lib.frame_order.rotor

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, sin, sqrt 30 from numpy import dot, inner, sinc, transpose 31 from numpy.linalg import norm 32 if dep_check.scipy_module: 33 from scipy.integrate import quad 34 35 # relax module imports. 36 from lib.frame_order.matrix_ops import rotate_daeg 37 38

39 -def compile_2nd_matrix_rotor(matrix, Rx2_eigen, smax):

40 """Generate the rotated 2nd degree Frame Order matrix for the rotor model. 41 42 The cone axis is assumed to be parallel to the z-axis in the eigenframe. 43 44 45 @param matrix: The Frame Order matrix, 2nd degree to be populated. 46 @type matrix: numpy 9D, rank-2 array 47 @param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself. 48 @type Rx2_eigen: numpy 9D, rank-2 array 49 @param smax: The maximum torsion angle. 50 @type smax: float 51 """ 52 53 # Zeros. 54 for i in range(9): 55 for j in range(9): 56 matrix[i, j] = 0.0 57 58 # Repetitive trig calculations. 59 sinc_smax = sinc(smax/pi) 60 sinc_2smax = sinc(2.0*smax/pi) 61 62 # Diagonal. 63 matrix[0, 0] = (sinc_2smax + 1.0) / 2.0 64 matrix[1, 1] = matrix[0, 0] 65 matrix[2, 2] = sinc_smax 66 matrix[3, 3] = matrix[0, 0] 67 matrix[4, 4] = matrix[0, 0] 68 matrix[5, 5] = matrix[2, 2] 69 matrix[6, 6] = matrix[2, 2] 70 matrix[7, 7] = matrix[2, 2] 71 matrix[8, 8] = 1.0 72 73 # Off diagonal set 1. 74 matrix[0, 4] = matrix[4, 0] = -(sinc_2smax - 1.0) / 2.0 75 76 # Off diagonal set 2. 77 matrix[1, 3] = matrix[3, 1] = -matrix[0, 4] 78 79 # Rotate and return the frame order matrix. 80 return rotate_daeg(matrix, Rx2_eigen)

81 82

83 -def pcs_numeric_int_rotor(sigma_max=None, c=None, r_pivot_atom=None, r_ln_pivot=None, A=None, R_eigen=None, RT_eigen=None, Ri_prime=None):

84 """Determine the averaged PCS value via numerical integration. 85 86 @keyword sigma_max: The maximum rotor angle. 87 @type sigma_max: float 88 @keyword c: The PCS constant (without the interatomic distance and in Angstrom units). 89 @type c: float 90 @keyword r_pivot_atom: The pivot point to atom vector. 91 @type r_pivot_atom: numpy rank-1, 3D array 92 @keyword r_ln_pivot: The lanthanide position to pivot point vector. 93 @type r_ln_pivot: numpy rank-1, 3D array 94 @keyword A: The full alignment tensor of the non-moving domain. 95 @type A: numpy rank-2, 3D array 96 @keyword R_eigen: The eigenframe rotation matrix. 97 @type R_eigen: numpy rank-2, 3D array 98 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 99 @type RT_eigen: numpy rank-2, 3D array 100 @keyword Ri_prime: The empty rotation matrix for the in-frame rotor motion, used to calculate the PCS for each state i in the numerical integration. 101 @type Ri_prime: numpy rank-2, 3D array 102 @return: The averaged PCS value. 103 @rtype: float 104 """ 105 106 # Preset the rotation matrix elements for state i. 107 Ri_prime[0, 2] = 0.0 108 Ri_prime[1, 2] = 0.0 109 Ri_prime[2, 0] = 0.0 110 Ri_prime[2, 1] = 0.0 111 Ri_prime[2, 2] = 1.0 112 113 # Perform numerical integration. 114 result = quad(pcs_pivot_motion_rotor, -sigma_max, sigma_max, args=(r_pivot_atom, r_ln_pivot, A, R_eigen, RT_eigen, Ri_prime)) 115 116 # The surface area normalisation factor. 117 SA = 2.0 * sigma_max 118 119 # Return the value. 120 return c * result[0] / SA

121 122

123 -def pcs_numeric_int_rotor_qrint(points=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):

