lib.frame_order.iso

1 ############################################################################### 2 # # 3 # Copyright (C) 2009-2013 Edward d'Auvergne # 4 # # 5 # This file is part of the program relax (http://www.nmr-relax.com). # 6 # # 7 # This program is free software: you can redistribute it and/or modify # 8 # it under the terms of the GNU General Public License as published by # 9 # the Free Software Foundation, either version 3 of the License, or # 10 # (at your option) any later version. # 11 # # 12 # This program is distributed in the hope that it will be useful, # 13 # but WITHOUT ANY WARRANTY; without even the implied warranty of # 14 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # 15 # GNU General Public License for more details. # 16 # # 17 # You should have received a copy of the GNU General Public License # 18 # along with this program. If not, see <http://www.gnu.org/licenses/>. # 19 # # 20 ############################################################################### 21 22 # Module docstring. 23 """Module for the handling of Frame Order.""" 24 25 # Dependency check module. 26 import dep_check 27 28 # Python module imports. 29 from math import cos, pi, sqrt 30 from numpy import sinc 31 if dep_check.scipy_module: 32 from scipy.integrate import tplquad 33 34 # relax module imports. 35 from lib.frame_order.matrix_ops import pcs_pivot_motion_full, pcs_pivot_motion_full_qrint, rotate_daeg 36 37

38 -def compile_2nd_matrix_iso_cone(matrix, Rx2_eigen, cone_theta, sigma_max):

39 """Generate the rotated 2nd degree Frame Order matrix for the isotropic cone. 40 41 The cone axis is assumed to be parallel to the z-axis in the eigenframe. 42 43 @param matrix: The Frame Order matrix, 2nd degree to be populated. 44 @type matrix: numpy 9D, rank-2 array 45 @param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself. 46 @type Rx2_eigen: numpy 9D, rank-2 array 47 @param cone_theta: The cone opening angle. 48 @type cone_theta: float 49 @param sigma_max: The maximum torsion angle. 50 @type sigma_max: float 51 """ 52 53 # Populate the Frame Order matrix in the eigenframe. 54 populate_2nd_eigenframe_iso_cone(matrix, cone_theta, sigma_max) 55 56 # Rotate and return the frame order matrix. 57 return rotate_daeg(matrix, Rx2_eigen)

58 59

60 -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):

61 """Determine the averaged PCS value via numerical integration. 62 63 @keyword theta_max: The half cone angle. 64 @type theta_max: float 65 @keyword sigma_max: The maximum torsion angle. 66 @type sigma_max: float 67 @keyword c: The PCS constant (without the interatomic distance and in Angstrom units). 68 @type c: float 69 @keyword r_pivot_atom: The pivot point to atom vector. 70 @type r_pivot_atom: numpy rank-1, 3D array 71 @keyword r_ln_pivot: The lanthanide position to pivot point vector. 72 @type r_ln_pivot: numpy rank-1, 3D array 73 @keyword A: The full alignment tensor of the non-moving domain. 74 @type A: numpy rank-2, 3D array 75 @keyword R_eigen: The eigenframe rotation matrix. 76 @type R_eigen: numpy rank-2, 3D array 77 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 78 @type RT_eigen: numpy rank-2, 3D array 79 @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. 80 @type Ri_prime: numpy rank-2, 3D array 81 @return: The averaged PCS value. 82 @rtype: float 83 """ 84 85 # Perform numerical integration. 86 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)) 87 88 # The surface area normalisation factor. 89 SA = 4.0 * pi * sigma_max * (1.0 - cos(theta_max)) 90 91 # Return the value. 92 return c * result[0] / SA

93 94

95 -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):

