functions.d2jw_mf

5 - def __init__(self):

6 "Function for creating the model-free spectral density hessians."

7 8

9 - def d2Jw(self):

10 """Function to create model-free spectral density hessians. 11 12 The spectral density hessians 13 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 14 15 Data structure: self.data.d2jw 16 Dimension: 4D, (number of NMR frequencies, 5 spectral density frequencies, model-free parameters, model-free parameters) 17 Type: Numeric 4D matrix, Float64 18 Dependencies: None 19 Required by: self.data.d2ri 20 21 22 Formulae 23 ~~~~~~~~ 24 25 Parameter transformations 26 ~~~~~~~~~~~~~~~~~~~~~~~~~ 27 ae = c.te 28 29 af = c.tf 30 31 as = c.ts 32 33 34 therefore: 35 36 tm.ae 37 te' = --------- 38 ae + c.tm 39 40 tm.af 41 tf' = --------- 42 af + c.tm 43 44 tm.as 45 ts' = --------- 46 as + c.tm 47 48 49 Original: Model-free parameter - Model-free parameter 50 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 51 52 d2J(w) 53 ------ = 0 54 dS2**2 55 56 57 d2J(w) 2 (ae + c.tm)**2 - (w.tm.ae)**2 58 ------- = - - . c . tm**2 . ---------------------------------- 59 dS2.dae 5 ((ae + c.tm)**2 + (w.tm.ae)**2)**2 60 61 62 d2J(w) 4 (ae + c.tm)**3 + 3.c.tm**3.ae.w**2.(ae + c.tm) - (1/ae).(w.tm.ae)**4 63 ------ = - - . c . tm**2 . (1 - S2) . -------------------------------------------------------------------- 64 dae**2 5 ((ae + c.tm)**2 + (w.tm.ae)**2)**3 65 66 67 Original: Other parameters 68 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ 69 70 d2J(w) d2J(w) d2J(w) 71 -------- = 0 , -------- = 0 , ------ = 0 72 dS2.dRex dS2.dcsa dS2.dr 73 74 75 d2J(w) d2J(w) d2J(w) 76 -------- = 0 , -------- = 0 , ------ = 0 77 dae.dRex dae.dcsa dae.dr 78 79 80 d2J(w) d2J(w) d2J(w) 81 ------- = 0 , --------- = 0 , ------- = 0 82 dRex**2 dRex.dcsa dRex.dr 83 84 85 d2J(w) d2J(w) 86 ------- = 0 , ------- = 0 87 dcsa**2 dcsa.dr 88 89 90 d2J(w) 91 ------ = 0 92 dr**2 93 94 95 Extended: Model-free parameter - Model-free parameter 96 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 97 98 d2J(w) 99 ------- = 0 100 dS2f**2 101 102 103 d2J(w) 2*tm / 1 (as + c.tm) . as \ 104 --------- = ---- | ------------- - ----------------------------- | 105 dS2f.dS2s 5 \ 1 + (w.tm)**2 (as + c.tm)**2 + (w.tm.as)**2 / 106 107 108 d2J(w) 2 (af + c.tm)**2 - (w.tm.af)**2 109 -------- = - - . c . tm**2 . ---------------------------------- 110 dS2f.daf 5 ((af + c.tm)**2 + (w.tm.af)**2)**2 111 112 113 d2J(w) 2 (as + c.tm)**2 - (w.tm.as)**2 114 -------- = - . c . tm**2 . (1 - S2s) . ---------------------------------- 115 dS2f.das 5 ((as + c.tm)**2 + (w.tm.as)**2)**2 116 117 118 d2J(w) d2J(w) 119 ------- = 0 , ------------- = 0 120 dS2s**2 dS2s.daf 121 122 123 d2J(w) 2 (as + c.tm)**2 - (w.tm.as)**2 124 -------- = - - . c . tm**2 . S2f . ---------------------------------- 125 dS2s.das 5 ((as + c.tm)**2 + (w.tm.as)**2)**2 126 127 128 d2J(w) 4 (af + c.tm)**3 + 3.c.tm**3.af.w**2.(af + c.tm) - (1/af).(w.tm.af)**4 129 ------ = - - . c . tm**2 . (1 - S2f) . -------------------------------------------------------------------- 130 daf**2 5 ((af + c.tm)**2 + (w.tm.af)**2)**3 131 132 133 d2J(w) 134 ------- = 0 135 daf.das 136 137 138 d2J(w) 4 (as + c.tm)**3 + 3.c.tm**3.as.w**2.(as + c.tm) - (1/as).