Subsections

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$ - τe partial derivative

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

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial \mathfrak{O}_k}}}$ = $\displaystyle {\frac{{2}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$$\displaystyle {\frac{{\partial^2 c_i}}{{\partial \mathfrak{O}_j \cdot \partial \mathfrak{O}_k}}}$τi$\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)$. (15.66)

$ \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

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial S^2}}}$ = $\displaystyle {\frac{{2}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$$\displaystyle {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}}$τi$\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)$. (15.67)

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

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

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial \tau_e}}}$ = $\displaystyle {\frac{{2}}{{5}}}$(1 - S2)$\displaystyle \sum_{{i=-k}}^{k}$$\displaystyle {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}}$τi2$\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}}}$. (15.68)

S2 - S2 partial derivative

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

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{(\partial S^2)^2}}}$ = 0. (15.69)

S2 - τe partial derivative

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

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial S^2 \cdot \partial \tau_e}}}$ = - $\displaystyle {\frac{{2}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\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}}}$. (15.70)

τe - τe partial derivative

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

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{{\partial \tau_e}^2}}}$ = - $\displaystyle {\frac{{4}}{{5}}}$(1 - S2)$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\displaystyle {\frac{{(\tau_e + \tau_i)^3 + 3 \omega^2 \tau_i^3 \tau_e (\tau_e ...
...4 \tau_e^3}}{{\left((\tau_e + \tau_i)^2 + (\omega \tau_e \tau_i)^2 \right)^3}}}$ (15.71)

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