The original model-free gradient

The model-free gradient of the original spectral density function (15.62) is the vector of partial derivatives of the function with respect to the geometric parameter $\mathfrak{G}_i$ , the orientational parameter $\mathfrak{O}_i$ , the order parameter S², and the internal correlation time τ_e. The positions in the vector correspond to the model parameters which are being optimised.

$\mathfrak{G}_j$ partial derivative

The partial derivative of (15.62) with respect to the geometric parameter $\mathfrak{G}_j$ is

$\begin{multline} \frac{\partial J(\omega)}{\partial \mathfrak{G}_j} = \frac{2}{... ...{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2} \Bigg) \Bigg). \end{multline}$

$\mathfrak{O}_j$ partial derivative

The partial derivative of (15.62) with respect to the orientational parameter $\mathfrak{O}_j$ is

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial \mathfrak{O}_j}}}$ = $\displaystyle {\frac{{2}}{{5}}}$ $\displaystyle \sum_{{i=-k}}^{k}$ $\displaystyle {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}}$ τ_i $\displaystyle \Bigg($ $\displaystyle {\frac{{S^2}}{{1 + (\omega \tau_i)^2}}}$ + $\displaystyle {\frac{{(1 - S^2)(\tau_e + \tau_i)\tau_e}}{{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}}}$ $\displaystyle \Bigg)$ .

(15.63)

S² partial derivative

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial S^2}}}$ = $\displaystyle {\frac{{2}}{{5}}}$ $\displaystyle \sum_{{i=-k}}^{k}$ c_iτ_i $\displaystyle \Bigg($ $\displaystyle {\frac{{1}}{{1 + (\omega \tau_i)^2}}}$ - $\displaystyle {\frac{{(\tau_e + \tau_i)\tau_e}}{{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}}}$ $\displaystyle \Bigg)$ .

(15.64)

τ_e partial derivative

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial \tau_e}}}$ = $\displaystyle {\frac{{2}}{{5}}}$ (1 - S²) $\displaystyle \sum_{{i=-k}}^{k}$ c_iτ_i² $\displaystyle {\frac{{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}}{{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}}}$ .

(15.65)