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    3 % Copyright (C) 2005-2006,2012,2014 Edward d'Auvergne                         %
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   21 
   22 
   23 % Values, gradients, and Hessians.
   24 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   25 
   26 \chapter{Optimisation of relaxation data -- values, gradients, and Hessians} \label{ch: values, gradients, and Hessians}
   27 
   28 
   29 
   30 % Introduction.
   31 %%%%%%%%%%%%%%%
   32 
   33 \section{Introduction to the mathematics behind the optimisation of relaxation data}
   34 
   35 
   36 A word of warning before reading this chapter, the topics covered here are quite advanced and are not necessary for understanding how to either use relax or to implement any of the data analysis techniques present within relax.
   37 The material of this chapter is intended as an in-depth explanation of the mathematics involved in the optimisation of the parameters of the model-free models, or any theory involving relaxation data.
   38 As such it contains the chi-squared equation, relaxation equations, spectral density functions, and diffusion tensor equations as well as their gradients (the vector of first partial derivatives) and Hessians (the matrix of second partial derivatives).
   39 All these equations are used in the optimisation of model-free models $m0$ to $m9$; models $tm0$ to $tm9$; the ellipsoidal, spheroidal, and spherical diffusion tensors; and the combination of the diffusion tensor and the model-free models.
   40 They also apply to all other theories involving the base $\Rone$, $\Rtwo$, and steady-state NOE relaxation rates.
   41 
   42 
   43 
   44 % The four parameter combinations.
   45 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   46 
   47 \section{The four parameter combinations}
   48 
   49 In model-free analysis four different combinations of parameters can be optimised, each of which requires a different approach to the construction of the chi-squared value, gradient, and Hessian.
   50 These categories depend on whether the model-free parameter set $\Mfset$, the diffusion tensor parameter set $\Diffset$, or both sets are simultaneously optimised.
   51 The addition of the local $\tau_m$ parameter to the model-free set $\Mfset$ creates a fourth parameter combination.
   52 
   53 
   54 
   55 
   56 % Optimisation of the model-free models.
   57 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   58 
   59 \subsection{Optimisation of the model-free models}
   60 
   61 This is the simplest category as it involves solely the optimisation of the model-free parameters of an individual residue while the diffusion tensor parameters are held constant.
   62 The model-free parameters belong to the set $\Mfset_i$ of the residue $i$.
   63 The models include $m0$ to $m9$ and the dimensionality is low with
   64 \begin{equation}
   65     \dim \Mfset_i = k \leqslant 5
   66 \end{equation}
   67 
   68 \noindent for the most complex model $m8 = \{S^2, \tau_f, S^2_f, \tau_s, R_{ex}\}$.
   69 The relaxation data of a single residue is used to build the chi-squared value, gradient, and Hessian.
   70 
   71 
   72 % Optimisation of the local tm models.
   73 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   74 
   75 \subsection{Optimisation of the local $\tau_m$ models}
   76 
   77 The addition of the local $\tau_m$ parameter to the set $\Mfset_i$ creates a new set of models which will be labelled $\Localset_i$.
   78 These include models $tm0$ to $tm9$.
   79 The local $\tau_m$ parameter is the single member of the set $\Diffset_i$ and in set notation
   80 \begin{equation}
   81     \Localset_i = \Diffset_i \cup \Mfset_i.
   82 \end{equation}
   83 
   84 Although the Brownian rotational diffusion parameter local $\tau_m$ is optimised, this category is residue specific.
   85 As such the complexity of the optimisation is lower than the next two categories.
   86 It is slightly greater than the optimisation of the set $\Mfset_i$ as
   87 \begin{equation}
   88     \dim \Localset_i = 1 + k \leqslant 6,
   89 \end{equation}
   90 
   91 \noindent where $k$ is the number of model-free parameters.
   92 
   93 
   94 
   95 % Optimisation of the diffusion tensor parameters.
   96 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   97 
   98 \subsection{Optimisation of the diffusion tensor parameters}
   99 
  100 The parameters of the Brownian rotational diffusion tensor belong to the set $\Diffset$.
  101 This set is the union of the geometric parameters $\Diffgeoset = \{\Diff_{iso}, \Diff_a, \Diff_r\}$ and the orientational parameters $\Difforiset$,
  102 \begin{equation}
  103     \Diffset = \Diffgeoset \cup \Difforiset.
  104 \end{equation}
  105 
  106 \noindent When diffusion is spherical solely the geometric parameter $\Diff_{iso}$ is optimised.
  107 When the molecule diffuses as a spheroid the geometric parameters $\Diff_{iso}$ and $\Diff_a$ and the orientational parameters $\theta$ (the polar angle) and $\phi$ (the azimuthal angle) are optimised.
  108 If the molecule diffuses as an ellipsoid the geometric parameters $\Diff_{iso}$, $\Diff_a$, and $\Diff_r$ are optimised together with the Euler angles $\alpha$, $\beta$, and $\gamma$.
  109 
  110 This category is defined as the optimisation of solely the parameters of $\Diffset$.
  111 The model-free parameters of $\Mfset$ are held constant.
  112 As all selected residues of the macromolecule are involved in the optimisation, this category is global and can be more complex than the optimisation of $\Mfset_i$ or $\Localset_i$.
  113 The dimensionality of the problem nevertheless low with
  114 \begin{equation}
  115     \dim \Diffset = 1, \qquad \dim \Diffset = 4, \qquad \dim \Diffset = 6,
  116 \end{equation}
  117 
  118 \noindent for the diffusion as a sphere, spheroid, and ellipsoid respectively.
  119 
  120 
  121 
  122 % Optimisation of the global model.
  123 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  124 
  125 \subsection{Optimisation of the global model $\Space$}
  126 
  127 The global model is defined as
  128 \begin{equation}
  129     \Space = \Diffset \cup \left( \bigcup_{i=1}^l \Mfset_i \right),
  130 \end{equation}
  131 
  132 \noindent where $i$ is the residue index and $l$ is the total number of residues used in the analysis.
  133 This is the most complex of the four categories as both diffusion tensor parameters and model-free parameters of all selected residues are optimised simultaneously.
  134 The dimensionality of the model $\Space$ is much greater than the other categories and is equal to
  135 \begin{equation}
  136     \dim \Space = \dim \Diffset + \sum_{i=1}^l k_i \leqslant 6 + 5l,
  137 \end{equation}
  138 
  139 \noindent where $k_i$ is the number of model-free parameters for the residue $i$ and is equal to $\dim \Mfset_i$, the number six corresponds to the maximum dimensionality of $\Diffset$, and the number five corresponds to the maximum dimensionality of $\Mfset_i$.
  140 
  141 
  142 
  143 
  144 % Construction of the values, gradients, and Hessians.
  145 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  146 
  147 \section{Construction of the values, gradients, and Hessians}
  148 
  149 
  150 % The sum of chi-squared values.
  151 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  152 
  153 \subsection{The sum of chi-squared values}
  154 
  155 For the single residue models of $\Mfset_i$ and $\Localset_i$ the chi-squared value $\chi^2_i$ which is optimised is simply Equation~\eqref{eq: maths: chi-squared} on page \pageref{eq: maths: chi-squared} in which the relaxation data is that of residue $i$.
  156 However for the global models $\Diffset$ and $\Space$ in which all selected residues are involved the optimised chi-squared value is the sum of those for each residue,
  157 \begin{equation}
  158     \chi^2 = \sum^l_{i=1} \chi^2_i,
  159 \end{equation}
  160 
  161 \noindent where $i$ is the residue index and $l$ is the total number of residues used in the analysis.
  162 This is equivalent to Equation~\eqref{eq: maths: chi-squared} when the index $i$ ranges over the relaxation data of all selected residues.
  163 
  164 
  165 
  166 % Construction of the gradient.
  167 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  168 
  169 \subsection{Construction of the gradient}
  170 
  171 % Gradient construction figure.
  172 \begin{figure}
  173   \centerline{
  174     \includegraphics[
  175       width=0.9\textwidth,
  176       bb=143 399 494 777
  177     ]
  178     {images/gradient}
  179   }
  180   \caption[The construction of the model-free gradient.]{
  181     The construction of the model-free gradient $\nabla \chi^2$ for the global model $\Space$.
  182     For each residue $i$ a different vector $\nabla \chi^2_i$ is constructed.
  183     The first element of the vector represented by the symbol $\partial \Diffset$ (the orange block) is the sub-vector of chi-squared partial derivatives with respect to each of the diffusion tensor parameters $\Diffset_j$.
  184     The rest of the elements, grouped into blocks for each residue denoted by the symbol $\partial \Mfset_i$, are the sub-vectors of chi-squared partial derivatives with respect to each of the model-free parameters $\Mfset_i^j$.
  185     For the residue dependent vector $\nabla \chi^2_i$ the partial derivatives with respect to the model-free parameters of $\Mfset_j$ where $i \ne j$ are zero.
  186     These blocks are left uncoloured.
  187     The complete gradient of $\Space$ is the sum of the vectors $\nabla \chi^2_i$.
  188   }
  189   \label{fig: gradient construction}
  190 \end{figure}
  191 
  192 The construction of the gradient is significantly different for the models $\Mfset_i$, $\Localset_i$, $\Diffset$, and $\Space$.
  193 In Figure~\ref{fig: gradient construction} the construction of the chi-squared gradient $\nabla \chi^2$ for the global model $\Space$ is demonstrated.
  194 In this case
  195 \begin{equation} \label{eq: spaceset gradient}
  196     \nabla \chi^2 = \sum_{i=1}^l \nabla \chi^2_i,
  197 \end{equation}
  198 
  199 \noindent where $\nabla \chi^2_i$ is the vector of partial derivatives of the chi-squared equation $\chi^2_i$ for the residue $i$.
  200 The length of this vector is
  201 \begin{equation}
  202     \lVert \nabla \chi^2_i \rVert = \dim \Space,
  203 \end{equation}
  204 
  205 \noindent with each position of the vector $j$ equal to $\frac{\partial \chi^2_i}{\partial \theta_j}$ where each $\theta_j$ is a parameter of the model.
  206 
  207 The construction of the gradient $\nabla \chi^2$ for the model $\Diffset$ is simply a subset of that of $\Space$.
  208 This is demonstrated in Figure~\ref{fig: gradient construction} by simply taking the component of the gradient $\nabla \chi^2_i$ denoted by the symbol $\partial \Diffset$ (the orange blocks) and summing these for all residues.
  209 This sum is given by \eqref{eq: spaceset gradient} and
  210 \begin{equation}
  211     \lVert \nabla \chi^2_i \rVert = \dim \Diffset.
  212 \end{equation}
  213 
  214 For the parameter set $\Localset_i$, which consists of the local $\tau_m$ parameter and the model-free parameters of a single residue, the gradient $\nabla \chi^2_i$ for the residue $i$ is simply the combination of the single orange block and single yellow block of the index $i$ (Figure~\ref{fig: gradient construction}).
  215 
  216 The model-free parameter set $\Mfset_i$ is even simpler.
  217 In Figure~\ref{fig: gradient construction} the gradient $\nabla \chi^2_i$ is simply the vector denoted by the single yellow block for the residue $i$.
  218 
  219 
  220 
  221 % Construction of the Hessian.
  222 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  223 
  224 \subsection{Construction of the Hessian}
  225 
  226 % Hessian kite figure.
  227 \begin{figure}
  228   \centerline{
  229     \includegraphics[
  230       width=0.8\textwidth,
  231       bb=61 11 585 789
  232     ]
  233     {images/kite}
  234   }
  235   \caption[The model-free Hessian kite.]{
  236     The model-free Hessian kite -- a demonstration of the construction of the model-free Hessian $\nabla^2 \chi^2$ for the global model $\Space$.
  237     For each residue $i$ a different matrix $\nabla^2 \chi^2_i$ is constructed.
  238     The first element of the matrix represented by the two symbols $\partial \Diffset$ (the red block) is the sub-matrix of chi-squared second partial derivatives with respect to the diffusion tensor parameters $\Diffset_j$ and $\Diffset_k$.
  239     The orange blocks are the sub-matrices of chi-squared second partial derivatives with respect to the diffusion parameter $\Diffset_j$ and the model-free parameter $\Mfset_i^k$.
  240     The yellow blocks are the sub-matrices of chi-squared second partial derivatives with respect to the model-free parameters $\Mfset_i^j$ and $\Mfset_i^k$.
  241     For the residue dependent matrix $\nabla^2 \chi^2_i$ the second partial derivatives with respect to the model-free parameters $\Mfset_l^j$ and $\Mfset_l^k$ where $i \ne l$ are zero.
  242     In addition, the second partial derivatives with respect to the model-free parameters $\Mfset_i^j$ and $\Mfset_l^k$ where $i \ne l$ are also zero.
  243     These blocks of sub-matrices are left uncoloured.
  244     The complete Hessian of $\Space$ is the sum of the matrices $\nabla^2 \chi^2_i$.
  245   }
  246   \label{fig: Hessian kite}
  247 \end{figure}
  248 
  249 The construction of the Hessian for the models $\Mfset_i$, $\Localset_i$, $\Diffset$, and $\Space$ is very similar to the procedure used for the gradient.
  250 The chi-squared Hessian for the global models $\Diffset$ and $\Space$ is
  251 \begin{equation} \label{eq: spaceset Hessian}
  252     \nabla^2 \chi^2 = \sum_{i=1}^l \nabla^2 \chi^2_i.
  253 \end{equation}
  254 
  255 \noindent Figure~\ref{fig: Hessian kite} demonstrates the construction of the full Hessian for the model $\Space$.
  256 The Hessian for the model $\Diffset$ is the sum of all the red blocks.
  257 The Hessian for the model $\Localset_i$ is the combination of the single red block for residue $i$, the two orange blocks representing the sub-matrices of chi-squared second partial derivatives with respect to the diffusion parameter $\Diffset_j$ and the model-free parameter $\Mfset_i^k$, and the single yellow block for that residue.
  258 The Hessian for the model-free model $\Mfset_i$ is simply the sub-matrix for the residue $i$ coloured yellow.
  259 
  260 
  261 
  262 % The value, gradient, and Hessian dependency chain.
  263 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  264 
  265 \section{The value, gradient, and Hessian dependency chain}
  266 
  267 The dependency chain which was outlined in the model-free chapter -- that the chi-squared function is dependent on the transformed relaxation equations which are dependent on the relaxation equations which themselves are dependent on the spectral density functions -- combine with the values, gradients, and Hessians to create a complex web of dependencies.
  268 The relationship between all the values, gradients, and Hessians are outlined in Figure~\ref{fig: dependencies}.
  269 
  270 % Dependency figure.
  271 \begin{figure}
  272   \centerline{
  273     \includegraphics[
  274       width=0.8\textwidth,
  275       bb=14 14 226 110
  276     ]
  277     {images/dependencies}
  278   }
  279   \caption[$\chi^2$ dependencies of the values, gradients, and Hessians.]{
  280     Dependencies between the $\chi^2$, transformed relaxation, relaxation, and spectral density equations, gradients, and Hessians.
  281   }
  282   \label{fig: dependencies}
  283 \end{figure}
  284 
  285 
  286 
  287 % The chi-squared value, gradient, and Hessian.
  288 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  289 
  290 \section{The $\chi^2$ value, gradient, and Hessian}
  291 
  292 % The chi-squared value.
  293 \subsection{The $\chi^2$ value}
  294 
  295 The $\chi^2$ value is defined as
  296 \begin{equation} \label{eq: maths: chi-squared}
  297  \chi^2(\theta) = \sum_{i=1}^n \frac{(\Ri - \Ri(\theta))^2}{\sigma_i^2},
  298 \end{equation}
  299 
  300 \noindent where the summation index $i$ ranges over all the relaxation data of all residues used in the analysis.
  301 
  302 
  303 
  304 % The chi-squared gradient.
  305 \subsection{The $\chi^2$ gradient}
  306 
  307 The $\chi^2$ gradient in vector notation is
  308 \begin{equation}
  309  \nabla \chi^2(\theta) = 2 \sum_{i=1}^n \frac{(\Ri - \Ri(\theta))^2}{\sigma_i^2} \nabla \Ri(\theta).
  310 \end{equation}
  311 
  312 
  313 
  314 % The chi-squared Hessian.
  315 \subsection{The $\chi^2$ Hessian}
  316 
  317 The $\chi^2$ Hessian in vector notation is
  318 \begin{equation}
  319  \nabla^2 \chi^2(\theta) = 2 \sum_{i=1}^n \frac{1}{\sigma_i^2} \left(\nabla \Ri(\theta) \cdot \nabla \Ri(\theta)^T - (\Ri - \Ri(\theta)) \nabla^2 \Ri(\theta) \right).
  320 \end{equation}
  321 
  322 
  323 
  324 % The Ri values, gradients, and Hessians.
  325 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  326 
  327 \newpage
  328 \section{The $\Ri(\theta)$ values, gradients, and Hessians}
  329 
  330 
  331 % The Ri values.
  332 \subsection{The $\Ri(\theta)$ values}
  333 
  334 The $\Ri(\theta)$ values are given by
  335 \begin{subequations}
  336 \begin{align}
  337     \Rone(\theta) & = \Rone'(\theta), \label{eq: Ri trans: R1} \\
  338     \Rtwo(\theta) & = \Rtwo'(\theta), \label{eq: Ri trans: R2} \\
  339     \mathrm{NOE}(\theta) & = 1 + \frac{\gH}{\gX} \frac{\crossrate(\theta)}{\Rone(\theta)}. \label{eq: Ri trans: NOE}
  340 \end{align}
  341 \end{subequations}
  342 
  343 
  344 % The Ri gradients.
  345 \subsection{The $\Ri(\theta)$ gradients}
  346 
  347 The $\Ri(\theta)$ gradients in vector notation are
  348 \begin{subequations}
  349 \begin{align}
  350     \nabla \Rone(\theta) & = \nabla \Rone'(\theta), \label{eq: Ri trans: dR1} \\
  351     \nabla \Rtwo(\theta) & = \nabla \Rtwo'(\theta), \label{eq: Ri trans: dR2} \\
  352     \nabla \mathrm{NOE}(\theta) & = \frac{\gH}{\gX} \frac{1}{\Rone(\theta)^2} \Big(
  353         \Rone(\theta) \nabla \crossrate(\theta) - \crossrate(\theta) \nabla \Rone(\theta)
  354     \Big). \label{eq: Ri trans: dNOE}
  355 \end{align}
  356 \end{subequations}
  357 
  358 
  359 % The Ri Hessians.
  360 \subsection{The $\Ri(\theta)$ Hessians}
  361 
  362 The $\Ri(\theta)$ Hessians in vector notation are
  363 \begin{subequations}
  364 \begin{align}
  365     \nabla^2 \Rone(\theta) & = \nabla^2 \Rone'(\theta), \label{eq: Ri trans: d2R1} \\
  366     \nabla^2 \Rtwo(\theta) & = \nabla^2 \Rtwo'(\theta), \label{eq: Ri trans: d2R2} \\
  367     \nabla^2 \mathrm{NOE}(\theta) & = \frac{\gH}{\gX} \frac{1}{\Rone(\theta)^3} \bigg[
  368         \crossrate(\theta) \Big( 2 \nabla \Rone(\theta) \cdot \nabla \Rone(\theta)^T - \Rone(\theta) \nabla^2 \Rone(\theta) \Big) \nonumber\\
  369         & \quad - \Rone(\theta) \Big( \nabla \crossrate(\theta) \cdot \nabla \Rone(\theta)^T - \Rone(\theta) \nabla^2 \crossrate(\theta) \Big)
  370     \bigg]. \label{eq: Ri trans: d2NOE}
  371 \end{align}
  372 \end{subequations}
  373 
  374 
  375 
  376 % Ri' values, gradients, and Hessians.
  377 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  378 
  379 \newpage
  380 \section{$\Ri'(\theta)$ values, gradients, and Hessians}
  381 
  382 The partial and second partial derivatives of the relaxation equations of the set R$'(\theta)$ are different for each parameter of the vector $\theta$.
  383 The vector representation of the gradient $\nabla \textrm{R}_i'(\theta)$ and the matrix representation of the Hessian $\nabla^2 \textrm{R}_i'(\theta)$ can be reconstructed from the individual elements presented in the next section.
  384 
  385 
  386 % Components.
  387 %~~~~~~~~~~~~
  388 
  389 \subsection{Components of the $\Ri'(\theta)$ equations}
  390 
  391 To simplify the calculations of the gradients and Hessians the $\Ri'(\theta)$ equations have been broken down into a number of components.
  392 These include the dipolar and CSA constants as well as the dipolar and CSA spectral density terms for each of the three transformed relaxation data types \{$\Rone$, $\Rtwo$, $\crossrate$\}.
  393 The segregation of these components simplifies the maths as many partial derivatives of the components are zero.
  394 
  395 
  396 % Dipolar comps.
  397 \subsubsection{Dipolar constant}
  398 
  399 The dipolar constant is defined as
  400 \begin{equation}
  401     d = \frac{1}{4} \left(\frac{\mu_0}{4\pi}\right)^2 \frac{\left( \gH \gX \hbar \right)^2}{<r^6>}. \label{eq: Ri': d}
  402 \end{equation}
  403 
  404 \noindent This component of the relaxation equations is independent of the parameter of the spectral density function $\theta_j$, the chemical exchange parameter $\rho_{ex}$, and the CSA parameter $\Delta\sigma$.
  405 Therefore the partial and second partial derivatives with respect to these parameters is zero.
  406 Only the derivative with respect to the bond length $r$ is non-zero being
  407 \begin{equation}
  408     d' \equiv \frac{\mathrm{d} d}{\mathrm{d} r} = - \frac{3}{2} \left(\frac{\mu_0}{4\pi}\right)^2 \frac{\left( \gH \gX \hbar \right)^2}{<r^7>}. \label{eq: Ri': d'}
  409 \end{equation}
  410 
  411 \noindent The second derivative with respect to the bond length is
  412 \begin{equation}
  413     d'' \equiv \frac{\mathrm{d}^2 d}{\mathrm{d} r^2} = \frac{21}{2} \left(\frac{\mu_0}{4\pi}\right)^2 \frac{\left( \gH \gX \hbar \right)^2}{<r^8>}. \label{eq: Ri': d"}
  414 \end{equation}
  415 
  416 
  417 % CSA comps.
  418 \subsubsection{CSA constant}
  419 
  420 The CSA constant is defined as
  421 \begin{equation}
  422     c = \frac{\left(\omega_X \cdot \Delta\sigma \right)^2}{3}. \label{eq: Ri': c}
  423 \end{equation}
  424 
  425 \noindent The partial derivative of this component with respect to all parameters but the CSA parameter $\Delta\sigma$ is zero.
