Source Code for Module lib.auto_relaxation.ri_comps

   1  ############################################################################### 
   2  #                                                                             # 
   3  # Copyright (C) 2003-2012 Edward d'Auvergne                                   # 
   4  #                                                                             # 
   5  # This file is part of the program relax (http://www.nmr-relax.com).          # 
   6  #                                                                             # 
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   8  # it under the terms of the GNU General Public License as published by        # 
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  10  # (at your option) any later version.                                         # 
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  15  # GNU General Public License for more details.                                # 
  16  #                                                                             # 
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  18  # along with this program.  If not, see <http://www.gnu.org/licenses/>.       # 
  19  #                                                                             # 
  20  ############################################################################### 
  21   
  22   
  23  # Constant formulae 
  24  # ================= 
  25  # 
  26  #    Dipolar constants 
  27  #    ================= 
  28  #                           1   / mu0  \ 2  (gH.gN.h_bar)**2 
  29  #        dip_const_func  =  - . | ---- |  . ---------------- 
  30  #                           4   \ 4.pi /         <r**6> 
  31  # 
  32  # 
  33  #                             3   / mu0  \ 2  (gH.gN.h_bar)**2 
  34  #        dip_const_grad  =  - - . | ---- |  . ---------------- 
  35  #                             2   \ 4.pi /         <r**7> 
  36  # 
  37  # 
  38  #                           21   / mu0  \ 2  (gH.gN.h_bar)**2 
  39  #        dip_const_hess  =  -- . | ---- |  . ---------------- 
  40  #                           2    \ 4.pi /         <r**8> 
  41  # 
  42  # 
  43  #    CSA constants 
  44  #    ============= 
  45  #                           (wN.csa)**2 
  46  #        csa_const_func  =  ----------- 
  47  #                                3 
  48  # 
  49  #                           2.wN**2.csa 
  50  #        csa_const_grad  =  ----------- 
  51  #                                3 
  52  # 
  53  #                           2.wN**2 
  54  #        csa_const_hess  =  ------- 
  55  #                              3 
  56  # 
  57  #    Rex constants 
  58  #    ============= 
  59  #        rex_const_func  =  rhoex * (2.pi.wH)**2 
  60  # 
  61  #        rex_const_grad  =  (2.pi.wH)**2 
  62  # 
  63  #        rex_const_hess  =  0 
  64  # 
  65  # 
  66  # Component formulae 
  67  # ================== 
  68  # 
  69  #    R1 components 
  70  #    ============= 
  71  # 
  72  #        Dipolar components 
  73  #        ================== 
  74  # 
  75  #            dip_R1_func     =  dip_const_func 
  76  # 
  77  #            dip_R1_grad     =  dip_const_grad 
  78  # 
  79  #            dip_R1_hess     =  dip_const_hess 
  80  # 
  81  # 
  82  #        Dipolar spectral density components 
  83  #        =================================== 
  84  # 
  85  #            dip_Jw_R1_func  =  J(wH-wN) + 3J(wN) + 6J(wH+wN) 
  86  # 
  87  #                               dJ(wH-wN)         dJ(wN)         dJ(wH+wN) 
  88  #            dip_Jw_R1_grad  =  ---------  +  3 . ------  +  6 . --------- 
  89  #                                  dJw             dJw              dJw 
  90  # 
  91  #                               d2J(wH-wN)          d2J(wN)          d2J(wH+wN) 
  92  #            dip_Jw_R1_hess  =  ----------  +  3 . ---------  +  6 . ---------- 
  93  #                               dJwi.dJwj          dJwi.dJwj         dJwi.dJwj 
  94  # 
  95  # 
  96  #        CSA components 
  97  #        ============== 
  98  # 
  99  #            csa_R1_func     =  csa_const_func 
 100  # 
 101  #            csa_R1_grad     =  csa_const_grad 
 102  # 
 103  #            csa_R1_hess     =  csa_const_hess 
 104  # 
 105  # 
 106  #        CSA spectral density components 
 107  #        =============================== 
 108  # 
 109  #            csa_Jw_R1_func  =  J(wN) 
 110  # 
 111  #                               dJ(wN) 
 112  #            csa_Jw_R1_grad  =  ------ 
 113  #                                dJw 
 114  # 
 115  #                                d2J(wN) 
 116  #            csa_Jw_R1_hess  =  --------- 
 117  #                               dJwi.dJwj 
 118  # 
 119  # 
 120  #        Rex components 
 121  #        ============== 
 122  # 
 123  #            rex_R1_func     =  0 
 124  # 
 125  #            rex_R1_grad     =  0 
 126  # 
 127  #            rex_R1_hess     =  0 
 128  # 
 129  # 
 130  #    R2 components 
 131  #    ============= 
 132  # 
 133  #        Dipolar components 
 134  #        ================== 
 135  # 
 136  #            dip_R2_func     =  dip_const_func / 2 
 137  # 
 138  #            dip_R2_grad     =  dip_const_grad / 2 
 139  # 
 140  #            dip_R2_hess     =  dip_const_hess / 2 
 141  # 
 142  # 
 143  #        Dipolar spectral density components 
 144  #        =================================== 
 145  # 
 146  #            dip_Jw_R2_func  =  4J(0) + J(wH-wN) + 3J(wN) + 6J(wH) + 6J(wH+wN) 
 147  # 
 148  #                                   dJ(0)     dJ(wH-wN)         dJ(wN)         dJ(wH)         dJ(wH+wN) 
 149  #            dip_Jw_R2_grad  =  4 . -----  +  ---------  +  3 . ------  +  6 . ------  +  6 . --------- 
 150  #                                    dJw         dJw             dJw            dJw              dJw 
 151  # 
 152  #                                     d2J(0)      d2J(wH-wN)          d2J(wN)           d2J(wH)          d2J(wH+wN) 
 153  #            dip_Jw_R2_hess  =  4 . ---------  +  ----------  +  3 . ---------  +  6 . ---------  +  6 . ---------- 
 154  #                                   dJwi.dJwj     dJwi.dJwj          dJwi.dJwj         dJwi.dJwj         dJwi.dJwj 
 155  # 
 156  # 
 157  #        CSA components 
 158  #        ============== 
 159  # 
 160  #            csa_R2_func     =  csa_const_func / 6 
 161  # 
 162  #            csa_R2_grad     =  csa_const_grad / 6 
 163  # 
