Source Code for Module lib.frame_order.iso_cone_torsionless

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  2  #                                                                             # 
  3  # Copyright (C) 2009-2014 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, sqrt 
 27  try: 
 28      from scipy.integrate import dblquad 
 29  except ImportError: 
 30      pass 
 31   
 32  # relax module imports. 
 33  from lib.frame_order.matrix_ops import pcs_pivot_motion_torsionless, pcs_pivot_motion_torsionless_qrint, rotate_daeg 
 34   
 35   
 36 -def compile_2nd_matrix_iso_cone_torsionless(matrix, Rx2_eigen, cone_theta): 
 37      """Generate the rotated 2nd degree Frame Order matrix for the torsionless isotropic cone. 
 38   
 39      The cone axis is assumed to be parallel to the z-axis in the eigenframe. 
 40   
 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      """ 
 49   
 50      # Zeros. 
 51      for i in range(9): 
 52          for j in range(9): 
 53              matrix[i, j] = 0.0 
 54   
 55      # Repetitive trig calculations. 
 56      cos_tmax = cos(cone_theta) 
 57      cos_tmax2 = cos_tmax**2 
 58   
 59      # Diagonal. 
 60      matrix[0, 0] = (3.0*cos_tmax2 + 6.0*cos_tmax + 15.0) / 24.0 
 61      matrix[1, 1] = (cos_tmax2 + 10.0*cos_tmax + 13.0) / 24.0 
 62      matrix[2, 2] = (4.0*cos_tmax2 + 10.0*cos_tmax + 10.0) / 24.0 
 63      matrix[3, 3] = matrix[1, 1] 
 64      matrix[4, 4] = matrix[0, 0] 
 65      matrix[5, 5] = matrix[2, 2] 
 66      matrix[6, 6] = matrix[2, 2] 
 67      matrix[7, 7] = matrix[2, 2] 
 68      matrix[8, 8] = (cos_tmax2 + cos_tmax + 1.0) / 3.0 
 69   
 70      # Off diagonal set 1. 
 71      matrix[0, 4] = matrix[4, 0] = (cos_tmax2 - 2.0*cos_tmax + 1.0) / 24.0 
 72      matrix[0, 8] = matrix[8, 0] = -(cos_tmax2 + cos_tmax - 2.0) / 6.0 
 73      matrix[4, 8] = matrix[8, 4] = matrix[0, 8] 
 74   
 75      # Off diagonal set 2. 
 76      matrix[1, 3] = matrix[3, 1] = matrix[0, 4] 
 77      matrix[2, 6] = matrix[6, 2] = -matrix[0, 8] 
 78      matrix[5, 7] = matrix[7, 5] = -matrix[0, 8] 
 79   
 80      # Rotate and return the frame order matrix. 
 81      return rotate_daeg(matrix, Rx2_eigen) 
 82   
 83   
 84 -def pcs_numeric_int_iso_cone_torsionless(theta_max=None, c=None, r_pivot_atom=None, r_ln_pivot=None, A=None, R_eigen=None, RT_eigen=None, Ri_prime=None): 
 85      """Determine the averaged PCS value via numerical integration. 
 86   
 87      @keyword theta_max:     The half cone angle. 
 88      @type theta_max:        float 
 89      @keyword c:             The PCS constant (without the interatomic distance and in Angstrom units). 
 90      @type c:                float 
 91      @keyword r_pivot_atom:  The pivot point to atom vector. 
 92      @type r_pivot_atom:     numpy rank-1, 3D array 
 93      @keyword r_ln_pivot:    The lanthanide position to pivot point vector. 
 94      @type r_ln_pivot:       numpy rank-1, 3D array 
 95      @keyword A:             The full alignment tensor of the non-moving domain. 
 96      @type A:                numpy rank-2, 3D array 
 97      @keyword R_eigen:       The eigenframe rotation matrix. 
 98      @type R_eigen:          numpy rank-2, 3D array 
 99      @keyword RT_eigen:      The transpose of the eigenframe rotation matrix (for faster calculations). 
100      @type RT_eigen:         numpy rank-2, 3D array 
101      @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. 
102      @type Ri_prime:         numpy rank-2, 3D array 
103      @return:                The averaged PCS value. 