  426 This derivative is
  427 \begin{equation}
  428     c' \equiv \frac{\mathrm{d} c}{\mathrm{d} \Delta\sigma} = \frac{2 \omega_X^2 \cdot \Delta\sigma}{3}. \label{eq: Ri': c'}
  429 \end{equation}
  430 
  431 \noindent The CSA constant second derivative with respect to $\Delta\sigma$ is
  432 \begin{equation}
  433     c'' \equiv \frac{\mathrm{d}^2 c}{\mathrm{d} \Delta\sigma^2} = \frac{2 \omega_X^2}{3}. \label{eq: Ri': c"}
  434 \end{equation}
  435 
  436 
  437 % Rex comps.
  438 \subsubsection{$R_{ex}$ constant}
  439 
  440 The $R_{ex}$ constant is defined as
  441 \begin{equation}
  442     R_{ex} = \rho_{ex} (2 \pi \omega_H)^2 . \label{eq: Ri': Rex}
  443 \end{equation}
  444 
  445 \noindent The partial derivative of this component with respect to all parameters but the chemical exchange parameter $\rho_{ex}$ is zero.
  446 This derivative is
  447 \begin{equation}
  448     R_{ex}' \equiv \frac{\mathrm{d} R_{ex}}{\mathrm{d} \rho_{ex}} = (2 \pi \omega_H)^2. \label{eq: Ri': Rex'}
  449 \end{equation}
  450 
  451 \noindent The $R_{ex}$ constant second derivative with respect to $\rho_{ex}$ is
  452 \begin{equation}
  453     R_{ex}'' \equiv \frac{\mathrm{d}^2 R_{ex}}{\mathrm{d} \rho_{ex}^2} = 0. \label{eq: Ri': Rex"}
  454 \end{equation}
  455 
  456 
  457 % R1 dip Spectral density terms.
  458 \subsubsection{Spectral density terms of the $\Rone$ dipolar component}
  459 
  460 For the dipolar component of the $\Rone$ equation~\eqref{eq: R1} on page~\pageref{eq: R1} the spectral density terms are
  461 \begin{equation}
  462     J_d^{\Rone} = J(\omega_H - \omega_X) + 3J(\omega_X) + 6J(\omega_H + \omega_X).  \label{eq: J terms: JR1d}
  463 \end{equation}
  464 
  465 \noindent The partial derivative of these terms with respect to the spectral density function parameter $\theta_j$ is
  466 \begin{equation}
  467     {J_d^{\Rone}}' \equiv \frac{\partial J_d^{\Rone}}{\partial \theta_j}
  468         = \frac{\partial J(\omega_H - \omega_X)}{\partial \theta_j}
  469         + 3 \frac{\partial J(\omega_X)}{\partial \theta_j}
  470         + 6 \frac{\partial J(\omega_H + \omega_X)}{\partial \theta_j}.  \label{eq: J terms: JR1d'}
  471 \end{equation}
  472 
  473 \noindent The second partial derivative with respect to the spectral density function parameters $\theta_j$ and $\theta_k$ is
  474 \begin{equation}
  475     {J_d^{\Rone}}'' \equiv \frac{\partial^2 J_d^{\Rone}}{\partial \theta_j \cdot \partial \theta_k}
  476         = \frac{\partial^2 J(\omega_H - \omega_X)}{\partial \theta_j \cdot \partial \theta_k}
  477         + 3 \frac{\partial^2 J(\omega_X)}{\partial \theta_j \cdot \partial \theta_k}
  478         + 6 \frac{\partial^2 J(\omega_H + \omega_X)}{\partial \theta_j \cdot \partial \theta_k}.  \label{eq: J terms: JR1d"}
  479 \end{equation}
  480 
  481 
  482 % R1 CSA Spectral density terms.
  483 \subsubsection{Spectral density terms of the $\Rone$ CSA component}
  484 
  485 For the CSA component of the $\Rone$ equation~\eqref{eq: R1} on page~\pageref{eq: R1} the spectral density terms are
  486 \begin{equation}
  487     J_c^{\Rone} = J(\omega_X).  \label{eq: J terms: JR1c}
  488 \end{equation}
  489 
  490 \noindent The partial derivative of these terms with respect to the spectral density function parameter $\theta_j$ is
  491 \begin{equation}
  492     {J_c^{\Rone}}' \equiv \frac{\partial J_c^{\Rone}}{\partial \theta_j}
  493         = \frac{\partial J(\omega_X)}{\partial \theta_j}.  \label{eq: J terms: JR1c'}
  494 \end{equation}
  495 
  496 \noindent The second partial derivative with respect to the spectral density function parameters $\theta_j$ and $\theta_k$ is
  497 \begin{equation}
  498     {J_c^{\Rone}}'' \equiv \frac{\partial^2 J_c^{\Rone}}{\partial \theta_j . \partial \theta_k}
  499         = \frac{\partial^2 J(\omega_X)}{\partial \theta_j \cdot \partial \theta_k}.  \label{eq: J terms: JR1c"}
  500 \end{equation}
  501 
  502 
  503 % R2 dip Spectral density terms.
  504 \subsubsection{Spectral density terms of the $\Rtwo$ dipolar component}
  505 
  506 For the dipolar component of the $\Rtwo$ equation~\eqref{eq: R2} on page~\pageref{eq: R2} the spectral density terms are
  507 \begin{equation}
  508     J_d^{\Rtwo} = 4J(0) + J(\omega_H - \omega_X) + 3J(\omega_X) + 6J(\omega_H) + 6J(\omega_H + \omega_X).  \label{eq: J terms: JR2d}
  509 \end{equation}
  510 
  511 \noindent The partial derivative of these terms with respect to the spectral density function parameter $\theta_j$ is
  512 \begin{equation}
  513     {J_d^{\Rtwo}}' \equiv \frac{\partial J_d^{\Rtwo}}{\partial \theta_j}
  514         = 4 \frac{\partial J(0)}{\partial \theta_j}
  515         + \frac{\partial J(\omega_H - \omega_X)}{\partial \theta_j}
  516         + 3 \frac{\partial J(\omega_X)}{\partial \theta_j}
  517         + 6 \frac{\partial J(\omega_H)}{\partial \theta_j}
  518         + 6 \frac{\partial J(\omega_H + \omega_X)}{\partial \theta_j}.  \label{eq: J terms: JR2d'}
  519 \end{equation}
  520 
  521 \noindent The second partial derivative with respect to the spectral density function parameters $\theta_j$ and $\theta_k$ is
  522 \begin{multline}
  523     {J_d^{\Rtwo}}'' \equiv \frac{\partial^2 J_d^{\Rtwo}}{\partial \theta_j \cdot \partial \theta_k}
  524         = 4 \frac{\partial^2 J(0)}{\partial \theta_j \cdot \partial \theta_k}
  525         + \frac{\partial^2 J(\omega_H - \omega_X)}{\partial \theta_j \cdot \partial \theta_k}
  526         + 3 \frac{\partial^2 J(\omega_X)}{\partial \theta_j \cdot \partial \theta_k} \\
  527         + 6 \frac{\partial^2 J(\omega_H)}{\partial \theta_j \cdot \partial \theta_k}
  528         + 6 \frac{\partial^2 J(\omega_H + \omega_X)}{\partial \theta_j \cdot \partial \theta_k}.  \label{eq: J terms: JR2d"}
  529 \end{multline}
  530 
  531 
  532 % R2 CSA Spectral density terms.
  533 \subsubsection{Spectral density terms of the $\Rtwo$ CSA component}
  534 
  535 For the CSA component of the $\Rtwo$ equation~\eqref{eq: R2} on page~\pageref{eq: R2} the spectral density terms are
  536 \begin{equation}
  537     J_c^{\Rtwo} = 4J(0) + 3J(\omega_X).  \label{eq: J terms: JR2c}
  538 \end{equation}
  539 
  540 \noindent The partial derivative of these terms with respect to the spectral density function parameter $\theta_j$ is
  541 \begin{equation}
  542     {J_c^{\Rtwo}}' \equiv \frac{\partial J_c^{\Rtwo}}{\partial \theta_j}
  543         = 4 \frac{\partial J(0)}{\partial \theta_j}
  544         + 3 \frac{\partial J(\omega_X)}{\partial \theta_j}.  \label{eq: J terms: JR2c'}
  545 \end{equation}
  546 
  547 \noindent The second partial derivative with respect to the spectral density function parameters $\theta_j$ and $\theta_k$ is
  548 \begin{equation}
  549     {J_c^{\Rtwo}}'' \equiv \frac{\partial^2 J_c^{\Rtwo}}{\partial \theta_j \cdot \partial \theta_k}
  550         = 4 \frac{\partial^2 J(0)}{\partial \theta_j \cdot \partial \theta_k}
  551         + 3 \frac{\partial^2 J(\omega_X)}{\partial \theta_j \cdot \partial \theta_k}.  \label{eq: J terms: JR2c"}
  552 \end{equation}
  553 
  554 
  555 % Sigma_NOE dip Spectral density terms.
  556 \subsubsection{Spectral density terms of the $\crossrate$ dipolar component}
  557 
  558 For the dipolar component of the $\crossrate$ equation~\eqref{eq: sigma_NOE} on page~\pageref{eq: sigma_NOE} the spectral density terms are
  559 \begin{equation}
  560     J_d^{\crossrate} = 6J(\omega_H + \omega_X) - J(\omega_H - \omega_X).  \label{eq: J terms: JsigmaNOEd}
  561 \end{equation}
  562 
  563 \noindent The partial derivative of these terms with respect to the spectral density function parameter $\theta_j$ is
  564 \begin{equation}
  565     {J_d^{\crossrate}}' \equiv \frac{\partial J_d^{\crossrate}}{\partial \theta_j}
  566         = 6 \frac{\partial J(\omega_H + \omega_X)}{\partial \theta_j}
  567           - \frac{\partial J(\omega_H - \omega_X)}{\partial \theta_j}.  \label{eq: J terms: JsigmaNOEd'}
  568 \end{equation}
  569 
  570 \noindent The second partial derivative with respect to the spectral density function parameters $\theta_j$ and $\theta_k$ is
  571 \begin{equation}
  572     {J_d^{\crossrate}}'' \equiv \frac{\partial^2 J_d^{\crossrate}}{\partial \theta_j \cdot \partial \theta_k}
  573         = 6 \frac{\partial^2 J(\omega_H + \omega_X)}{\partial \theta_j \cdot \partial \theta_k}
  574           - \frac{\partial^2 J(\omega_H - \omega_X)}{\partial \theta_j \cdot \partial \theta_k}.  \label{eq: J terms: JsigmaNOEd"}
  575 \end{equation}
  576 
  577 
  578 
  579 % Ri' values.
  580 %~~~~~~~~~~~~
  581 
  582 \subsection{$\Ri'(\theta)$ values}
  583 
  584 Using the components of the relaxation equations defined above the three relaxation equations can be re-expressed as
  585 \begin{subequations}
  586 \begin{align}
  587     \Rone(\theta) & = d J_d^{\Rone} + c J_c^{\Rone},                          \label{eq: Ri': R1} \\
  588     \Rtwo(\theta) & = \frac{d}{2} J_d^{\Rtwo} + \frac{c}{6} J_c^{\Rtwo},      \label{eq: Ri': R2} \\
  589     \crossrate(\theta) & = d J_d^{\crossrate}.                          \label{eq: Ri': sigmaNOE}
  590 \end{align}
  591 \end{subequations}
  592 
  593 
  594 
  595 % Ri' gradients.
  596 %~~~~~~~~~~~~~~~
  597 
  598 \subsection{$\Ri'(\theta)$ gradients}
  599 
  600 A different partial derivative exists for the spectral density function parameter $\theta_j$, the chemical exchange parameter $\rho_{ex}$, CSA parameter $\Delta\sigma$, and bond length parameter $r$.
  601 In model-free analysis the spectral density parameters include both the parameters of the diffusion tensor and the parameters of the various model-free models.
  602 
  603 
  604 % Spectral density function parameter.
  605 \subsubsection{$\theta_j$ partial derivative}
  606 
  607 The partial derivatives of the relaxation equations with respect to the spectral density function parameter $\theta_j$ are
  608 \begin{subequations}
  609 \begin{align}
  610     \frac{\partial \Rone(\theta)}{\partial \theta_j} &= d {J_d^{\Rone}}' + c {J_c^{\Rone}}',                      \label{eq: Ri': dR1/dmf} \\
  611     \frac{\partial \Rtwo(\theta)}{\partial \theta_j} &= \frac{d}{2} {J_d^{\Rtwo}}' + \frac{c}{6} {J_c^{\Rtwo}}',  \label{eq: Ri': dR2/dmf} \\
  612     \frac{\partial \crossrate(\theta)}{\partial \theta_j} &= d {J_d^{\crossrate}}'.                         \label{eq: Ri': dsigmaNOE/dmf}
  613 \end{align}
  614 \end{subequations}
  615 
  616 
  617 % Chemical exchange parameter.
  618 \subsubsection{$\rho_{ex}$ partial derivative}
  619 
  620 The partial derivatives of the relaxation equations with respect to the chemical exchange parameter $\rho_{ex}$ are
  621 \begin{subequations}
  622 \begin{align}
  623     \frac{\partial \Rone(\theta)}{\partial \rho_{ex}} &= 0,          \label{eq: Ri': dR1/dRex} \\
  624     \frac{\partial \Rtwo(\theta)}{\partial \rho_{ex}} &= (2 \pi \omega_H)^2,          \label{eq: Ri': dR2/dRex} \\
  625     \frac{\partial \crossrate(\theta)}{\partial \rho_{ex}} &= 0.   \label{eq: Ri': dsigmaNOE/dRex}
  626 \end{align}
  627 \end{subequations}
  628 
  629 
  630 % CSA parameter.
  631 \subsubsection{$\Delta\sigma$ partial derivative}
  632 
  633 The partial derivatives of the relaxation equations with respect to the CSA parameter $\Delta\sigma$ are
  634 \begin{subequations}
  635 \begin{align}
  636     \frac{\partial \Rone(\theta)}{\partial \Delta\sigma} &= c' J_c^{\Rone},             \label{eq: Ri': dR1/dCSA} \\
  637     \frac{\partial \Rtwo(\theta)}{\partial \Delta\sigma} &= \frac{c'}{6} J_c^{\Rtwo},   \label{eq: Ri': dR2/dCSA} \\
  638     \frac{\partial \crossrate(\theta)}{\partial \Delta\sigma} &= 0.                 \label{eq: Ri': dsigmaNOE/dCSA}
  639 \end{align}
  640 \end{subequations}
  641 
  642 
  643 % Bond length parameter.
  644 \subsubsection{$r$ partial derivative}
  645 
  646 The partial derivatives of the relaxation equations with respect to the bond length parameter $r$ are
  647 \begin{subequations}
  648 \begin{align}
  649     \frac{\partial \Rone(\theta)}{\partial r} &= d' J_d^{\Rone},                \label{eq: Ri': dR1/dr} \\
  650     \frac{\partial \Rtwo(\theta)}{\partial r} &= \frac{d'}{2} J_d^{\Rtwo},      \label{eq: Ri': dR2/dr} \\
  651     \frac{\partial \crossrate(\theta)}{\partial r} &= d' J_d^{\crossrate}.  \label{eq: Ri': dsigmaNOE/dr}
  652 \end{align}
  653 \end{subequations}
  654 
  655 
  656 
  657 % Ri' Hessians.
  658 %~~~~~~~~~~~~~~
  659 
  660 \subsection{$\Ri'(\theta)$ Hessians}
  661 
  662 Again different second partial derivatives with respect to the spectral density function parameters $\theta_j$ and $\theta_k$, the chemical exchange parameter $\rho_{ex}$, CSA parameter $\Delta\sigma$, and bond length parameter $r$.
  663 These second partial derivatives are the components of the $\Ri'(\theta)$ Hessian matrices.
  664 
  665 
  666 % Spectral density function parameter -- Spectral density function parameter.
  667 \subsubsection{$\theta_j$ -- $\theta_k$ partial derivative}
  668 
  669 The second partial derivatives of the relaxation equations with respect to the spectral density function parameters $\theta_j$ and $\theta_k$ are
  670 \begin{subequations}
  671 \begin{align}
  672     \frac{\partial^2 \Rone(\theta)}{\partial \theta_j \cdot \partial \theta_k} &= d {J_d^{\Rone}}'' + c {J_c^{\Rone}}'',                      \label{eq: Ri': d2R1/dmfj.dmfk} \\
  673     \frac{\partial^2 \Rtwo(\theta)}{\partial \theta_j \cdot \partial \theta_k} &= \frac{d}{2} {J_d^{\Rtwo}}'' + \frac{c}{6} {J_c^{\Rtwo}}'',  \label{eq: Ri': d2R2/dmfj.dmfk} \\
  674     \frac{\partial^2 \crossrate(\theta)}{\partial \theta_j \cdot \partial \theta_k} &= d {J_d^{\crossrate}}''.                          \label{eq: Ri': d2sigmaNOE/dmfj.dmfk}
  675 \end{align}
  676 \end{subequations}
  677 
  678 
  679 % Spectral density function parameter -- Chemical exchange parameter.
  680 \subsubsection{$\theta_j$ -- $\rho_{ex}$ partial derivative}
  681 
  682 The second partial derivatives of the relaxation equations with respect to the spectral density function parameter $\theta_j$ and the chemical exchange parameter $\rho_{ex}$ are
  683 \begin{subequations}
  684 \begin{align}
  685     \frac{\partial^2 \Rone(\theta)}{\partial \theta_j \cdot \partial \rho_{ex}} &= 0,        \label{eq: Ri': d2R1/dmfj.dRex} \\
  686     \frac{\partial^2 \Rtwo(\theta)}{\partial \theta_j \cdot \partial \rho_{ex}} &= 0,        \label{eq: Ri': d2R2/dmfj.dRex} \\
  687     \frac{\partial^2 \crossrate(\theta)}{\partial \theta_j \cdot \partial \rho_{ex}} &= 0. \label{eq: Ri': d2sigmaNOE/dmfj.dRex}
  688 \end{align}
  689 \end{subequations}
  690 
  691 
  692 % Spectral density function parameter -- CSA parameter.
  693 \subsubsection{$\theta_j$ -- $\Delta\sigma$ partial derivative}
  694 
  695 The second partial derivatives of the relaxation equations with respect to the spectral density function parameter $\theta_j$ and the CSA parameter $\Delta\sigma$ are
  696 \begin{subequations}
  697 \begin{align}
  698     \frac{\partial^2 \Rone(\theta)}{\partial \theta_j \cdot \partial \Delta\sigma} &= c' {J_c^{\Rone}}',            \label{eq: Ri': d2R1/dmfj.dCSA} \\
  699     \frac{\partial^2 \Rtwo(\theta)}{\partial \theta_j \cdot \partial \Delta\sigma} &= \frac{c'}{6} {J_c^{\Rtwo}}',  \label{eq: Ri': d2R2/dmfj.dCSA} \\
  700     \frac{\partial^2 \crossrate(\theta)}{\partial \theta_j \cdot \partial \Delta\sigma} &= 0.                   \label{eq: Ri': d2sigmaNOE/dmfj.dCSA}
  701 \end{align}
  702 \end{subequations}
  703 
  704 
  705 % Spectral density function parameter -- Bond length parameter.
  706 \subsubsection{$\theta_j$ -- $r$ partial derivative}
  707 
  708 The second partial derivatives of the relaxation equations with respect to the spectral density function parameter $\theta_j$ and the bond length parameter $r$ are
  709 \begin{subequations}
  710 \begin{align}
  711     \frac{\partial^2 \Rone(\theta)}{\partial \theta_j \cdot \partial r} &= d' {J_d^{\Rone}}',               \label{eq: Ri': d2R1/dmfj.dr} \\
  712     \frac{\partial^2 \Rtwo(\theta)}{\partial \theta_j \cdot \partial r} &= \frac{d'}{2} {J_d^{\Rtwo}}',     \label{eq: Ri': d2R2/dmfj.dr} \\
  713     \frac{\partial^2 \crossrate(\theta)}{\partial \theta_j \cdot \partial r} &= d' {J_d^{\crossrate}}'. \label{eq: Ri': d2sigmaNOE/dmfj.dr}
  714 \end{align}
  715 \end{subequations}
  716 
  717 
  718 % Chemical exchange parameter -- Chemical exchange parameter.
  719 \subsubsection{$\rho_{ex}$ -- $\rho_{ex}$ partial derivative}
  720 
  721 The second partial derivatives of the relaxation equations with respect to the chemical exchange parameter $\rho_{ex}$ twice are
  722 \begin{subequations}
  723 \begin{align}
  724     \frac{\partial^2 \Rone(\theta)}{{\partial \rho_{ex}}^2} &= 0,        \label{eq: Ri': d2R1/dRex2} \\
  725     \frac{\partial^2 \Rtwo(\theta)}{{\partial \rho_{ex}}^2} &= 0,        \label{eq: Ri': d2R2/dRex2} \\
  726     \frac{\partial^2 \crossrate(\theta)}{{\partial \rho_{ex}}^2} &= 0. \label{eq: Ri': d2sigmaNOE/dRex2}
  727 \end{align}
  728 \end{subequations}
  729 
  730 
  731 % Chemical exchange parameter -- CSA parameter.
  732 \subsubsection{$\rho_{ex}$ -- $\Delta\sigma$ partial derivative}
  733 
  734 The second partial derivatives of the relaxation equations with respect to the chemical exchange parameter $\rho_{ex}$ and the CSA parameter $\Delta\sigma$ are
  735 \begin{subequations}
  736 \begin{align}
  737     \frac{\partial^2 \Rone(\theta)}{\partial \rho_{ex} \cdot \partial \Delta\sigma} &= 0,        \label{eq: Ri': d2R1/dRex.dCSA} \\
  738     \frac{\partial^2 \Rtwo(\theta)}{\partial \rho_{ex} \cdot \partial \Delta\sigma} &= 0,        \label{eq: Ri': d2R2/dRex.dCSA} \\
  739     \frac{\partial^2 \crossrate(\theta)}{\partial \rho_{ex} \cdot \partial \Delta\sigma} &= 0. \label{eq: Ri': d2sigmaNOE/dRex.dCSA}
  740 \end{align}
  741 \end{subequations}
  742 
  743 
  744 % Chemical exchange parameter -- Bond length parameter.
  745 \subsubsection{$\rho_{ex}$ -- $r$ partial derivative}
  746 
  747 The second partial derivatives of the relaxation equations with respect to the chemical exchange parameter $\rho_{ex}$ and the bond length parameter $r$ are
  748 \begin{subequations}
  749 \begin{align}
  750     \frac{\partial^2 \Rone(\theta)}{\partial \rho_{ex} \cdot \partial r} &= 0,           \label{eq: Ri': d2R1/dRex.dr} \\
  751     \frac{\partial^2 \Rtwo(\theta)}{\partial \rho_{ex} \cdot \partial r} &= 0,           \label{eq: Ri': d2R2/dRex.dr} \\
  752     \frac{\partial^2 \crossrate(\theta)}{\partial \rho_{ex} \cdot \partial r} &= 0.    \label{eq: Ri': d2sigmaNOE/dRex.dr}
  753 \end{align}
  754 \end{subequations}
  755 
  756 
  757 % CSA parameter -- CSA parameter.