 164  #            csa_R2_hess     =  csa_const_hess / 6 
 165  # 
 166  # 
 167  #        CSA spectral density components 
 168  #        =============================== 
 169  # 
 170  #            csa_Jw_R2_func  =  4J(0) + 3J(wN) 
 171  # 
 172  #                                   dJ(0)         dJ(wN) 
 173  #            csa_Jw_R2_grad  =  4 . -----  +  3 . ------ 
 174  #                                    dJw           dJw 
 175  # 
 176  #                                     d2J(0)           d2J(wN) 
 177  #            csa_Jw_R2_hess  =  4 . ---------  +  3 . --------- 
 178  #                                   dJwi.dJwj         dJwi.dJwj 
 179  # 
 180  # 
 181  #        Rex components 
 182  #        ============== 
 183  # 
 184  #            rex_R2_func     =  rex_const_func 
 185  # 
 186  #            rex_R2_grad     =  rex_const_grad 
 187  # 
 188  #            rex_R2_hess     =  0 
 189  # 
 190  # 
 191  #    sigma_noe components 
 192  #    ==================== 
 193  # 
 194  #        Dipolar components 
 195  #        ================== 
 196  # 
 197  #            dip_sigma_noe_func      =  dip_const_func 
 198  # 
 199  #            dip_sigma_noe_grad      =  dip_const_grad 
 200  # 
 201  #            dip_sigma_noe_hess      =  dip_const_hess 
 202  # 
 203  # 
 204  #        Dipolar spectral density components 
 205  #        =================================== 
 206  # 
 207  #            dip_Jw_sigma_noe_func  =  6J(wH+wN) - J(wH-wN) 
 208  # 
 209  #                                          dJ(wH+wN)     dJ(wH-wN) 
 210  #            dip_Jw_sigma_noe_grad  =  6 . ---------  -  --------- 
 211  #                                             dJw           dJw 
 212  # 
 213  #                                          d2J(wH+wN)     d2J(wH-wN) 
 214  #            dip_Jw_sigma_noe_hess  =  6 . ----------  -  ---------- 
 215  #                                          dJwi.dJwj      dJwi.dJwj 
 216  # 
 217  # 
 218  #        CSA components 
 219  #        ============== 
 220  # 
 221  #            csa_sigma_noe_func      =  0 
 222  # 
 223  #            csa_sigma_noe_grad      =  0 
 224  # 
 225  #            csa_sigma_noe_hess      =  0 
 226  # 
 227  # 
 228  #        CSA spectral density components 
 229  #        =============================== 
 230  # 
 231  #            csa_Jw_sigma_noe_func   =  0 
 232  # 
 233  #            csa_Jw_sigma_noe_grad   =  0 
 234  # 
 235  #            csa_Jw_sigma_noe_hess   =  0 
 236  # 
 237  # 
 238  #        Rex components 
 239  #        ============== 
 240  # 
 241  #            rex_sigma_noe_func      =  0 
 242  # 
 243  #            rex_sigma_noe_grad      =  0 
 244  # 
 245  #            rex_sigma_noe_hess      =  0 
 246  # 
 247   
 248  # Python module imports. 
 249  from math import pi 
 250   
 251   
 252  # The main functions for the calculation of the Ri components. 
 253  ############################################################## 
 254   
 255  # These functions are duplicated many times for all combinations of Rex, bond length, and CSA as model parameters 
 256  # to make the code more efficient. 
 257   
 258  # Ri. 
 259 -def ri_comps(data, params): 
 260      """Calculate the ri function components. 
 261   
 262      Calculated: 
 263          - Dipolar J(w) components. 
 264          - CSA J(w) components. 
 265      Pre-calculated: 
 266          - Rex constant components. 
 267          - Dipolar constant components. 
 268          - CSA constant components. 
 269      """ 
 270   
 271      # Loop over the relaxation values. 
 272      for i in range(data.num_ri): 
 273          # Dipolar J(w) components 
 274          data.dip_jw_comps_func[i] = data.create_dip_jw_func[i](data.jw, data.remap_table[i]) 
 275   
 276          # CSA J(w) components. 
 277          if data.create_csa_jw_func[i]: 
 278              data.csa_jw_comps_func[i] = data.create_csa_jw_func[i](data.jw, data.remap_table[i]) 
 279   
 280   
 281  # Ri (Rex). 
 282 -def ri_comps_rex(data, params): 
 283      """Calculate the ri function components. 
 284   
 285      Calculated: 
 286          - Dipolar J(w) components. 
 287          - CSA J(w) components. 
 288          - Rex constant components. 
 289      Pre-calculated: 
 290          - Dipolar constant components. 
 291          - CSA constant components. 
 292      """ 
 293   
 294      # Loop over the relaxation values. 
 295      for i in range(data.num_ri): 
 296          # Dipolar J(w) components 
 297          data.dip_jw_comps_func[i] = data.create_dip_jw_func[i](data.jw, data.remap_table[i]) 
 298   
 299          # CSA J(w) components. 
 300          if data.create_csa_jw_func[i]: 
 301              data.csa_jw_comps_func[i] = data.create_csa_jw_func[i](data.jw, data.remap_table[i]) 
 302   
 303          # Rex components 
 304          if data.create_rex_func[i]: 
 305              data.rex_comps_func[i] = data.create_rex_func[i](params[data.rex_i], data.frq[data.remap_table[i]]) 
 306   
 307   
 308  # Ri (Bond length). 
 309 -def ri_comps_r(data, params): 
 310      """Calculate the ri function components. 
 311   
 312      Calculated: 
 313          - Dipolar constant components. 
 314          - Dipolar J(w) components. 
 315          - CSA J(w) components. 
 316      Pre-calculated: 
 317          - Rex constant components. 
 318          - CSA constant components. 
 319      """ 
 320   
 321      # Dipolar constant function value. 
 322      comp_dip_const_func(data, params[data.r_i]) 
 323   
 324      # Loop over the relaxation values. 
 325      for i in range(data.num_ri): 
 326          # Dipolar constant components. 
 327          data.dip_comps_func[i] = data.dip_const_func 
 328          if data.create_dip_func[i]: 
 329              data.dip_comps_func[i] = data.create_dip_func[i](data.dip_const_func) 
 330   
 331          # Dipolar J(w) components 
 332          data.dip_jw_comps_func[i] = data.create_dip_jw_func[i](data.jw, data.remap_table[i]) 
 333   
 334          # CSA J(w) components. 
 335          if data.create_csa_jw_func[i]: 
 336              data.csa_jw_comps_func[i] = data.create_csa_jw_func[i](data.jw, data.remap_table[i]) 
 337   
 338   
 339  # Ri (CSA). 
 340 -def ri_comps_csa(data, params): 
 341      """Calculate the ri function components. 
 342   
 343      Calculated: 
 344          - Dipolar J(w) components. 
 345          - CSA constant components. 
 346          - CSA J(w) components. 
