lib.frame_order.rotor

1 ############################################################################### 2 # # 3 # Copyright (C) 2009-2014 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 # Python module imports. 26 from math import cos, pi, sin, sqrt 27 from numpy import dot, inner, sinc, transpose 28 from numpy.linalg import norm 29 try: 30 from scipy.integrate import quad 31 except ImportError: 32 pass 33 34 # relax module imports. 35 from lib.frame_order.matrix_ops import rotate_daeg 36 37

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

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

80 81

82 -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):

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

120 121

122 -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):

123 """Determine the averaged PCS value via numerical integration. 124 125 @keyword points: The Sobol points in the torsion-tilt angle space. 126 @type points: numpy rank-2, 3D array 127 @keyword sigma_max: The maximum rotor angle. 128 @type sigma_max: float 129 @keyword c: The PCS constant (without the interatomic distance and in Angstrom units). 130 @type c: numpy rank-1 array 131 @keyword full_in_ref_frame: An array of flags specifying if the tensor in the reference frame is the full or reduced tensor. 132 @type full_in_ref_frame: numpy rank-1 array 133 @keyword r_pivot_atom: The pivot point to atom vector. 134 @type r_pivot_atom: numpy rank-2, 3D array 135 @keyword r_pivot_atom_rev: The reversed pivot point to atom vector. 136 @type r_pivot_atom_rev: numpy rank-2, 3D array 137 @keyword r_ln_pivot: The lanthanide position to pivot point vector. 138 @type r_ln_pivot: numpy rank-2, 3D array 139 @keyword A: The full alignment tensor of the non-moving domain. 140 @type A: numpy rank-2, 3D array 141 @keyword R_eigen: The eigenframe rotation matrix. 142 @type R_eigen: numpy rank-2, 3D array 143 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 144 @type RT_eigen: numpy rank-2, 3D array 145 @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. 146 @type Ri_prime: numpy rank-2, 3D array 147 @keyword pcs_theta: The storage structure for the back-calculated PCS values. 148 @type pcs_theta: numpy rank-2 array 149 @keyword pcs_theta_err: The storage structure for the back-calculated PCS errors. 150 @type pcs_theta_err: numpy rank-2 array 151 @keyword missing_pcs: A structure used to indicate which PCS values are missing. 152 @type missing_pcs: numpy rank-2 array 153 @keyword error_flag: A flag which if True will cause the PCS errors to be estimated and stored in pcs_theta_err. 154 @type error_flag: bool 155 """ 156 157 # Clear the data structures. 158 for i in range(len(pcs_theta)): 159 for j in range(len(pcs_theta[i])): 160 pcs_theta[i, j] = 0.0 161 pcs_theta_err[i, j] = 0.0 162 163 # Loop over the samples. 164 num = 0 165 for i in range(len(points)): 166 # Unpack the point. 167 sigma = points[i] 168 169 # Outside of the distribution, so skip the point. 170 if sigma > sigma_max or sigma < -sigma_max: 171 continue 172 173 # Calculate the PCSs for this state. 174 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) 175 176 # Increment the number of points. 177 num += 1 178 179 # Calculate the PCS and error. 180 for i in range(len(pcs_theta)): 181 for j in range(len(pcs_theta[i])): 182 if num == 0: 183 continue 184 185 # The average PCS. 186 pcs_theta[i, j] = c[i] * pcs_theta[i, j] / float(num) 187 188 # The error. 189 if error_flag: 190 pcs_theta_err[i, j] = abs(pcs_theta_err[i, j] / float(num) - pcs_theta[i, j]**2) / float(num) 191 pcs_theta_err[i, j] = c[i] * sqrt(pcs_theta_err[i, j]) 192 print("%8.3f +/- %-8.3f" % (pcs_theta[i, j]*1e6, pcs_theta_err[i, j]*1e6))

193 194

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

196 """Calculate the PCS value after a pivoted motion for the rotor model. 197 198 @param sigma_i: The rotor angle for state i. 199 @type sigma_i: float 200 @param r_pivot_atom: The pivot point to atom vector. 201 @type r_pivot_atom: numpy rank-1, 3D array 202 @param r_ln_pivot: The lanthanide position to pivot point vector. 203 @type r_ln_pivot: numpy rank-1, 3D array 204 @param A: The full alignment tensor of the non-moving domain. 205 @type A: numpy rank-2, 3D array 206 @param R_eigen: The eigenframe rotation matrix. 207 @type R_eigen: numpy rank-2, 3D array 208 @param RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 209 @type RT_eigen: numpy rank-2, 3D array 210 @param Ri_prime: The empty rotation matrix for the in-frame rotor motion for state i. 211 @type Ri_prime: numpy rank-2, 3D array 212 @return: The PCS value for the changed position. 213 @rtype: float 214 """ 215 216 # The rotation matrix. 217 c_sigma = cos(sigma_i) 218 s_sigma = sin(sigma_i) 219 Ri_prime[0, 0] = c_sigma 220 Ri_prime[0, 1] = -s_sigma 221 Ri_prime[1, 0] = s_sigma 222 Ri_prime[1, 1] = c_sigma 223 224 # The rotation. 225 R_i = dot(R_eigen, dot(Ri_prime, RT_eigen)) 226 227 # Calculate the new vector. 228 vect = dot(R_i, r_pivot_atom) + r_ln_pivot 229 230 # The vector length. 231 length = norm(vect) 232 233 # The projection. 234 proj = dot(vect, dot(A, vect)) 235 236 # The PCS. 237 pcs = proj / length**5 238 239 # Return the PCS value (without the PCS constant). 240 return pcs