(w.tm.as)**4 139 ------ = - - . c . tm**2 . S2f . (1 - S2s) . -------------------------------------------------------------------- 140 das**2 5 ((as + c.tm)**2 + (w.tm.as)**2)**3 141 142 143 Extended: Other parameters 144 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ 145 146 d2J(w) d2J(w) d2J(w) 147 --------- = 0 , --------- = 0 , ------- = 0 148 dS2f.dRex dS2f.dcsa dS2f.dr 149 150 151 d2J(w) d2J(w) d2J(w) 152 --------- = 0 , --------- = 0 , ------- = 0 153 dS2s.dRex dS2s.dcsa dS2s.dr 154 155 156 d2J(w) d2J(w) d2J(w) 157 -------- = 0 , -------- = 0 , ------ = 0 158 daf.dRex daf.dcsa daf.dr 159 160 161 d2J(w) d2J(w) d2J(w) 162 -------- = 0 , -------- = 0 , ------ = 0 163 das.dRex das.dcsa das.dr 164 165 166 d2J(w) d2J(w) d2J(w) 167 ------- = 0 , --------- = 0 , ------- = 0 168 dRex**2 dRex.dcsa dRex.dr 169 170 171 d2J(w) d2J(w) 172 ------- = 0 , ------- = 0 173 dcsa**2 dcsa.dr 174 175 176 d2J(w) 177 ------ = 0 178 dr**2 179 180 181 182 """ 183 184 # Initialise the spectral density hessians. 185 self.data.d2jw = zeros((self.mf.data.num_frq, 5, len(self.data.params), len(self.data.params)), Float64) 186 187 # Isotropic rotational diffusion. 188 if match(self.data.diff_type, 'iso'): 189 # Model 1 and 3 hessians are zero. 190 if match('m[24]', self.data.model): 191 for i in range(self.mf.data.num_frq): 192 for param1 in range(len(self.data.jw_param_types)): 193 for param2 in range(param1 + 1): 194 if (self.data.jw_param_types[param1] == 'S2' and self.data.jw_param_types[param2] == 'te') \ 195 or (self.data.jw_param_types[param1] == 'te' and self.data.jw_param_types[param2] == 'S2'): 196 # Calculate the S2/te partial derivatives. 197 self.data.d2jw[i, 0, param1, param2] = self.calc_d2jw_dS2dae_iso_m24(i, 0) 198 self.data.d2jw[i, 1, param1, param2] = self.calc_d2jw_dS2dae_iso_m24(i, 1) 199 self.data.d2jw[i, 2, param1, param2] = self.calc_d2jw_dS2dae_iso_m24(i, 2) 200 self.data.d2jw[i, 3, param1, param2] = self.calc_d2jw_dS2dae_iso_m24(i, 3) 201 self.data.d2jw[i, 4, param1, param2] = self.calc_d2jw_dS2dae_iso_m24(i, 4) 202 # Off diagonal hessian components are symmetric. 203 self.data.d2jw[i, 0, param2, param1] = self.data.d2jw[i, 0, param1, param2] 204 self.data.d2jw[i, 1, param2, param1] = self.data.d2jw[i, 1, param1, param2] 205 self.data.d2jw[i, 2, param2, param1] = self.data.d2jw[i, 2, param1, param2] 206 self.data.d2jw[i, 3, param2, param1] = self.data.d2jw[i, 3, param1, param2] 207 self.data.d2jw[i, 4, param2, param1] = self.data.d2jw[i, 4, param1, param2] 208 elif self.data.jw_param_types[param1] == 'te' and self.data.jw_param_types[param2] == 'te': 209 # Calculate the te/te partial derivatives. 210 self.data.d2jw[i, 0, param1, param2] = self.calc_d2jw_dae2_iso_m24(i, 0) 211 self.data.d2jw[i, 1, param1, param2] = self.calc_d2jw_dae2_iso_m24(i, 1) 212 self.data.d2jw[i, 2, param1, param2] = self.calc_d2jw_dae2_iso_m24(i, 2) 213 self.data.d2jw[i, 3, param1, param2] = self.calc_d2jw_dae2_iso_m24(i, 3) 214 self.data.d2jw[i, 4, param1, param2] = self.calc_d2jw_dae2_iso_m24(i, 4) 215 elif match('m5', self.data.model): 216 for i in range(self.mf.data.num_frq): 217 for param1 in range(len(self.data.jw_param_types)): 218 for param2 in range(param1 + 1): 219 if (self.data.jw_param_types[param1] == 'S2f' and self.data.jw_param_types[param2] == 'S2s') \ 220 or (self.data.jw_param_types[param1] == 'S2s' and self.data.jw_param_types[param2] == 'S2f'): 221 # Calculate the S2f/S2s partial derivatives. 222 self.data.d2jw[i, 0, param1, param2] = self.calc_d2jw_dS2fdS2s_iso_m5(i, 0) 223 self.data.d2jw[i, 1, param1, param2] = self.calc_d2jw_dS2fdS2s_iso_m5(i, 1) 224 self.data.d2jw[i, 2, param1, param2] = self.calc_d2jw_dS2fdS2s_iso_m5(i, 2) 225 self.data.d2jw[i, 3, param1, param2] = self.calc_d2jw_dS2fdS2s_iso_m5(i, 3) 226 self.data.d2jw[i, 4, param1, param2] = self.calc_d2jw_dS2fdS2s_iso_m5(i, 4) 227 # Off diagonal hessian components are symmetric. 