 347      Pre-calculated: 
 348          - Rex constant components. 
 349          - Dipolar constant components. 
 350      """ 
 351   
 352      # CSA constant function value. 
 353      comp_csa_const_func(data, params[data.csa_i]) 
 354   
 355      # Loop over the relaxation values. 
 356      for i in range(data.num_ri): 
 357          # Dipolar J(w) components 
 358          data.dip_jw_comps_func[i] = data.create_dip_jw_func[i](data.jw, data.remap_table[i]) 
 359   
 360          # CSA constant components. 
 361          if data.create_csa_func[i]: 
 362              data.csa_comps_func[i] = data.create_csa_func[i](data.csa_const_func[data.remap_table[i]]) 
 363   
 364          # CSA J(w) components. 
 365          if data.create_csa_jw_func[i]: 
 366              data.csa_jw_comps_func[i] = data.create_csa_jw_func[i](data.jw, data.remap_table[i]) 
 367   
 368   
 369  # Ri (Bond length, CSA). 
 370 -def ri_comps_r_csa(data, params): 
 371      """Calculate the ri function components. 
 372   
 373      Calculated: 
 374          - Dipolar constant components. 
 375          - Dipolar J(w) components. 
 376          - CSA constant components. 
 377          - CSA J(w) components. 
 378      Pre-calculated: 
 379          - Rex constant components. 
 380      """ 
 381   
 382      # Dipolar constant function value. 
 383      comp_dip_const_func(data, params[data.r_i]) 
 384   
 385      # CSA constant function value. 
 386      comp_csa_const_func(data, params[data.csa_i]) 
 387   
 388      # Loop over the relaxation values. 
 389      for i in range(data.num_ri): 
 390          # Dipolar constant components. 
 391          data.dip_comps_func[i] = data.dip_const_func 
 392          if data.create_dip_func[i]: 
 393              data.dip_comps_func[i] = data.create_dip_func[i](data.dip_const_func) 
 394   
 395          # Dipolar J(w) components 
 396          data.dip_jw_comps_func[i] = data.create_dip_jw_func[i](data.jw, data.remap_table[i]) 
 397   
 398          # CSA constant components. 
 399          if data.create_csa_func[i]: 
 400              data.csa_comps_func[i] = data.create_csa_func[i](data.csa_const_func[data.remap_table[i]]) 
 401   
 402          # CSA J(w) components. 
 403          if data.create_csa_jw_func[i]: 
 404              data.csa_jw_comps_func[i] = data.create_csa_jw_func[i](data.jw, data.remap_table[i]) 
 405   
 406   
 407  # Ri (Bond length, Rex). 
 408 -def ri_comps_r_rex(data, params): 
 409      """Calculate the ri function components. 
 410   
 411      Calculated: 
 412          - Dipolar constant components. 
 413          - Dipolar J(w) components. 
 414          - CSA J(w) components. 
 415          - Rex constant components. 
 416      Pre-calculated: 
 417          - CSA constant components. 
 418      """ 
 419   
 420      # Dipolar constant function value. 
 421      comp_dip_const_func(data, params[data.r_i]) 
 422   
 423      # Loop over the relaxation values. 
 424      for i in range(data.num_ri): 
 425          # Dipolar constant components. 
 426          data.dip_comps_func[i] = data.dip_const_func 
 427          if data.create_dip_func[i]: 
 428              data.dip_comps_func[i] = data.create_dip_func[i](data.dip_const_func) 
 429   
 430          # Dipolar J(w) components 
 431          data.dip_jw_comps_func[i] = data.create_dip_jw_func[i](data.jw, data.remap_table[i]) 
 432   
 433          # CSA J(w) components. 
 434          if data.create_csa_jw_func[i]: 
 435              data.csa_jw_comps_func[i] = data.create_csa_jw_func[i](data.jw, data.remap_table[i]) 
 436   
 437          # Rex components 
 438          if data.create_rex_func[i]: 
 439              data.rex_comps_func[i] = data.create_rex_func[i](params[data.rex_i], data.frq[data.remap_table[i]]) 
 440   
 441   
 442  # Ri (CSA, Rex). 
 443 -def ri_comps_csa_rex(data, params): 
 444      """Calculate the ri function components. 
 445   
 446      Calculated: 
 447          - Dipolar J(w) components. 
 448          - CSA constant components. 
 449          - CSA J(w) components. 
 450          - Rex constant components. 
 451      Pre-calculated: 
 452          - Dipolar constant components. 
 453      """ 
 454   
 455      # CSA constant function value. 
 456      comp_csa_const_func(data, params[data.csa_i]) 
 457   
 458      # Loop over the relaxation values. 
 459      for i in range(data.num_ri): 
 460          # Dipolar J(w) components 
 461          data.dip_jw_comps_func[i] = data.create_dip_jw_func[i](data.jw, data.remap_table[i]) 
 462   
 463          # CSA constant components. 
 464          if data.create_csa_func[i]: 
 465              data.csa_comps_func[i] = data.create_csa_func[i](data.csa_const_func[data.remap_table[i]]) 
 466   
 467          # CSA J(w) components. 
 468          if data.create_csa_jw_func[i]: 
 469              data.csa_jw_comps_func[i] = data.create_csa_jw_func[i](data.jw, data.remap_table[i]) 
 470   
 471          # Rex components 
 472          if data.create_rex_func[i]: 
 473              data.rex_comps_func[i] = data.create_rex_func[i](params[data.rex_i], data.frq[data.remap_table[i]]) 
 474   
 475   
 476  # Ri (Bond length, CSA, Rex). 
 477 -def ri_comps_r_csa_rex(data, params): 
 478      """Calculate the ri function components. 
 479   
 480      Calculated: 
 481          - Dipolar constant components. 
 482          - Dipolar J(w) components. 
 483          - CSA constant components. 
 484          - CSA J(w) components. 
 485          - Rex constant components. 
 486      Pre-calculated: 
 487          - None. 
 488      """ 
 489   
 490      # Dipolar constant function value. 
 491      comp_dip_const_func(data, params[data.r_i]) 
 492   
 493      # CSA constant function value. 
 494      comp_csa_const_func(data, params[data.csa_i]) 
 495   
 496      # Loop over the relaxation values. 
 497      for i in range(data.num_ri): 
 498          # Dipolar constant components. 
 499          data.dip_comps_func[i] = data.dip_const_func 
 500          if data.create_dip_func[i]: 
 501              data.dip_comps_func[i] = data.create_dip_func[i](data.dip_const_func) 
 502   
 503          # Dipolar J(w) components 
 504          data.dip_jw_comps_func[i] = data.create_dip_jw_func[i](data.jw, data.remap_table[i]) 
 505   
 506          # CSA constant components. 
