However, in describing the spatial arrangements of more general patterns of atoms, we can extend these descriptors to include general distances, angles and torsions derived from the coordinates of 2, 3 or 4 atoms which may not be bonded.

In these cases, the **atom** is the key **geometric** **object** from
which the parameters are calculated.

For many years, structural chemists (and crystallographers in particular) have used many non-standard geometrical parameters to describe specific structural features. Calculation of these parameters often requires the use of geometric objects which are more complex than the individual atom positions.

Examples, illustrated below, are:

- distances, angles and torsions calculated using the coordinates of the
**centroid**of some group of atoms. - distances, angles and torsions calculated using the coordinates of a
**dummy****atom**, positioned at some point relative to the fragment, perhaps along a non-bonded vector X-Y, at some specified distance from Y. - angles between pairs of
**vectors**, where the vectors are themselves defined in terms of atomic positions. - distances of specific atoms, centroids or dummy atoms from a mean
**plane**through several atoms. - angles between vectors and
**normals**to mean planes, or between two plane**normals**(dihedral angles). - special geometric parameters used to describe the geometry of specific
types of structural units: the
**puckering****parameters**used to describe ring conformations are an example of special parameters. A number of**directionality****parameters**are also provided for examining the angular orientations of non-bonded vectors with respect to special planes in the molecule.**Ex.**In the morphine example use of the following instructions:SETUP X1 7 8 9 10 11 12 SETUP P1 7 8 9 10 11 12 SETUP P2 1 2 3 4 5 6 DEFINE NANG 1 6 5 DEFINE N - C1 1 6 DEFINE N - C2 5 6 DEFINE DIST X1 6 DEFINE ?ANG P1 P2

has defined geometric objects as follows:

- the atoms of the fragment (atoms being regarded as "default" geometric
objects")
- the centroid X1 of the aromatic ring
- the mean planes P1 and P2 through the aromatic and heterocyclic rings
respectively.

- the intra-annular valence angle NANG at the heterocyclic N atom
- the two N-C bond lengths N - C1 and N - C2
- the distance DIST between the centroid X1 and atom 6 of the fragment (the N
atom)
- the angle ?ANG between the normals to the two planes P1 and P2 (the
significance of ? will be noted later)

- the atoms of the fragment (atoms being regarded as "default" geometric
objects")

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Volume 1 Chapter 7 Composite Geometric Parameters.