228 self.data.d2jw[i, 0, param2, param1] = self.data.d2jw[i, 0, param1, param2] 229 self.data.d2jw[i, 1, param2, param1] = self.data.d2jw[i, 1, param1, param2] 230 self.data.d2jw[i, 2, param2, param1] = self.data.d2jw[i, 2, param1, param2] 231 self.data.d2jw[i, 3, param2, param1] = self.data.d2jw[i, 3, param1, param2] 232 self.data.d2jw[i, 4, param2, param1] = self.data.d2jw[i, 4, param1, param2] 233 elif (self.data.jw_param_types[param1] == 'S2f' and self.data.jw_param_types[param2] == 'ts') \ 234 or (self.data.jw_param_types[param1] == 'ts' and self.data.jw_param_types[param2] == 'S2f'): 235 # Calculate the S2f/ts partial derivatives. 236 self.data.d2jw[i, 0, param1, param2] = self.calc_d2jw_dS2fdas_iso_m5(i, 0) 237 self.data.d2jw[i, 1, param1, param2] = self.calc_d2jw_dS2fdas_iso_m5(i, 1) 238 self.data.d2jw[i, 2, param1, param2] = self.calc_d2jw_dS2fdas_iso_m5(i, 2) 239 self.data.d2jw[i, 3, param1, param2] = self.calc_d2jw_dS2fdas_iso_m5(i, 3) 240 self.data.d2jw[i, 4, param1, param2] = self.calc_d2jw_dS2fdas_iso_m5(i, 4) 241 # Off diagonal hessian components are symmetric. 242 self.data.d2jw[i, 0, param2, param1] = self.data.d2jw[i, 0, param1, param2] 243 self.data.d2jw[i, 1, param2, param1] = self.data.d2jw[i, 1, param1, param2] 244 self.data.d2jw[i, 2, param2, param1] = self.data.d2jw[i, 2, param1, param2] 245 self.data.d2jw[i, 3, param2, param1] = self.data.d2jw[i, 3, param1, param2] 246 self.data.d2jw[i, 4, param2, param1] = self.data.d2jw[i, 4, param1, param2] 247 elif (self.data.jw_param_types[param1] == 'S2s' and self.data.jw_param_types[param2] == 'ts') \ 248 or (self.data.jw_param_types[param1] == 'ts' and self.data.jw_param_types[param2] == 'S2s'): 249 # Calculate the S2s/ts partial derivatives. 250 self.data.d2jw[i, 0, param1, param2] = self.calc_d2jw_dS2sdas_iso_m5(i, 0) 251 self.data.d2jw[i, 1, param1, param2] = self.calc_d2jw_dS2sdas_iso_m5(i, 1) 252 self.data.d2jw[i, 2, param1, param2] = self.calc_d2jw_dS2sdas_iso_m5(i, 2) 253 self.data.d2jw[i, 3, param1, param2] = self.calc_d2jw_dS2sdas_iso_m5(i, 3) 254 self.data.d2jw[i, 4, param1, param2] = self.calc_d2jw_dS2sdas_iso_m5(i, 4) 255 # Off diagonal hessian components are symmetric. 256 self.data.d2jw[i, 0, param2, param1] = self.data.d2jw[i, 0, param1, param2] 257 self.data.d2jw[i, 1, param2, param1] = self.data.d2jw[i, 1, param1, param2] 258 self.data.d2jw[i, 2, param2, param1] = self.data.d2jw[i, 2, param1, param2] 259 self.data.d2jw[i, 3, param2, param1] = self.data.d2jw[i, 3, param1, param2] 260 self.data.d2jw[i, 4, param2, param1] = self.data.d2jw[i, 4, param1, param2] 261 elif self.data.jw_param_types[param1] == 'ts' and self.data.jw_param_types[param2] == 'ts': 262 # Calculate the ts/ts partial derivatives. 263 self.data.d2jw[i, 0, param1, param2] = self.calc_d2jw_das2_iso_m5(i, 0) 264 self.data.d2jw[i, 1, param1, param2] = self.calc_d2jw_das2_iso_m5(i, 1) 265 self.data.d2jw[i, 2, param1, param2] = self.calc_d2jw_das2_iso_m5(i, 2) 266 self.data.d2jw[i, 3, param1, param2] = self.calc_d2jw_das2_iso_m5(i, 3) 267 self.data.d2jw[i, 4, param1, param2] = self.calc_d2jw_das2_iso_m5(i, 4) 268 269 270 # Axially symmetric rotational diffusion. 271 elif match(self.data.diff_type, 'axail'): 272 raise NameError, "Axially symetric diffusion not implemented yet, quitting program." 273 274 # Anisotropic rotational diffusion. 275 elif match(self.data.diff_type, 'aniso'): 276 raise NameError, "Anisotropic diffusion not implemented yet, quitting program." 277 278 else: 279 raise NameError, "Function option not set correctly, quitting program."

