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

$${\frac{{\partial J(\omega)}}{{\partial \mathfrak{O}_j}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}} \Bigg( {\frac{{S^2}}{{1 + (\omega \tau_i)^2}}} {\frac{{(1 - S^2)(\tau_e + \tau_i)\tau_e}}{{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}}} \Bigg)$$ (15.63)

S² partial derivative

The partial derivative of (15.62) with respect to the order parameter S² is

$${\frac{{\partial J(\omega)}}{{\partial S^2}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} \Bigg( {\frac{{1}}{{1 + (\omega \tau_i)^2}}} {\frac{{(\tau_e + \tau_i)\tau_e}}{{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}}} \Bigg)$$ (15.64)

τ_e partial derivative

The partial derivative of (15.62) with respect to the correlation time τ_e is

$${\frac{{\partial J(\omega)}}{{\partial \tau_e}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}}{{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}}}$$ (15.65)