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Symmetry restraint

The asymmetry penalty added to the objective function is defined as


\begin{displaymath}
F_{symm} = \sum_{i<j} \omega_i \omega_j (d_{ij} - d'_{ij})^2
\end{displaymath} (7.71)

where the sum runs over all pairs of equivalent atoms $ij$, $\omega_i$ is an atom weight for atom $i$, $d_{ij}$ is an intra-molecular distance between atoms $ij$ in the first segment, and $d'_{ij}$ is the equivalent distance in the second segment.

For each $i<j$, the first derivatives are:

$\displaystyle \frac{\partial c}{\partial \vec{d}_{ij}}$ $\textstyle =$ $\displaystyle 2 \omega_i \omega_j (d_{ij} - d'_{ij})
\frac{\vec{d}_{ij}}{d_{ij}}$ (7.72)
$\displaystyle \frac{\partial c}{\partial \vec{d}'_{ij}}$ $\textstyle =$ $\displaystyle -
2 \omega_i \omega_j (d_{ij} - d'_{ij})
\frac{\vec{d}'_{ij}}{d'_{ij}}$ (7.73)

Thus, the total first derivatives are obtained by summing the two expressions above for all $i$ and $j>i$ distances.



Bozidar BJ Jerkovic 2001-12-21