Source Code for Module lib.frame_order.iso_cone

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  2  #                                                                             # 
  3  # Copyright (C) 2009-2012,2014 Edward d'Auvergne                              # 
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  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 
 27  from numpy import divide, dot, eye, float64, multiply, sinc, swapaxes, tensordot 
 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_qr_int, pcs_pivot_motion_full_quad_int, rotate_daeg 
 35   
 36   
 37 -def compile_1st_matrix_iso_cone(matrix, R_eigen, cone_theta, sigma_max): 
 38      """Generate the 1st degree Frame Order matrix for the isotropic cone. 
 39   
 40      @param matrix:      The Frame Order matrix, 1st degree to be populated. 
 41      @type matrix:       numpy 3D, rank-2 array 
 42      @param R_eigen:     The eigenframe rotation matrix. 
 43      @type R_eigen:      numpy 3D, rank-2 array 
 44      @param cone_theta:  The cone opening angle. 
 45      @type cone_theta:   float 
 46      @param sigma_max:   The maximum torsion angle. 
 47      @type sigma_max:    float 
 48      """ 
 49   
 50      # Zeros. 
 51      matrix[:] = 0.0 
 52   
 53      # Pre-calculate trig values. 
 54      sinc_sigma_max = sinc(sigma_max/pi) 
 55      cos_theta = cos(cone_theta) 
 56   
 57      # Diagonal values. 
 58      matrix[0, 0] = sinc_sigma_max * (cos_theta + 3.0) 
 59      matrix[1, 1] = matrix[0, 0] 
 60      matrix[2, 2] = 2.0*cos_theta + 2.0 
 61   
 62      # Rotate and return the frame order matrix. 
 63      return 0.25 * rotate_daeg(matrix, R_eigen) 
 64   
 65   
 66 -def compile_2nd_matrix_iso_cone(matrix, Rx2_eigen, cone_theta, sigma_max): 
 67      """Generate the rotated 2nd degree Frame Order matrix for the isotropic cone. 
 68   
 69      The cone axis is assumed to be parallel to the z-axis in the eigenframe. 
 70   
 71      @param matrix:      The Frame Order matrix, 2nd degree to be populated. 
 72      @type matrix:       numpy 9D, rank-2 array 
 73      @param Rx2_eigen:   The Kronecker product of the eigenframe rotation matrix with itself. 
 74      @type Rx2_eigen:    numpy 9D, rank-2 array 
 75      @param cone_theta:  The cone opening angle. 
 76      @type cone_theta:   float 
 77      @param sigma_max:   The maximum torsion angle. 
 78      @type sigma_max:    float 
 79      """ 
 80   
 81      # Zeros. 
 82      matrix[:] = 0.0 
 83   
 84      # Repetitive trig calculations. 
 85      sinc_smax = sinc(sigma_max/pi) 
 86      sinc_2smax = sinc(2.0*sigma_max/pi) 
 87      cos_tmax = cos(cone_theta) 
 88      cos_tmax2 = cos_tmax**2 
 89   
 90      # Larger factors. 
 91      fact_sinc_2smax = sinc_2smax*(cos_tmax2 + 4.0*cos_tmax + 7.0) / 24.0 
 92      fact_cos_tmax2 = (cos_tmax2 + cos_tmax + 4.0) / 12.0 
 93      fact_cos_tmax = (cos_tmax + 1.0) / 4.0 
 94   
 95      # Diagonal. 
 96      matrix[0, 0] = fact_sinc_2smax  +  fact_cos_tmax2 
 97      matrix[1, 1] = fact_sinc_2smax  +  fact_cos_tmax 
 98      matrix[2, 2] = sinc_smax * (2.0*cos_tmax2 + 5.0*cos_tmax + 5.0) / 12.0 
 99      matrix[3, 3] = matrix[1, 1] 
100      matrix[4, 4] = matrix[0, 0] 
101      matrix[5, 5] = matrix[2, 2] 
102      matrix[6, 6] = matrix[2, 2] 
103      matrix[7, 7] = matrix[2, 2] 
104      matrix[8, 8] = (cos_tmax2 + cos_tmax + 1.0) / 3.0 
105   
106      # Off diagonal set 1. 
107      matrix[0, 4] = matrix[4, 0] = -fact_sinc_2smax  +  fact_cos_tmax2 
108      matrix[0, 8] = matrix[8, 0] = -(cos_tmax2 + cos_tmax - 2.0) / 6.0 
109      matrix[4, 8] = matrix[8, 4] = matrix[0, 8] 
110   
111      # Off diagonal set 2. 
112      matrix[1, 3] = matrix[3, 1] = fact_sinc_2smax  -  fact_cos_tmax 
113      matrix[2, 6] = matrix[6, 2] = sinc_smax * (cos_tmax2 + cos_tmax - 2.0) / 6.0 
114      matrix[5, 7] = matrix[7, 5] = matrix[2, 6] 
115   
116      # Rotate and return the frame order matrix. 
117      return rotate_daeg(matrix, Rx2_eigen) 
118   
119   
120 -def pcs_numeric_qr_int_iso_cone(points=None, max_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): 
121      """Determine the averaged PCS value via numerical integration. 
122   
123      @keyword points:            The Sobol points in the torsion-tilt angle space. 
124      @type points:               numpy rank-2, 3D array 
125      @keyword max_points:        The maximum number of Sobol' points to use.  Once this number is reached, the loop over the Sobol' torsion-tilt angles is terminated. 
126      @type max_points:           int 
127      @keyword theta_max:         The half cone angle. 
128      @type theta_max:            float 
129      @keyword sigma_max:         The maximum torsion angle. 
