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Spline restraint

Any restraint form can be represented by a cubic spline [#!PreTeuVet92!#]:


$\displaystyle c$ $\textstyle =$ $\displaystyle A c_j + B c_{j+1} + C c''_j + D c''_{j+1}$ (7.65)
$\displaystyle A$ $\textstyle =$ $\displaystyle \frac{f_{j+1} - f}{f_{j+1} - f_j}$ (7.66)
$\displaystyle B$ $\textstyle =$ $\displaystyle 1 - A$ (7.67)
$\displaystyle C$ $\textstyle =$ $\displaystyle \frac{1}{6}(A^3-A)(f_{j+1} - f_j)^2$ (7.68)
$\displaystyle D$ $\textstyle =$ $\displaystyle \frac{1}{6}(B^3-B)(f_{j+1} - f_j)^2$ (7.69)

where $f_j \le f \le f_{j+1}$.

The first derivatives are:

\begin{displaymath}
\frac{ \; {d}c}{ \; {d}f} = \frac{c_{j+1} - c_j}{f_{j+1} - f...
...{j+1}-f_j) c''_j +
\frac{3B^2 - 1}{6} (f_{j+1}-f_j) c''_{j+1}
\end{displaymath} (7.70)

The values of $c$ and $c'$ beyond $f_1$ and $f_n$ are obtained by linear interpolation from the termini. A violation of the restraint is calculated by finding the global minimum. A relative violation is estimated by using a standard deviation (, force constant) obtained by fitting a parabola to the global minimum.

Variable spacing of spline points could be used to save on memory. However, this would increase the execution time, so it is not used.


next up previous contents index
Next: Symmetry restraint Up: Restraints and their derivatives Previous: Lennard-Jones restraint   Contents   Index
Bozidar BJ Jerkovic 2001-12-21