Atom 1 (centre)

Atom 2 (centre)

Joint face in both atoms’ polyhedra

Fig. 3. Voronoi face between two atoms;it lies midway in between

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

Fig. 1. Example of 2D Voronoi tessellation

Acknowledgements The authors thank the EPSRC for funding through its e-science programme

Fig. 2. Example of 3D Voronoi polyhedron

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).

Individual observations of O-S interactions

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

Atom-atom separation /A

Fa

ce a

rea

/A

2

Bonded interactions

Fig. 4. O-S interactions

Individual observations of Cl-O interactions

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

Atom-atom separation /A

Fa

ce a

rea

/A

2

Bonded interactions

Fig. 5. Cl-O interactions

"Realm" of the possible for O-S interactions

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

Atom-atom separation /A

Fac

e ar

ea /

A2

Fig. 6. The accessible area in Voronoi feature space for O-S

"Realm" of the possible for Cl-O interactions

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

Atom-atom separation /A

Fa

ce a

rea

/A

2

Fig. 7. The accessible area in Voronoi feature space for Cl-O

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.