  758 \subsubsection{$\Delta\sigma$ -- $\Delta\sigma$ partial derivative}
  759 
  760 The second partial derivatives of the relaxation equations with respect to the CSA parameter $\Delta\sigma$ twice are
  761 \begin{subequations}
  762 \begin{align}
  763     \frac{\partial^2 \Rone(\theta)}{{\partial \Delta\sigma}^2} &= c'' J_c^{\Rone},              \label{eq: Ri': d2R1/dCSA2} \\
  764     \frac{\partial^2 \Rtwo(\theta)}{{\partial \Delta\sigma}^2} &= \frac{c''}{6} J_c^{\Rtwo},    \label{eq: Ri': d2R2/dCSA2} \\
  765     \frac{\partial^2 \crossrate(\theta)}{{\partial \Delta\sigma}^2} &= 0.                   \label{eq: Ri': d2sigmaNOE/dCSA2}
  766 \end{align}
  767 \end{subequations}
  768 
  769 
  770 % CSA parameter -- Bond length parameter.
  771 \subsubsection{$\Delta\sigma$ -- $r$ partial derivative}
  772 
  773 The second partial derivatives of the relaxation equations with respect to the CSA parameter $\Delta\sigma$ and the bond length parameter $r$ are
  774 \begin{subequations}
  775 \begin{align}
  776     \frac{\partial^2 \Rone(\theta)}{\partial \Delta\sigma \cdot \partial r} &= 0,         \label{eq: Ri': d2R1/dCSA.dr} \\
  777     \frac{\partial^2 \Rtwo(\theta)}{\partial \Delta\sigma \cdot \partial r} &= 0,         \label{eq: Ri': d2R2/dCSA.dr} \\
  778     \frac{\partial^2 \crossrate(\theta)}{\partial \Delta\sigma \cdot \partial r} &= 0.  \label{eq: Ri': d2sigmaNOE/dCSA.dr}
  779 \end{align}
  780 \end{subequations}
  781 
  782 
  783 % Bond length parameter -- Bond length parameter.
  784 \subsubsection{$r$ -- $r$ partial derivative}
  785 
  786 The second partial derivatives of the relaxation equations with respect to the bond length parameter $r$ twice are
  787 \begin{subequations}
  788 \begin{align}
  789     \frac{\partial^2 \Rone(\theta)}{{\partial r}^2} &= d'' J_d^{\Rone},                 \label{eq: Ri': d2R1/dr2} \\
  790     \frac{\partial^2 \Rtwo(\theta)}{{\partial r}^2} &= \frac{d''}{2} J_d^{\Rtwo},       \label{eq: Ri': d2R2/dr2} \\
  791     \frac{\partial^2 \crossrate(\theta)}{{\partial r}^2} &= d'' J_d^{\crossrate}.   \label{eq: Ri': d2sigmaNOE/dr2}
  792 \end{align}
  793 \end{subequations}
  794 
  795 
  796 
  797 
  798 % Model-free analysis.
  799 %%%%%%%%%%%%%%%%%%%%%%
  800 
  801 \newpage
  802 \section{Optimisation equations for the model-free analysis}
  803 
  804 
  805 
  806 % The model-free equations.
  807 %~~~~~~~~~~~~~~~~~~~~~~~~~~
  808 
  809 \subsection{The model-free equations}
  810 
  811 In the original model-free analysis of \cite{LipariSzabo82a} the correlation function $C(\tau)$ of the XH bond vector is approximated by decoupling the internal fluctuations of the bond vector $C_{\mathrm{I}}(\tau)$ from the correlation function of the overall Brownian rotational diffusion $C_{\mathrm{O}}(\tau)$ by the equation
  812 \begin{equation}
  813     C(\tau) = C_{\mathrm{O}}(\tau) \cdot C_{\mathrm{I}}(\tau).
  814 \end{equation}
  815 
  816 \noindent The overall correlation functions of the diffusion of a sphere, spheroid, and ellipsoid are presented respectively in section~\ref{ellipsoid equation} on page~\pageref{ellipsoid equation}, section~\ref{spheroid equation} on page~\pageref{spheroid equation}, and section~\ref{sphere equation} on page~\pageref{sphere equation}.
  817 These three different equations can be combined into one generic correlation function which is independent of the type of diffusion.
  818 This generic correlation function is
  819 \begin{equation}
  820     C_{\mathrm{O}}(\tau) = \frac{1}{5} \sum_{i=-k}^k c_i \cdot e^{-\tau/\tau_i},
  821 \end{equation}
  822 
  823 \noindent where $c_i$ are the weights and $\tau_i$ are correlation times of the exponential terms.
  824 In the original model-free analysis of \citet{LipariSzabo82a,LipariSzabo82b} the internal motions are modelled by the correlation function
  825 \begin{equation}
  826     C_{\mathrm{I}}(\tau) = S^2 + (1 - S^2) e^{-\tau / \tau_e},
  827 \end{equation}
  828 
  829 \noindent where $S^2$ is the generalised Lipari and Szabo order parameter which is related to the amplitude of the motion and $\tau_e$ is the effective correlation time which is an indicator of the timescale of the motion, albeit being dependent on the value of the order parameter.
  830 The order parameter ranges from one for complete rigidity to zero for unrestricted motions.
  831 Model-free theory was extended by \citet{Clore90a} to include motions on two timescales by the correlation function
  832 \begin{equation}
  833     C_{\mathrm{I}}(\tau) = S^2 + (1 - S^2_f) e^{-\tau / \tau_f} + (S^2_f - S^2) e^{-\tau / \tau_s},
  834 \end{equation}
  835 
  836 \noindent where the faster of the motions is defined by the order parameter $S^2_f$ and the correlation time $\tau_f$, the slower by the parameters $S^2_s$ and $\tau_s$, and the two order parameter are related by the equation $S^2 = S^2_f \cdot S^2_s$.
  837 
  838 The relaxation equations of \citet{Abragam61} are composed of a sum of power spectral density functions $J(\omega)$ at five frequencies.
  839 The spectral density function is related to the correlation function as the two are a Fourier pair.
  840 Applying the Fourier transform to the correlation function composed of the generic diffusion equation and the original model-free correlation function results in the equation
  841 \begin{equation} \label{eq: maths: J(w) model-free generic}
  842     J(\omega) = \frac{2}{5} \sum_{i=-k}^k c_i \cdot \tau_i \Bigg(
  843         \frac{S^2}{1 + (\omega \tau_i)^2}
  844         + \frac{(1 - S^2)(\tau_e + \tau_i)\tau_e}{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}
  845     \Bigg).
  846 \end{equation}
  847 
  848 The Fourier transform using the extended model-free correlation function is
  849 \begin{equation} \label{eq: maths: J(w) model-free ext generic}
  850     J(\omega) = \frac{2}{5} \sum_{i=-k}^k c_i \cdot \tau_i \Bigg(
  851         \frac{S^2}{1 + (\omega \tau_i)^2}
  852         + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
  853         + \frac{(S^2_f - S^2)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
  854     \Bigg).
  855 \end{equation}
  856 
  857 
  858 
  859 % The original model-free gradient.
  860 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  861 
  862 \subsection{The original model-free gradient}
  863 
  864 The model-free gradient of the original spectral density function~\eqref{eq: maths: J(w) model-free generic} is the vector of partial derivatives of the function with respect to the geometric parameter $\Diffgeoset_i$, the orientational parameter $\Difforiset_i$, the order parameter $S^2$, and the internal correlation time $\tau_e$.
  865 The positions in the vector correspond to the model parameters which are being optimised.
  866 
  867 
  868 
  869 % Gj partial derivative.
  870 \subsubsection{$\Diffgeoset_j$ partial derivative}
  871 
  872 The partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the geometric parameter $\Diffgeoset_j$ is
  873 \begin{multline}
  874     \frac{\partial J(\omega)}{\partial \Diffgeoset_j} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
  875         c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \Bigg(
  876             S^2 \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2}
  877             + (1 - S^2) \tau_e^2 \frac{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}
  878         \Bigg) \\
  879         +  \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i \Bigg(
  880             \frac{S^2}{1 + (\omega \tau_i)^2}
  881             + \frac{(1 - S^2)(\tau_e + \tau_i)\tau_e}{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}
  882         \Bigg)
  883     \Bigg).
  884 \end{multline}
  885 
  886 
  887 
  888 % Oj partial derivative.
  889 \subsubsection{$\Difforiset_j$ partial derivative}
  890 
  891 The partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the orientational parameter $\Difforiset_j$ is
  892 \begin{equation}
  893     \frac{\partial J(\omega)}{\partial \Difforiset_j} = \frac{2}{5} \sum_{i=-k}^k \frac{\partial c_i}{\partial \Difforiset_j} \tau_i \Bigg(
  894         \frac{S^2}{1 + (\omega \tau_i)^2}
  895         + \frac{(1 - S^2)(\tau_e + \tau_i)\tau_e}{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}
  896     \Bigg).
  897 \end{equation}
  898 
  899 
  900 
  901 % S2 partial derivative.
  902 \subsubsection{$S^2$ partial derivative}
  903 
  904 The partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the order parameter $S^2$ is
  905 \begin{equation}
  906     \frac{\partial J(\omega)}{\partial S^2} = \frac{2}{5} \sum_{i=-k}^k c_i \tau_i \Bigg(
  907         \frac{1}{1 + (\omega \tau_i)^2}
  908         - \frac{(\tau_e + \tau_i)\tau_e}{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}
  909     \Bigg).
  910 \end{equation}
  911 
  912 
  913 
  914 % te partial derivative.
  915 \subsubsection{$\tau_e$ partial derivative}
  916 
  917 The partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the correlation time $\tau_e$ is
  918 \begin{equation}
  919     \frac{\partial J(\omega)}{\partial \tau_e} = \frac{2}{5} (1 - S^2) \sum_{i=-k}^k c_i \tau_i^2
  920         \frac{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}.
  921 \end{equation}
  922 
  923 
  924 
  925 % The original model-free Hessian.
  926 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  927 
  928 \newpage
  929 \subsection{The original model-free Hessian}
  930 
  931 The model-free Hessian of the original spectral density function~\eqref{eq: maths: J(w) model-free generic} is the matrix of second partial derivatives.
  932 The matrix coordinates correspond to the model parameters which are being optimised.
  933 
  934 
  935 
  936 % Gj-Gk partial derivative.
  937 \subsubsection{$\Diffgeoset_j$ -- $\Diffgeoset_k$ partial derivative}
  938 
  939 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the geometric parameters $\Diffgeoset_j$ and $\Diffgeoset_k$ is
  940 \begin{multline}
  941     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \Diffgeoset_k} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
  942         -2 c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \cdot \frac{\partial \tau_i}{\partial \Diffgeoset_k} \Bigg(
  943             S^2 \omega^2 \tau_i \frac{3 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^3}  \\
  944             + (1 - S^2) \tau_e^2 \frac{(\tau_e + \tau_i)^3  +  3 \omega^2 \tau_e^3 \tau_i (\tau_e + \tau_i)  -  (\omega \tau_e)^4 \tau_i^3}
  945                 {\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^3}
  946         \Bigg) \\
  947         + \Bigg(
  948             \frac{\partial \tau_i}{\partial \Diffgeoset_j} \cdot \frac{\partial c_i}{\partial \Diffgeoset_k}
  949             + \frac{\partial \tau_i}{\partial \Diffgeoset_k} \cdot \frac{\partial c_i}{\partial \Diffgeoset_j}
  950             + c_i \frac{\partial^2 \tau_i}{\partial \Diffgeoset_j \cdot \partial \Diffgeoset_k}
  951         \Bigg)
  952         \Bigg(
  953             S^2 \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2} \\
  954             + (1 - S^2) \tau_e^2 \frac{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}
  955         \Bigg) \\
  956         + \Bigg(
  957             \frac{\partial^2 c_i}{\partial \Diffgeoset_j \cdot \partial \Diffgeoset_k} \tau_i \Bigg(
  958                 \frac{S^2}{1 + (\omega \tau_i)^2}
  959                 + \frac{(1 - S^2)(\tau_e + \tau_i)\tau_e}{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}
  960             \Bigg)
  961         \Bigg)
  962     \Bigg).
  963 \end{multline}
  964                 
  965 
  966 
  967 % Gj-Ok partial derivative.
  968 \subsubsection{$\Diffgeoset_j$ -- $\Difforiset_k$ partial derivative}
  969 
  970 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the geometric parameter $\Diffgeoset_j$ and the orientational parameter $\Difforiset_k$ is
  971 \begin{multline}
  972     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \Difforiset_k} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
  973         \frac{\partial \tau_i}{\partial \Diffgeoset_j} \frac{\partial c_i}{\partial \Difforiset_k} \Bigg(
  974             S^2 \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2}
  975             + (1 - S^2) \tau_e^2 \frac{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}
  976         \Bigg) \\
  977         +  \frac{\partial^2 c_i}{\partial \Diffgeoset_j \cdot \partial \Difforiset_k} \tau_i \Bigg(
  978             \frac{S^2}{1 + (\omega \tau_i)^2}
  979             + \frac{(1 - S^2)(\tau_e + \tau_i)\tau_e}{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}
  980         \Bigg)
  981     \Bigg).
  982 \end{multline}
  983 
  984 
  985 
  986 % Gj-S2 partial derivative.
  987 \subsubsection{$\Diffgeoset_j$ -- $S^2$ partial derivative}
  988 
  989 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the geometric parameter $\Diffgeoset_j$ and the order parameter $S^2$ is
  990 \begin{multline}
  991     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial S^2} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
  992         c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \Bigg(
  993             \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2}
  994             - \tau_e^2 \frac{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}
  995         \Bigg) \\
  996         +  \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i \Bigg(
  997             \frac{1}{1 + (\omega \tau_i)^2}
  998             - \frac{(\tau_e + \tau_i)\tau_e}{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}
  999         \Bigg)
 1000     \Bigg).
 1001 \end{multline}
 1002 
 1003 
 1004 
 1005 % Gj-te partial derivative.
 1006 \subsubsection{$\Diffgeoset_j$ -- $\tau_e$ partial derivative}
 1007 
 1008 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the geometric parameter $\Diffgeoset_j$ and the correlation time $\tau_e$ is
 1009 \begin{multline}
 1010     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \tau_e} = \frac{2}{5} (1 - S^2) \sum_{i=-k}^k \Bigg(
 1011         2 c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \tau_e \tau_i (\tau_e + \tau_i)
 1012             \frac{(\tau_e + \tau_i)^2 - 3(\omega \tau_e \tau_i)^2}{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^3}  \\
 1013         + \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i^2 \frac{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}
 1014     \Bigg).
 1015 \end{multline}
 1016 
 1017 
 1018 % Oj-Ok partial derivative.
 1019 \subsubsection{$\Difforiset_j$ -- $\Difforiset_k$ partial derivative}
 1020 
 1021 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the orientational parameters $\Difforiset_j$ and $\Difforiset_k$ is
 1022 \begin{equation}
 1023     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial \Difforiset_k} = \frac{2}{5} \sum_{i=-k}^k
 1024         \frac{\partial^2 c_i}{\partial \Difforiset_j \cdot \partial \Difforiset_k} \tau_i \Bigg(
 1025             \frac{S^2}{1 + (\omega \tau_i)^2}
 1026             + \frac{(1 - S^2)(\tau_e + \tau_i)\tau_e}{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}
 1027         \Bigg).
 1028 \end{equation}
 1029 
 1030 
 1031 
 1032 % Oj-S2 partial derivative.
 1033 \subsubsection{$\Difforiset_j$ -- $S^2$ partial derivative}
 1034 
 1035 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the orientational parameter $\Difforiset_j$ and the order parameter $S^2$ is
 1036 \begin{equation}
 1037     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial S^2} = \frac{2}{5} \sum_{i=-k}^k \frac{\partial c_i}{\partial \Difforiset_j} \tau_i \Bigg(
 1038         \frac{1}{1 + (\omega \tau_i)^2}
 1039         - \frac{(\tau_e + \tau_i)\tau_e}{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}
 1040     \Bigg).
 1041 \end{equation}
 1042 
 1043 
 1044 
 1045 % Oj-te partial derivative.
 1046 \subsubsection{$\Difforiset_j$ -- $\tau_e$ partial derivative}
 1047 
 1048 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the orientational parameter $\Difforiset_j$ and the correlation time $\tau_e$ is
 1049 \begin{equation}
 1050     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial \tau_e} = \frac{2}{5} (1 - S^2) \sum_{i=-k}^k
 1051         \frac{\partial c_i}{\partial \Difforiset_j} \tau_i^2
 1052         \frac{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}.
 1053 \end{equation}
 1054 
 1055 
 1056 
 1057 % S2-S2 partial derivative.
 1058 \subsubsection{$S^2$ -- $S^2$ partial derivative}
 1059 
 1060 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the order parameter $S^2$ twice is
 1061 \begin{equation}
 1062     \frac{\partial^2 J(\omega)}{(\partial S^2)^2} = 0.
 1063 \end{equation}
 1064 
 1065 
 1066 
 1067 % S2-te partial derivative.
 1068 \subsubsection{$S^2$ -- $\tau_e$ partial derivative}
 1069 
 1070 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the order parameter $S^2$ and correlation time $\tau_e$ is
 1071 \begin{equation}
 1072     \frac{\partial^2 J(\omega)}{\partial S^2 \cdot \partial \tau_e} = -\frac{2}{5} \sum_{i=-k}^k c_i \tau_i^2
 1073         \frac{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}.
 1074 \end{equation}
 1075 
 1076 
 1077 
 1078 % te-te partial derivative.
 1079 \subsubsection{$\tau_e$ -- $\tau_e$ partial derivative}
 1080 
 1081 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the correlation time $\tau_e$ twice is
 1082 \begin{equation}
 1083     \frac{\partial^2 J(\omega)}{{\partial \tau_e}^2} = -\frac{4}{5} (1 - S^2) \sum_{i=-k}^k c_i \tau_i^2
 1084         \frac{(\tau_e + \tau_i)^3  +  3 \omega^2 \tau_i^3 \tau_e (\tau_e + \tau_i)  -  (\omega \tau_i)^4 \tau_e^3}
 1085             {\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^3}
 1086 \end{equation}
 1087 
 1088 
 1089 
 1090 
 1091 % The extended model-free gradient.
 1092 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 1093 
 1094 \newpage
 1095 \subsection{The extended model-free gradient}
 1096 
 1097 The model-free gradient of the extended spectral density function~\eqref{eq: maths: J(w) model-free ext generic} is the vector of partial derivatives of the function with respect to the geometric parameter $\Diffgeoset_i$, the orientational parameter $\Difforiset_i$, the order parameters $S^2$ and $S^2_f$, and the internal correlation times $\tau_f$ and $\tau_s$.
 1098 The positions in the vector correspond to the model parameters which are being optimised.
 1099 
 1100 
 1101 
 1102 % Gj partial derivative.
 1103 \subsubsection{$\Diffgeoset_j$ partial derivative}
 1104 
 1105 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ is
 1106 \begin{multline}
 1107     \frac{\partial J(\omega)}{\partial \Diffgeoset_j} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
 1108         c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \Bigg(
 1109             S^2 \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2} \\
 1110             + (1 - S^2_f) \tau_f^2 \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2} \\
 1111             + (S^2_f - S^2) \tau_s^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1112         \Bigg) \\
 1113         +  \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i \Bigg(
 1114             \frac{S^2}{1 + (\omega \tau_i)^2}
 1115             + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1116             + \frac{(S^2_f - S^2)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1117         \Bigg)
 1118     \Bigg).
 1119 \end{multline}
 1120 
 1121 
 1122 
 1123 % Oj partial derivative.
 1124 \subsubsection{$\Difforiset_j$ partial derivative}
 1125 
 1126 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameter $\Difforiset_j$ is
 1127 \begin{equation}
 1128     \frac{\partial J(\omega)}{\partial \Difforiset_j} = \frac{2}{5} \sum_{i=-k}^k \frac{\partial c_i}{\partial \Difforiset_j} \tau_i \Bigg(
 1129         \frac{S^2}{1 + (\omega \tau_i)^2}
 1130         + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1131         + \frac{(S^2_f - S^2)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1132     \Bigg).
 1133 \end{equation}
 1134 
 1135 
 1136 
 1137 % S2 partial derivative.
 1138 \subsubsection{$S^2$ partial derivative}
 1139 
 1140 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2$ is
 1141 \begin{equation}
 1142     \frac{\partial J(\omega)}{\partial S^2} = \frac{2}{5} \sum_{i=-k}^k c_i \tau_i \Bigg(
 1143         \frac{1}{1 + (\omega \tau_i)^2}
 1144         - \frac{(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1145     \Bigg).
 1146 \end{equation}
 1147 
 1148 
 1149 
 1150 % S2f partial derivative.
 1151 \subsubsection{$S^2_f$ partial derivative}
 1152 
 1153 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_f$ is
 1154 \begin{equation}
 1155     \frac{\partial J(\omega)}{\partial S^2_f} = -\frac{2}{5} \sum_{i=-k}^k c_i \tau_i \Bigg(
 1156         \frac{(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1157         - \frac{(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1158     \Bigg).
 1159 \end{equation}
 1160 
 1161 
 1162 
 1163 % tf partial derivative.
 1164 \subsubsection{$\tau_f$ partial derivative}
 1165 
 1166 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the correlation time $\tau_f$ is
 1167 \begin{equation}
 1168     \frac{\partial J(\omega)}{\partial \tau_f} = \frac{2}{5} (1 - S^2_f) \sum_{i=-k}^k c_i \tau_i^2
 1169         \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}.
 1170 \end{equation}
 1171 
 1172 
 1173 
 1174 % ts partial derivative.
 1175 \subsubsection{$\tau_s$ partial derivative}
 1176 
 1177 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the correlation time $\tau_s$ is
 1178 \begin{equation}
 1179     \frac{\partial J(\omega)}{\partial \tau_s} = \frac{2}{5} (S^2_f - S^2) \sum_{i=-k}^k c_i \tau_i^2
 1180         \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}.
 1181 \end{equation}
 1182 
 1183 
 1184 
 1185 
 1186 % The extended model-free Hessian.
 1187 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 1188 
 1189 \newpage
 1190 \subsection{The extended model-free Hessian}
 1191 
 1192 The model-free Hessian of the extended spectral density function~\eqref{eq: maths: J(w) model-free ext generic} is the matrix of second partial derivatives.
 1193 The matrix coordinates correspond to the model parameters which are being optimised.
 1194 
 1195 
 1196 
 1197 % Gj-Gk partial derivative.