 507          if data.create_csa_func[i]: 
 508              data.csa_comps_func[i] = data.create_csa_func[i](data.csa_const_func[data.remap_table[i]]) 
 509   
 510          # CSA J(w) components. 
 511          if data.create_csa_jw_func[i]: 
 512              data.csa_jw_comps_func[i] = data.create_csa_jw_func[i](data.jw, data.remap_table[i]) 
 513   
 514          # Rex components 
 515          if data.create_rex_func[i]: 
 516              data.rex_comps_func[i] = data.create_rex_func[i](params[data.rex_i], data.frq[data.remap_table[i]]) 
 517   
 518   
 519  # R1 comps. 
 520 -def r1_comps(data, i, params): 
 521      """Calculate the r1 function components.""" 
 522   
 523      # Dipolar constant function value. 
 524      if data.r_i: 
 525          comp_dip_const_func(data, params[data.r_i]) 
 526   
 527      # CSA constant function value. 
 528      if data.csa_i: 
 529          comp_csa_const_func(data, params[data.csa_i]) 
 530   
 531      # Dipolar constant components. 
 532      data.dip_comps_func[i] = data.dip_const_func 
 533   
 534      # Dipolar J(w) components 
 535      data.dip_jw_comps_func[i] = comp_r1_dip_jw(data.jw, data.remap_table[i]) 
 536   
 537      # CSA constant components. 
 538      data.csa_comps_func[i] = data.csa_const_func[data.remap_table[i]] 
 539   
 540      # CSA J(w) components. 
 541      data.csa_jw_comps_func[i] = comp_r1_csa_jw(data.jw, data.remap_table[i]) 
 542   
 543   
 544   
 545  # The main functions for the calculation of the dRi components. 
 546  ############################################################### 
 547   
 548  # These functions are duplicated many times for all combinations of Rex, bond length, and CSA as model parameters 
 549  # to make the code more efficient. 
 550   
 551   
 552  # dRi. 
 553 -def dri_comps(data, params): 
 554      """Calculate the dri gradient components. 
 555   
 556      Calculated: 
 557          - Dipolar J(w) components. 
 558          - CSA J(w) components. 
 559      Pre-calculated: 
 560          - Rex constant components. 
 561          - Dipolar constant components. 
 562          - CSA constant components. 
 563      """ 
 564   
 565      # Loop over the relaxation values. 
 566      for i in range(data.num_ri): 
 567          # Dipolar J(w) components 
 568          data.dip_jw_comps_grad[i] = data.create_dip_jw_grad[i](data.djw, data.remap_table[i]) 
 569   
 570          # CSA J(w) components. 
 571          if data.create_csa_jw_grad[i]: 
 572              data.csa_jw_comps_grad[i] = data.create_csa_jw_grad[i](data.djw, data.remap_table[i]) 
 573   
 574   
 575  # dRi (Rex). 
 576 -def dri_comps_rex(data, params): 
 577      """Calculate the dri gradient components. 
 578   
 579      Calculated: 
 580          - Dipolar J(w) components. 
 581          - CSA J(w) components. 
 582          - Rex constant components. 
 583      Pre-calculated: 
 584          - Dipolar constant components. 
 585          - CSA constant components. 
 586      """ 
 587   
 588      # Loop over the relaxation values. 
 589      for i in range(data.num_ri): 
 590          # Dipolar J(w) components 
 591          data.dip_jw_comps_grad[i] = data.create_dip_jw_grad[i](data.djw, data.remap_table[i]) 
 592   
 593          # CSA J(w) components. 
 594          if data.create_csa_jw_grad[i]: 
 595              data.csa_jw_comps_grad[i] = data.create_csa_jw_grad[i](data.djw, data.remap_table[i]) 
 596   
 597          # Rex components 
 598          if data.create_rex_grad[i]: 
 599              data.rex_comps_grad[i] = data.create_rex_grad[i](data.frq[data.remap_table[i]]) 
 600   
 601   
 602  # dRi (Bond length). 
 603 -def dri_comps_r(data, params): 
 604      """Calculate the dri gradient components. 
 605   
 606      Calculated: 
 607          - Dipolar constant components. 
 608          - Dipolar J(w) components. 
 609          - CSA J(w) components. 
 610      Pre-calculated: 
 611          - Rex constant components. 
 612          - CSA constant components. 
 613      """ 
 614   
 615      # Dipolar constant gradient value. 
 616      comp_dip_const_grad(data, params[data.r_i]) 
 617   
 618      # Loop over the relaxation values. 
 619      for i in range(data.num_ri): 
 620          # Dipolar constant components. 
 621          data.dip_comps_grad[i] = data.dip_const_grad 
 622          if data.create_dip_grad[i]: 
 623              data.dip_comps_grad[i] = data.create_dip_grad[i](data.dip_const_grad) 
 624   
 625          # Dipolar J(w) components 
 626          data.dip_jw_comps_grad[i] = data.create_dip_jw_grad[i](data.djw, data.remap_table[i]) 
 627   
 628          # CSA J(w) components. 
 629          if data.create_csa_jw_grad[i]: 
 630              data.csa_jw_comps_grad[i] = data.create_csa_jw_grad[i](data.djw, data.remap_table[i]) 
 631   
 632   
 633  # dRi (CSA). 
 634 -def dri_comps_csa(data, params): 
 635      """Calculate the dri gradient components. 
 636   
 637      Calculated: 
 638          - Dipolar J(w) components. 
 639          - CSA constant components. 
 640          - CSA J(w) components. 
 641      Pre-calculated: 
 642          - Rex constant components. 
 643          - Dipolar constant components. 
 644      """ 
 645   
 646      # CSA constant gradient value. 
 647      comp_csa_const_grad(data, params[data.csa_i]) 
 648   
 649      # Loop over the relaxation values. 
 650      for i in range(data.num_ri): 
 651          # Dipolar J(w) components 
 652          data.dip_jw_comps_grad[i] = data.create_dip_jw_grad[i](data.djw, data.remap_table[i]) 
 653   
 654          # CSA constant components. 
 655          if data.create_csa_grad[i]: 
 656              data.csa_comps_grad[i] = data.create_csa_grad[i](data.csa_const_grad[data.remap_table[i]]) 
 657   
 658          # CSA J(w) components. 
 659          if data.create_csa_jw_grad[i]: 
 660              data.csa_jw_comps_grad[i] = data.create_csa_jw_grad[i](data.djw, data.remap_table[i]) 
 661   
 662   
 663  # dRi (Bond length, CSA). 