104      @rtype:                 float 
105      """ 
106   
107      # Perform numerical integration. 
108      result = dblquad(pcs_pivot_motion_torsionless, -pi, pi, lambda phi: 0.0, lambda phi: theta_max, args=(r_pivot_atom, r_ln_pivot, A, R_eigen, RT_eigen, Ri_prime)) 
109   
110      # The surface area normalisation factor. 
111      SA = 2.0 * pi * (1.0 - cos(theta_max)) 
112   
113      # Return the value. 
114      return c * result[0] / SA 
115   
116   
117 -def pcs_numeric_int_iso_cone_torsionless_qrint(points=None, theta_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): 
118      """Determine the averaged PCS value via numerical integration. 
119   
120      @keyword points:            The Sobol points in the torsion-tilt angle space. 
121      @type points:               numpy rank-2, 3D array 
122      @keyword theta_max:         The half cone angle. 
123      @type theta_max:            float 
124      @keyword c:                 The PCS constant (without the interatomic distance and in Angstrom units). 
125      @type c:                    numpy rank-1 array 
126      @keyword full_in_ref_frame: An array of flags specifying if the tensor in the reference frame is the full or reduced tensor. 
127      @type full_in_ref_frame:    numpy rank-1 array 
128      @keyword r_pivot_atom:      The pivot point to atom vector. 
129      @type r_pivot_atom:         numpy rank-2, 3D array 
130      @keyword r_pivot_atom_rev:  The reversed pivot point to atom vector. 
131      @type r_pivot_atom_rev:     numpy rank-2, 3D array 
132      @keyword r_ln_pivot:        The lanthanide position to pivot point vector. 
133      @type r_ln_pivot:           numpy rank-2, 3D array 
134      @keyword A:                 The full alignment tensor of the non-moving domain. 
135      @type A:                    numpy rank-2, 3D array 
136      @keyword R_eigen:           The eigenframe rotation matrix. 
137      @type R_eigen:              numpy rank-2, 3D array 
138      @keyword RT_eigen:          The transpose of the eigenframe rotation matrix (for faster calculations). 
139      @type RT_eigen:             numpy rank-2, 3D array 
140      @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. 
141      @type Ri_prime:             numpy rank-2, 3D array 
142      @keyword pcs_theta:         The storage structure for the back-calculated PCS values. 
143      @type pcs_theta:            numpy rank-2 array 
144      @keyword pcs_theta_err:     The storage structure for the back-calculated PCS errors. 
145      @type pcs_theta_err:        numpy rank-2 array 
146      @keyword missing_pcs:       A structure used to indicate which PCS values are missing. 
147      @type missing_pcs:          numpy rank-2 array 
148      @keyword error_flag:        A flag which if True will cause the PCS errors to be estimated and stored in pcs_theta_err. 
149      @type error_flag:           bool 
150      """ 
151   
152      # Clear the data structures. 
153      for i in range(len(pcs_theta)): 
154          for j in range(len(pcs_theta[i])): 
155              pcs_theta[i, j] = 0.0 
156              pcs_theta_err[i, j] = 0.0 
157   
158      # Loop over the samples. 
159      num = 0 
160      for i in range(len(points)): 
161          # Unpack the point. 
162          theta, phi = points[i] 
163   
164          # Outside of the distribution, so skip the point. 
165          if theta > theta_max: 
166              continue 
167   
168          # Calculate the PCSs for this state. 
169          pcs_pivot_motion_torsionless_qrint(theta_i=theta, phi_i=phi, 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) 
170   
171          # Increment the number of points. 
172          num += 1 
173   
174      # Calculate the PCS and error. 
175      for i in range(len(pcs_theta)): 
176          for j in range(len(pcs_theta[i])): 
177              # The average PCS. 
178              pcs_theta[i, j] = c[i] * pcs_theta[i, j] / float(num) 
179   
180              # The error. 
181              if error_flag: 
182                  pcs_theta_err[i, j] = abs(pcs_theta_err[i, j] / float(num)  -  pcs_theta[i, j]**2) / float(num) 
183                  pcs_theta_err[i, j] = c[i] * sqrt(pcs_theta_err[i, j]) 
184                  print("%8.3f +/- %-8.3f" % (pcs_theta[i, j]*1e6, pcs_theta_err[i, j]*1e6)) 
185