The original model-free Hessian

The model-free Hessian of the original spectral density function (15.62) is the matrix of second partial derivatives. The matrix coordinates correspond to the model parameters which are being optimised.

$\mathfrak{G}_j$ - $\mathfrak{G}_k$ partial derivative

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

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

$\mathfrak{G}_j$ - $\mathfrak{O}_k$ partial derivative

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

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

$\mathfrak{G}_j$ - S² partial derivative

The second partial derivative of (15.62) with respect to the geometric parameter $\mathfrak{G}_j$ and the order parameter S² is

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

$\mathfrak{G}_j$ - τ_e partial derivative

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

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

$\mathfrak{O}_j$ - $\mathfrak{O}_k$ partial derivative

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

$${\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial \mathfrak{O}_k}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{\partial^2 c_i}}{{\partial \mathfrak{O}_j \cdot \partial \mathfrak{O}_k}}} \Bigg( {\frac{{S^2}}{{1 + (\omega \tau_i)^2}}} {\frac{{(1 - S^2)(\tau_e + \tau_i)\tau_e}}{{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}}} \Bigg)$$ (15.66)

$\mathfrak{O}_j$ - S² partial derivative

The second partial derivative of (15.62) with respect to the orientational parameter $\mathfrak{O}_j$ and the order parameter S² is

$${\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial S^2}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}} \Bigg( {\frac{{1}}{{1 + (\omega \tau_i)^2}}} {\frac{{(\tau_e + \tau_i)\tau_e}}{{(\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2}}} \Bigg)$$ (15.67)

$\mathfrak{O}_j$ - τ_e partial derivative

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

$${\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial \tau_e}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}} {\frac{{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}}{{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}}}$$ (15.68)

S² - S² partial derivative

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

$${\frac{{\partial^2 J(\omega)}}{{(\partial S^2)^2}}}$$ (15.69)

S² - τ_e partial derivative

The second partial derivative of (15.62) with respect to the order parameter S² and correlation time τ_e is

$${\frac{{\partial^2 J(\omega)}}{{\partial S^2 \cdot \partial \tau_e}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{(\tau_e + \tau_i)^2 - (\omega \tau_e \tau_i)^2}}{{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^2}}}$$ (15.70)

τ_e - τ_e partial derivative

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

$${\frac{{\partial^2 J(\omega)}}{{{\partial \tau_e}^2}}} {\frac{{4}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{(\tau_e + \tau_i)^3 + 3 \omega^2 \tau_i^3 \tau_e (\tau_e ... ...4 \tau_e^3}}{{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^3}}}$$ (15.71)