs: numpy rank-2 array 127 @keyword error_flag: A flag which if True will cause the PCS errors to be estimated and stored in pcs_theta_err. 128 @type error_flag: bool 129 """ 130 131 # Clear the data structures. 132 for i in range(len(pcs_theta)): 133 for j in range(len(pcs_theta[i])): 134 pcs_theta[i, j] = 0.0 135 pcs_theta_err[i, j] = 0.0 136 137 # Loop over the samples. 138 num = 0 139 for i in range(len(points)): 140 # Unpack the point. 141 theta, phi, sigma = points[i] 142 143 # Outside of the distribution, so skip the point. 144 if theta > theta_max: 145 continue 146 if sigma > sigma_max or sigma < -sigma_max: 147 continue 148 149 # Calculate the PCSs for this state. 150 pcs_pivot_motion_full_qrint(theta_i=theta, phi_i=phi, sigma_i=sigma, 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) 151 152 # Increment the number of points. 153 num += 1 154 155 # Calculate the PCS and error. 156 for i in range(len(pcs_theta)): 157 for j in range(len(pcs_theta[i])): 158 # The average PCS. 159 pcs_theta[i, j] = c[i] * pcs_theta[i, j] / float(num) 160 161 # The error. 162 if error_flag: 163 pcs_theta_err[i, j] = abs(pcs_theta_err[i, j] / float(num) - pcs_theta[i, j]**2) / float(num) 164 pcs_theta_err[i, j] = c[i] * sqrt(pcs_theta_err[i, j]) 165 print("%8.3f +/- %-8.3f" % (pcs_theta[i, j]*1e6, pcs_theta_err[i, j]*1e6))
166 167
168 -def populate_1st_eigenframe_iso_cone(matrix, angle):
169 """Populate the 1st degree Frame Order matrix in the eigenframe for an isotropic cone. 170 171 The cone axis is assumed to be parallel to the z-axis in the eigenframe. 172 173 @param matrix: The Frame Order matrix, 1st degree. 174 @type matrix: numpy 3D, rank-2 array 175 @param angle: The cone angle. 176 @type angle: float 177 """ 178 179 # Zeros. 180 for i in range(3): 181 for j in range(3): 182 matrix[i, j] = 0.0 183 184 # The c33 element. 185 matrix[2, 2] = (cos(angle) + 1.0) / 2.0
186 187
188 -def populate_2nd_eigenframe_iso_cone(matrix, tmax, smax):
189 """Populate the 2nd degree Frame Order matrix in the eigenframe for the isotropic cone. 190 191 The cone axis is assumed to be parallel to the z-axis in the eigenframe. 192 193 194 @param matrix: The Frame Order matrix, 2nd degree. 195 @type matrix: numpy 9D, rank-2 array 196 @param tmax: The cone opening angle. 197 @type tmax: float 198 @param smax: The maximum torsion angle. 199 @type smax: float 200 """ 201 202 # Zeros. 203 for i in range(9): 204 for j in range(9): 205 matrix[i, j] = 0.0 206 207 # Repetitive trig calculations. 208 sinc_smax = sinc(smax/pi) 209 sinc_2smax = sinc(2.0*smax/pi) 210 cos_tmax = cos(tmax) 211 cos_tmax2 = cos_tmax**2 212 213 # Larger factors. 214 fact_sinc_2smax = sinc_2smax*(cos_tmax2 + 4.0*cos_tmax + 7.0) / 24.0 215 fact_cos_tmax2 = (cos_tmax2 + cos_tmax + 4.0) / 12.0 216 fact_cos_tmax = (cos_tmax + 1.0) / 4.0 217 218 # Diagonal. 219 matrix[0, 0] = fact_sinc_2smax + fact_cos_tmax2 220 matrix[1, 1] = fact_sinc_2smax + fact_cos_tmax 221 matrix[2, 2] = sinc_smax * (2.0*cos_tmax2 + 5.0*cos_tmax + 5.0) / 12.0 222 matrix[3, 3] = matrix[1, 1] 223 matrix[4, 4] = matrix[0, 0] 224 matrix[5, 5] = matrix[2, 2] 225 matrix[6, 6] = matrix[2, 2] 226 matrix[7, 7] = matrix[2, 2] 227 matrix[8, 8] = (cos_tmax2 + cos_tmax + 1.0) / 3.0 228 229 # Off diagonal set 1. 230 matrix[0, 4] = matrix[4, 0] = -fact_sinc_2smax + fact_cos_tmax2 231 matrix[0, 8] = matrix[8, 0] = -(cos_tmax2 + cos_tmax - 2.0) / 6.0 232 matrix[4, 8] = matrix[8, 4] = matrix[0, 8] 233 234 # Off diagonal set 2. 235 matrix[1, 3] = matrix[3, 1] = fact_sinc_2smax - fact_cos_tmax 236 matrix[2, 6] = matrix[6, 2] = sinc_smax * (cos_tmax2 + cos_tmax - 2.0) / 6.0 237 matrix[5, 7] = matrix[7, 5] = matrix[2, 6]
238