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Analysis of atom-atom interactions via the
Voronoi tessellation S. W. Christensen M. B.
Hursthouse, Department of Chemistry, University
of Southampton, Highfield, Southampton, SO17 1BJ,
UK
Introduction Though much is known about the
local spatial arrangement of atoms that are
bonded, very little is known where non-bonded
atom-atom interactions are prominent.
Specifically, the geometry and topology of
interactions between atoms in adjacent molecules,
in molecular crystals, is poorly understood. This
is a consequence of the lack of (i) an
unambiguous method of identifying which atoms any
given atom may be interacting with, and (ii) an
objective measure of the geometry of
interactions. The Voronoi tessellation provides
an immediate solution to both these questions. It
is based on a set of points (for crystallography
atom coordinates) each of which is enclosed in
its own polyhedron (see fig. 1 for a 2D
illustration and fig. 2 for a--highly
symmetric--3D polyhedron). Neighbour polyhedra
share a face, such that all the polyhedra
together completely fill space. Neighbour atoms
are therefore those whose polyhedra share a face,
and the faces, representing the geometry of the
interactions, are available for detailed
analysis.

• Atom-atom interactions If an atom-atom
  interaction is given by the face shared by the
  two atoms' polyhedra, the geometry of the
  interaction may be quantified in several
  different ways. The most obvious one is the
  interatomic distance, which for bonded
  interactions is the bond length. But in addition,
  the face may be quantified via unique measures
  e.g.
• its area
• the volume between the face and one of the
  atoms (see fig. 3)
• the solid angle under which the face is seen
  from either of the atoms
• the circumference
• the average length of the edges
• and more ...
• A high degree of correlation must be expected to
  exist between some of these e.g. the volume will
  be very nearly proportional to the product of
  area and distance.

Experimental Results A total of 1053
structures, all published in the randomly chosen
year 1980, and spanning across all kinds of
compounds, were obtained from the Cambridge
Structural Database and their Voronoi
tessellations calculated. In all, 370460 unique
atom-atom interactions were present, distributed
across 315 different types (e.g. C-C, H-O, Cl-S
etc.) The by far most common types were H-H
(33), C-H (29), H-O (10) and C-C (8). Many
different Voronoi features were considered, but
for purposes of illustration only two are used
here interatomic distance, and face area. As an
example are shown all instances of O-S and Cl-O
interactions (figs. 4 and 5). It is readily
apparent that although both types of interaction
may occupy the same position in the
separation-area diagram, there are areas where
one type "may go" but the other may not. I.e. the
two types of interaction experience different
geometric constraints - this implies a chemical
difference between them vis-a-vis their local
neighbour atoms. For the short interaction
distances (e.g. the bonded interactions) this is
unsurprising for the long distances it is an
novel insight. Based on the data, models were
derived delimiting the areas in Voronoi feature
space that the interactions may be expected to
reside in (figs. 6 and 7).
Outlook Besides supplying a graphic means of
furthering the understanding of the differences
between different types of interactions, the
Voronoi tessellation features quantifying
interactions may be used in identifying possible
crystal structures from impossible ones (e.g.
when predicting molecular packings from theory),
and more generally they provide a means to
quantify the surface characteristics of molecules
in crystals, which may subsequently be used in
predicting crystal properties.
Acknowledgements The authors thank the EPSRC
for funding through its e-science programme