96 """Determine the averaged PCS value via numerical integration. 97 98 @keyword points: The Sobol points in the torsion-tilt angle space. 99 @type points: numpy rank-2, 3D array 100 @keyword theta_max: The half cone angle. 101 @type theta_max: float 102 @keyword sigma_max: The maximum torsion angle. 103 @type sigma_max: float 104 @keyword c: The PCS constant (without the interatomic distance and in Angstrom units). 105 @type c: numpy rank-1 array 106 @keyword full_in_ref_frame: An array of flags specifying if the tensor in the reference frame is the full or reduced tensor. 107 @type full_in_ref_frame: numpy rank-1 array 108 @keyword r_pivot_atom: The pivot point to atom vector. 109 @type r_pivot_atom: numpy rank-2, 3D array 110 @keyword r_pivot_atom_rev: The reversed pivot point to atom vector. 111 @type r_pivot_atom_rev: numpy rank-2, 3D array 112 @keyword r_ln_pivot: The lanthanide position to pivot point vector. 113 @type r_ln_pivot: numpy rank-2, 3D array 114 @keyword A: The full alignment tensor of the non-moving domain. 115 @type A: numpy rank-2, 3D array 116 @keyword R_eigen: The eigenframe rotation matrix. 117 @type R_eigen: numpy rank-2, 3D array 118 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 119 @type RT_eigen: numpy rank-2, 3D array 120 @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. 121 @type Ri_prime: numpy rank-2, 3D array 122 @keyword pcs_theta: The storage structure for the back-calculated PCS values. 123 @type pcs_theta: numpy rank-2 array 124 @keyword pcs_theta_err: The storage structure for the back-calculated PCS errors. 125 @type pcs_theta_err: numpy rank-2 array 126 @keyword missing_pcs: A structure used to indicate which PCS values are missing. 127 @type missing_pcs: numpy rank-2 array 128 @keyword error_flag: A flag which if True will cause the PCS errors to be estimated and stored in pcs_theta_err. 129 @type error_flag: bool 130 """ 131 132 # Clear the data structures. 133 for i in range(len(pcs_theta)): 134 for j in range(len(pcs_theta[i])): 135 pcs_theta[i, j] = 0.0 136 pcs_theta_err[i, j] = 0.0 137 138 # Loop over the samples. 139 num = 0 140 for i in range(len(points)): 141 # Unpack the point. 142 theta, phi, sigma = points[i] 143 144 # Outside of the distribution, so skip the point. 145 if theta > theta_max: 146 continue 147 if sigma > sigma_max or sigma < -sigma_max: 148 continue 149 150 # Calculate the PCSs for this state. 151 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) 152 153 # Increment the number of points. 154 num += 1 155 156 # Calculate the PCS and error. 157 for i in range(len(pcs_theta)): 158 for j in range(len(pcs_theta[i])): 159 # The average PCS. 160 pcs_theta[i, j] = c[i] * pcs_theta[i, j] / float(num) 161 162 # The error. 163 if error_flag: 164 pcs_theta_err[i, j] = abs(pcs_theta_err[i, j] / float(num) - pcs_theta[i, j]**2) / float(num) 165 pcs_theta_err[i, j] = c[i] * sqrt(pcs_theta_err[i, j]) 166 print("%8.3f +/- %-8.3f" % (pcs_theta[i, j]*1e6, pcs_theta_err[i, j]*1e6))

167 168

169 -def populate_1st_eigenframe_iso_cone(matrix, angle):

170 """Populate the 1st degree Frame Order matrix in the eigenframe for an isotropic cone. 171 172 The cone axis is assumed to be parallel to the z-axis in the eigenframe. 173 174 @param matrix: The Frame Order matrix, 1st degree. 175 @type matrix: numpy 3D, rank-2 array 176 @param angle: The cone angle. 177 @type angle: float 178 """ 179 180 # Zeros. 181 for i in range(3): 182 for j in range(3): 183 matrix[i, j] = 0.0 184 185 # The c33 element. 186 matrix[2, 2] = (cos(angle) + 1.0) / 2.0

187 188

189 -def populate_2nd_eigenframe_iso_cone(matrix, tmax, smax):

190 """Populate the 2nd degree Frame Order matrix in the eigenframe for the isotropic cone. 191 192 The cone axis is assumed to be parallel to the z-axis in the eigenframe. 193 194 195 @param matrix: The Frame Order matrix, 2nd degree. 196 @type matrix: numpy 9D, rank-2 array 197 @param tmax: The cone opening angle. 198 @type tmax: float 199 @param smax: The maximum torsion angle. 200 @type smax: float 201 """ 202 203 # Zeros. 204 for i in range(9): 205 for j in range(9): 206 matrix[i, j] = 0.0 207 208 # Repetitive trig calculations. 209 sinc_smax = sinc(smax/pi) 210 sinc_2smax = sinc(2.0*smax/pi) 211 cos_tmax = cos(tmax) 212 cos_tmax2 = cos_tmax**2 213 214 # Larger factors. 215 fact_sinc_2smax = sinc_2smax*(cos_tmax2 + 4.0*cos_tmax + 7.0) / 24.0 216 fact_cos_tmax2 = (cos_tmax2 + cos_tmax + 4.0) / 12.0 217 fact_cos_tmax = (cos_tmax + 1.0) / 4.0 218 219 # Diagonal. 220 matrix[0, 0] = fact_sinc_2smax + fact_cos_tmax2 221 matrix[1, 1] = fact_sinc_2smax + fact_cos_tmax 222 matrix[2, 2] = sinc_smax * (2.0*cos_tmax2 + 5.0*cos_tmax + 5.0) / 12.0 223 matrix[3, 3] = matrix[1, 1] 224 matrix[4, 4] = matrix[0, 0] 225 matrix[5, 5] = matrix[2, 2] 226 matrix[6, 6] = matrix[2, 2] 227 matrix[7, 7] = matrix[2, 2] 228 matrix[8, 8] = (cos_tmax2 + cos_tmax + 1.0) / 3.0 229 230 # Off diagonal set 1. 231 matrix[0, 4] = matrix[4, 0] = -fact_sinc_2smax + fact_cos_tmax2 232 matrix[0, 8] = matrix[8, 0] = -(cos_tmax2 + cos_tmax - 2.0) / 6.0 233 matrix[4, 8] = matrix[8, 4] = matrix[0, 8] 234 235 # Off diagonal set 2. 236 matrix[1, 3] = matrix[3, 1] = fact_sinc_2smax - fact_cos_tmax 237 matrix[2, 6] = matrix[6, 2] = sinc_smax * (cos_tmax2 + cos_tmax - 2.0) / 6.0 238 matrix[5, 7] = matrix[7, 5] = matrix[2, 6]

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Source Code for Module lib.frame_order.iso_cone