 1198 \subsubsection{$\Diffgeoset_j$ -- $\Diffgeoset_k$ partial derivative}
 1199 
 1200 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameters $\Diffgeoset_j$ and $\Diffgeoset_k$ is
 1201 \begin{multline}
 1202     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \Diffgeoset_k} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
 1203         -2 c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \cdot \frac{\partial \tau_i}{\partial \Diffgeoset_k} \Bigg(
 1204             S^2 \omega^2 \tau_i \frac{3 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^3}  \\
 1205             + (1 - S^2_f) \tau_f^2 \frac{(\tau_f + \tau_i)^3  +  3 \omega^2 \tau_f^3 \tau_i (\tau_f + \tau_i)  -  (\omega \tau_f)^4 \tau_i^3}
 1206                 {\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^3} \\
 1207             + (S^2_f - S^2) \tau_s^2 \frac{(\tau_s + \tau_i)^3  +  3 \omega^2 \tau_s^3 \tau_i (\tau_s + \tau_i)  -  (\omega \tau_s)^4 \tau_i^3}
 1208                 {\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^3}
 1209         \Bigg) \\
 1210         + \Bigg(
 1211             \frac{\partial \tau_i}{\partial \Diffgeoset_j} \cdot \frac{\partial c_i}{\partial \Diffgeoset_k}
 1212             + \frac{\partial \tau_i}{\partial \Diffgeoset_k} \cdot \frac{\partial c_i}{\partial \Diffgeoset_j}
 1213             + c_i \frac{\partial^2 \tau_i}{\partial \Diffgeoset_j \cdot \partial \Diffgeoset_k}
 1214         \Bigg)
 1215         \Bigg(
 1216             S^2 \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2} \\
 1217             + (1 - S^2_f) \tau_f^2 \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2} \\
 1218             + (S^2_f - S^2) \tau_s^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1219         \Bigg) \\
 1220         + \Bigg(
 1221             \frac{\partial^2 c_i}{\partial \Diffgeoset_j \cdot \partial \Diffgeoset_k} \tau_i \Bigg(
 1222                 \frac{S^2}{1 + (\omega \tau_i)^2}
 1223                 + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1224                 + \frac{(S^2_f - S^2)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1225             \Bigg)
 1226         \Bigg)
 1227     \Bigg).
 1228 \end{multline}
 1229                 
 1230 
 1231 
 1232 % Gj-Ok partial derivative.
 1233 \subsubsection{$\Diffgeoset_j$ -- $\Difforiset_k$ partial derivative}
 1234 
 1235 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ and the orientational parameter $\Difforiset_k$ is
 1236 \begin{multline}
 1237     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \Difforiset_k} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
 1238         \frac{\partial \tau_i}{\partial \Diffgeoset_j} \frac{\partial c_i}{\partial \Difforiset_k} \Bigg(
 1239             S^2 \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2} \\
 1240             + (1 - S^2_f) \tau_f^2 \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2} \\
 1241             + (S^2_f - S^2) \tau_s^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1242         \Bigg) \\
 1243         +  \frac{\partial^2 c_i}{\partial \Diffgeoset_j \cdot \partial \Difforiset_k} \tau_i \Bigg(
 1244             \frac{S^2}{1 + (\omega \tau_i)^2}
 1245             + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1246             + \frac{(S^2_f - S^2)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1247         \Bigg)
 1248     \Bigg).
 1249 \end{multline}
 1250 
 1251 
 1252 
 1253 % Gj-S2 partial derivative.
 1254 \subsubsection{$\Diffgeoset_j$ -- $S^2$ partial derivative}
 1255 
 1256 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ and the order parameter $S^2$ is
 1257 \begin{multline}
 1258     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial S^2} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
 1259         c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \Bigg(
 1260             \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2}
 1261             - \tau_s^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1262         \Bigg) \\
 1263         +  \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i \Bigg(
 1264             \frac{1}{1 + (\omega \tau_i)^2}
 1265             - \frac{(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1266         \Bigg)
 1267     \Bigg).
 1268 \end{multline}
 1269 
 1270 
 1271 
 1272 % Gj-S2f partial derivative.
 1273 \subsubsection{$\Diffgeoset_j$ -- $S^2_f$ partial derivative}
 1274 
 1275 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ and the order parameter $S^2_f$ is
 1276 \begin{multline}
 1277     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial S^2_f} = -\frac{2}{5} \sum_{i=-k}^k \Bigg(
 1278         c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \Bigg(
 1279             \tau_f^2 \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}
 1280             - \tau_s^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1281         \Bigg) \\
 1282         +  \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i \Bigg(
 1283             \frac{(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1284             - \frac{(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1285         \Bigg)
 1286     \Bigg).
 1287 \end{multline}
 1288 
 1289 
 1290 
 1291 % Gj-tf partial derivative.
 1292 \subsubsection{$\Diffgeoset_j$ -- $\tau_f$ partial derivative}
 1293 
 1294 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ and the correlation time $\tau_f$ is
 1295 \begin{multline}
 1296     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \tau_f} = \frac{2}{5} (1 - S^2_f) \sum_{i=-k}^k \Bigg(
 1297         2 c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \tau_f \tau_i (\tau_f + \tau_i)
 1298             \frac{(\tau_f + \tau_i)^2 - 3(\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^3}  \\
 1299         + \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i^2 \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}
 1300     \Bigg).
 1301 \end{multline}
 1302 
 1303 
 1304 
 1305 % Gj-ts partial derivative.
 1306 \subsubsection{$\Diffgeoset_j$ -- $\tau_s$ partial derivative}
 1307 
 1308 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ and the correlation time $\tau_s$ is
 1309 \begin{multline}
 1310     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \tau_s} = \frac{2}{5} (S^2_f - S^2) \sum_{i=-k}^k \Bigg(
 1311         2 c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \tau_s \tau_i (\tau_s + \tau_i)
 1312             \frac{(\tau_s + \tau_i)^2 - 3(\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^3}  \\
 1313         + \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1314     \Bigg).
 1315 \end{multline}
 1316 
 1317 
 1318 
 1319 % Oj-Ok partial derivative.
 1320 \subsubsection{$\Difforiset_j$ -- $\Difforiset_k$ partial derivative}
 1321 
 1322 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameters $\Difforiset_j$ and $\Difforiset_k$ is
 1323 \begin{multline}
 1324     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial \Difforiset_k} = \frac{2}{5} \sum_{i=-k}^k
 1325         \frac{\partial^2 c_i}{\partial \Difforiset_j \cdot \partial \Difforiset_k} \tau_i \Bigg(
 1326             \frac{S^2}{1 + (\omega \tau_i)^2}
 1327             + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2} \\
 1328             + \frac{(S^2_f - S^2)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1329         \Bigg).
 1330 \end{multline}
 1331 
 1332 
 1333 
 1334 % Oj-S2 partial derivative.
 1335 \subsubsection{$\Difforiset_j$ -- $S^2$ partial derivative}
 1336 
 1337 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameter $\Difforiset_j$ and the order parameter $S^2$ is
 1338 \begin{equation}
 1339     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial S^2} = \frac{2}{5} \sum_{i=-k}^k \frac{\partial c_i}{\partial \Difforiset_j} \tau_i \Bigg(
 1340         \frac{1}{1 + (\omega \tau_i)^2}
 1341         - \frac{(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1342     \Bigg).
 1343 \end{equation}
 1344 
 1345 
 1346 
 1347 % Oj-S2f partial derivative.
 1348 \subsubsection{$\Difforiset_j$ -- $S^2_f$ partial derivative}
 1349 
 1350 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameter $\Difforiset_j$ and the order parameter $S^2_f$ is
 1351 \begin{equation}
 1352     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial S^2_f} = -\frac{2}{5} \sum_{i=-k}^k \frac{\partial c_i}{\partial \Difforiset_j} \tau_i \Bigg(
 1353         \frac{(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1354         - \frac{(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1355     \Bigg).
 1356 \end{equation}
 1357 
 1358 
 1359 
 1360 % Oj-tf partial derivative.
 1361 \subsubsection{$\Difforiset_j$ -- $\tau_f$ partial derivative}
 1362 
 1363 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameter $\Difforiset_j$ and the correlation time $\tau_f$ is
 1364 \begin{equation}
 1365     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial \tau_f} = \frac{2}{5} (1 - S^2_f) \sum_{i=-k}^k
 1366         \frac{\partial c_i}{\partial \Difforiset_j} \tau_i^2
 1367         \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}.
 1368 \end{equation}
 1369 
 1370 
 1371 
 1372 % Oj-ts partial derivative.
 1373 \subsubsection{$\Difforiset_j$ -- $\tau_s$ partial derivative}
 1374 
 1375 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameter $\Difforiset_j$ and the correlation time $\tau_s$ is
 1376 \begin{equation}
 1377     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial \tau_s} = \frac{2}{5} (S^2_f - S^2) \sum_{i=-k}^k
 1378         \frac{\partial c_i}{\partial \Difforiset_j} \tau_i^2
 1379         \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}.
 1380 \end{equation}
 1381 
 1382 
 1383 
 1384 % S2-S2 partial derivative.
 1385 \subsubsection{$S^2$ -- $S^2$ partial derivative}
 1386 
 1387 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2$ twice is
 1388 \begin{equation}
 1389     \frac{\partial^2 J(\omega)}{(\partial S^2)^2} = 0.
 1390 \end{equation}
 1391 
 1392 
 1393 
 1394 % S2-S2f partial derivative.
 1395 \subsubsection{$S^2$ -- $S^2_f$ partial derivative}
 1396 
 1397 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameters $S^2$ and $S^2_f$ is
 1398 \begin{equation}
 1399     \frac{\partial^2 J(\omega)}{\partial S^2 \cdot \partial S^2_f} = 0.
 1400 \end{equation}
 1401 
 1402 
 1403 
 1404 % S2-tf partial derivative.
 1405 \subsubsection{$S^2$ -- $\tau_f$ partial derivative}
 1406 
 1407 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2$ and correlation time $\tau_f$ is
 1408 \begin{equation}
 1409     \frac{\partial^2 J(\omega)}{\partial S^2 \cdot \partial \tau_f} = 0.
 1410 \end{equation}
 1411 
 1412 
 1413 % S2-ts partial derivative.
 1414 \subsubsection{$S^2$ -- $\tau_s$ partial derivative}
 1415 
 1416 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2$ and correlation time $\tau_s$ is
 1417 \begin{equation}
 1418     \frac{\partial^2 J(\omega)}{\partial S^2 \cdot \partial \tau_s} = -\frac{2}{5} \sum_{i=-k}^k c_i \tau_i^2
 1419         \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}.
 1420 \end{equation}
 1421 
 1422 
 1423 
 1424 % S2f-S2f partial derivative.
 1425 \subsubsection{$S^2_f$ -- $S^2_f$ partial derivative}
 1426 
 1427 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_f$ twice is
 1428 \begin{equation}
 1429     \frac{\partial^2 J(\omega)}{(\partial S^2_f)^2} = 0.
 1430 \end{equation}
 1431 
 1432 
 1433 
 1434 % S2f-tf partial derivative.
 1435 \subsubsection{$S^2_f$ -- $\tau_f$ partial derivative}
 1436 
 1437 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_f$ and correlation time $\tau_f$ is
 1438 \begin{equation}
 1439     \frac{\partial^2 J(\omega)}{\partial S^2_f \cdot \partial \tau_f} = -\frac{2}{5} \sum_{i=-k}^k c_i \tau_i^2
 1440         \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}.
 1441 \end{equation}
 1442 
 1443 
 1444 
 1445 % S2f-ts partial derivative.
 1446 \subsubsection{$S^2_f$ -- $\tau_s$ partial derivative}
 1447 
 1448 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_f$ and correlation time $\tau_s$ is
 1449 \begin{equation}
 1450     \frac{\partial^2 J(\omega)}{\partial S^2_f \cdot \partial \tau_s} = \frac{2}{5} \sum_{i=-k}^k c_i \tau_i^2
 1451         \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}.
 1452 \end{equation}
 1453 
 1454 
 1455 
 1456 % tf-tf partial derivative.
 1457 \subsubsection{$\tau_f$ -- $\tau_f$ partial derivative}
 1458 
 1459 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the correlation time $\tau_f$ twice is
 1460 \begin{equation}
 1461     \frac{\partial^2 J(\omega)}{{\partial \tau_f}^2} = -\frac{4}{5} (1 - S^2_f) \sum_{i=-k}^k c_i \tau_i^2
 1462         \frac{(\tau_f + \tau_i)^3  +  3 \omega^2 \tau_i^3 \tau_f (\tau_f + \tau_i)  -  (\omega \tau_i)^4 \tau_f^3}
 1463             {\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^3}
 1464 \end{equation}
 1465 
 1466 
 1467 
 1468 % tf-ts partial derivative.
 1469 \subsubsection{$\tau_f$ -- $\tau_s$ partial derivative}
 1470 
 1471 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the correlation times $\tau_f$ and $\tau_s$ is
 1472 \begin{equation}
 1473     \frac{\partial^2 J(\omega)}{\partial \tau_f \cdot \partial \tau_s} = 0.
 1474 \end{equation}
 1475 
 1476 
 1477 
 1478 % ts-ts partial derivative.
 1479 \subsubsection{$\tau_s$ -- $\tau_s$ partial derivative}
 1480 
 1481 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the correlation time $\tau_s$ twice is
 1482 \begin{equation}
 1483     \frac{\partial^2 J(\omega)}{{\partial \tau_s}^2} = -\frac{4}{5} (S^2_f - S^2) \sum_{i=-k}^k c_i \tau_i^2
 1484         \frac{(\tau_s + \tau_i)^3  +  3 \omega^2 \tau_i^3 \tau_s (\tau_s + \tau_i)  -  (\omega \tau_i)^4 \tau_s^3}
 1485             {\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^3}
 1486 \end{equation}
 1487 
 1488 
 1489 
 1490 
 1491 % The alternative extended model-free gradient.
 1492 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 1493 
 1494 \newpage
 1495 \subsection{The alternative extended model-free gradient}
 1496 
 1497 Because of the equation $S^2 = S^2_f \cdot S^2_s$ and the form of the extended spectral density function~\eqref{eq: maths: J(w) model-free ext generic} a convolution\index{parameter convolution} of the model-free space occurs if the model-free parameters \{$S^2_f$, $S^2_s$, $\tau_f$, $\tau_s$\} are optimised rather than the parameters \{$S^2$, $S^2_f$, $\tau_f$, $\tau_s$\}.  This convolution increases the complexity of the gradient.  For completeness the first partial derivatives are presented below.
 1498 
 1499 
 1500 
 1501 % Gj partial derivative.
 1502 \subsubsection{$\Diffgeoset_j$ partial derivative}
 1503 
 1504 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ is
 1505 \begin{multline}
 1506     \frac{\partial J(\omega)}{\partial \Diffgeoset_j} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
 1507         c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \Bigg(
 1508             S^2_f \cdot S^2_s \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2} \\
 1509             + (1 - S^2_f) \tau_f^2 \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2} \\
 1510             + S^2_f(1 - S^2_s) \tau_s^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1511         \Bigg) \\
 1512         +  \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i \Bigg(
 1513             \frac{S^2_f \cdot S^2_s}{1 + (\omega \tau_i)^2}
 1514             + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1515             + \frac{S^2_f(1 - S^2_s)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1516         \Bigg)
 1517     \Bigg).
 1518 \end{multline}
 1519 
 1520 
 1521 
 1522 % Oj partial derivative.
 1523 \subsubsection{$\Difforiset_j$ partial derivative}
 1524 
 1525 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameter $\Difforiset_j$ is
 1526 \begin{equation}
 1527     \frac{\partial J(\omega)}{\partial \Difforiset_j} = \frac{2}{5} \sum_{i=-k}^k \frac{\partial c_i}{\partial \Difforiset_j} \tau_i \Bigg(
 1528         \frac{S^2_f \cdot S^2_s}{1 + (\omega \tau_i)^2}
 1529         + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1530         + \frac{S^2_f(1 - S^2_s)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1531     \Bigg).
 1532 \end{equation}
 1533 
 1534 
 1535 
 1536 % S2f partial derivative.
 1537 \subsubsection{$S^2_f$ partial derivative}
 1538 
 1539 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_f$ is
 1540 \begin{equation}
 1541     \frac{\partial J(\omega)}{\partial S^2_f} = \frac{2}{5} \sum_{i=-k}^k c_i \tau_i \Bigg(
 1542         \frac{S^2_s}{1 + (\omega \tau_i)^2}
 1543         - \frac{(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1544         + \frac{(1 - S^2_s) (\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1545     \Bigg).
 1546 \end{equation}
 1547 
 1548 
 1549 
 1550 % S2s partial derivative.
 1551 \subsubsection{$S^2_s$ partial derivative}
 1552 
 1553 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_s$ is
 1554 \begin{equation}
 1555     \frac{\partial J(\omega)}{\partial S^2_s} = \frac{2}{5} S^2_f \sum_{i=-k}^k c_i \tau_i \Bigg(
 1556         \frac{1}{1 + (\omega \tau_i)^2}
 1557         - \frac{(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1558     \Bigg).
 1559 \end{equation}
 1560 
 1561 
 1562 
 1563 % tf partial derivative.
 1564 \subsubsection{$\tau_f$ partial derivative}
 1565 
 1566 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the correlation time $\tau_f$ is
 1567 \begin{equation}
 1568     \frac{\partial J(\omega)}{\partial \tau_f} = \frac{2}{5} (1 - S^2_f) \sum_{i=-k}^k c_i \tau_i^2
 1569         \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}.
 1570 \end{equation}
 1571 
 1572 
 1573 
 1574 % ts partial derivative.
 1575 \subsubsection{$\tau_s$ partial derivative}
 1576 
 1577 The partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the correlation time $\tau_s$ is
 1578 \begin{equation}
 1579     \frac{\partial J(\omega)}{\partial \tau_s} = \frac{2}{5} S^2_f(1 - S^2_s) \sum_{i=-k}^k c_i \tau_i^2
 1580         \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}.
 1581 \end{equation}
 1582 
 1583 
 1584 
 1585 
 1586 % The alternative extended model-free Hessian.
 1587 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 1588 
 1589 \newpage
 1590 \subsection{The alternative extended model-free Hessian}
 1591 
 1592 The model-free Hessian of the extended spectral density function~\eqref{eq: maths: J(w) model-free ext generic} is also complicated by the convolution resulting from the use of the parameters \{$S^2_f$, $S^2_s$, $\tau_f$, $\tau_s$\}.  The second partial derivatives with respect to these parameters are presented below.
 1593 
 1594 
 1595 
 1596 % Gj-Gk partial derivative.
 1597 \subsubsection{$\Diffgeoset_j$ -- $\Diffgeoset_k$ partial derivative}
 1598 
 1599 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameters $\Diffgeoset_j$ and $\Diffgeoset_k$ is
 1600 \begin{multline}
 1601     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \Diffgeoset_k} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
 1602         -2 c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \cdot \frac{\partial \tau_i}{\partial \Diffgeoset_k} \Bigg(
 1603             S^2_f \cdot S^2_s \omega^2 \tau_i \frac{3 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^3}  \\
 1604             + (1 - S^2_f) \tau_f^2 \frac{(\tau_f + \tau_i)^3  +  3 \omega^2 \tau_f^3 \tau_i (\tau_f + \tau_i)  -  (\omega \tau_f)^4 \tau_i^3}
 1605                 {\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^3} \\
 1606             + S^2_f(1 - S^2_s) \tau_s^2 \frac{(\tau_s + \tau_i)^3  +  3 \omega^2 \tau_s^3 \tau_i (\tau_s + \tau_i)  -  (\omega \tau_s)^4 \tau_i^3}
 1607                 {\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^3}
 1608         \Bigg) \\
 1609         + \Bigg(
 1610             \frac{\partial \tau_i}{\partial \Diffgeoset_j} \cdot \frac{\partial c_i}{\partial \Diffgeoset_k}
 1611             + \frac{\partial \tau_i}{\partial \Diffgeoset_k} \cdot \frac{\partial c_i}{\partial \Diffgeoset_j}
 1612             + c_i \frac{\partial^2 \tau_i}{\partial \Diffgeoset_j \cdot \partial \Diffgeoset_k}
 1613         \Bigg)
 1614         \Bigg(
 1615             S^2_f \cdot S^2_s \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2} \\
 1616             + (1 - S^2_f) \tau_f^2 \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2} \\
 1617             + S^2_f(1 - S^2_s) \tau_s^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1618         \Bigg) \\
 1619         + \Bigg(
 1620             \frac{\partial^2 c_i}{\partial \Diffgeoset_j \cdot \partial \Diffgeoset_k} \tau_i \Bigg(
 1621                 \frac{S^2_f \cdot S^2_s}{1 + (\omega \tau_i)^2}
 1622                 + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1623                 + \frac{S^2_f(1 - S^2_s)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1624             \Bigg)
 1625         \Bigg)
 1626     \Bigg).
 1627 \end{multline}
 1628                 
 1629 
 1630 
 1631 % Gj-Ok partial derivative.
 1632 \subsubsection{$\Diffgeoset_j$ -- $\Difforiset_k$ partial derivative}
 1633 
 1634 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ and the orientational parameter $\Difforiset_k$ is
 1635 \begin{multline}
 1636     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \Difforiset_k} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
 1637         \frac{\partial \tau_i}{\partial \Diffgeoset_j} \frac{\partial c_i}{\partial \Difforiset_k} \Bigg(
 1638             S^2_f \cdot S^2_s \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2} \\
 1639             + (1 - S^2_f) \tau_f^2 \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2} \\
 1640             + S^2_f(1 - S^2_s) \tau_s^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1641         \Bigg) \\
 1642         +  \frac{\partial^2 c_i}{\partial \Diffgeoset_j \cdot \partial \Difforiset_k} \tau_i \Bigg(
 1643             \frac{S^2_f \cdot S^2_s}{1 + (\omega \tau_i)^2}
 1644             + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1645             + \frac{S^2_f(1 - S^2_s)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1646         \Bigg)
 1647     \Bigg).
 1648 \end{multline}
 1649 
 1650 
 1651 
 1652 % Gj-S2f partial derivative.
 1653 \subsubsection{$\Diffgeoset_j$ -- $S^2_f$ partial derivative}
 1654 
 1655 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ and the order parameter $S^2_f$ is
 1656 \begin{multline}
 1657     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial S^2_f} = \frac{2}{5} \sum_{i=-k}^k \Bigg(
 1658         c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \Bigg(
 1659             S^2_s \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2}
 1660             - \tau_f^2 \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2} \\
 1661             + (1 - S^2_s) \tau_s^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1662         \Bigg) \\
 1663         +  \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i \Bigg(
 1664             \frac{S^2_s}{1 + (\omega \tau_i)^2}
 1665             - \frac{(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2} \\
 1666             + \frac{(1 - S^2_s)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1667         \Bigg)
 1668     \Bigg).
