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 {S²_f, S²_s, τ_f, τ_s}. 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

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

$\mathfrak{G}_j$ - S²_f partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter $\mathfrak{G}_j$ and the order parameter S²_f is

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

$\mathfrak{G}_j$ - S²_s partial derivative

The second partial derivative of (15.63) with respect to the geometric parameter $\mathfrak{G}_j$ and the order parameter S²_s is

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

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

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

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

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

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

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

$\mathfrak{O}_j$ - S²_f partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter $\mathfrak{O}_j$ and the order parameter S²_f is

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

$\mathfrak{O}_j$ - S²_s partial derivative

The second partial derivative of (15.63) with respect to the orientational parameter $\mathfrak{O}_j$ and the order parameter S²_s is

$${\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial S^2_s}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}} \Bigg( {\frac{{1}}{{1 + (\omega \tau_i)^2}}} {\frac{{(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}} \Bigg)$$ (15.97)

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

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

$${\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial \tau_f}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}} {\frac{{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}}{{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}}}$$ (15.98)

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

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

$${\frac{{\partial^2 J(\omega)}}{{\partial \mathfrak{O}_j \cdot \partial \tau_s}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}} {\frac{{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}}{{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}}}$$ (15.99)

S²_f - S²_f partial derivative

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

$${\frac{{\partial^2 J(\omega)}}{{(\partial S^2_f)^2}}}$$ (15.100)

S²_f - S²_s partial derivative

The second partial derivative of (15.63) with respect to the order parameters S²_f and S²_s is

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

S²_f - τ_f partial derivative

The second partial derivative of (15.63) with respect to the order parameter S²_f and correlation time τ_f is

$${\frac{{\partial^2 J(\omega)}}{{\partial S^2_f \cdot \partial \tau_f}}} {\frac{{2}}{{5}}} \sum_{{i=-k}}^{k} {\frac{{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}}{{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}}}$$ (15.102)

S²_f - τ_s partial derivative

The second partial derivative of (15.63) with respect to the order parameter S²_f and correlation time τ_s is

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

S²_s - S²_s partial derivative

The second partial derivative of (15.63) with respect to the order parameter S²_s twice is

$${\frac{{\partial^2 J(\omega)}}{{(\partial S^2_s)^2}}}$$ (15.104)

S²_s - τ_f partial derivative

The second partial derivative of (15.63) with respect to the order parameter S²_s and correlation time τ_f is

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

S²_s - τ_s partial derivative

The second partial derivative of (15.63) with respect to the order parameter S²_s and correlation time τ_s is

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

τ_f - τ_f partial derivative

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

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

τ_f - τ_s partial derivative

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

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

τ_s - τ_s partial derivative

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

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