 664 -def dri_comps_r_csa(data, params): 
 665      """Calculate the dri gradient components. 
 666   
 667      Calculated: 
 668          - Dipolar constant components. 
 669          - Dipolar J(w) components. 
 670          - CSA constant components. 
 671          - CSA J(w) components. 
 672      Pre-calculated: 
 673          - Rex constant components. 
 674      """ 
 675   
 676      # Dipolar constant gradient value. 
 677      comp_dip_const_grad(data, params[data.r_i]) 
 678   
 679      # CSA constant gradient value. 
 680      comp_csa_const_grad(data, params[data.csa_i]) 
 681   
 682      # Loop over the relaxation values. 
 683      for i in range(data.num_ri): 
 684          # Dipolar constant components. 
 685          data.dip_comps_grad[i] = data.dip_const_grad 
 686          if data.create_dip_grad[i]: 
 687              data.dip_comps_grad[i] = data.create_dip_grad[i](data.dip_const_grad) 
 688   
 689          # Dipolar J(w) components 
 690          data.dip_jw_comps_grad[i] = data.create_dip_jw_grad[i](data.djw, data.remap_table[i]) 
 691   
 692          # CSA constant components. 
 693          if data.create_csa_grad[i]: 
 694              data.csa_comps_grad[i] = data.create_csa_grad[i](data.csa_const_grad[data.remap_table[i]]) 
 695   
 696          # CSA J(w) components. 
 697          if data.create_csa_jw_grad[i]: 
 698              data.csa_jw_comps_grad[i] = data.create_csa_jw_grad[i](data.djw, data.remap_table[i]) 
 699   
 700   
 701  # dRi (Bond length, Rex). 
 702 -def dri_comps_r_rex(data, params): 
 703      """Calculate the dri gradient components. 
 704   
 705      Calculated: 
 706          - Dipolar constant components. 
 707          - Dipolar J(w) components. 
 708          - CSA J(w) components. 
 709          - Rex constant components. 
 710      Pre-calculated: 
 711          - CSA constant components. 
 712      """ 
 713   
 714      # Dipolar constant gradient value. 
 715      comp_dip_const_grad(data, params[data.r_i]) 
 716   
 717      # Loop over the relaxation values. 
 718      for i in range(data.num_ri): 
 719          # Dipolar constant components. 
 720          data.dip_comps_grad[i] = data.dip_const_grad 
 721          if data.create_dip_grad[i]: 
 722              data.dip_comps_grad[i] = data.create_dip_grad[i](data.dip_const_grad) 
 723   
 724          # Dipolar J(w) components 
 725          data.dip_jw_comps_grad[i] = data.create_dip_jw_grad[i](data.djw, data.remap_table[i]) 
 726   
 727          # CSA J(w) components. 
 728          if data.create_csa_jw_grad[i]: 
 729              data.csa_jw_comps_grad[i] = data.create_csa_jw_grad[i](data.djw, data.remap_table[i]) 
 730   
 731          # Rex components 
 732          if data.create_rex_grad[i]: 
 733              data.rex_comps_grad[i] = data.create_rex_grad[i](data.frq[data.remap_table[i]]) 
 734   
 735   
 736  # dRi (CSA, Rex). 
 737 -def dri_comps_csa_rex(data, params): 
 738      """Calculate the dri gradient components. 
 739   
 740      Calculated: 
 741          - Dipolar J(w) components. 
 742          - CSA constant components. 
 743          - CSA J(w) components. 
 744          - Rex constant components. 
 745      Pre-calculated: 
 746          - Dipolar constant components. 
 747      """ 
 748   
 749      # CSA constant gradient value. 
 750      comp_csa_const_grad(data, params[data.csa_i]) 
 751   
 752      # Loop over the relaxation values. 
 753      for i in range(data.num_ri): 
 754          # Dipolar J(w) components 
 755          data.dip_jw_comps_grad[i] = data.create_dip_jw_grad[i](data.djw, data.remap_table[i]) 
 756   
 757          # CSA constant components. 
 758          if data.create_csa_grad[i]: 
 759              data.csa_comps_grad[i] = data.create_csa_grad[i](data.csa_const_grad[data.remap_table[i]]) 
 760   
 761          # CSA J(w) components. 
 762          if data.create_csa_jw_grad[i]: 
 763              data.csa_jw_comps_grad[i] = data.create_csa_jw_grad[i](data.djw, data.remap_table[i]) 
 764   
 765          # Rex components 
 766          if data.create_rex_grad[i]: 
 767              data.rex_comps_grad[i] = data.create_rex_grad[i](data.frq[data.remap_table[i]]) 
 768   
 769   
 770  # dRi (Bond length, CSA, Rex). 
 771 -def dri_comps_r_csa_rex(data, params): 
 772      """Calculate the dri gradient components. 
 773   
 774      Calculated: 
 775          - Dipolar constant components. 
 776          - Dipolar J(w) components. 
 777          - CSA constant components. 
 778          - CSA J(w) components. 
 779          - Rex constant components. 
 780      Pre-calculated: 
 781          - None. 
 782      """ 
 783   
 784      # Dipolar constant gradient value. 
 785      comp_dip_const_grad(data, params[data.r_i]) 
 786   
 787      # CSA constant gradient value. 
 788      comp_csa_const_grad(data, params[data.csa_i]) 
 789   
 790      # Loop over the relaxation values. 
 791      for i in range(data.num_ri): 
 792          # Dipolar constant components. 
 793          data.dip_comps_grad[i] = data.dip_const_grad 
 794          if data.create_dip_grad[i]: 
 795              data.dip_comps_grad[i] = data.create_dip_grad[i](data.dip_const_grad) 
 796   
 797          # Dipolar J(w) components 
 798          data.dip_jw_comps_grad[i] = data.create_dip_jw_grad[i](data.djw, data.remap_table[i]) 
 799   
 800          # CSA constant components. 
 801          if data.create_csa_grad[i]: 
 802              data.csa_comps_grad[i] = data.create_csa_grad[i](data.csa_const_grad[data.remap_table[i]]) 
 803   
 804          # CSA J(w) components. 
 805          if data.create_csa_jw_grad[i]: 
 806              data.csa_jw_comps_grad[i] = data.create_csa_jw_grad[i](data.djw, data.remap_table[i]) 
 807   
 808          # Rex components 
 809          if data.create_rex_grad[i]: 
 810              data.rex_comps_grad[i] = data.create_rex_grad[i](data.frq[data.remap_table[i]]) 
 811   
 812   
 813  # dR1 comps. 
 814 -def dr1_comps(data, i, params): 
 815      """Calculate the dr1 gradient components.""" 
