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Optimal superposition.
We consider a given time frame in which the atomic positions of a (sub)molecule are given by ${\bf x}_\alpha, \alpha = 1\ldots N$. The corresponding positions in the reference structure are denoted as ${\bf x}^{(0)}_\alpha, \alpha = 1\ldots N$. For both the given structure and the reference structure we introduce the yet undetermined centroids ${\bf X}$ and ${\bf X}^{(0)}$, respectively, and define the deviation
\begin{displaymath}
{\bf\Delta}_\alpha \doteq
{\bf D}({\bf q})\left[{\bf x}^{(0)...
...bf X}^{(0)}\right] -
\left[{\bf x}_\alpha - {\bf X}\right].
\end{displaymath} (4.45)

Here ${\bf D}({\bf q})$ is a rotation matrix which depends on also yet undetermined angular coordinates which we chose to be quaternion parameters, abbreviated as vector ${\bf q} = (q_0,q_1,q_2,q_3)$. The quaternion parameters fulfill the normalization condition ${\bf q}\cdot{\bf q} = 1$ [56]. The target function to be minimized is now defined as
\begin{displaymath}
m({\bf q};{\bf X},{\bf X}^{(0)}) =
\sum_\alpha \omega_\alpha \vert{\bf\Delta}\vert^2_\alpha.
\end{displaymath} (4.46)

where $\omega_\alpha$ are atomic weights (see Section 4.2.1). The minimization with respect to the centroids is decoupled from the minimization with respect to the quaternion parameters and yields
$\displaystyle {\bf X}$ $\textstyle =$ $\displaystyle \sum_\alpha \omega_\alpha {\bf x}_\alpha,$ (4.47)
$\displaystyle {\bf X}^{(0)}$ $\textstyle =$ $\displaystyle \sum_\alpha \omega_\alpha {\bf x}^{(0)}_\alpha.$ (4.48)

We are now left with a minimization problem for the rotational part which can be written as
\begin{displaymath}
m({\bf q}) = \sum_\alpha \omega_\alpha
\left[{\bf D}({\bf q...
...r}^{(0)}_\alpha - {\bf r}_\alpha\right]^2
\stackrel{!}{=} Min.
\end{displaymath} (4.49)

The relative position vectors
$\displaystyle {\bf r}_\alpha$ $\textstyle =$ $\displaystyle {\bf x}_\alpha - {\bf X},$ (4.50)
$\displaystyle {\bf r}^{(0)}_\alpha$ $\textstyle =$ $\displaystyle {\bf x}^{(0)}_\alpha - {\bf X}^{(0)},$ (4.51)

are fixed and the rotation matrix reads [56]
\begin{displaymath}
{\bf D}({\bf q}) =
\left(
\begin{array}{ccc}
q_0^2+q_1^2-q...
...2(q_0q_1+q_2q_3) &q_0^2+q_3^2-q^2_1-q^2_2
\end{array}\right).
\end{displaymath} (4.52)


next up previous contents
Next: Quaternions and rotations. Up: Theory and implementation Previous: Theory and implementation   Contents
pellegrini eric 2009-10-06