Subsections

The dot product Hessian of the ellipsoid

The second partial derivative of the dot product δi with respect to the orientational parameters $ \mathfrak{O}_j$ and $ \mathfrak{O}_k$ is

$\displaystyle {\frac{{\partial^2 \delta_i}}{{\partial \mathfrak{O}_j \cdot \partial \mathfrak{O}_k}}}$ = $\displaystyle {\frac{{\partial^2}}{{\partial \mathfrak{O}_j \cdot \partial \mathfrak{O}_k}}}$$\displaystyle \left(\vphantom{ \widehat{XH} \cdot \widehat{\mathfrak{D}_i} }\right.$$\displaystyle \widehat{{XH}}$$\displaystyle \widehat{{\mathfrak{D}_i}}$$\displaystyle \left.\vphantom{ \widehat{XH} \cdot \widehat{\mathfrak{D}_i} }\right)$
= $\displaystyle \widehat{{XH}}$$\displaystyle {\frac{{\partial^2 \widehat{\mathfrak{D}_i}}}{{\partial \mathfrak{O}_j \cdot \partial \mathfrak{O}_k}}}$.
(15.177)

The Dx Hessian

The second partial derivatives of the unit vector $ \widehat{{\mathfrak{D}_x}}$ with respect to the Euler angles are
\begin{subequations}\begin{align}\frac{\partial^2 \widehat{\mathfrak{D}_x}}{\par...
...lpha \cos \beta \sin \gamma \\ 0 \\ \end{pmatrix}. \end{align}\end{subequations}

The Dy Hessian

The second partial derivatives of the unit vector $ \widehat{{\mathfrak{D}_y}}$ with respect to the Euler angles are
\begin{subequations}\begin{align}\frac{\partial^2 \widehat{\mathfrak{D}_y}}{\par...
...lpha \cos \beta \sin \gamma \\ 0 \\ \end{pmatrix}. \end{align}\end{subequations}

The Dz Hessian

The second partial derivatives of the unit vector $ \widehat{{\mathfrak{D}_z}}$ with respect to the Euler angles are
\begin{subequations}\begin{align}\frac{\partial^2 \widehat{\mathfrak{D}_z}}{\par...
... \\ -\sin \beta \sin \gamma \\ 0 \\ \end{pmatrix}. \end{align}\end{subequations}

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