241 242

243 -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):

244 """Calculate the PCS value after a pivoted motion for the rotor model. 245 246 @keyword sigma_i: The rotor angle for state i. 247 @type sigma_i: float 248 @keyword full_in_ref_frame: An array of flags specifying if the tensor in the reference frame is the full or reduced tensor. 249 @type full_in_ref_frame: numpy rank-1 array 250 @keyword r_pivot_atom: The pivot point to atom vector. 251 @type r_pivot_atom: numpy rank-2, 3D array 252 @keyword r_pivot_atom_rev: The reversed pivot point to atom vector. 253 @type r_pivot_atom_rev: numpy rank-2, 3D array 254 @keyword r_ln_pivot: The lanthanide position to pivot point vector. 255 @type r_ln_pivot: numpy rank-2, 3D array 256 @keyword A: The full alignment tensor of the non-moving domain. 257 @type A: numpy rank-2, 3D array 258 @keyword R_eigen: The eigenframe rotation matrix. 259 @type R_eigen: numpy rank-2, 3D array 260 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations). 261 @type RT_eigen: numpy rank-2, 3D array 262 @keyword Ri_prime: The empty rotation matrix for the in-frame rotor motion for state i. 263 @type Ri_prime: numpy rank-2, 3D array 264 @keyword pcs_theta: The storage structure for the back-calculated PCS values. 265 @type pcs_theta: numpy rank-2 array 266 @keyword pcs_theta_err: The storage structure for the back-calculated PCS errors. 267 @type pcs_theta_err: numpy rank-2 array 268 @keyword missing_pcs: A structure used to indicate which PCS values are missing. 269 @type missing_pcs: numpy rank-2 array 270 @keyword error_flag: A flag which if True will cause the PCS errors to be estimated and stored in pcs_theta_err. 271 @type error_flag: bool 272 """ 273 274 # The rotation matrix. 275 c_sigma = cos(sigma_i) 276 s_sigma = sin(sigma_i) 277 Ri_prime[0, 0] = c_sigma 278 Ri_prime[0, 1] = -s_sigma 279 Ri_prime[0, 2] = 0.0 280 Ri_prime[1, 0] = s_sigma 281 Ri_prime[1, 1] = c_sigma 282 Ri_prime[1, 2] = 0.0 283 Ri_prime[2, 0] = 0.0 284 Ri_prime[2, 1] = 0.0 285 Ri_prime[2, 2] = 1.0 286 287 # The rotation. 288 R_i = dot(R_eigen, dot(Ri_prime, RT_eigen)) 289 290 # Pre-calculate all the new vectors (forwards and reverse). 291 rot_vect_rev = transpose(dot(R_i, r_pivot_atom_rev) + r_ln_pivot) 292 rot_vect = transpose(dot(R_i, r_pivot_atom) + r_ln_pivot) 293 294 # Loop over the atoms. 295 for j in range(len(r_pivot_atom[0])): 296 # The vector length (to the 5th power). 297 length_rev = 1.0 / sqrt(inner(rot_vect_rev[j], rot_vect_rev[j]))**5 298 length = 1.0 / sqrt(inner(rot_vect[j], rot_vect[j]))**5 299 300 # Loop over the alignments. 301 for i in range(len(pcs_theta)): 302 # Skip missing data. 303 if missing_pcs[i, j]: 304 continue 305 306 # The projection. 307 if full_in_ref_frame[i]: 308 proj = dot(rot_vect[j], dot(A[i], rot_vect[j])) 309 length_i = length 310 else: 311 proj = dot(rot_vect_rev[j], dot(A[i], rot_vect_rev[j])) 312 length_i = length_rev 313 314 # The PCS. 315 pcs_theta[i, j] += proj * length_i 316 317 # The square. 318 if error_flag: 319 pcs_theta_err[i, j] += (proj * length_i)**2

320

Source Code for Module lib.frame_order.rotor