 816   
 817      # Dipolar constant gradient value. 
 818      if data.r_i: 
 819          comp_dip_const_grad(data, params[data.r_i]) 
 820   
 821      # CSA constant gradient value. 
 822      if data.csa_i: 
 823          comp_csa_const_grad(data, params[data.csa_i]) 
 824   
 825      # Dipolar constant components. 
 826      data.dip_comps_grad[i] = data.dip_const_grad 
 827   
 828      # Dipolar J(w) components 
 829      data.dip_jw_comps_grad[i] = comp_r1_dip_jw(data.djw, data.remap_table[i]) 
 830   
 831      # CSA constant components. 
 832      data.csa_comps_grad[i] = data.csa_const_grad[data.remap_table[i]] 
 833   
 834      # CSA J(w) components. 
 835      data.csa_jw_comps_grad[i] = comp_r1_csa_jw(data.djw, data.remap_table[i]) 
 836   
 837   
 838  # The main functions for the calculation of the d2Ri components. 
 839  ################################################################ 
 840   
 841  # These functions are duplicated many times for all combinations of Rex, bond length, and CSA as model parameters 
 842  # to make the code more efficient. 
 843   
 844   
 845  # d2Ri. 
 846 -def d2ri_comps(data, params): 
 847      """Calculate the d2ri Hessian components. 
 848   
 849      Calculated: 
 850          - Dipolar J(w) components. 
 851          - CSA J(w) components. 
 852      Pre-calculated: 
 853          - Rex constant components. 
 854          - Dipolar constant components. 
 855          - CSA constant components. 
 856      """ 
 857   
 858      # Loop over the relaxation values. 
 859      for i in range(data.num_ri): 
 860          # Dipolar J(w) components 
 861          data.dip_jw_comps_hess[i] = data.create_dip_jw_hess[i](data.d2jw, data.remap_table[i]) 
 862   
 863          # CSA J(w) components. 
 864          if data.create_csa_jw_hess[i]: 
 865              data.csa_jw_comps_hess[i] = data.create_csa_jw_hess[i](data.d2jw, data.remap_table[i]) 
 866   
 867   
 868  # d2Ri (Bond length). 
 869 -def d2ri_comps_r(data, params): 
 870      """Calculate the d2ri Hessian components. 
 871   
 872      Calculated: 
 873          - Dipolar constant components. 
 874          - Dipolar J(w) components. 
 875          - CSA J(w) components. 
 876      Pre-calculated: 
 877          - Rex constant components. 
 878          - CSA constant components. 
 879      """ 
 880   
 881      # Dipolar constant Hessian value. 
 882      comp_dip_const_hess(data, params[data.r_i]) 
 883   
 884      # Loop over the relaxation values. 
 885      for i in range(data.num_ri): 
 886          # Dipolar constant components. 
 887          data.dip_comps_hess[i] = data.dip_const_hess 
 888          if data.create_dip_hess[i]: 
 889              data.dip_comps_hess[i] = data.create_dip_hess[i](data.dip_const_hess) 
 890   
 891          # Dipolar J(w) components 
 892          data.dip_jw_comps_hess[i] = data.create_dip_jw_hess[i](data.d2jw, data.remap_table[i]) 
 893   
 894          # CSA J(w) components. 
 895          if data.create_csa_jw_hess[i]: 
 896              data.csa_jw_comps_hess[i] = data.create_csa_jw_hess[i](data.d2jw, data.remap_table[i]) 
 897   
 898   
 899  # d2Ri (CSA). 
 900 -def d2ri_comps_csa(data, params): 
 901      """Calculate the d2ri Hessian components. 
 902   
 903      Calculated: 
 904          - Dipolar J(w) components. 
 905          - CSA constant components. 
 906          - CSA J(w) components. 
 907      Pre-calculated: 
 908          - Rex constant components. 
 909          - Dipolar constant components. 
 910      """ 
 911   
 912      # CSA constant Hessian value. 
 913      comp_csa_const_hess(data, params) 
 914   
 915      # Loop over the relaxation values. 
 916      for i in range(data.num_ri): 
 917          # Dipolar J(w) components 
 918          data.dip_jw_comps_hess[i] = data.create_dip_jw_hess[i](data.d2jw, data.remap_table[i]) 
 919   
 920          # CSA constant components. 
 921          if data.create_csa_hess[i]: 
 922              data.csa_comps_hess[i] = data.create_csa_hess[i](data.csa_const_hess[data.remap_table[i]]) 
 923   
 924          # CSA J(w) components. 
 925          if data.create_csa_jw_hess[i]: 
 926              data.csa_jw_comps_hess[i] = data.create_csa_jw_hess[i](data.d2jw, data.remap_table[i]) 
 927   
 928   
 929  # d2Ri (Bond length, CSA). 
 930 -def d2ri_comps_r_csa(data, params): 
 931      """Calculate the d2ri Hessian components. 
 932   
 933      Calculated: 
 934          - Dipolar constant components. 
 935          - Dipolar J(w) components. 
 936          - CSA constant components. 
 937          - CSA J(w) components. 
 938      Pre-calculated: 
 939          - Rex constant components. 
 940      """ 
 941   
 942      # Dipolar constant Hessian value. 
 943      comp_dip_const_hess(data, params[data.r_i]) 
 944   
 945      # CSA constant Hessian value. 
 946      comp_csa_const_hess(data, params) 
 947   
 948      # Loop over the relaxation values. 
 949      for i in range(data.num_ri): 
 950          # Dipolar constant components. 
 951          data.dip_comps_hess[i] = data.dip_const_hess 
 952          if data.create_dip_hess[i]: 
 953              data.dip_comps_hess[i] = data.create_dip_hess[i](data.dip_const_hess) 
 954   
 955          # Dipolar J(w) components 
 956          data.dip_jw_comps_hess[i] = data.create_dip_jw_hess[i](data.d2jw, data.remap_table[i]) 
 957   
 958          # CSA constant components. 
 959          if data.create_csa_hess[i]: 
 960              data.csa_comps_hess[i] = data.create_csa_hess[i](data.csa_const_hess[data.remap_table[i]]) 
 961   
 962          # CSA J(w) components. 
 963          if data.create_csa_jw_hess[i]: 
 964              data.csa_jw_comps_hess[i] = data.create_csa_jw_hess[i](data.d2jw, data.remap_table[i]) 
 965   
 966   
 967  # d2R1 comps. 
 968 -def d2r1_comps(data, i, params): 
 969      """Calculate the d2ri Hessian components.""" 
 970   
 971      # Dipolar constant gradient value. 
