Subsections

The alternative extended model-free Hessian

The model-free Hessian of the extended spectral density function (15.63) is also complicated by the convolution resulting from the use of the parameters {S2f, S2s, τf, τs}. The second partial derivatives with respect to these parameters are presented below.

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

The second partial derivative of (15.63) 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_s \tau_i)^2}
\Bigg)
\Bigg)
\Bigg).
\end{multline}

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

The second partial derivative of (15.63) 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_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
\Bigg)
\Bigg).
\end{multline}

$ \mathfrak{G}_j$ - S2f partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter $ \mathfrak{G}_j$ and the order parameter S2f is

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

$ \mathfrak{G}_j$ - S2s partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter $ \mathfrak{G}_j$ and the order parameter S2s is

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

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

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

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

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

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

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

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

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

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

$ \mathfrak{O}_j$ - S2f partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter $ \mathfrak{O}_j$ and the order parameter S2f is

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial S^2_f}}}$ = $\displaystyle {\frac{{2}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$$\displaystyle {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}}$τi$\displaystyle \Bigg($$\displaystyle {\frac{{S^2_s}}{{1 + (\omega \tau_i)^2}}}$ - $\displaystyle {\frac{{(\tau_f + \tau_i)\tau_f}}{{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}}}$ + $\displaystyle {\frac{{(1 - S^2_s) (\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}}$$\displaystyle \Bigg)$. (15.96)

$ \mathfrak{O}_j$ - S2s partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter $ \mathfrak{O}_j$ and the order parameter S2s is

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial S^2_s}}}$ = $\displaystyle {\frac{{2}}{{5}}}$S2f$\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_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}}$$\displaystyle \Bigg)$. (15.97)

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

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

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial \tau_f}}}$ = $\displaystyle {\frac{{2}}{{5}}}$(1 - S2f)$\displaystyle \sum_{{i=-k}}^{k}$$\displaystyle {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}}$τi2$\displaystyle {\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}}}$. (15.98)

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

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

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial \tau_s}}}$ = $\displaystyle {\frac{{2}}{{5}}}$S2f(1 - S2s)$\displaystyle \sum_{{i=-k}}^{k}$$\displaystyle {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}}$τi2$\displaystyle {\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}}}$. (15.99)

S2f - S2f partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2f twice is

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{(\partial S^2_f)^2}}}$ = 0. (15.100)

S2f - S2s partial derivative

The second partial derivative of (15.63) with respect to the order parameters S2f and S2s is

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial S^2_f \cdot \partial S^2_s}}}$ = $\displaystyle {\frac{{2}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$ciτi$\displaystyle \Bigg($$\displaystyle {\frac{{1}}{{1 + (\omega \tau_i)^2}}}$ - $\displaystyle {\frac{{(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}}$$\displaystyle \Bigg)$. (15.101)

S2f - τf partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2f and correlation time τf is

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial S^2_f \cdot \partial \tau_f}}}$ = - $\displaystyle {\frac{{2}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\displaystyle {\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}}}$. (15.102)

S2f - τs partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2f and correlation time τs is

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial S^2_f \cdot \partial \tau_s}}}$ = $\displaystyle {\frac{{2}}{{5}}}$(1 - S2s)$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\displaystyle {\frac{{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}}{{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}}}$. (15.103)

S2s - S2s partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2s twice is

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{(\partial S^2_s)^2}}}$ = 0. (15.104)

S2s - τf partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2s and correlation time τf is

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial S^2_s \cdot \partial \tau_f}}}$ = 0. (15.105)

S2s - τs partial derivative

The second partial derivative of (15.63) with respect to the order parameter S2s and correlation time τs is

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial S^2_s \cdot \partial \tau_s}}}$ = - $\displaystyle {\frac{{2}}{{5}}}$S2f$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\displaystyle {\frac{{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}}{{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}}}$. (15.106)

τf - τf partial derivative

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

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{{\partial \tau_f}^2}}}$ = - $\displaystyle {\frac{{4}}{{5}}}$(1 - S2f)$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\displaystyle {\frac{{(\tau_f + \tau_i)^3 + 3 \omega^2 \tau_i^3 \tau_f (\tau_f ...
...4 \tau_f^3}}{{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^3}}}$ (15.107)

τf - τs partial derivative

The second partial derivative of (15.62) with respect to the correlation times τf and τs is

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{\partial \tau_f \cdot \partial \tau_s}}}$ = 0. (15.108)

τs - τs partial derivative

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

$\displaystyle {\frac{{\partial^2 J(\omega)}}{{{\partial \tau_s}^2}}}$ = - $\displaystyle {\frac{{4}}{{5}}}$S2f(1 - S2s)$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\displaystyle {\frac{{(\tau_s + \tau_i)^3 + 3 \omega^2 \tau_i^3 \tau_s (\tau_s ...
...4 \tau_s^3}}{{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^3}}}$ (15.109)

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