124 """Determine the averaged PCS value via numerical integration. 125 126 @keyword points: The Sobol points in the torsion-tilt angle space. 127 @type points: numpy rank-2, 3D array 128 @keyword sigma_max: The maximum rotor angle. 129 @type sigma_max: float 130 @keyword c: The PCS constant (without the interatomic distance and in Angstrom units). 131 @type c: numpy rank-1 array 132 @keyword full_in_ref_frame: An array of flags specifying if the tensor in the reference frame is the full or reduced tensor. 133 @type full_in_ref_frame: numpy rank-1 array 134 @keyword r_pivot_atom: The pivot point to atom vector. 135 @type r_pivot_atom: numpy rank-2, 3D array 136 @keyword r_pivot_atom_rev: The reversed pivot point to atom vector. 137 @type r_pivot_atom_rev: numpy rank-2, 3D array 138 @keyword r_ln_pivot: The lanthanide position to pivot point vector. 139 @type r_ln_pivot: numpy rank-2, 3D array 140 @keyword A: The full alignment tensor of the non-moving domain. 141 @type A: numpy rank-2, 3D array 142 @keyword R_eigen: The eigenframe rotation matrix. 143 @type R_eigen: numpy rank-2, 3D array 144 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 145 @type RT_eigen: numpy rank-2, 3D array 146 @keyword Ri_prime: The empty rotation matrix for the in-frame rotor motion, used to calculate the PCS for each state i in the numerical integration. 147 @type Ri_prime: numpy rank-2, 3D array 148 @keyword pcs_theta: The storage structure for the back-calculated PCS values. 149 @type pcs_theta: numpy rank-2 array 150 @keyword pcs_theta_err: The storage structure for the back-calculated PCS errors. 151 @type pcs_theta_err: numpy rank-2 array 152 @keyword missing_pcs: A structure used to indicate which PCS values are missing. 153 @type missing_pcs: numpy rank-2 array 154 @keyword error_flag: A flag which if True will cause the PCS errors to be estimated and stored in pcs_theta_err. 155 @type error_flag: bool 156 """ 157 158 # Clear the data structures. 159 for i in range(len(pcs_theta)): 160 for j in range(len(pcs_theta[i])): 161 pcs_theta[i, j] = 0.0 162 pcs_theta_err[i, j] = 0.0 163 164 # Loop over the samples. 165 num = 0 166 for i in range(len(points)): 167 # Unpack the point. 168 sigma = points[i] 169 170 # Outside of the distribution, so skip the point. 171 if sigma > sigma_max or sigma < -sigma_max: 172 continue 173 174 # Calculate the PCSs for this state. 175 pcs_pivot_motion_rotor_qrint(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) 176 177 # Increment the number of points. 178 num += 1 179 180 # Calculate the PCS and error. 181 for i in range(len(pcs_theta)): 182 for j in range(len(pcs_theta[i])): 183 # The average PCS. 184 pcs_theta[i, j] = c[i] * pcs_theta[i, j] / float(num) 185 186 # The error. 187 if error_flag: 188 pcs_theta_err[i, j] = abs(pcs_theta_err[i, j] / float(num) - pcs_theta[i, j]**2) / float(num) 189 pcs_theta_err[i, j] = c[i] * sqrt(pcs_theta_err[i, j]) 190 print("%8.3f +/- %-8.3f" % (pcs_theta[i, j]*1e6, pcs_theta_err[i, j]*1e6))

191 192

193 -def pcs_pivot_motion_rotor(sigma_i, r_pivot_atom, r_ln_pivot, A, R_eigen, RT_eigen, Ri_prime):

194 """Calculate the PCS value after a pivoted motion for the rotor model. 195 196 @param sigma_i: The rotor angle for state i. 197 @type sigma_i: float 198 @param r_pivot_atom: The pivot point to atom vector. 199 @type r_pivot_atom: numpy rank-1, 3D array 200 @param r_ln_pivot: The lanthanide position to pivot point vector. 201 @type r_ln_pivot: numpy rank-1, 3D array 202 @param A: The full alignment tensor of the non-moving domain. 203 @type A: numpy rank-2, 3D array 204 @param R_eigen: The eigenframe rotation matrix. 205 @type R_eigen: numpy rank-2, 3D array 206 @param RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 207 @type RT_eigen: numpy rank-2, 3D array 208 @param Ri_prime: The empty rotation matrix for the in-frame rotor motion for state i. 209 @type Ri_prime: numpy rank-2, 3D array 210 @return: The PCS value for the changed position. 211 @rtype: float 212 """ 213 214 # The rotation matrix. 215 c_sigma = cos(sigma_i) 216 s_sigma = sin(sigma_i) 217 Ri_prime[0, 0] = c_sigma 218 Ri_prime[0, 1] = -s_sigma 219 Ri_prime[1, 0] = s_sigma 220 Ri_prime[1, 1] = c_sigma 221 222 # The rotation. 223 R_i = dot(R_eigen, dot(Ri_prime, RT_eigen)) 224 225 # Calculate the new vector. 226 vect = dot(R_i, r_pivot_atom) + r_ln_pivot 227 228 # The vector length. 229 length = norm(vect) 230 231 # The projection. 232 proj = dot(vect, dot(A, vect)) 233 234 # The PCS. 235 pcs = proj / length**5 236 237 # Return the PCS value (without the PCS constant). 238 return pcs