 972      if data.r_i: 
 973          comp_dip_const_hess(data, params[data.r_i]) 
 974   
 975      # CSA constant gradient value. 
 976      if data.csa_i: 
 977          comp_csa_const_hess(data, params) 
 978   
 979      # Dipolar constant components. 
 980      data.dip_comps_hess[i] = data.dip_const_hess 
 981   
 982      # Dipolar J(w) components 
 983      data.dip_jw_comps_hess[i] = comp_r1_dip_jw(data.d2jw, data.remap_table[i]) 
 984   
 985      # CSA constant components. 
 986      data.csa_comps_hess[i] = data.csa_const_hess[data.remap_table[i]] 
 987   
 988      # CSA J(w) components. 
 989      data.csa_jw_comps_hess[i] = comp_r1_csa_jw(data.d2jw, data.remap_table[i]) 
 990   
 991   
 992   
 993  # Functions to calculate the invariant part of the dipolar and CSA constants. 
 994  ############################################################################# 
 995   
 996   
 997  # Dipolar. 
 998 -def calc_fixed_dip(data): 
 999      """Calculate the fixed component of the dipolar constant. 
1000   
1001      The equation is:: 
1002   
1003                              / mu0  \ 2 
1004          dip_const_fixed  =  | ---- |  . (gH.gN.h_bar)**2 
1005                              \ 4.pi / 
1006      """ 
1007   
1008      data.dip_const_fixed = ((data.mu0 / (4.0*pi)) * data.h_bar * data.gh * data.gx) ** 2 
1009   
1010   
1011  # CSA. 
1012 -def calc_fixed_csa(data): 
1013      """Calculate the fixed component of the CSA constants. 
1014   
1015      The equation is:: 
1016   
1017                              wN**2 
1018          csa_const_fixed  =  ----- 
1019                                3 
1020      """ 
1021   
1022      for j in range(data.num_frq): 
1023          data.csa_const_fixed[j] = data.frq_sqrd_list[j, 1] / 3.0 
1024   
1025   
1026   
1027  # Functions to calculate the dipolar constant values, gradients, and Hessians. 
1028  ############################################################################## 
1029   
1030   
1031  # Function. 
1032 -def comp_dip_const_func(data, bond_length): 
1033      """Calculate the dipolar constant. 
1034   
1035      Dipolar constant 
1036      ================ 
1037   
1038      The equation is:: 
1039   
1040                             1   / mu0  \ 2  (gH.gN.h_bar)**2 
1041          dip_const_func  =  - . | ---- |  . ---------------- 
1042                             4   \ 4.pi /         <r**6> 
1043      """ 
1044   
1045      if bond_length == 0.0: 
1046          data.dip_const_func = 1e99 
1047      else: 
1048          data.dip_const_func = 0.25 * data.dip_const_fixed * bond_length**-6 
1049   
1050   
1051  # Gradient. 
1052 -def comp_dip_const_grad(data, bond_length): 
1053      """Calculate the derivative of the dipolar constant. 
1054   
1055      Dipolar constant gradient 
1056      ========================= 
1057   
1058      The equation is:: 
1059   
1060                               3   / mu0  \ 2  (gH.gN.h_bar)**2 
1061          dip_const_grad  =  - - . | ---- |  . ---------------- 
1062                               2   \ 4.pi /         <r**7> 
1063      """ 
1064   
1065      if bond_length == 0.0: 
1066          data.dip_const_grad = 1e99 
1067      else: 
1068          data.dip_const_grad = -1.5 * data.dip_const_fixed * bond_length**-7 
1069   
1070   
1071  # Hessian. 
1072 -def comp_dip_const_hess(data, bond_length): 
1073      """Calculate the second derivative of the dipolar constant. 
1074   
1075      Dipolar constant Hessian 
1076      ======================== 
1077   
1078      The equation is:: 
1079   
1080                             21   / mu0  \ 2  (gH.gN.h_bar)**2 
1081          dip_const_hess  =  -- . | ---- |  . ---------------- 
1082                             2    \ 4.pi /         <r**8> 
1083      """ 
1084   
1085      if bond_length == 0.0: 
1086          data.dip_const_hess = 1e99 
1087      else: 
1088          data.dip_const_hess = 10.5 * data.dip_const_fixed * bond_length**-8 
1089   
1090   
1091   
1092  # Functions to calculate the CSA constant values, gradients, and Hessians. 
1093  ########################################################################## 
1094   
1095   
1096  # Function. 
1097 -def comp_csa_const_func(data, csa): 
1098      """Calculate the CSA constant. 
1099   
1100      CSA constant 
1101      ============ 
1102   
1103      The equation is:: 
1104   
1105                             (wN.csa)**2 
1106          csa_const_func  =  ----------- 
1107                                  3 
1108      """ 
1109   
1110      for i in range(data.num_frq): 
1111          data.csa_const_func[i] = data.csa_const_fixed[i] * csa**2 
1112   
1113   
1114  # Gradient. 
1115 -def comp_csa_const_grad(data, csa): 
1116      """Calculate the derivative of the CSA constant. 
1117   
1118      CSA constant gradient 
1119      ===================== 
1120   
1121      The equation is:: 
1122   
1123                             2.wN**2.csa 
1124          csa_const_grad  =  ----------- 
1125                                  3 
1126      """ 
1127   
1128      for i in range(data.num_frq): 
1129          data.csa_const_grad[i] = 2.0 * data.csa_const_fixed[i] * csa 
1130   
1131   
1132  # Hessian. 
1133 -def comp_csa_const_hess(data, params): 
1134      """Calculate the second derivative of the CSA constant. 
1135   
1136      CSA constant Hessian 
1137      ==================== 
1138   
1139      The equation is:: 
1140   
1141                             2.wN**2 
1142          csa_const_hess  =  ------- 
1143                                3 
1144      """ 
1145   
1146      for i in range(data.num_frq): 
1147          data.csa_const_hess[i] = 2.0 * data.csa_const_fixed[i] 
1148   
1149   
1150   
1151  # Functions to calculate the Rex constant values and gradients. 
1152  ############################################################### 
1153   
1154   
1155  # Function. 
1156 -def comp_rex_const_func(rhoex, frq): 
1157      """Calculate the Rex value. 
1158   
1159      Rex constant 
1160      ============ 
1161   
1162      The equation is:: 
1163   
1164          rex_const_func  =  rhoex * wH**2 
1165      """ 
1166   
1167      return rhoex * (2.0 * pi * frq) ** 2 
1168   
1169   
1170  # Gradient. 
1171 -def comp_rex_const_grad(frq): 
1172      """Calculate the Rex gradient. 
