Subsections

The extended model-free gradient

The model-free gradient of the extended spectral density function (15.63) 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 parameters S2 and S2f, and the internal correlation times τf and τs. The positions in the vector correspond to the model parameters which are being optimised.

$ \mathfrak{G}_j$ partial derivative

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

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

$ \mathfrak{O}_j$ partial derivative

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

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial \mathfrak{O}_j}}}$ = $\displaystyle {\frac{{2}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$$\displaystyle {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}}$τi$\displaystyle \Bigg($$\displaystyle {\frac{{S^2}}{{1 + (\omega \tau_i)^2}}}$ + $\displaystyle {\frac{{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}}{{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}}}$ + $\displaystyle {\frac{{(S^2_f - S^2)(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}}$$\displaystyle \Bigg)$. (15.72)

S2 partial derivative

The partial derivative of (15.63) with respect to the order parameter S2 is

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial S^2}}}$ = $\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)$. (15.73)

S2f partial derivative

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

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial S^2_f}}}$ = - $\displaystyle {\frac{{2}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$ciτi$\displaystyle \Bigg($$\displaystyle {\frac{{(\tau_f + \tau_i)\tau_f}}{{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}}}$ - $\displaystyle {\frac{{(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}}$$\displaystyle \Bigg)$. (15.74)

τf partial derivative

The partial derivative of (15.63) with respect to the correlation time τf is

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial \tau_f}}}$ = $\displaystyle {\frac{{2}}{{5}}}$(1 - S2f)$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\displaystyle {\frac{{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}}{{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}}}$. (15.75)

τs partial derivative

The partial derivative of (15.63) with respect to the correlation time τs is

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial \tau_s}}}$ = $\displaystyle {\frac{{2}}{{5}}}$(S2f - S2)$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\displaystyle {\frac{{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}}{{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}}}$. (15.76)

The relax user manual (PDF), created 2016-10-28.