 1669 \end{multline}
 1670 
 1671 
 1672 
 1673 % Gj-S2s partial derivative.
 1674 \subsubsection{$\Diffgeoset_j$ -- $S^2_s$ partial derivative}
 1675 
 1676 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ and the order parameter $S^2_s$ is
 1677 \begin{multline}
 1678     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial S^2_s} = \frac{2}{5} S^2_f \sum_{i=-k}^k \Bigg(
 1679         c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \Bigg(
 1680             \frac{1 - (\omega \tau_i)^2}{\left(1 + (\omega \tau_i)^2 \right)^2}
 1681             - \tau_s^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1682         \Bigg) \\
 1683         +  \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i \Bigg(
 1684             \frac{1}{1 + (\omega \tau_i)^2}
 1685             - \frac{(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1686         \Bigg)
 1687     \Bigg).
 1688 \end{multline}
 1689 
 1690 
 1691 
 1692 % Gj-tf partial derivative.
 1693 \subsubsection{$\Diffgeoset_j$ -- $\tau_f$ partial derivative}
 1694 
 1695 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ and the correlation time $\tau_f$ is
 1696 \begin{multline}
 1697     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \tau_f} = \frac{2}{5} (1 - S^2_f) \sum_{i=-k}^k \Bigg(
 1698         2 c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \tau_f \tau_i (\tau_f + \tau_i)
 1699             \frac{(\tau_f + \tau_i)^2 - 3(\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^3}  \\
 1700         + \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i^2 \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}
 1701     \Bigg).
 1702 \end{multline}
 1703 
 1704 
 1705 
 1706 % Gj-ts partial derivative.
 1707 \subsubsection{$\Diffgeoset_j$ -- $\tau_s$ partial derivative}
 1708 
 1709 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the geometric parameter $\Diffgeoset_j$ and the correlation time $\tau_s$ is
 1710 \begin{multline}
 1711     \frac{\partial^2 J(\omega)}{\partial \Diffgeoset_j \cdot \partial \tau_s} = \frac{2}{5} S^2_f(1 - S^2_s) \sum_{i=-k}^k \Bigg(
 1712         2 c_i \frac{\partial \tau_i}{\partial \Diffgeoset_j} \tau_s \tau_i (\tau_s + \tau_i)
 1713             \frac{(\tau_s + \tau_i)^2 - 3(\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^3}  \\
 1714         + \frac{\partial c_i}{\partial \Diffgeoset_j} \tau_i^2 \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}
 1715     \Bigg).
 1716 \end{multline}
 1717 
 1718 
 1719 
 1720 % Oj-Ok partial derivative.
 1721 \subsubsection{$\Difforiset_j$ -- $\Difforiset_k$ partial derivative}
 1722 
 1723 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameters $\Difforiset_j$ and $\Difforiset_k$ is
 1724 \begin{multline}
 1725     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial \Difforiset_k} = \frac{2}{5} \sum_{i=-k}^k
 1726         \frac{\partial^2 c_i}{\partial \Difforiset_j \cdot \partial \Difforiset_k} \tau_i \Bigg(
 1727             \frac{S^2_f \cdot S^2_s}{1 + (\omega \tau_i)^2}
 1728             + \frac{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2} \\
 1729             + \frac{S^2_f(1 - S^2_s)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1730         \Bigg).
 1731 \end{multline}
 1732 
 1733 
 1734 
 1735 % Oj-S2f partial derivative.
 1736 \subsubsection{$\Difforiset_j$ -- $S^2_f$ partial derivative}
 1737 
 1738 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameter $\Difforiset_j$ and the order parameter $S^2_f$ is
 1739 \begin{equation}
 1740     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial S^2_f} = \frac{2}{5} \sum_{i=-k}^k \frac{\partial c_i}{\partial \Difforiset_j} \tau_i \Bigg(
 1741         \frac{S^2_s}{1 + (\omega \tau_i)^2}
 1742         - \frac{(\tau_f + \tau_i)\tau_f}{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}
 1743         + \frac{(1 - S^2_s)(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1744     \Bigg).
 1745 \end{equation}
 1746 
 1747 
 1748 
 1749 % Oj-S2s partial derivative.
 1750 \subsubsection{$\Difforiset_j$ -- $S^2_s$ partial derivative}
 1751 
 1752 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameter $\Difforiset_j$ and the order parameter $S^2_s$ is
 1753 \begin{equation}
 1754     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial S^2_s} = \frac{2}{5} S^2_f \sum_{i=-k}^k \frac{\partial c_i}{\partial \Difforiset_j} \tau_i \Bigg(
 1755         \frac{1}{1 + (\omega \tau_i)^2}
 1756         - \frac{(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1757     \Bigg).
 1758 \end{equation}
 1759 
 1760 
 1761 
 1762 % Oj-tf partial derivative.
 1763 \subsubsection{$\Difforiset_j$ -- $\tau_f$ partial derivative}
 1764 
 1765 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameter $\Difforiset_j$ and the correlation time $\tau_f$ is
 1766 \begin{equation}
 1767     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial \tau_f} = \frac{2}{5} (1 - S^2_f) \sum_{i=-k}^k
 1768         \frac{\partial c_i}{\partial \Difforiset_j} \tau_i^2
 1769         \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}.
 1770 \end{equation}
 1771 
 1772 
 1773 
 1774 % Oj-ts partial derivative.
 1775 \subsubsection{$\Difforiset_j$ -- $\tau_s$ partial derivative}
 1776 
 1777 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the orientational parameter $\Difforiset_j$ and the correlation time $\tau_s$ is
 1778 \begin{equation}
 1779     \frac{\partial^2 J(\omega)}{\partial \Difforiset_j \cdot \partial \tau_s} = \frac{2}{5} S^2_f(1 - S^2_s) \sum_{i=-k}^k
 1780         \frac{\partial c_i}{\partial \Difforiset_j} \tau_i^2
 1781         \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}.
 1782 \end{equation}
 1783 
 1784 
 1785 
 1786 % S2f-S2f partial derivative.
 1787 \subsubsection{$S^2_f$ -- $S^2_f$ partial derivative}
 1788 
 1789 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_f$ twice is
 1790 \begin{equation}
 1791     \frac{\partial^2 J(\omega)}{(\partial S^2_f)^2} = 0.
 1792 \end{equation}
 1793 
 1794 
 1795 
 1796 % S2f-S2s partial derivative.
 1797 \subsubsection{$S^2_f$ -- $S^2_s$ partial derivative}
 1798 
 1799 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameters $S^2_f$ and $S^2_s$ is
 1800 \begin{equation}
 1801     \frac{\partial^2 J(\omega)}{\partial S^2_f \cdot \partial S^2_s} = \frac{2}{5} \sum_{i=-k}^k c_i \tau_i \Bigg(
 1802         \frac{1}{1 + (\omega \tau_i)^2}
 1803         - \frac{(\tau_s + \tau_i)\tau_s}{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
 1804     \Bigg).
 1805 \end{equation}
 1806 
 1807 
 1808 
 1809 % S2f-tf partial derivative.
 1810 \subsubsection{$S^2_f$ -- $\tau_f$ partial derivative}
 1811 
 1812 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_f$ and correlation time $\tau_f$ is
 1813 \begin{equation}
 1814     \frac{\partial^2 J(\omega)}{\partial S^2_f \cdot \partial \tau_f} = -\frac{2}{5} \sum_{i=-k}^k c_i \tau_i^2
 1815         \frac{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}.
 1816 \end{equation}
 1817 
 1818 
 1819 
 1820 % S2f-ts partial derivative.
 1821 \subsubsection{$S^2_f$ -- $\tau_s$ partial derivative}
 1822 
 1823 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_f$ and correlation time $\tau_s$ is
 1824 \begin{equation}
 1825     \frac{\partial^2 J(\omega)}{\partial S^2_f \cdot \partial \tau_s} = \frac{2}{5} (1 - S^2_s) \sum_{i=-k}^k c_i \tau_i^2
 1826         \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}.
 1827 \end{equation}
 1828 
 1829 
 1830 
 1831 % S2s-S2s partial derivative.
 1832 \subsubsection{$S^2_s$ -- $S^2_s$ partial derivative}
 1833 
 1834 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_s$ twice is
 1835 \begin{equation}
 1836     \frac{\partial^2 J(\omega)}{(\partial S^2_s)^2} = 0.
 1837 \end{equation}
 1838 
 1839 
 1840 
 1841 % S2s-tf partial derivative.
 1842 \subsubsection{$S^2_s$ -- $\tau_f$ partial derivative}
 1843 
 1844 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_s$ and correlation time $\tau_f$ is
 1845 \begin{equation}
 1846     \frac{\partial^2 J(\omega)}{\partial S^2_s \cdot \partial \tau_f} = 0.
 1847 \end{equation}
 1848 
 1849 
 1850 
 1851 % S2s-ts partial derivative.
 1852 \subsubsection{$S^2_s$ -- $\tau_s$ partial derivative}
 1853 
 1854 The second partial derivative of~\eqref{eq: maths: J(w) model-free ext generic} with respect to the order parameter $S^2_s$ and correlation time $\tau_s$ is
 1855 \begin{equation}
 1856     \frac{\partial^2 J(\omega)}{\partial S^2_s \cdot \partial \tau_s} = -\frac{2}{5} S^2_f \sum_{i=-k}^k c_i \tau_i^2
 1857         \frac{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}.
 1858 \end{equation}
 1859 
 1860 
 1861 
 1862 % tf-tf partial derivative.
 1863 \subsubsection{$\tau_f$ -- $\tau_f$ partial derivative}
 1864 
 1865 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the correlation time $\tau_f$ twice is
 1866 \begin{equation}
 1867     \frac{\partial^2 J(\omega)}{{\partial \tau_f}^2} = -\frac{4}{5} (1 - S^2_f) \sum_{i=-k}^k c_i \tau_i^2
 1868         \frac{(\tau_f + \tau_i)^3  +  3 \omega^2 \tau_i^3 \tau_f (\tau_f + \tau_i)  -  (\omega \tau_i)^4 \tau_f^3}
 1869             {\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^3}
 1870 \end{equation}
 1871 
 1872 
 1873 
 1874 % tf-ts partial derivative.
 1875 \subsubsection{$\tau_f$ -- $\tau_s$ partial derivative}
 1876 
 1877 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the correlation times $\tau_f$ and $\tau_s$ is
 1878 \begin{equation}
 1879     \frac{\partial^2 J(\omega)}{\partial \tau_f \cdot \partial \tau_s} = 0.
 1880 \end{equation}
 1881 
 1882 
 1883 
 1884 % ts-ts partial derivative.
 1885 \subsubsection{$\tau_s$ -- $\tau_s$ partial derivative}
 1886 
 1887 The second partial derivative of~\eqref{eq: maths: J(w) model-free generic} with respect to the correlation time $\tau_s$ twice is
 1888 \begin{equation}
 1889     \frac{\partial^2 J(\omega)}{{\partial \tau_s}^2} = -\frac{4}{5} S^2_f(1 - S^2_s) \sum_{i=-k}^k c_i \tau_i^2
 1890         \frac{(\tau_s + \tau_i)^3  +  3 \omega^2 \tau_i^3 \tau_s (\tau_s + \tau_i)  -  (\omega \tau_i)^4 \tau_s^3}
 1891             {\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^3}
 1892 \end{equation}
 1893 
 1894 
 1895 
 1896 
 1897 % Ellipsoidal diffusion tensor.
 1898 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 1899 
 1900 \newpage
 1901 \section{Ellipsoidal diffusion tensor}
 1902 
 1903 \index{diffusion!ellipsoid (asymmetric)|textbf}
 1904 
 1905 
 1906 
 1907 % The diffusion equation of the ellipsoid.
 1908 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 1909 
 1910 \subsection{The diffusion equation of the ellipsoid} \label{ellipsoid equation}
 1911 
 1912 The correlation function of the Brownian rotational diffusion of an ellipsoid is
 1913 \begin{equation} \label{eq: ellipsoid correlation function}
 1914     C_{\mathrm{O}}(\tau) = \frac{1}{5} \sum^2_{i=-2} c_i e^{-\frac{\tau}{\tau_i}}.
 1915 \end{equation}
 1916 
 1917 \noindent where $c_i$ are the weights of the five exponential terms which are dependent on the orientation of the XH bond vector and $\tau_i$ are the correlation times of the five exponential terms.
 1918 
 1919 
 1920 
 1921 % The weights of the ellipsoid.
 1922 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 1923 
 1924 \subsection{The weights of the ellipsoid}
 1925 
 1926 
 1927 % Definitions.
 1928 \subsubsection{Definitions}
 1929 
 1930 The three direction cosines\index{direction cosine} defining the XH bond vector within the diffusion frame are
 1931 \begin{subequations}
 1932 \begin{align}
 1933     \delta_x &= \widehat{XH} \cdot \widehat{\Diff_x}, \\
 1934     \delta_y &= \widehat{XH} \cdot \widehat{\Diff_y}, \\
 1935     \delta_z &= \widehat{XH} \cdot \widehat{\Diff_z}.
 1936 \end{align}
 1937 \end{subequations}
 1938 
 1939 \noindent Let the set of geometric parameters be
 1940 \begin{equation}
 1941     \Diffgeoset = \{\Diff_{iso}, \Diff_a, \Diff_r\},
 1942 \end{equation}
 1943 
 1944 \noindent and the set of orientational parameters be the Euler angles\index{Euler angles}
 1945 \begin{equation}
 1946     \Difforiset = \{\alpha, \beta, \gamma\}.
 1947 \end{equation}
 1948 
 1949 
 1950 
 1951 % The weights.
 1952 \subsubsection{The weights}
 1953 
 1954 The five weights $c_i$ in the correlation function of the Brownian rotational diffusion of an ellipsoid~\eqref{eq: ellipsoid correlation function} are
 1955 \begin{subequations}
 1956 \begin{align}
 1957  c_{-2} &= \tfrac{1}{4}(d - e),     \\
 1958  c_{-1} &= 3\delta_y^2\delta_z^2,   \\
 1959  c_{0}  &= 3\delta_x^2\delta_z^2,   \\
 1960  c_{1}  &= 3\delta_x^2\delta_y^2,   \\
 1961  c_{2}  &= \tfrac{1}{4}(d + e),
 1962 \end{align}
 1963 \end{subequations}
 1964 
 1965 \noindent where
 1966 \begin{align}
 1967  d &= 3 \left( \delta_x^4 + \delta_y^4 + \delta_z^4 \right) - 1, \\
 1968  e &= \frac{1}{\mathfrak{R}} \bigg[ (1 + 3\Diff_r) \left(\delta_x^4 + 2\delta_y^2\delta_z^2\right)
 1969    + (1 - 3\Diff_r) \left(\delta_y^4 + 2\delta_x^2\delta_z^2\right) - 2 \left(\delta_z^4 + 2\delta_x^2\delta_y^2\right) \bigg].
 1970 \end{align}
 1971 
 1972 \noindent The factor $\mathfrak{R}$ is defined as
 1973 \begin{equation} \label{eq: R}
 1974  \mathfrak{R} = \sqrt{1 + 3\Diff_r^2}.
 1975 \end{equation}
 1976 
 1977 
 1978 
 1979 % The weight gradients of the ellipsoid.
 1980 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 1981 
 1982 \subsection{The weight gradients of the ellipsoid}
 1983 
 1984 
 1985 % Oi partial derivative.
 1986 \subsubsection{$\Difforiset_i$ partial derivative}
 1987 
 1988 The partial derivatives with respect to the orientational parameter $\Difforiset_i$ are
 1989 \begin{subequations}
 1990 \begin{align}
 1991     \frac{\partial c_{-2}}{\partial \Difforiset_i} &= 3 \left( \delta_x^3 \frac{\partial \delta_x}{\partial \Difforiset_i}  +  \delta_y^3 \frac{\partial \delta_y}{\partial \Difforiset_i}  +  \delta_z^3 \frac{\partial \delta_z}{\partial \Difforiset_i} \right) - \frac{\partial e}{\partial \Difforiset_i}, \\
 1992     \frac{\partial c_{-1}}{\partial \Difforiset_i} &= 6 \delta_y \delta_z \left( \delta_y \frac{\partial \delta_z}{\partial \Difforiset_i}  +  \delta_z \frac{\partial \delta_y}{\partial \Difforiset_i} \right), \\
 1993     \frac{\partial c_{0}}{\partial \Difforiset_i}  &= 6 \delta_x \delta_z \left( \delta_x \frac{\partial \delta_z}{\partial \Difforiset_i}  +  \delta_z \frac{\partial \delta_x}{\partial \Difforiset_i} \right), \\
 1994     \frac{\partial c_{1}}{\partial \Difforiset_i}  &= 6 \delta_x \delta_y \left( \delta_x \frac{\partial \delta_y}{\partial \Difforiset_i}  +  \delta_y \frac{\partial \delta_x}{\partial \Difforiset_i} \right), \\
 1995     \frac{\partial c_{2}}{\partial \Difforiset_i}  &= 3 \left( \delta_x^3 \frac{\partial \delta_x}{\partial \Difforiset_i}  +  \delta_y^3 \frac{\partial \delta_y}{\partial \Difforiset_i}  +  \delta_z^3 \frac{\partial \delta_z}{\partial \Difforiset_i} \right) + \frac{\partial e}{\partial \Difforiset_i},
 1996 \end{align}
 1997 \end{subequations}
 1998 
 1999 \noindent where
 2000 \begin{align}
 2001     \frac{\partial e}{\partial \Difforiset_i}  =  \frac{1}{\mathfrak{R}} \Bigg[
 2002         (1 + 3\Diff_r) \left(
 2003             \delta_x^3 \frac{\partial \delta_x}{\partial \Difforiset_i}
 2004             +  \delta_y \delta_z \left( \delta_y \frac{\partial \delta_z}{\partial \Difforiset_i}  +  \delta_z \frac{\partial \delta_y}{\partial \Difforiset_i} \right) \right) & \nonumber \\
 2005         + (1 - 3\Diff_r) \left(
 2006             \delta_y^3 \frac{\partial \delta_y}{\partial \Difforiset_i}
 2007             +  \delta_x \delta_z \left( \delta_x \frac{\partial \delta_z}{\partial \Difforiset_i}  +  \delta_z \frac{\partial \delta_x}{\partial \Difforiset_i} \right) \right) & \nonumber \\
 2008         - 2 \left(
 2009             \delta_z^3 \frac{\partial \delta_z}{\partial \Difforiset_i}
 2010             +  \delta_x \delta_y \left( \delta_x \frac{\partial \delta_y}{\partial \Difforiset_i}  +  \delta_y \frac{\partial \delta_x}{\partial \Difforiset_i} \right) \right) &
 2011     \Bigg].
 2012 \end{align}
 2013 
 2014 
 2015 % tm partial derivative.
 2016 \subsubsection{$\tau_m$ partial derivative}
 2017 
 2018 The partial derivatives with respect to the $\tau_m$ geometric parameter are
 2019 \begin{subequations}
 2020 \begin{align}
 2021     \frac{\partial c_{-2}}{\partial \tau_m} &= 0, \\
 2022     \frac{\partial c_{-1}}{\partial \tau_m} &= 0, \\
 2023     \frac{\partial c_{0}}{\partial \tau_m}  &= 0, \\
 2024     \frac{\partial c_{1}}{\partial \tau_m}  &= 0, \\
 2025     \frac{\partial c_{2}}{\partial \tau_m}  &= 0.
 2026 \end{align}
 2027 \end{subequations}
 2028 
 2029 
 2030 % Da partial derivative.
 2031 \subsubsection{$\Diff_a$ partial derivative}
 2032 
 2033 The partial derivatives with respect to the $\Diff_a$ geometric parameter are
 2034 \begin{subequations}
 2035 \begin{align}
 2036     \frac{\partial c_{-2}}{\partial \Diff_a} &= 0, \\
 2037     \frac{\partial c_{-1}}{\partial \Diff_a} &= 0, \\
 2038     \frac{\partial c_{0}}{\partial \Diff_a}  &= 0, \\
 2039     \frac{\partial c_{1}}{\partial \Diff_a}  &= 0, \\
 2040     \frac{\partial c_{2}}{\partial \Diff_a}  &= 0.
 2041 \end{align}
 2042 \end{subequations}
 2043 
 2044 
 2045 % Dr partial derivative.
 2046 \subsubsection{$\Diff_r$ partial derivative}
 2047 
 2048 The partial derivatives with respect to the $\Diff_r$ geometric parameter are
 2049 \begin{subequations}
 2050 \begin{align}
 2051     \frac{\partial c_{-2}}{\partial \Diff_r} &= -\frac{3}{4} \frac{\partial e}{\partial \Diff_r}, \\
 2052     \frac{\partial c_{-1}}{\partial \Diff_r} &= 0, \\
 2053     \frac{\partial c_{0}}{\partial \Diff_r}  &= 0, \\
 2054     \frac{\partial c_{1}}{\partial \Diff_r}  &= 0, \\
 2055     \frac{\partial c_{2}}{\partial \Diff_r}  &= \frac{3}{4} \frac{\partial e}{\partial \Diff_r},
 2056 \end{align}
 2057 \end{subequations}
 2058 
 2059 \noindent where
 2060 \begin{equation}
 2061     \frac{\partial e}{\partial \Diff_r} = \frac{1}{\mathfrak{R}^3} \bigg[ (1 - \Diff_r) \left(\delta_x^4 + 2\delta_y^2\delta_z^2\right)
 2062         - (1 + \Diff_r) \left(\delta_y^4 + 2\delta_x^2\delta_z^2\right) + 2 \Diff_r \left(\delta_z^4 + 2\delta_x^2\delta_y^2\right) \bigg].
 2063 \end{equation}
 2064 
 2065 
 2066 
 2067 % The weight Hessians of the ellipsoid.
 2068 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2069 
 2070 \newpage
 2071 \subsection{The weight Hessians of the ellipsoid}
 2072 
 2073 
 2074 % Oi-Oj partial derivative.
