Surface Properties

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de and di

The simplest and most immediately useful property to map onto the surface is the distance from the surface to the nearest nucleus external to the surface, which we call de. This property provides an immediate picture of the nature of intermolecular contacts in the crystal.

In a similar fashion, we can define di, the distance from the surface to the nearest nucleus internal to the surface, which is valuable when used in conjunction with de.

The range of de and di across the Hirshfeld surface varies considerably depending on the atoms in the molecule (size dependence) and the particular type of intermolecular interaction experienced (interaction dependence). Mapping de over the same range of all molecules would reduce the colour contrast for molecules which feature only a small range of contact distances (such as hydrocarbons). To gain the maximum benefit from the colour mapping on the surface, we choose a de range most suited to the each group of molecules under direct comparison.

dnorm

dnorm is a normalised contact distance. di is normalised by the van der Waals radius of the atom involved; de is similarly normalised, and the sum of these two quantities is the dnorm property:[1]

$d_{norm} = \frac{d_{i} - r_{i}^{vdW}}{r_{i}^{vdW}} + \frac{d_{e} - r_{e}^{vdW}}{r_{e}^{vdW}}$

Where atoms make intermolecular contacts closer than the sum of their van der Waals radii, these contacts will be highlighted in red on the dnorm surface. Longer contacts are blue, and contacts around the sum of van der Waals radii are white, as shown in the diagram at right.

Note that the definition of dnorm above means that close contacts, which show up on the Hirshfeld surface as red spots, must occur in pairs of identical size, either on another region of the same Hirshfeld surface, or on a neighbouring one.

Curvedness and Shape Index

A molecular surface defines the shape of a molecule. Because the Hirshfeld surface defines the shape of the molecule in terms of its surrounding crystalline environment, we think that the local shape of the surface may provide some chemical insight.

Schematic diagram of a surface normal n, and the two principal directions u and v
At any point p on the surface, we can determine the outward normal, and there exist two principal directions u and v (see right) along which the principal curvatures κ1 and κ2 are calculated

The two conventional measures of curvature, the mean curvature, H, and the Gaussian curvature, K, do not provide much physical information. Koenderink[2],[3] has introduced two more useful measures of surface curvature, the curvedness, C, and the shape index, S:

$C = \frac{2}{\pi}\ln\sqrt{\frac{\kappa_{1}^{2}+\kappa_{2}^{2}}{2}}$        $S = -\frac{2}{\pi}\arctan\frac{\kappa_{1}+\kappa_{2}}{\kappa_{1}-\kappa_{2}}$

Curvedness is a function of the root-mean-square curvature of the surface, with flat areas of the surface having a low curvedness and areas of sharp curvature having a high curvedness. Areas on the Hirshfeld surface with high curvedness tend to divide the surface into contact patches with each neighbouring molecule, so that the curvedness of the Hirshfeld surface could be used to define a coordination number in the crystal.

Shape index is a qualitative measure of shape and can be sensitive to very subtle changes in surface shape, particularly in regions where the total curvature (or the curvedness) is very low. One important attribute of the shape index is that two regions where the shape index differs only by a sign represent complementary "stamp" and "mould" pairs. This means that maps of shape index on the Hirshfeld surface can be used to identify complementary hollows (with shape index < 0) and bumps (with shape index > 0).

Key references provide numerous examples of the application of shape index and curvature to a wide variety of molecular crystals.[4][5]

Fragment patch surface for rubrene with one adjacent molecule

Fragment Patches

This mapping colours patches on the Hirshfeld surface differently depending on their closeness to adjacent molecules. It provides a convenient way to identify the nearest neighbour coordination environment of a molecule (and hence its "coordination number"). Molecules external to each patch can be generated by right clicking on the coloured patch.

References

1. J.J. McKinnon, D. Jayatilaka, M.A. Spackman, Chem Commun., 2007 ,3814 - 3816:
Towards quantitative analysis of intermolecular interactions with Hirshfeld surfaces
2. J.J. Koenderink, Solid Shape, Cambridge MA, MIT Press, 1990
3. J. J. Koenderink, A. J. Van Doorn, Image and Vision Computing, 1992, 10, 557.
4. J.J. McKinnon, M.A. Spackman, A.S. Mitchell, Acta Cryst. B, 2004, 60 ,627-668:
Novel tools for visualizing and exploring intermolecular interactions in molecular crystals
5. M.A. Spackman, D. Jayatilaka, CrystEngComm, 2009, 11 ,19-32:
Hirshfeld surface analysis