130      @type sigma_max:            float 
131      @keyword c:                 The PCS constant (without the interatomic distance and in Angstrom units). 
132      @type c:                    numpy rank-1 array 
133      @keyword full_in_ref_frame: An array of flags specifying if the tensor in the reference frame is the full or reduced tensor. 
134      @type full_in_ref_frame:    numpy rank-1 array 
135      @keyword r_pivot_atom:      The pivot point to atom vector. 
136      @type r_pivot_atom:         numpy rank-2, 3D array 
137      @keyword r_pivot_atom_rev:  The reversed pivot point to atom vector. 
138      @type r_pivot_atom_rev:     numpy rank-2, 3D array 
139      @keyword r_ln_pivot:        The lanthanide position to pivot point vector. 
140      @type r_ln_pivot:           numpy rank-2, 3D array 
141      @keyword A:                 The full alignment tensor of the non-moving domain. 
142      @type A:                    numpy rank-2, 3D array 
143      @keyword R_eigen:           The eigenframe rotation matrix. 
144      @type R_eigen:              numpy rank-2, 3D array 
145      @keyword RT_eigen:          The transpose of the eigenframe rotation matrix (for faster calculations). 
146      @type RT_eigen:             numpy rank-2, 3D array 
147      @keyword Ri_prime:          The array of pre-calculated rotation matrices for the in-frame isotropic cone motion, used to calculate the PCS for each state i in the numerical integration. 
148      @type Ri_prime:             numpy rank-3, array of 3D arrays 
149      @keyword pcs_theta:         The storage structure for the back-calculated PCS values. 
150      @type pcs_theta:            numpy rank-2 array 
151      @keyword pcs_theta_err:     The storage structure for the back-calculated PCS errors. 
152      @type pcs_theta_err:        numpy rank-2 array 
153      @keyword missing_pcs:       A structure used to indicate which PCS values are missing. 
154      @type missing_pcs:          numpy rank-2 array 
155      """ 
156   
157      # Clear the data structures. 
158      pcs_theta[:] = 0.0 
159      pcs_theta_err[:] = 0.0 
160   
161      # Fast frame shift. 
162      Ri = dot(R_eigen, tensordot(Ri_prime, RT_eigen, axes=1)) 
163      Ri = swapaxes(Ri, 0, 1) 
164   
165      # Unpack the points. 
166      theta, phi, sigma = points 
167   
168      # Loop over the samples. 
169      num = 0 
170      for i in range(len(points[0])): 
171          # The maximum number of points has been reached (well, surpassed by one so exit the loop before it is used). 
172          if num == max_points: 
173              break 
174   
175          # Outside of the distribution, so skip the point. 
176          if theta[i] > theta_max: 
177              continue 
178          if abs(sigma[i]) > sigma_max: 
179              continue 
180   
181          # Calculate the PCSs for this state. 
182          pcs_pivot_motion_full_qr_int(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, Ri=Ri[i], pcs_theta=pcs_theta, pcs_theta_err=pcs_theta_err, missing_pcs=missing_pcs) 
183   
184          # Increment the number of points. 
185          num += 1 
186   
187      # Default to the rigid state if no points lie in the distribution. 
188      if num == 0: 
189          # Fast identity frame shift. 
190          Ri_prime = eye(3, dtype=float64) 
191          Ri = dot(R_eigen, tensordot(Ri_prime, RT_eigen, axes=1)) 
192          Ri = swapaxes(Ri, 0, 1) 
193   
194          # Calculate the PCSs for this state. 
195          pcs_pivot_motion_full_qr_int(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, Ri=Ri, pcs_theta=pcs_theta, pcs_theta_err=pcs_theta_err, missing_pcs=missing_pcs) 
196   
197          # Multiply the constant. 
198          multiply(c, pcs_theta, pcs_theta) 
199   
200      # Average the PCS. 
201      else: 
202          multiply(c, pcs_theta, pcs_theta) 
203          divide(pcs_theta, float(num), pcs_theta) 
204   
205   
206 -def pcs_numeric_quad_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): 
207      """Determine the averaged PCS value via numerical integration. 
208   
209      @keyword theta_max:     The half cone angle. 
210      @type theta_max:        float 
211      @keyword sigma_max:     The maximum torsion angle. 
212      @type sigma_max:        float 
213      @keyword c:             The PCS constant (without the interatomic distance and in Angstrom units). 
214      @type c:                float 
215      @keyword r_pivot_atom:  The pivot point to atom vector. 
216      @type r_pivot_atom:     numpy rank-1, 3D array 
217      @keyword r_ln_pivot:    The lanthanide position to pivot point vector. 
218      @type r_ln_pivot:       numpy rank-1, 3D array 
219      @keyword A:             The full alignment tensor of the non-moving domain. 
220      @type A:                numpy rank-2, 3D array 
221      @keyword R_eigen:       The eigenframe rotation matrix. 
222      @type R_eigen:          numpy rank-2, 3D array 
223      @keyword RT_eigen:      The transpose of the eigenframe rotation matrix (for faster calculations). 
224      @type RT_eigen:         numpy rank-2, 3D array 
225      @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. 
226      @type Ri_prime:         numpy rank-2, 3D array 
227      @return:                The averaged PCS value. 
228      @rtype:                 float 
229      """ 
230   
231      # Perform numerical integration. 
232      result = tplquad(pcs_pivot_motion_full_quad_int, -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)) 
233   
234      # The surface area normalisation factor. 
235      SA = 4.0 * pi * sigma_max * (1.0 - cos(theta_max)) 
236   
237      # Return the value. 
238      return c * result[0] / SA 
239