1173   
1174      Rex gradient 
1175      ============ 
1176   
1177      The equation is:: 
1178   
1179          rex_const_grad  =  wH**2 
1180      """ 
1181   
1182      return (2.0 * pi * frq) ** 2 
1183   
1184   
1185   
1186  # Functions to calculate the dipolar constant components. 
1187  ######################################################### 
1188   
1189   
1190 -def comp_r2_dip_const(dip_const_data): 
1191      """Calculate the R1 dipolar constant components. 
1192   
1193      The equations are:: 
1194   
1195          dip_const_func / 2 
1196   
1197          dip_const_grad / 2 
1198   
1199          dip_const_hess / 2 
1200      """ 
1201   
1202      return dip_const_data / 2.0 
1203   
1204   
1205   
1206  # Functions to calculate the CSA constant components. 
1207  ##################################################### 
1208   
1209  # sigma_noe has no CSA components! 
1210   
1211 -def comp_r1_csa_const(csa_const_data): 
1212      """Calculate the R1 CSA constant components. 
1213   
1214      The equations are:: 
1215   
1216          csa_const_func 
1217   
1218          csa_const_grad 
1219   
1220          csa_const_hess 
1221      """ 
1222   
1223      return csa_const_data 
1224   
1225   
1226 -def comp_r2_csa_const(csa_const_data): 
1227      """Calculate the R2 CSA constant. components 
1228   
1229      The equations are:: 
1230   
1231          csa_const_func / 6 
1232   
1233          csa_const_grad / 6 
1234   
1235          csa_const_hess / 6 
1236      """ 
1237   
1238      return csa_const_data / 6.0 
1239   
1240   
1241   
1242  # Functions to calculate the dipolar J(w) components. 
1243  ##################################################### 
1244   
1245   
1246  # R1. 
1247 -def comp_r1_dip_jw(jw_data, frq_num): 
1248      """Calculate the R1 dipolar J(w) components. 
1249   
1250      The equations are:: 
1251   
1252          dip_Jw_R1_func  =  J(wH-wN) + 3J(wN) + 6J(wH+wN) 
1253   
1254                             dJ(wH-wN)         dJ(wN)         dJ(wH+wN) 
1255          dip_Jw_R1_grad  =  ---------  +  3 . ------  +  6 . --------- 
1256                                dJw             dJw              dJw 
1257   
1258                             d2J(wH-wN)          d2J(wN)          d2J(wH+wN) 
1259          dip_Jw_R1_hess  =  ----------  +  3 . ---------  +  6 . ---------- 
1260                             dJwi.dJwj          dJwi.dJwj         dJwi.dJwj 
1261      """ 
1262   
1263      return jw_data[frq_num, 2] + 3.0*jw_data[frq_num, 1] + 6.0*jw_data[frq_num, 4] 
1264   
1265   
1266  # R2. 
1267 -def comp_r2_dip_jw(jw_data, frq_num): 
1268      """Calculate the R2 dipolar J(w) components. 
1269   
1270      The equations are:: 
1271   
1272          dip_Jw_R2_func  =  4J(0) + J(wH-wN) + 3J(wN) + 6J(wH) + 6J(wH+wN) 
1273   
1274                                 dJ(0)     dJ(wH-wN)         dJ(wN)         dJ(wH)         dJ(wH+wN) 
1275          dip_Jw_R2_grad  =  4 . -----  +  ---------  +  3 . ------  +  6 . ------  +  6 . --------- 
1276                                  dJw         dJw             dJw            dJw              dJw 
1277   
1278                                   d2J(0)      d2J(wH-wN)          d2J(wN)           d2J(wH)          d2J(wH+wN) 
1279          dip_Jw_R2_hess  =  4 . ---------  +  ----------  +  3 . ---------  +  6 . ---------  +  6 . ---------- 
1280                                 dJwi.dJwj     dJwi.dJwj          dJwi.dJwj         dJwi.dJwj         dJwi.dJwj 
1281      """ 
1282   
1283      return 4.0*jw_data[frq_num, 0] + jw_data[frq_num, 2] + 3.0*jw_data[frq_num, 1] + 6.0*jw_data[frq_num, 3] + 6.0*jw_data[frq_num, 4] 
1284   
1285   
1286  # sigma_noe. 
1287 -def comp_sigma_noe_dip_jw(jw_data, frq_num): 
1288      """Calculate the sigma_noe dipolar J(w) components. 
1289   
1290      The equations are:: 
1291   
1292          dip_Jw_sigma_noe_func  =  6J(wH+wN) - J(wH-wN) 
1293   
1294                                        dJ(wH+wN)     dJ(wH-wN) 
1295          dip_Jw_sigma_noe_grad  =  6 . ---------  -  --------- 
1296                                           dJw           dJw 
1297   
1298                                        d2J(wH+wN)     d2J(wH-wN) 
1299          dip_Jw_sigma_noe_hess  =  6 . ----------  -  ---------- 
1300                                        dJwi.dJwj      dJwi.dJwj 
1301      """ 
1302   
1303      return 6.0*jw_data[frq_num, 4] - jw_data[frq_num, 2] 
1304   
1305   
1306   
1307  # Functions to calculate the CSA J(w) components. 
1308  ################################################# 
1309   
1310  # sigma_noe has no CSA J(w) components! 
1311   
1312  # R1. 
1313 -def comp_r1_csa_jw(jw_data, frq_num): 
1314      """Calculate the R1 CSA J(w) components. 
1315   
1316      The equations are:: 
1317   
1318          csa_Jw_R1_func  =  J(wN) 
1319   
1320                             dJ(wN) 
1321          csa_Jw_R1_grad  =  ------ 
1322                              dJw 
1323   
1324                              d2J(wN) 
1325          csa_Jw_R1_hess  =  --------- 
1326                             dJwi.dJwj 
1327      """ 
1328   
1329      return jw_data[frq_num, 1] 
1330   
1331   
1332  # R2. 
1333 -def comp_r2_csa_jw(jw_data, frq_num): 
1334      """Calculate the R1 CSA J(w) components. 
1335   
1336      The equations are:: 
1337   
1338          csa_Jw_R2_func  =  4J(0) + 3J(wN) 
1339   
1340                                 dJ(0)         dJ(wN) 
1341          csa_Jw_R2_grad  =  4 . -----  +  3 . ------ 
1342                                  dJw           dJw 
1343   
1344                                   d2J(0)           d2J(wN) 
1345          csa_Jw_R2_hess  =  4 . ---------  +  3 . --------- 
1346                                 dJwi.dJwj         dJwi.dJwj 
1347      """ 
1348   
1349      return 4.0*jw_data[frq_num, 0] + 3.0*jw_data[frq_num, 1] 
1350