239 240

241 -def pcs_pivot_motion_rotor_qrint(sigma_i=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):

242 """Calculate the PCS value after a pivoted motion for the rotor model. 243 244 @keyword sigma_i: The rotor angle for state i. 245 @type sigma_i: float 246 @keyword full_in_ref_frame: An array of flags specifying if the tensor in the reference frame is the full or reduced tensor. 247 @type full_in_ref_frame: numpy rank-1 array 248 @keyword r_pivot_atom: The pivot point to atom vector. 249 @type r_pivot_atom: numpy rank-2, 3D array 250 @keyword r_pivot_atom_rev: The reversed pivot point to atom vector. 251 @type r_pivot_atom_rev: numpy rank-2, 3D array 252 @keyword r_ln_pivot: The lanthanide position to pivot point vector. 253 @type r_ln_pivot: numpy rank-2, 3D array 254 @keyword A: The full alignment tensor of the non-moving domain. 255 @type A: numpy rank-2, 3D array 256 @keyword R_eigen: The eigenframe rotation matrix. 257 @type R_eigen: numpy rank-2, 3D array 258 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 259 @type RT_eigen: numpy rank-2, 3D array 260 @keyword Ri_prime: The empty rotation matrix for the in-frame rotor motion for state i. 261 @type Ri_prime: numpy rank-2, 3D array 262 @keyword pcs_theta: The storage structure for the back-calculated PCS values. 263 @type pcs_theta: numpy rank-2 array 264 @keyword pcs_theta_err: The storage structure for the back-calculated PCS errors. 265 @type pcs_theta_err: numpy rank-2 array 266 @keyword missing_pcs: A structure used to indicate which PCS values are missing. 267 @type missing_pcs: numpy rank-2 array 268 @keyword error_flag: A flag which if True will cause the PCS errors to be estimated and stored in pcs_theta_err. 269 @type error_flag: bool 270 """ 271 272 # The rotation matrix. 273 c_sigma = cos(sigma_i) 274 s_sigma = sin(sigma_i) 275 Ri_prime[0, 0] = c_sigma 276 Ri_prime[0, 1] = -s_sigma 277 Ri_prime[0, 2] = 0.0 278 Ri_prime[1, 0] = s_sigma 279 Ri_prime[1, 1] = c_sigma 280 Ri_prime[1, 2] = 0.0 281 Ri_prime[2, 0] = 0.0 282 Ri_prime[2, 1] = 0.0 283 Ri_prime[2, 2] = 1.0 284 285 # The rotation. 286 R_i = dot(R_eigen, dot(Ri_prime, RT_eigen)) 287 288 # Pre-calculate all the new vectors (forwards and reverse). 289 rot_vect_rev = transpose(dot(R_i, r_pivot_atom_rev) + r_ln_pivot) 290 rot_vect = transpose(dot(R_i, r_pivot_atom) + r_ln_pivot) 291 292 # Loop over the atoms. 293 for j in range(len(r_pivot_atom[0])): 294 # The vector length (to the 5th power). 295 length_rev = 1.0 / sqrt(inner(rot_vect_rev[j], rot_vect_rev[j]))**5 296 length = 1.0 / sqrt(inner(rot_vect[j], rot_vect[j]))**5 297 298 # Loop over the alignments. 299 for i in range(len(pcs_theta)): 300 # Skip missing data. 301 if missing_pcs[i, j]: 302 continue 303 304 # The projection. 305 if full_in_ref_frame[i]: 306 proj = dot(rot_vect[j], dot(A[i], rot_vect[j])) 307 length_i = length 308 else: 309 proj = dot(rot_vect_rev[j], dot(A[i], rot_vect_rev[j])) 310 length_i = length_rev 311 312 # The PCS. 313 pcs_theta[i, j] += proj * length_i 314 315 # The square. 316 if error_flag: 317 pcs_theta_err[i, j] += (proj * length_i)**2

318

Source Code for Module lib.frame_order.rotor