 2075 \subsubsection{$\Difforiset_i$ -- $\Difforiset_j$ partial derivative}
 2076 
 2077 The second partial derivatives with respect to the orientational parameters $\Difforiset_i$ and $\Difforiset_j$ are
 2078 \begin{subequations}
 2079 \begin{align}
 2080     \frac{\partial^2 c_{-2}}{\partial \Difforiset_i \cdot \partial \Difforiset_j}  =  3 \Bigg(
 2081         \delta_x^2 \left( \delta_x \frac{\partial^2 \delta_x}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2082             +  3 \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right) & \nonumber \\
 2083         +  \delta_y^2 \left( \delta_y \frac{\partial^2 \delta_y}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2084             +  3 \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j} \right) & \nonumber \\
 2085         +  \delta_z^2 \left( \delta_z \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2086             +  3 \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j} \right) &
 2087         \Bigg)  -  \frac{\partial^2 e}{\partial \Difforiset_i \cdot \partial \Difforiset_j},
 2088 \end{align}
 2089 \begin{multline}
 2090     \frac{\partial^2 c_{-1}}{\partial \Difforiset_i \cdot \partial \Difforiset_j} = 
 2091         6 \delta_y^2 \left( \delta_z \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2092             +  \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j} \right) \\
 2093         +  12 \delta_y \delta_z \left( \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j}
 2094             +  \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j} \right) \\
 2095         +  6 \delta_z^2 \left( \delta_y \frac{\partial^2 \delta_y}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2096             +  \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j} \right),
 2097 \end{multline}
 2098 \begin{multline}
 2099     \frac{\partial^2 c_{0}}{\partial \Difforiset_i \cdot \partial \Difforiset_j}  = 
 2100         6 \delta_x^2 \left( \delta_z \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2101             +  \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j} \right) \\
 2102         +  12 \delta_x \delta_z \left( \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j}
 2103             +  \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right) \\
 2104         +  6 \delta_z^2 \left( \delta_x \frac{\partial^2 \delta_x}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2105             +  \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right),
 2106 \end{multline}
 2107 \begin{multline}
 2108     \frac{\partial^2 c_{1}}{\partial \Difforiset_i \cdot \partial \Difforiset_j}  = 
 2109         6 \delta_x^2 \left( \delta_y \frac{\partial^2 \delta_y}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2110             +  \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j} \right) \\
 2111         +  12 \delta_x \delta_y \left( \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j}
 2112             +  \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right) \\
 2113         +  6 \delta_y^2 \left( \delta_x \frac{\partial^2 \delta_x}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2114             +  \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right),
 2115 \end{multline}
 2116 \begin{align}
 2117     \frac{\partial^2 c_{2}}{\partial \Difforiset_i \cdot \partial \Difforiset_j}  =  3 \Bigg(
 2118         \delta_x^2 \left( \delta_x \frac{\partial^2 \delta_x}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2119             +  3 \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right) & \nonumber \\
 2120         +  \delta_y^2 \left( \delta_y \frac{\partial^2 \delta_y}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2121             +  3 \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j} \right) & \nonumber \\
 2122         +  \delta_z^2 \left( \delta_z \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2123             +  3 \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j} \right) &
 2124         \Bigg)  +  \frac{\partial^2 e}{\partial \Difforiset_i \cdot \partial \Difforiset_j},
 2125 \end{align}
 2126 \end{subequations}
 2127 
 2128 \noindent where
 2129 \begin{align}
 2130     \frac{\partial^2 e}{\partial \Difforiset_i \cdot \partial \Difforiset_j}  =  \frac{1}{\mathfrak{R}} \Bigg[
 2131         (1 + 3\Diff_r) \Bigg(
 2132             \delta_x^2 \left( \delta_x \frac{\partial^2 \delta_x}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2133                 +  3 \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right) \phantom{\Bigg)} & \nonumber \\
 2134             +  \delta_y^2 \left( \delta_z \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2135                 +  \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j} \right) \phantom{\Bigg)} & \nonumber \\
 2136             +  \delta_z^2 \left( \delta_y \frac{\partial^2 \delta_y}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2137                 +  \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j} \right) \phantom{\Bigg)} & \nonumber \\
 2138             +  2 \delta_y \delta_z \left( \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j}
 2139                 +  \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j} \right)
 2140         \Bigg) & \nonumber \\
 2141         +  (1 - 3\Diff_r) \Bigg(
 2142             \delta_y^2 \left( \delta_y \frac{\partial^2 \delta_y}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2143                 +  3 \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j} \right) \phantom{\Bigg)} & \nonumber \\
 2144             +  \delta_x^2 \left( \delta_z \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2145                 +  \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j} \right) \phantom{\Bigg)} & \nonumber \\
 2146             +  \delta_z^2 \left( \delta_x \frac{\partial^2 \delta_x}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2147                 +  \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right) \phantom{\Bigg)} & \nonumber \\
 2148             +  2 \delta_x \delta_z \left( \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j}
 2149                 +  \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right)
 2150         \Bigg) & \nonumber \\
 2151         -  2 \Bigg(
 2152             \delta_z^2 \left( \delta_z \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2153                 +  3 \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j} \right) \phantom{\Bigg)} & \nonumber \\
 2154             +  \delta_x^2 \left( \delta_y \frac{\partial^2 \delta_y}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2155                 +  \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j} \right) \phantom{\Bigg)} & \nonumber \\
 2156             +  \delta_y^2 \left( \delta_x \frac{\partial^2 \delta_x}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 2157                 +  \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right) \phantom{\Bigg)} & \nonumber \\
 2158             +  2 \delta_x \delta_y \left( \frac{\partial \delta_x}{\partial \Difforiset_i} \cdot \frac{\partial \delta_y}{\partial \Difforiset_j}
 2159                 +  \frac{\partial \delta_y}{\partial \Difforiset_i} \cdot \frac{\partial \delta_x}{\partial \Difforiset_j} \right)
 2160         \Bigg) &
 2161     \Bigg].
 2162 \end{align}
 2163 
 2164 
 2165 
 2166 % Oi-tm partial derivative.
 2167 \subsubsection{$\Difforiset_i$ -- $\tau_m$ partial derivative}
 2168 
 2169 The second partial derivatives with respect to the orientational parameter $\Difforiset_i$ and the geometric parameter $\tau_m$ are
 2170 \begin{subequations}
 2171 \begin{align}
 2172     \frac{\partial^2 c_{-2}}{\partial \Difforiset_i \cdot \partial \tau_m}  &=  0, \\
 2173     \frac{\partial^2 c_{-1}}{\partial \Difforiset_i \cdot \partial \tau_m} &= 0, \\
 2174     \frac{\partial^2 c_{0}}{\partial \Difforiset_i \cdot \partial \tau_m}  &= 0, \\
 2175     \frac{\partial^2 c_{1}}{\partial \Difforiset_i \cdot \partial \tau_m}  &= 0, \\
 2176     \frac{\partial^2 c_{2}}{\partial \Difforiset_i \cdot \partial \tau_m}  &= 0.
 2177 \end{align}
 2178 \end{subequations}
 2179 
 2180 
 2181 
 2182 % Oi-Da partial derivative.
 2183 \subsubsection{$\Difforiset_i$ -- $\Diff_a$ partial derivative}
 2184 
 2185 The second partial derivatives with respect to the orientational parameter $\Difforiset_i$ and the geometric parameter $\Diff_a$ are
 2186 \begin{subequations}
 2187 \begin{align}
 2188     \frac{\partial^2 c_{-2}}{\partial \Difforiset_i \cdot \partial \Diff_a}  &=  0, \\
 2189     \frac{\partial^2 c_{-1}}{\partial \Difforiset_i \cdot \partial \Diff_a} &= 0, \\
 2190     \frac{\partial^2 c_{0}}{\partial \Difforiset_i \cdot \partial \Diff_a}  &= 0, \\
 2191     \frac{\partial^2 c_{1}}{\partial \Difforiset_i \cdot \partial \Diff_a}  &= 0, \\
 2192     \frac{\partial^2 c_{2}}{\partial \Difforiset_i \cdot \partial \Diff_a}  &= 0.
 2193 \end{align}
 2194 \end{subequations}
 2195 
 2196 
 2197 
 2198 % Oi-Dr partial derivative.
 2199 \subsubsection{$\Difforiset_i$ -- $\Diff_r$ partial derivative}
 2200 
 2201 The second partial derivatives with respect to the orientational parameter $\Difforiset_i$ and the geometric parameter $\Diff_r$ are
 2202 \begin{subequations}
 2203 \begin{align}
 2204     \frac{\partial^2 c_{-2}}{\partial \Difforiset_i \cdot \partial \Diff_r}  &=  -3 \frac{\partial^2 e}{\partial \Difforiset_i \cdot \partial \Diff_r}, \\
 2205     \frac{\partial^2 c_{-1}}{\partial \Difforiset_i \cdot \partial \Diff_r} &= 0, \\
 2206     \frac{\partial^2 c_{0}}{\partial \Difforiset_i \cdot \partial \Diff_r}  &= 0, \\
 2207     \frac{\partial^2 c_{1}}{\partial \Difforiset_i \cdot \partial \Diff_r}  &= 0, \\
 2208     \frac{\partial^2 c_{2}}{\partial \Difforiset_i \cdot \partial \Diff_r}  &= 3 \frac{\partial^2 e}{\partial \Difforiset_i \cdot \partial \Diff_r},
 2209 \end{align}
 2210 \end{subequations}
 2211 
 2212 \noindent where
 2213 \begin{align}
 2214     \frac{\partial^2 e}{\partial \Difforiset_i \cdot \partial \Diff_r}  =  \frac{1}{\mathfrak{R}^3} \Bigg[
 2215         (1 - \Diff_r) \left(
 2216             \delta_x^3 \frac{\partial \delta_x}{\partial \Difforiset_i}
 2217             +  \delta_y \delta_z \left( \delta_y \frac{\partial \delta_z}{\partial \Difforiset_i}  +  \delta_z \frac{\partial \delta_y}{\partial \Difforiset_i} \right) \right) & \nonumber \\
 2218         -  (1 + \Diff_r) \left(
 2219             \delta_y^3 \frac{\partial \delta_y}{\partial \Difforiset_i}
 2220             +  \delta_x \delta_z \left( \delta_x \frac{\partial \delta_z}{\partial \Difforiset_i}  +  \delta_z \frac{\partial \delta_x}{\partial \Difforiset_i} \right) \right) & \nonumber \\
 2221         +  2 \Diff_r \left(
 2222             \delta_z^3 \frac{\partial \delta_z}{\partial \Difforiset_i}
 2223             +  \delta_x \delta_y \left( \delta_x \frac{\partial \delta_y}{\partial \Difforiset_i}  +  \delta_y \frac{\partial \delta_x}{\partial \Difforiset_i} \right) \right) &
 2224     \Bigg].
 2225 \end{align}
 2226 
 2227 
 2228 
 2229 % tm-tm partial derivative.
 2230 \subsubsection{$\tau_m$ -- $\tau_m$ partial derivative}
 2231 
 2232 The second partial derivatives with respect to the geometric parameter $\tau_m$ twice are
 2233 \begin{subequations}
 2234 \begin{align}
 2235     \frac{\partial^2 c_{-2}}{{\partial \tau_m}^2}  &=  0, \\
 2236     \frac{\partial^2 c_{-1}}{{\partial \tau_m}^2} &= 0, \\
 2237     \frac{\partial^2 c_{0}}{{\partial \tau_m}^2}  &= 0, \\
 2238     \frac{\partial^2 c_{1}}{{\partial \tau_m}^2}  &= 0, \\
 2239     \frac{\partial^2 c_{2}}{{\partial \tau_m}^2}  &= 0.
 2240 \end{align}
 2241 \end{subequations}
 2242 
 2243 
 2244 
 2245 % tm-Da partial derivative.
 2246 \subsubsection{$\tau_m$ -- $\Diff_a$ partial derivative}
 2247 
 2248 The second partial derivatives with respect to the geometric parameters $\tau_m$ and $\Diff_a$ are
 2249 \begin{subequations}
 2250 \begin{align}
 2251     \frac{\partial^2 c_{-2}}{\partial \tau_m \cdot \partial \Diff_a}  &=  0, \\
 2252     \frac{\partial^2 c_{-1}}{\partial \tau_m \cdot \partial \Diff_a} &= 0, \\
 2253     \frac{\partial^2 c_{0}}{\partial \tau_m \cdot \partial \Diff_a}  &= 0, \\
 2254     \frac{\partial^2 c_{1}}{\partial \tau_m \cdot \partial \Diff_a}  &= 0, \\
 2255     \frac{\partial^2 c_{2}}{\partial \tau_m \cdot \partial \Diff_a}  &= 0.
 2256 \end{align}
 2257 \end{subequations}
 2258 
 2259 
 2260 
 2261 % tm-Dr partial derivative.
 2262 \subsubsection{$\tau_m$ -- $\Diff_r$ partial derivative}
 2263 
 2264 The second partial derivatives with respect to the geometric parameters $\tau_m$ and $\Diff_r$ are
 2265 \begin{subequations}
 2266 \begin{align}
 2267     \frac{\partial^2 c_{-2}}{\partial \tau_m \cdot \partial \Diff_r}  &=  0, \\
 2268     \frac{\partial^2 c_{-1}}{\partial \tau_m \cdot \partial \Diff_r} &= 0, \\
 2269     \frac{\partial^2 c_{0}}{\partial \tau_m \cdot \partial \Diff_r}  &= 0, \\
 2270     \frac{\partial^2 c_{1}}{\partial \tau_m \cdot \partial \Diff_r}  &= 0, \\
 2271     \frac{\partial^2 c_{2}}{\partial \tau_m \cdot \partial \Diff_r}  &= 0.
 2272 \end{align}
 2273 \end{subequations}
 2274 
 2275 
 2276 
 2277 % Da-Da partial derivative.
 2278 \subsubsection{$\Diff_a$ -- $\Diff_a$ partial derivative}
 2279 
 2280 The second partial derivatives with respect to the geometric parameter $\Diff_a$ twice are
 2281 \begin{subequations}
 2282 \begin{align}
 2283     \frac{\partial^2 c_{-2}}{{\partial \Diff_a}^2}  &=  0, \\
 2284     \frac{\partial^2 c_{-1}}{{\partial \Diff_a}^2} &= 0, \\
 2285     \frac{\partial^2 c_{0}}{{\partial \Diff_a}^2}  &= 0, \\
 2286     \frac{\partial^2 c_{1}}{{\partial \Diff_a}^2}  &= 0, \\
 2287     \frac{\partial^2 c_{2}}{{\partial \Diff_a}^2}  &= 0.
 2288 \end{align}
 2289 \end{subequations}
 2290 
 2291 
 2292 
 2293 % Da-Dr partial derivative.
 2294 \subsubsection{$\Diff_a$ -- $\Diff_r$ partial derivative}
 2295 
 2296 The second partial derivatives with respect to the geometric parameters $\Diff_a$ and $\Diff_r$ are
 2297 \begin{subequations}
 2298 \begin{align}
 2299     \frac{\partial^2 c_{-2}}{\partial \Diff_a \cdot \partial \Diff_r}  &=  0, \\
 2300     \frac{\partial^2 c_{-1}}{\partial \Diff_a \cdot \partial \Diff_r} &= 0, \\
 2301     \frac{\partial^2 c_{0}}{\partial \Diff_a \cdot \partial \Diff_r}  &= 0, \\
 2302     \frac{\partial^2 c_{1}}{\partial \Diff_a \cdot \partial \Diff_r}  &= 0, \\
 2303     \frac{\partial^2 c_{2}}{\partial \Diff_a \cdot \partial \Diff_r}  &= 0.
 2304 \end{align}
 2305 \end{subequations}
 2306 
 2307 
 2308 
 2309 % Dr-Dr partial derivative.
 2310 \subsubsection{$\Diff_r$ -- $\Diff_r$ partial derivative}
 2311 
 2312 The second partial derivatives with respect to the geometric parameter $\Diff_r$ twice are
 2313 \begin{subequations}
 2314 \begin{align}
 2315     \frac{\partial^2 c_{-2}}{{\partial \Diff_r}^2}  &=  - \frac{3}{4} \frac{\partial^2 e}{\partial \Diff_r^2}, \\
 2316     \frac{\partial^2 c_{-1}}{{\partial \Diff_r}^2} &= 0, \\
 2317     \frac{\partial^2 c_{0}}{{\partial \Diff_r}^2}  &= 0, \\
 2318     \frac{\partial^2 c_{1}}{{\partial \Diff_r}^2}  &= 0, \\
 2319     \frac{\partial^2 c_{2}}{{\partial \Diff_r}^2}  &= \frac{3}{4} \frac{\partial^2 e}{\partial \Diff_r^2},
 2320 \end{align}
 2321 \end{subequations}
 2322 
 2323 \noindent where
 2324 \begin{align}
 2325     \frac{\partial^2 e}{\partial \Diff_r^2}  =  \frac{1}{\mathfrak{R}^5} \bigg[
 2326         (6\Diff_r^2 - 9\Diff_r - 1) \left(\delta_x^4 + 2\delta_y^2\delta_z^2\right) & \nonumber \\
 2327         + (6\Diff_r^2 + 9\Diff_r - 1) \left(\delta_y^4 + 2\delta_x^2\delta_z^2\right) & \nonumber \\
 2328         - 2 (6\Diff_r^2 - 1) \left(\delta_z^4 + 2\delta_x^2\delta_y^2\right) & \bigg].
 2329 \end{align}
 2330 
 2331 
 2332 
 2333 % The correlation times of the ellipsoid.
 2334 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2335 
 2336 \newpage
 2337 \subsection{The correlation times of the ellipsoid}
 2338 
 2339 The five correlation times $\tau_i$ in the correlation function of the Brownian rotational diffusion of an ellipsoid~\eqref{eq: ellipsoid correlation function} on page~\pageref{eq: ellipsoid correlation function} are
 2340 \begin{subequations}
 2341 \begin{align}
 2342     \tau_{-2} &= (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-1}, \\
 2343     \tau_{-1} &= (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-1}, \\
 2344     \tau_{0}  &= (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-1}, \\
 2345     \tau_{1}  &= (6\Diff_{iso} + 2\Diff_a)^{-1}, \\
 2346     \tau_{2}  &= (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-1},
 2347 \end{align}
 2348 \end{subequations}
 2349 
 2350 \noindent where $\mathfrak{R}$ is defined in Equation~\eqref{eq: R} on page~\pageref{eq: R}.
 2351 
 2352 
 2353 
 2354 % The correlation time gradients of the ellipsoid.
 2355 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2356 
 2357 \subsection{The correlation time gradients of the ellipsoid}
 2358 
 2359 
 2360 % tm partial derivative.
 2361 \subsubsection{$\tau_m$ partial derivative}
 2362 
 2363 The partial derivatives with respect to the geometric parameter $\tau_m$ are
 2364 \begin{subequations}
 2365 \begin{align}
 2366     \frac{\partial \tau_{-2}}{\partial \tau_m} &= {\tau_m}^{-2} (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-2}, \\
 2367     \frac{\partial \tau_{-1}}{\partial \tau_m} &= {\tau_m}^{-2} (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-2}, \\
 2368     \frac{\partial \tau_{0}}{\partial \tau_m}  &= {\tau_m}^{-2} (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-2}, \\
 2369     \frac{\partial \tau_{1}}{\partial \tau_m}  &= {\tau_m}^{-2} (6\Diff_{iso} + 2\Diff_a)^{-2}, \\
 2370     \frac{\partial \tau_{2}}{\partial \tau_m}  &= {\tau_m}^{-2} (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-2}.
 2371 \end{align}
 2372 \end{subequations}
 2373 
 2374 
 2375 
 2376 % Da partial derivative.
 2377 \subsubsection{$\Diff_a$ partial derivative}
 2378 
 2379 The partial derivatives with respect to the geometric parameter $\Diff_a$ are
 2380 \begin{subequations}
 2381 \begin{align}
 2382     \frac{\partial \tau_{-2}}{\partial \Diff_a} &= 2\mathfrak{R} (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-2}, \\
 2383     \frac{\partial \tau_{-1}}{\partial \Diff_a} &= (1 + 3\Diff_r) (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-2}, \\
 2384     \frac{\partial \tau_{0}}{\partial \Diff_a}  &= (1 - 3\Diff_r) (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-2}, \\
 2385     \frac{\partial \tau_{1}}{\partial \Diff_a}  &= -2 (6\Diff_{iso} + 2\Diff_a)^{-2}, \\
 2386     \frac{\partial \tau_{2}}{\partial \Diff_a}  &= -2\mathfrak{R} (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-2}.
 2387 \end{align}
 2388 \end{subequations}
 2389 
 2390 
 2391 
 2392 % Dr partial derivative.
 2393 \subsubsection{$\Diff_r$ partial derivative}
 2394 
 2395 The partial derivatives with respect to the geometric parameter $\Diff_r$ are
 2396 \begin{subequations}
 2397 \begin{align}
 2398     \frac{\partial \tau_{-2}}{\partial \Diff_r} &= 6\frac{\Diff_a \Diff_r}{\mathfrak{R}} (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-2}, \\
 2399     \frac{\partial \tau_{-1}}{\partial \Diff_r} &= 3\Diff_a (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-2}, \\
 2400     \frac{\partial \tau_{0}}{\partial \Diff_r}  &= -3\Diff_a (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-2}, \\
 2401     \frac{\partial \tau_{1}}{\partial \Diff_r}  &= 0, \\
 2402     \frac{\partial \tau_{2}}{\partial \Diff_r}  &= -6\frac{\Diff_a \Diff_r}{\mathfrak{R}} (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-2}.
 2403 \end{align}
 2404 \end{subequations}
 2405 
 2406 
 2407 
 2408 % The correlation time Hessians of the ellipsoid.
 2409 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2410 
 2411 \newpage
 2412 \subsection{The correlation time Hessians of the ellipsoid}
 2413 
 2414 
 2415 % tm-tm partial derivative.
 2416 \subsubsection{$\tau_m$ -- $\tau_m$ partial derivative}
 2417 
 2418 The second partial derivatives with respect to the geometric parameter $\tau_m$ twice are
 2419 \begin{subequations}
 2420 \begin{align}
 2421     \frac{\partial^2 \tau_{-2}}{{\partial \tau_m}^2} &= 2{\tau_m}^{-4} (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-3}
 2422         - 2{\tau_m}^{-3} (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-2}, \\
 2423     \frac{\partial^2 \tau_{-1}}{{\partial \tau_m}^2} &= 2{\tau_m}^{-4} (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-3}
 2424         - 2{\tau_m}^{-3} (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-2}, \\
 2425     \frac{\partial^2 \tau_{0}}{{\partial \tau_m}^2}  &= 2{\tau_m}^{-4} (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-3}
 2426         - 2{\tau_m}^{-3} (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-2}, \\
 2427     \frac{\partial^2 \tau_{1}}{{\partial \tau_m}^2}  &= 2{\tau_m}^{-4} (6\Diff_{iso} + 2\Diff_a)^{-3}
 2428         - 2{\tau_m}^{-3} (6\Diff_{iso} + 2\Diff_a)^{-2}, \\
 2429     \frac{\partial^2 \tau_{2}}{{\partial \tau_m}^2}  &= 2{\tau_m}^{-4} (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-3}
 2430         - 2{\tau_m}^{-3} (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-2}.
 2431 \end{align}
 2432 \end{subequations}
 2433 
 2434 
 2435 
 2436 % tm-Da partial derivative.
