The extended model-free gradient

The model-free gradient of the extended spectral density function (15.63) 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 parameters S² and S²_f, and the internal correlation times τ_f and τ_s. The positions in the vector correspond to the model parameters which are being optimised.

$\mathfrak{G}_j$ partial derivative

The partial derivative of (15.63) with respect to the geometric parameter $\mathfrak{G}_j$ is

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

$\mathfrak{O}_j$ partial derivative

The partial derivative of (15.63) with respect to the orientational parameter $\mathfrak{O}_j$ is

$${\frac{{\partial J(\omega)}}{{\partial \mathfrak{O}_j}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}} \Bigg( {\frac{{S^2}}{{1 + (\omega \tau_i)^2}}} {\frac{{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}}{{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}}} {\frac{{(S^2_f - S^2)(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}} \Bigg)$$ (15.72)

S² partial derivative

The partial derivative of (15.63) with respect to the order parameter S² is

$${\frac{{\partial J(\omega)}}{{\partial S^2}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} \Bigg( {\frac{{1}}{{1 + (\omega \tau_i)^2}}} {\frac{{(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}} \Bigg)$$ (15.73)

S²_f partial derivative

The partial derivative of (15.63) with respect to the order parameter S²_f is

$${\frac{{\partial J(\omega)}}{{\partial S^2_f}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} \Bigg( {\frac{{(\tau_f + \tau_i)\tau_f}}{{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}}} {\frac{{(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}} \Bigg)$$ (15.74)

τ_f partial derivative

The partial derivative of (15.63) with respect to the correlation time τ_f is

$${\frac{{\partial J(\omega)}}{{\partial \tau_f}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}}{{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}}}$$ (15.75)

τ_s partial derivative

The partial derivative of (15.63) with respect to the correlation time τ_s is

$${\frac{{\partial J(\omega)}}{{\partial \tau_s}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}}{{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}}}$$ (15.76)