280 281

282 - def calc_d2jw_dS2dae_iso_m24(self, i, frq_index):

283 "Calculate the model 2 and 4 S2/te partial derivative of the spectral density function for isotropic rotational diffusion." 284 285 a = (self.data.ae_plus_c_tm_sqrd - self.data.omega_tm_ae_sqrd[i, frq_index]) 286 b = (self.data.ae_plus_c_tm_sqrd + self.data.omega_tm_ae_sqrd[i, frq_index])**2 287 temp = -0.4 * self.data.c * self.data.tm_sqrd * (a/b) 288 return temp

289 290

291 - def calc_d2jw_dae2_iso_m24(self, i, frq_index):

292 "Calculate the model 2 and 4 te/te partial derivative of the spectral density function for isotropic rotational diffusion." 293 294 a = self.data.ae_plus_c_tm**3 295 b = 3.0 * self.data.c * self.data.tm**3 * self.data.ae * self.mf.data.frq_sqrd_list[i][frq_index] * self.data.ae_plus_c_tm 296 c = (self.mf.data.frq_list[i][frq_index] * self.data.tm)**4 * self.data.ae**3 297 d = (self.data.ae_plus_c_tm_sqrd + self.data.omega_tm_ae_sqrd[i, frq_index])**3 298 temp = -0.8 * self.data.c * self.data.tm_sqrd * (1.0 - self.data.s2) * (a + b - c) / d 299 return temp

300 301

302 - def calc_d2jw_dS2fdS2s_iso_m5(self, i, frq_index):

303 "Calculate the model 5 S2f/S2s partial derivative of the spectral density function for isotropic rotational diffusion." 304 305 a = 1.0 / (1.0 + self.data.omega_tm_sqrd[i, frq_index]) 306 b = self.data.as_plus_c_tm * self.data.as / (self.data.as_plus_c_tm_sqrd + self.data.omega_tm_as_sqrd[i, frq_index]) 307 temp = 0.4 * self.data.tm * (a - b) 308 return temp

309 310

311 - def calc_d2jw_dS2fdas_iso_m5(self, i, frq_index):

312 "Calculate the model 5 S2f/ts partial derivative of the spectral density function for isotropic rotational diffusion." 313 314 a = (self.data.as_plus_c_tm_sqrd - self.data.omega_tm_as_sqrd[i, frq_index]) 315 b = (self.data.as_plus_c_tm_sqrd + self.data.omega_tm_as_sqrd[i, frq_index])**2 316 temp = -0.4 * self.data.c * self.data.tm_sqrd * (a/b) 317 return temp

318 319

320 - def calc_d2jw_dS2sdas_iso_m5(self, i, frq_index):

321 "Calculate the model 5 S2s/ts partial derivative of the spectral density function for isotropic rotational diffusion." 322 323 a = (self.data.as_plus_c_tm_sqrd - self.data.omega_tm_as_sqrd[i, frq_index]) 324 b = (self.data.as_plus_c_tm_sqrd + self.data.omega_tm_as_sqrd[i, frq_index])**2 325 temp = 0.4 * self.data.c * self.data.tm_sqrd * (1.0 - self.data.s2s) * (a/b) 326 return temp

327 328

329 - def calc_d2jw_das2_iso_m5(self, i, frq_index):

330 "Calculate the model 5 ts/ts partial derivative of the spectral density function for isotropic rotational diffusion." 331 332 a = self.data.as_plus_c_tm**3 333 b = 3.0 * self.data.c * self.data.tm**3 * self.data.as * self.mf.data.frq_sqrd_list[i][frq_index] * self.data.as_plus_c_tm 334 c = (self.mf.data.frq_list[i][frq_index] * self.data.tm)**4 * self.data.as**3 335 d = (self.data.as_plus_c_tm_sqrd + self.data.omega_tm_as_sqrd[i, frq_index])**3 336 temp = -0.8 * self.data.c * self.data.tm_sqrd * self.data.s2f * (1.0 - self.data.s2s) * (a + b - c) / d 337 return temp

Source Code for Module functions.d2jw_mf_trans