 2437 \subsubsection{$\tau_m$ -- $\Diff_a$ partial derivative}
 2438 
 2439 The second partial derivatives with respect to the geometric parameters $\tau_m$ and $\Diff_a$ are
 2440 \begin{subequations}
 2441 \begin{align}
 2442     \frac{\partial^2 \tau_{-2}}{\partial \tau_m \cdot \partial \Diff_a} &= 4\mathfrak{R} {\tau_m}^{-2} (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-3}, \\
 2443     \frac{\partial^2 \tau_{-1}}{\partial \tau_m \cdot \partial \Diff_a} &= 2(1 + 3\Diff_r){\tau_m}^{-2} (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-3}, \\
 2444     \frac{\partial^2 \tau_{0}}{\partial \tau_m \cdot \partial \Diff_a}  &= 2(1 - 3\Diff_r){\tau_m}^{-2} (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-3}, \\
 2445     \frac{\partial^2 \tau_{1}}{\partial \tau_m \cdot \partial \Diff_a}  &= -4{\tau_m}^{-2} (6\Diff_{iso} + 2\Diff_a)^{-3}, \\
 2446     \frac{\partial^2 \tau_{2}}{\partial \tau_m \cdot \partial \Diff_a}  &= -4\mathfrak{R} {\tau_m}^{-2} (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-3}.
 2447 \end{align}
 2448 \end{subequations}
 2449 
 2450 
 2451 
 2452 % tm-Dr partial derivative.
 2453 \subsubsection{$\tau_m$ -- $\Diff_r$ partial derivative}
 2454 
 2455 The second partial derivatives with respect to the geometric parameters $\tau_m$ and $\Diff_r$ are
 2456 \begin{subequations}
 2457 \begin{align}
 2458     \frac{\partial^2 \tau_{-2}}{\partial \tau_m \cdot \partial \Diff_r} &= 12 \frac{\Diff_a \Diff_r}{\mathfrak{R}} {\tau_m}^{-2} (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-3}, \\
 2459     \frac{\partial^2 \tau_{-1}}{\partial \tau_m \cdot \partial \Diff_r} &= 6 \Diff_a {\tau_m}^{-2} (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-3}, \\
 2460     \frac{\partial^2 \tau_{0}}{\partial \tau_m \cdot \partial \Diff_r}  &= -6 \Diff_a {\tau_m}^{-2} (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-3}, \\
 2461     \frac{\partial^2 \tau_{1}}{\partial \tau_m \cdot \partial \Diff_r}  &= 0, \\
 2462     \frac{\partial^2 \tau_{2}}{\partial \tau_m \cdot \partial \Diff_r}  &= -12 \frac{\Diff_a \Diff_r}{\mathfrak{R}} {\tau_m}^{-2} (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-3}.
 2463 \end{align}
 2464 \end{subequations}
 2465 
 2466 
 2467 
 2468 % Da-Da partial derivative.
 2469 \subsubsection{$\Diff_a$ -- $\Diff_a$ partial derivative}
 2470 
 2471 The second partial derivatives with respect to the geometric parameter $\Diff_a$ twice are
 2472 \begin{subequations}
 2473 \begin{align}
 2474     \frac{\partial^2 \tau_{-2}}{{\partial \Diff_a}^2} &= 8\mathfrak{R}^2 (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-3}, \\
 2475     \frac{\partial^2 \tau_{-1}}{{\partial \Diff_a}^2} &= 2(1 + 3\Diff_r)^2 (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-3}, \\
 2476     \frac{\partial^2 \tau_{0}}{{\partial \Diff_a}^2}  &= 2(1 - 3\Diff_r)^2 (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-3}, \\
 2477     \frac{\partial^2 \tau_{1}}{{\partial \Diff_a}^2}  &= 8 (6\Diff_{iso} + 2\Diff_a)^{-3}, \\
 2478     \frac{\partial^2 \tau_{2}}{{\partial \Diff_a}^2}  &= 8\mathfrak{R}^2 (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-3}.
 2479 \end{align}
 2480 \end{subequations}
 2481 
 2482 
 2483 
 2484 % Da-Dr partial derivative.
 2485 \subsubsection{$\Diff_a$ -- $\Diff_r$ partial derivative}
 2486 
 2487 The second partial derivatives with respect to the geometric parameters $\Diff_a$ and $\Diff_r$ are
 2488 \begin{subequations}
 2489 \begin{align}
 2490     \frac{\partial^2 \tau_{-2}}{\partial \Diff_a \cdot \partial \Diff_r} &= 24 \Diff_a \Diff_r (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-3}
 2491         + 6 \frac{\Diff_r}{\mathfrak{R}} (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-2}, \\
 2492     \frac{\partial^2 \tau_{-1}}{\partial \Diff_a \cdot \partial \Diff_r} &= 6\Diff_a(1 + 3\Diff_r) (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-3}
 2493         + 3 (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-2}, \\
 2494     \frac{\partial^2 \tau_{0}}{\partial \Diff_a \cdot \partial \Diff_r}  &= -6\Diff_a(1 - 3\Diff_r) (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-3}
 2495         - 3 (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-2}, \\
 2496     \frac{\partial^2 \tau_{1}}{\partial \Diff_a \cdot \partial \Diff_r}  &= 0, \\
 2497     \frac{\partial^2 \tau_{2}}{\partial \Diff_a \cdot \partial \Diff_r}  &= 24 \Diff_a \Diff_r (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-3}
 2498         - 6 \frac{\Diff_r}{\mathfrak{R}} (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-2}.
 2499 \end{align}
 2500 \end{subequations}
 2501 
 2502 
 2503 
 2504 % Dr-Dr partial derivative.
 2505 \subsubsection{$\Diff_r$ -- $\Diff_r$ partial derivative}
 2506 
 2507 The second partial derivatives with respect to the geometric parameter $\Diff_r$ twice are
 2508 \begin{subequations}
 2509 \begin{align}
 2510     \frac{\partial^2 \tau_{-2}}{{\partial \Diff_r}^2} &=
 2511         72 \left( \frac{\Diff_a \Diff_r}{\mathfrak{R}} \right)^2 (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-3}
 2512         + 6 \frac{\Diff_a}{\mathfrak{R}^3} (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-2}, \\
 2513     \frac{\partial^2 \tau_{-1}}{{\partial \Diff_r}^2} &= 18\Diff_a^2 (6\Diff_{iso} - \Diff_a (1 + 3\Diff_r))^{-3}, \\
 2514     \frac{\partial^2 \tau_{0}}{{\partial \Diff_r}^2}  &= 18\Diff_a^2 (6\Diff_{iso} - \Diff_a (1 - 3\Diff_r))^{-3}, \\
 2515     \frac{\partial^2 \tau_{1}}{{\partial \Diff_r}^2}  &= 0, \\
 2516     \frac{\partial^2 \tau_{2}}{{\partial \Diff_r}^2}  &= 
 2517         72 \left( \frac{\Diff_a \Diff_r}{\mathfrak{R}} \right)^2 (6\Diff_{iso} - 2\Diff_a\mathfrak{R})^{-3}
 2518         - 6 \frac{\Diff_a}{\mathfrak{R}^3} (6\Diff_{iso} + 2\Diff_a\mathfrak{R})^{-2}.
 2519 \end{align}
 2520 \end{subequations}
 2521 
 2522 
 2523 
 2524 
 2525 % Spheroidal diffusion tensor.
 2526 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 2527 
 2528 \newpage
 2529 \section{Spheroidal diffusion tensor}
 2530 
 2531 \index{diffusion!spheroid (axially symmetric)|textbf}
 2532 
 2533 
 2534 
 2535 % The diffusion equation of the spheroid.
 2536 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2537 
 2538 \subsection{The diffusion equation of the spheroid} \label{spheroid equation}
 2539 
 2540 The correlation function of the Brownian rotational diffusion of a spheroid is
 2541 \begin{equation} \label{eq: spheroid correlation function}
 2542     C_{\mathrm{O}}(\tau) = \frac{1}{5} \sum^1_{i=-1} c_i e^{-\frac{\tau}{\tau_i}}.
 2543 \end{equation}
 2544 
 2545 \noindent where $c_i$ are the weights of the three exponential terms which are dependent on the orientation of the XH bond vector and $\tau_i$ are the correlation times of the three exponential terms.
 2546 
 2547 
 2548 
 2549 % The weights of the spheroid.
 2550 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2551 
 2552 \subsection{The weights of the spheroid}
 2553 
 2554 
 2555 % Definitions.
 2556 \subsubsection{Definitions}
 2557 
 2558 The direction cosine\index{direction cosine} defining the XH bond vector within the spheroidal diffusion frame is
 2559 \begin{equation}
 2560     \delta_z = \widehat{XH} \cdot \widehat{\Diff_z}.
 2561 \end{equation}
 2562 
 2563 \noindent Let the set of geometric parameters be
 2564 \begin{equation}
 2565     \Diffgeoset = \{\Diff_{iso}, \Diff_a\},
 2566 \end{equation}
 2567 
 2568 \noindent and the set of orientational parameters be the spherical angles\index{spherical angles}
 2569 \begin{equation}
 2570     \Difforiset = \{\theta, \phi\}.
 2571 \end{equation}
 2572 
 2573 
 2574 
 2575 % The weights.
 2576 \subsubsection{The weights}
 2577 
 2578 The three spheroid weights $c_i$ in the correlation function of the Brownian rotational diffusion of a spheroid~\eqref{eq: spheroid correlation function} are
 2579 \begin{subequations}
 2580 \begin{align}
 2581  c_{-1} &= \tfrac{1}{4} (3\delta_z^2 - 1)^2,   \\
 2582  c_{0}  &= 3\delta_z^2(1 - \delta_z^2),   \\
 2583  c_{1}  &= \tfrac{3}{4} (\delta_z^2 - 1)^2.
 2584 \end{align}
 2585 \end{subequations}
 2586 
 2587 
 2588 
 2589 % The weight gradients of the spheroid.
 2590 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2591 
 2592 \subsection{The weight gradients of the spheroid}
 2593 
 2594 
 2595 % Oi partial derivative.
 2596 \subsubsection{$\Difforiset_i$ partial derivative}
 2597 
 2598 The partial derivatives with respect to the orientational parameter $\Difforiset_i$ are
 2599 \begin{subequations}
 2600 \begin{align}
 2601     \frac{\partial c_{-1}}{\partial \Difforiset_i} &= 3\delta_z (3\delta_z^2 - 1) \frac{\partial \delta_z}{\partial \Difforiset_i}, \\
 2602     \frac{\partial c_{0}}{\partial \Difforiset_i}  &= 6\delta_z (1 - 2\delta_z^2) \frac{\partial \delta_z}{\partial \Difforiset_i}, \\
 2603     \frac{\partial c_{1}}{\partial \Difforiset_i}  &= 3\delta_z (\delta_z^2 - 1)  \frac{\partial \delta_z}{\partial \Difforiset_i}.
 2604 \end{align}
 2605 \end{subequations}
 2606 
 2607 
 2608 
 2609 % The weight Hessians of the spheroid.
 2610 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2611 
 2612 \subsection{The weight Hessians of the spheroid}
 2613 
 2614 
 2615 % Oi-Oj partial derivative.
 2616 \subsubsection{$\Difforiset_i$ -- $\Difforiset_j$ partial derivative}
 2617 
 2618 The second partial derivatives with respect to the orientational parameters $\Difforiset_i$ and $\Difforiset_j$ are
 2619 \begin{subequations}
 2620 \begin{align}
 2621     \frac{\partial^2 c_{-1}}{\partial \Difforiset_i \cdot \partial \Difforiset_j}  &=  3 \left(
 2622         (9\delta_z^2 - 1) \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j}
 2623         +  \delta_z (3\delta_z^2 - 1) \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j} \right), \\
 2624     \frac{\partial^2 c_{0}}{\partial \Difforiset_i \cdot \partial \Difforiset_j}  &=  6 \left(
 2625         (1 - 6\delta_z^2) \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j}
 2626         +  \delta_z (1 - 2\delta_z^2) \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j} \right), \\
 2627     \frac{\partial^2 c_{1}}{\partial \Difforiset_i \cdot \partial \Difforiset_j}  &=  3 \left(
 2628         (3\delta_z^2 - 1) \frac{\partial \delta_z}{\partial \Difforiset_i} \cdot \frac{\partial \delta_z}{\partial \Difforiset_j}
 2629         +  \delta_z (\delta_z^2 - 1) \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j} \right).
 2630 \end{align}
 2631 \end{subequations}
 2632 
 2633 
 2634 
 2635 % The correlation times of the spheroid.
 2636 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2637 
 2638 \newpage
 2639 \subsection{The correlation times of the spheroid}
 2640 
 2641 The three spheroid correlation times $\tau_i$ in the correlation function of the Brownian rotational diffusion of a spheroid~\eqref{eq: spheroid correlation function} are
 2642 \begin{subequations}
 2643 \begin{align}
 2644     \tau_{-1} &= (6\Diff_{iso} - 2\Diff_a)^{-1}, \\
 2645     \tau_{0}  &= (6\Diff_{iso} - \Diff_a)^{-1}, \\
 2646     \tau_{1}  &= (6\Diff_{iso} + 2\Diff_a)^{-1}.
 2647 \end{align}
 2648 \end{subequations}
 2649 
 2650 
 2651 
 2652 % The correlation time gradients of the spheroid.
 2653 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2654 
 2655 \subsection{The correlation time gradients of the spheroid}
 2656 
 2657 
 2658 % tm partial derivative.
 2659 \subsubsection{$\tau_m$ partial derivative}
 2660 
 2661 The partial derivatives with respect to the geometric parameter $\tau_m$ are
 2662 \begin{subequations}
 2663 \begin{align}
 2664     \frac{\partial \tau_{-1}}{\partial \tau_m} &= {\tau_m}^{-2} (6\Diff_{iso} - 2\Diff_a)^{-2}, \\
 2665     \frac{\partial \tau_{0}}{\partial \tau_m}  &= {\tau_m}^{-2} (6\Diff_{iso} - \Diff_a)^{-2}, \\
 2666     \frac{\partial \tau_{1}}{\partial \tau_m}  &= {\tau_m}^{-2} (6\Diff_{iso} + 2\Diff_a)^{-2}.
 2667 \end{align}
 2668 \end{subequations}
 2669 
 2670 
 2671 
 2672 % Da partial derivative.
 2673 \subsubsection{$\Diff_a$ partial derivative}
 2674 
 2675 The partial derivatives with respect to the geometric parameter $\Diff_a$ are
 2676 \begin{subequations}
 2677 \begin{align}
 2678     \frac{\partial \tau_{-1}}{\partial \Diff_a} &= 2(6\Diff_{iso} - 2\Diff_a)^{-2}, \\
 2679     \frac{\partial \tau_{0}}{\partial \Diff_a}  &= (6\Diff_{iso} - \Diff_a)^{-2}, \\
 2680     \frac{\partial \tau_{1}}{\partial \Diff_a}  &= -2(6\Diff_{iso} + 2\Diff_a)^{-2}.
 2681 \end{align}
 2682 \end{subequations}
 2683 
 2684 
 2685 
 2686 % The correlation time Hessians of the spheroid.
 2687 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2688 
 2689 \subsection{The correlation time Hessians of the spheroid}
 2690 
 2691 
 2692 % tm-tm partial derivative.
 2693 \subsubsection{$\tau_m$ -- $\tau_m$ partial derivative}
 2694 
 2695 The second partial derivatives with respect to the geometric parameter $\tau_m$ twice are
 2696 \begin{subequations}
 2697 \begin{align}
 2698     \frac{\partial^2 \tau_{-1}}{{\partial \tau_m}^2} &= 2{\tau_m}^{-4} (6\Diff_{iso} - 2\Diff_a)^{-3}
 2699         - 2{\tau_m}^{-3} (6\Diff_{iso} - 2\Diff_a)^{-2}, \\
 2700     \frac{\partial^2 \tau_{0}}{{\partial \tau_m}^2}  &= 2{\tau_m}^{-4} (6\Diff_{iso} - \Diff_a)^{-3}
 2701         - 2{\tau_m}^{-3} (6\Diff_{iso} - \Diff_a)^{-2}, \\
 2702     \frac{\partial^2 \tau_{1}}{{\partial \tau_m}^2}  &= 2{\tau_m}^{-4} (6\Diff_{iso} + 2\Diff_a)^{-3}
 2703         - 2{\tau_m}^{-3} (6\Diff_{iso} + 2\Diff_a)^{-2}.
 2704 \end{align}
 2705 \end{subequations}
 2706 
 2707 
 2708 
 2709 % tm-Da partial derivative.
 2710 \subsubsection{$\tau_m$ -- $\Diff_a$ partial derivative}
 2711 
 2712 The second partial derivatives with respect to the geometric parameters $\tau_m$ and $\Diff_a$ are
 2713 \begin{subequations}
 2714 \begin{align}
 2715     \frac{\partial^2 \tau_{-1}}{\partial \tau_m \cdot \partial \Diff_a} &= 4{\tau_m}^{-2} (6\Diff_{iso} - 2\Diff_a)^{-3}, \\
 2716     \frac{\partial^2 \tau_{0}}{\partial \tau_m \cdot \partial \Diff_a}  &= 2{\tau_m}^{-2} (6\Diff_{iso} - \Diff_a)^{-3}, \\
 2717     \frac{\partial^2 \tau_{1}}{\partial \tau_m \cdot \partial \Diff_a}  &= -4{\tau_m}^{-2} (6\Diff_{iso} + 2\Diff_a)^{-3}.
 2718 \end{align}
 2719 \end{subequations}
 2720 
 2721 
 2722 
 2723 % Da-Da partial derivative.
 2724 \subsubsection{$\Diff_a$ -- $\Diff_a$ partial derivative}
 2725 
 2726 The second partial derivatives with respect to the geometric parameter $\Diff_a$ twice are
 2727 \begin{subequations}
 2728 \begin{align}
 2729     \frac{\partial^2 \tau_{-1}}{{\partial \Diff_a}^2} &= 8 (6\Diff_{iso} - 2\Diff_a)^{-3}, \\
 2730     \frac{\partial^2 \tau_{0}}{{\partial \Diff_a}^2}  &= 2 (6\Diff_{iso} - \Diff_a)^{-3}, \\
 2731     \frac{\partial^2 \tau_{1}}{{\partial \Diff_a}^2}  &= 8 (6\Diff_{iso} + 2\Diff_a)^{-3}.
 2732 \end{align}
 2733 \end{subequations}
 2734 
 2735 
 2736 
 2737 
 2738 % Spherical diffusion tensor.
 2739 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 2740 
 2741 \newpage
 2742 \section{Spherical diffusion tensor}
 2743 
 2744 \index{diffusion!sphere (isotropic)|textbf}
 2745 
 2746 
 2747 
 2748 % The diffusion equation of the sphere.
 2749 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2750 
 2751 \subsection{The diffusion equation of the sphere} \label{sphere equation}
 2752 
 2753 The correlation function of the Brownian rotational diffusion of a sphere is
 2754 \begin{align} \label{eq: spherical correlation function}
 2755     C_{\mathrm{O}}(\tau) &= \frac{1}{5} e^{-\frac{\tau}{\tau_m}}, \\
 2756               &= \frac{1}{5} \sum^0_{i=0} c_i e^{-\frac{\tau}{\tau_i}}.
 2757 \end{align}
 2758 
 2759 \noindent where $c_i$ is the weight of the single exponential term and $\tau_i$ is the correlation time of the single exponential term.
 2760 
 2761 
 2762 
 2763 % The weight of the sphere.
 2764 %~~~~~~~~~~~~~~~~~~~~~~~~~~
 2765 
 2766 \subsection{The weight of the sphere}
 2767 
 2768 
 2769 % Definitions.
 2770 \subsubsection{Definitions}
 2771 
 2772 The entire diffusion parameter set consists of a single geometric parameter and is
 2773 \begin{equation}
 2774     \Diffset = \{ \tau_m \}.
 2775 \end{equation}
 2776 
 2777 
 2778 
 2779 % Summation terms.
 2780 \subsubsection{Summation terms}
 2781 
 2782 The summation indices of the correlation function of the Brownian rotational diffusion of a sphere~\eqref{eq: spheroid correlation function} range from $k = 0$ to $k = 0$ therefore
 2783 \begin{equation}
 2784     i \in \{ 0 \}.
 2785 \end{equation}
 2786 
 2787 
 2788 % The weights.
 2789 \subsubsection{The weights}
 2790 
 2791 The single weight $c_i$ in the correlation function of the Brownian rotational diffusion of a sphere~\eqref{eq: spheroid correlation function} is
 2792 \begin{equation}
 2793     c_{0} = 1.
 2794 \end{equation}
 2795 
 2796 
 2797 
 2798 % The weight gradient of the sphere.
 2799 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2800 
 2801 \subsection{The weight gradient of the sphere}
 2802 
 2803 
 2804 % tm partial derivative.
 2805 \subsubsection{$\tau_m$ partial derivative}
 2806 
 2807 The partial derivative with respect to the geometric parameter $\tau_m$ is
 2808 \begin{equation}
 2809     \frac{\partial c_{0}}{\partial \tau_m} = 0.
 2810 \end{equation}
 2811 
 2812 
 2813 
 2814 % The weight Hessian of the sphere.
 2815 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2816 
 2817 \subsection{The weight Hessian of the sphere}
 2818 
 2819 
 2820 % tm-tm partial derivative.
 2821 \subsubsection{$\tau_m$ -- $\tau_m$ partial derivative}
 2822 
 2823 The second partial derivatives with respect to the geometric parameter $\tau_m$ twice is
 2824 \begin{equation}
 2825     \frac{\partial^2 c_{0}}{{\partial \tau_m}^2} = 0.
 2826 \end{equation}
 2827 
 2828 
 2829 
 2830 % The correlation time of the sphere.
 2831 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2832 
 2833 \subsection{The correlation time of the sphere}
 2834 
 2835 The single correlation time $\tau_i$ of the correlation function of the Brownian rotational diffusion of a sphere~\eqref{eq: spheroid correlation function} is
 2836 \begin{equation}
 2837     \tau_{0} = \tau_m.
 2838 \end{equation}
 2839 
 2840 
 2841 
 2842 % The correlation time gradient of the sphere.
 2843 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2844 
 2845 \subsection{The correlation time gradient of the sphere}
 2846 
 2847 
 2848 % tm partial derivative.
 2849 \subsubsection{$\tau_m$ partial derivative}
 2850 
 2851 The partial derivative with respect to the geometric parameter $\tau_m$ is
 2852 \begin{equation}
 2853     \frac{\partial \tau_{0}}{\partial \tau_m} = 1.
 2854 \end{equation}
 2855 
 2856 
 2857 
 2858 % The correlation time Hessian of the sphere.
 2859 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2860 
 2861 \subsection{The correlation time Hessian of the sphere}
 2862 
 2863 
 2864 % tm-tm partial derivative.
