Crystallographic Symmetry Interactions

Crystallographic symmetry interactions (e.g., packing interactions) between the molecule(s) located in the asymmetric unit and all symmetry-related molecules surrounding the asymmetric unit are computed by
E_{PVDW} = \sum^{n_{S}}_{S=1} \sum_{i<j} f_{VDW}(R_{iSj})
\end{displaymath} (4.19)

E_{PELE} = \sum^{n_{S}}_{S=1} \sum_{i<j} f_{ELEC}(R_{iSj})
\end{displaymath} (4.20)

where the first sum extends over all cystallographic symmetry operators ( ${\cal O}_{s}, \vec{t}_{s}$) and $R_{iSj}$ is defined by
R_{iSj}= \vert {\cal F}^{-1} \cdot MinG(
{\cal F} \cdot \v...
...l O}_s \cdot {\cal F} \cdot \vec{r}_{j} + \vec{t}_s
) \vert
\end{displaymath} (4.21)

${\cal F}$ is the matrix that converts orthogonal coordinates into fractional coordinates, and the second sum extends over all pairs of atoms $(i,j)$ for which $R_{i}s_{j}$ is less then a specified cutoff selected by the constraints interaction statement $r_{cut}$ and $r_i$ and $r_j$ are the coordinates. The function $MinG(\vec{r})$ defines the minimum image distance in fractional coordinate space. It operates separately on each component of the three-dimensional vector $\vec{r}$, where the operation on each component $x$ is given by
MinG(x)= sign(-x) \; int(\vert x\vert + 1/2) + x
\end{displaymath} (4.22)

The function $int(x)$ is defined as the integer part of x, and $sign(x)$ is defined as the sign of x. The nonbonding interaction energy between two atoms is independent of whether it is an intermolecular or an intramolecular interaction. The atom/group search and the update of the interaction list are carried out analogously to the intramolecular case. (Refer to Section 13.3 for the definition of symmetry operators and the unit-cell constants that define ${\cal F}$.)

Xplor-NIH 2023-11-10