 2865 \subsubsection{$\tau_m$ -- $\tau_m$ partial derivative}
 2866 
 2867 The second partial derivative with respect to the geometric parameter $\tau_m$ twice is
 2868 \begin{equation}
 2869     \frac{\partial^2 \tau_{0}}{{\partial \tau_m}^2} = 0.
 2870 \end{equation}
 2871 
 2872 
 2873 
 2874 
 2875 % Ellipsoidal dot product derivatives.
 2876 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 2877 
 2878 \newpage
 2879 \section{Ellipsoidal dot product derivatives}
 2880 
 2881 
 2882 
 2883 % The dot product of the ellipsoid.
 2884 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2885 
 2886 \subsection{The dot product of the ellipsoid}
 2887 
 2888 The dot product is defined as
 2889 \begin{equation}
 2890     \delta_i = \widehat{XH} \cdot \widehat{\Diff_i},
 2891 \end{equation}
 2892 
 2893 \noindent where $i$ is one of \{$x$, $y$, $z$\}, $\widehat{XH}$ is a unit vector parallel to the XH bond vector, and $\widehat{\Diff_i}$ is one of the unit vectors defining the diffusion frame.
 2894 The three diffusion frame unit vectors can be expressed using the Euler angles $\alpha$, $\beta$, and $\gamma$ as
 2895 \begin{subequations}
 2896 \begin{align}
 2897     \widehat{\Diff_x} &= \begin{pmatrix}
 2898         -\sin \alpha \sin \gamma + \cos \alpha \cos \beta \cos \gamma \\
 2899         -\sin \alpha \cos \gamma - \cos \alpha \cos \beta \sin \gamma \\
 2900         \cos \alpha \sin \beta \\
 2901     \end{pmatrix}, \\
 2902     \widehat{\Diff_y} &= \begin{pmatrix}
 2903         \cos \alpha \sin \gamma + \sin \alpha \cos \beta \cos \gamma \\
 2904         \cos \alpha \cos \gamma - \sin \alpha \cos \beta \sin \gamma \\
 2905         \sin \alpha \sin \beta \\
 2906     \end{pmatrix}, \\
 2907     \widehat{\Diff_z} &= \begin{pmatrix}
 2908         -\sin \beta \cos \gamma \\
 2909         \sin \beta \sin \gamma \\
 2910         \cos \beta \\
 2911     \end{pmatrix}.
 2912 \end{align}
 2913 \end{subequations}
 2914 
 2915 
 2916 
 2917 % The dot product gradient of the ellipsoid.
 2918 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 2919 
 2920 \subsection{The dot product gradient of the ellipsoid}
 2921 
 2922 The partial derivative of the dot product $\delta_i$ with respect to the orientational parameter $\Difforiset_j$ is
 2923 \begin{equation}
 2924     \frac{\partial \delta_i}{\partial \Difforiset_j}
 2925         = \frac{\partial}{\partial \Difforiset_j} \left( \widehat{XH} \cdot \widehat{\Diff_i} \right) \\
 2926         = \widehat{XH} \frac{\partial \widehat{\Diff_i}}{\partial \Difforiset_j}  +  \frac{\partial \widehat{XH}}{\partial \Difforiset_j} \widehat{\Diff_i}.
 2927 \end{equation}
 2928 
 2929 \noindent Because $\widehat{XH}$ is constant and not dependent on the Euler angles its derivative is zero.
 2930 Therefore
 2931 \begin{equation}
 2932     \frac{\partial \delta_i}{\partial \Difforiset_j} = \widehat{XH} \frac{\partial \widehat{\Diff_i}}{\partial \Difforiset_j}.
 2933 \end{equation}
 2934 
 2935 
 2936 
 2937 % The Dx gradient.
 2938 \begin{latexonly}
 2939     \subsubsection{The $\Diff_x$ gradient}
 2940 \end{latexonly}
 2941 \begin{htmlonly}
 2942     \subsubsection{The $D_x$ gradient}
 2943 \end{htmlonly}
 2944 
 2945 The partial derivatives of the unit vector $\widehat{\Diff_x}$ with respect to the Euler angles are
 2946 \begin{subequations}
 2947 \begin{align}
 2948     \frac{\partial \widehat{\Diff_x}}{\partial \alpha} &= \begin{pmatrix}
 2949         -\cos \alpha \sin \gamma - \sin \alpha \cos \beta \cos \gamma \\
 2950         -\cos \alpha \cos \gamma + \sin \alpha \cos \beta \sin \gamma \\
 2951         -\sin \alpha \sin \beta \\
 2952     \end{pmatrix}, \\
 2953     \frac{\partial \widehat{\Diff_x}}{\partial \beta} &= \begin{pmatrix}
 2954         -\cos \alpha \sin \beta \cos \gamma \\
 2955         \cos \alpha \sin \beta \sin \gamma \\
 2956         \cos \alpha \cos \beta \\
 2957     \end{pmatrix}, \\
 2958     \frac{\partial \widehat{\Diff_x}}{\partial \gamma} &= \begin{pmatrix}
 2959         -\sin \alpha \cos \gamma - \cos \alpha \cos \beta \sin \gamma \\
 2960         \sin \alpha \sin \gamma - \cos \alpha \cos \beta \cos \gamma \\
 2961         0 \\
 2962     \end{pmatrix}.
 2963 \end{align}
 2964 \end{subequations}
 2965 
 2966 
 2967 
 2968 % The Dy gradient.
 2969 \begin{latexonly}
 2970     \subsubsection{The $\Diff_y$ gradient}
 2971 \end{latexonly}
 2972 \begin{htmlonly}
 2973     \subsubsection{The $D_y$ gradient}
 2974 \end{htmlonly}
 2975 
 2976 The partial derivatives of the unit vector $\widehat{\Diff_y}$ with respect to the Euler angles are
 2977 \begin{subequations}
 2978 \begin{align}
 2979     \frac{\partial \widehat{\Diff_y}}{\partial \alpha} &= \begin{pmatrix}
 2980         -\sin \alpha \sin \gamma + \cos \alpha \cos \beta \cos \gamma \\
 2981         -\sin \alpha \cos \gamma - \cos \alpha \cos \beta \sin \gamma \\
 2982         \cos \alpha \sin \beta \\
 2983     \end{pmatrix}, \\
 2984     \frac{\partial \widehat{\Diff_y}}{\partial \beta} &= \begin{pmatrix}
 2985         -\sin \alpha \sin \beta \cos \gamma \\
 2986         \sin \alpha \sin \beta \sin \gamma \\
 2987         \sin \alpha \cos \beta \\
 2988     \end{pmatrix}, \\
 2989     \frac{\partial \widehat{\Diff_y}}{\partial \gamma} &= \begin{pmatrix}
 2990         \cos \alpha \cos \gamma - \sin \alpha \cos \beta \sin \gamma \\
 2991         -\cos \alpha \sin \gamma - \sin \alpha \cos \beta \cos \gamma \\
 2992         0 \\
 2993     \end{pmatrix}.
 2994 \end{align}
 2995 \end{subequations}
 2996 
 2997 
 2998 
 2999 % The Dz gradient.
 3000 \begin{latexonly}
 3001     \subsubsection{The $\Diff_z$ gradient}
 3002 \end{latexonly}
 3003 \begin{htmlonly}
 3004     \subsubsection{The $D_z$ gradient}
 3005 \end{htmlonly}
 3006 
 3007 The partial derivatives of the unit vector $\widehat{\Diff_z}$ with respect to the Euler angles are
 3008 \begin{subequations}
 3009 \begin{align}
 3010     \frac{\partial \widehat{\Diff_z}}{\partial \alpha} &= \begin{pmatrix}
 3011         0 \\
 3012         0 \\
 3013         0 \\
 3014     \end{pmatrix}, \\
 3015     \frac{\partial \widehat{\Diff_z}}{\partial \beta} &= \begin{pmatrix}
 3016         -\cos \beta \cos \gamma \\
 3017         \cos \beta \sin \gamma \\
 3018         -\sin \beta \\
 3019     \end{pmatrix}, \\
 3020     \frac{\partial \widehat{\Diff_z}}{\partial \gamma} &= \begin{pmatrix}
 3021         \sin \beta \sin \gamma \\
 3022         \sin \beta \cos \gamma \\
 3023         0 \\
 3024     \end{pmatrix}.
 3025 \end{align}
 3026 \end{subequations}
 3027 
 3028 
 3029 
 3030 % The dot product Hessian of the ellipsoid.
 3031 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 3032 
 3033 \newpage
 3034 \subsection{The dot product Hessian of the ellipsoid}
 3035 
 3036 The second partial derivative of the dot product $\delta_i$ with respect to the orientational parameters $\Difforiset_j$ and $\Difforiset_k$ is
 3037 \begin{equation}
 3038     \frac{\partial^2 \delta_i}{\partial \Difforiset_j \cdot \partial \Difforiset_k}
 3039         = \frac{\partial^2}{\partial \Difforiset_j \cdot \partial \Difforiset_k} \left( \widehat{XH} \cdot \widehat{\Diff_i} \right) \\
 3040         = \widehat{XH} \frac{\partial^2 \widehat{\Diff_i}}{\partial \Difforiset_j \cdot \partial \Difforiset_k}.
 3041 \end{equation}
 3042 
 3043 
 3044 
 3045 % The Dx Hessian.
 3046 \begin{latexonly}
 3047     \subsubsection{The $\Diff_x$ Hessian}
 3048 \end{latexonly}
 3049 \begin{htmlonly}
 3050     \subsubsection{The $D_x$ Hessian}
 3051 \end{htmlonly}
 3052 
 3053 The second partial derivatives of the unit vector $\widehat{\Diff_x}$ with respect to the Euler angles are
 3054 \begin{subequations}
 3055 \begin{align}
 3056     \frac{\partial^2 \widehat{\Diff_x}}{\partial \alpha^2} &= \begin{pmatrix}
 3057         \sin \alpha \sin \gamma - \cos \alpha \cos \beta \cos \gamma \\
 3058         \sin \alpha \cos \gamma + \cos \alpha \cos \beta \sin \gamma \\
 3059         -\cos \alpha \sin \beta \\
 3060     \end{pmatrix}, \\
 3061     \frac{\partial^2 \widehat{\Diff_x}}{\partial \alpha \cdot \partial \beta} &= \begin{pmatrix}
 3062         \sin \alpha \sin \beta \cos \gamma \\
 3063         - \sin \alpha \sin \beta \sin \gamma \\
 3064         -\sin \alpha \cos \beta \\
 3065     \end{pmatrix}, \\
 3066     \frac{\partial^2 \widehat{\Diff_x}}{\partial \alpha \cdot \partial \gamma} &= \begin{pmatrix}
 3067         -\cos \alpha \cos \gamma + \sin \alpha \cos \beta \sin \gamma \\
 3068         \cos \alpha \sin \gamma + \sin \alpha \cos \beta \cos \gamma \\
 3069         0 \\
 3070     \end{pmatrix}, \\
 3071     \frac{\partial^2 \widehat{\Diff_x}}{\partial \beta^2} &= \begin{pmatrix}
 3072         -\cos \alpha \cos \beta \cos \gamma \\
 3073         \cos \alpha \cos \beta \sin \gamma \\
 3074         -\cos \alpha \sin \beta \\
 3075     \end{pmatrix}, \\
 3076     \frac{\partial^2 \widehat{\Diff_x}}{\partial \beta \cdot \partial \gamma} &= \begin{pmatrix}
 3077         \cos \alpha \sin \beta \sin \gamma \\
 3078         \cos \alpha \sin \beta \cos \gamma \\
 3079         0 \\
 3080     \end{pmatrix}, \\
 3081     \frac{\partial^2 \widehat{\Diff_x}}{\partial \gamma^2} &= \begin{pmatrix}
 3082         \sin \alpha \sin \gamma - \cos \alpha \cos \beta \cos \gamma \\
 3083         \sin \alpha \cos \gamma + \cos \alpha \cos \beta \sin \gamma \\
 3084         0 \\
 3085     \end{pmatrix}.
 3086 \end{align}
 3087 \end{subequations}
 3088 
 3089 
 3090 
 3091 % The Dy Hessian.
 3092 \begin{latexonly}
 3093     \subsubsection{The $\Diff_y$ Hessian}
 3094 \end{latexonly}
 3095 \begin{htmlonly}
 3096     \subsubsection{The $D_y$ Hessian}
 3097 \end{htmlonly}
 3098 
 3099 The second partial derivatives of the unit vector $\widehat{\Diff_y}$ with respect to the Euler angles are
 3100 \begin{subequations}
 3101 \begin{align}
 3102     \frac{\partial^2 \widehat{\Diff_y}}{\partial \alpha^2} &= \begin{pmatrix}
 3103         -\cos \alpha \sin \gamma - \sin \alpha \cos \beta \cos \gamma \\
 3104         -\cos \alpha \cos \gamma + \sin \alpha \cos \beta \sin \gamma \\
 3105         -\sin \alpha \sin \beta \\
 3106     \end{pmatrix}, \\
 3107     \frac{\partial^2 \widehat{\Diff_y}}{\partial \alpha \cdot \partial \beta} &= \begin{pmatrix}
 3108         -\cos \alpha \sin \beta \cos \gamma \\
 3109         \cos \alpha \sin \beta \sin \gamma \\
 3110         \cos \alpha \cos \beta \\
 3111     \end{pmatrix}, \\
 3112     \frac{\partial^2 \widehat{\Diff_y}}{\partial \alpha \cdot \partial \gamma} &= \begin{pmatrix}
 3113         -\sin \alpha \cos \gamma - \cos \alpha \cos \beta \sin \gamma \\
 3114         \sin \alpha \sin \gamma - \cos \alpha \cos \beta \cos \gamma \\
 3115         0 \\
 3116     \end{pmatrix}, \\
 3117     \frac{\partial^2 \widehat{\Diff_y}}{\partial \beta^2} &= \begin{pmatrix}
 3118         -\sin \alpha \cos \beta \cos \gamma \\
 3119         \sin \alpha \cos \beta \sin \gamma \\
 3120         -\sin \alpha \sin \beta \\
 3121     \end{pmatrix}, \\
 3122     \frac{\partial^2 \widehat{\Diff_y}}{\partial \beta \cdot \partial \gamma} &= \begin{pmatrix}
 3123         \sin \alpha \sin \beta \sin \gamma \\
 3124         \sin \alpha \sin \beta \cos \gamma \\
 3125         0 \\
 3126     \end{pmatrix}, \\
 3127     \frac{\partial^2 \widehat{\Diff_y}}{\partial \gamma^2} &= \begin{pmatrix}
 3128         -\cos \alpha \sin \gamma - \sin \alpha \cos \beta \cos \gamma \\
 3129         -\cos \alpha \cos \gamma + \sin \alpha \cos \beta \sin \gamma \\
 3130         0 \\
 3131     \end{pmatrix}.
 3132 \end{align}
 3133 \end{subequations}
 3134 
 3135 
 3136 
 3137 % The Dz Hessian.
 3138 \begin{latexonly}
 3139     \subsubsection{The $\Diff_z$ Hessian}
 3140 \end{latexonly}
 3141 \begin{htmlonly}
 3142     \subsubsection{The $D_z$ Hessian}
 3143 \end{htmlonly}
 3144 
 3145 The second partial derivatives of the unit vector $\widehat{\Diff_z}$ with respect to the Euler angles are
 3146 \begin{subequations}
 3147 \begin{align}
 3148     \frac{\partial^2 \widehat{\Diff_z}}{\partial \alpha^2} &= \begin{pmatrix}
 3149         0 \\
 3150         0 \\
 3151         0 \\
 3152     \end{pmatrix}, \\
 3153     \frac{\partial^2 \widehat{\Diff_z}}{\partial \alpha \cdot \partial \beta} &= \begin{pmatrix}
 3154         0 \\
 3155         0 \\
 3156         0 \\
 3157     \end{pmatrix}, \\
 3158     \frac{\partial^2 \widehat{\Diff_z}}{\partial \alpha \cdot \partial \gamma} &= \begin{pmatrix}
 3159         0 \\
 3160         0 \\
 3161         0 \\
 3162     \end{pmatrix}, \\
 3163     \frac{\partial^2 \widehat{\Diff_z}}{\partial \beta^2} &= \begin{pmatrix}
 3164         \sin \beta \cos \gamma \\
 3165         -\sin \beta \sin \gamma \\
 3166         -\cos \beta \\
 3167     \end{pmatrix}, \\
 3168     \frac{\partial^2 \widehat{\Diff_z}}{\partial \beta \cdot \partial \gamma} &= \begin{pmatrix}
 3169         \cos \beta \sin \gamma \\
 3170         \cos \beta \cos \gamma \\
 3171         0 \\
 3172     \end{pmatrix}, \\
 3173     \frac{\partial^2 \widehat{\Diff_z}}{\partial \gamma^2} &= \begin{pmatrix}
 3174         \sin \beta \cos \gamma \\
 3175         -\sin \beta \sin \gamma \\
 3176         0 \\
 3177     \end{pmatrix}.
 3178 \end{align}
 3179 \end{subequations}
 3180 
 3181 
 3182 
 3183 
 3184 % Spheroidal dot product derivatives.
 3185 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 3186 
 3187 \newpage
 3188 \section{Spheroidal dot product derivatives}
 3189 
 3190 
 3191 
 3192 % The dot product of the spheroid.
 3193 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 3194 
 3195 \subsection{The dot product of the spheroid}
 3196 
 3197 The single dot product of the spheroid is defined as
 3198 \begin{equation}
 3199     \delta_z = \widehat{XH} \cdot \widehat{\Diff_\Par},
 3200 \end{equation}
 3201 
 3202 \noindent where $\widehat{XH}$ is a unit vector parallel to the XH vector.
 3203 $\widehat{\Diff_\Par}$ is a unit vector parallel to the unique axis of the diffusion tensor and can be expressed using the spherical angles where $\theta$ is the polar angle and $\phi$ is the azimuthal angle as
 3204 \begin{equation}
 3205     \widehat{\Diff_\Par} = \begin{pmatrix}
 3206         \sin \theta \cos \phi \\
 3207         \sin \theta \sin \phi \\
 3208         \cos \theta \\
 3209     \end{pmatrix}.
 3210 \end{equation}
 3211 
 3212 
 3213 
 3214 % The dot product gradient of the spheroid.
 3215 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 3216 
 3217 \subsection{The dot product gradient of the spheroid}
 3218 
 3219 The partial derivative of the dot product with respect to the orientational parameter $\Difforiset_i$ is
 3220 \begin{equation}
 3221     \frac{\partial \delta_z}{\partial \Difforiset_i}
 3222         = \frac{\partial}{\partial \Difforiset_i} \left( \widehat{XH} \cdot \widehat{\Diff_\Par} \right) \\
 3223         = \widehat{XH} \frac{\partial \widehat{\Diff_\Par}}{\partial \Difforiset_i}  +  \frac{\partial \widehat{XH}}{\partial \Difforiset_i} \widehat{\Diff_\Par}.
 3224 \end{equation}
 3225 
 3226 \noindent Because the XH bond vector is constant and not dependent on the spherical angles its derivative is zero.
 3227 Therefore
 3228 \begin{equation}
 3229     \frac{\partial \delta_z}{\partial \Difforiset_i} = \widehat{XH} \frac{\partial \widehat{\Diff_\Par}}{\partial \Difforiset_i}.
 3230 \end{equation}
 3231 
 3232 
 3233 
 3234 % The Dpar gradient.
 3235 \begin{latexonly}
 3236     \subsubsection{The $\Diff_\Par$ gradient}
 3237 \end{latexonly}
 3238 \begin{htmlonly}
 3239     \subsubsection{The $D_{par}$ gradient}
 3240 \end{htmlonly}
 3241 
 3242 The partial derivatives of the unit vector $\widehat{\Diff_\Par}$ with respect to the spherical angles are
 3243 \begin{subequations}
 3244 \begin{align}
 3245     \frac{\partial \widehat{\Diff_\Par}}{\partial \theta} &= \begin{pmatrix}
 3246         \cos \theta \cos \phi \\
 3247         \cos \theta \sin \phi \\
 3248         -\sin \theta \\
 3249     \end{pmatrix}, \\
 3250     \frac{\partial \widehat{\Diff_\Par}}{\partial \phi} &= \begin{pmatrix}
 3251         -\sin \theta \sin \phi \\
 3252         \sin \theta \cos \phi \\
 3253         0 \\
 3254     \end{pmatrix}.
 3255 \end{align}
 3256 \end{subequations}
 3257 
 3258 
 3259 
 3260 % The dot product Hessian of the spheroid.
 3261 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 3262 
 3263 \subsection{The dot product Hessian of the spheroid}
 3264 
 3265 The second partial derivative of the single spheroidal dot product $\delta_z$ with respect to the orientational parameters $\Difforiset_i$ and $\Difforiset_j$ is
 3266 \begin{equation}
 3267     \frac{\partial^2 \delta_z}{\partial \Difforiset_i \cdot \partial \Difforiset_j}
 3268         = \frac{\partial^2}{\partial \Difforiset_i \cdot \partial \Difforiset_j} \left( \widehat{XH} \cdot \widehat{\Diff_\Par} \right) \\
 3269         = \widehat{XH} \frac{\partial^2 \widehat{\Diff_\Par}}{\partial \Difforiset_i \cdot \partial \Difforiset_j}.
 3270 \end{equation}
 3271 
 3272 
 3273 
 3274 % The Dpar Hessian.
 3275 \begin{latexonly}
 3276     \subsubsection{The $\Diff_\Par$ Hessian}
 3277 \end{latexonly}
 3278 \begin{htmlonly}
 3279     \subsubsection{The $D_{par}$ Hessian}
 3280 \end{htmlonly}
 3281 
 3282 The second partial derivatives of the unit vector $\widehat{\Diff_\Par}$ with respect to the spherical angles are
 3283 \begin{subequations}
 3284 \begin{align}
 3285     \frac{\partial^2 \widehat{\Diff_\Par}}{\partial \theta^2} &= \begin{pmatrix}
 3286         -\sin \theta \cos \phi \\
 3287         -\sin \theta \sin \phi \\
 3288         -\cos \theta \\
 3289     \end{pmatrix}, \\
 3290     \frac{\partial^2 \widehat{\Diff_\Par}}{\partial \theta \cdot \partial \phi} &= \begin{pmatrix}
 3291         -\cos \theta \sin \phi \\
 3292         \cos \theta \cos \phi \\
 3293         0 \\
 3294     \end{pmatrix}, \\
 3295     \frac{\partial^2 \widehat{\Diff_\Par}}{\partial \phi^2} &= \begin{pmatrix}
 3296         -\sin \theta \cos \phi \\
 3297         -\sin \theta \sin \phi \\
 3298         0 \\
 3299     \end{pmatrix}.
 3300 \end{align}
 3301 \end{subequations}
 3302 
 3303 
 3304 
 3305 
 3306