# ¶ The Hirshfeld Surface

The Hirshfeld surface of a molecule in a crystal is constructed by partitioning space in the crystal into regions where the electron distribution of a sum of spherical atoms for the molecule (the promolecule) dominates the corresponding sum over the crystal (the procrystal). Following Hirshfeld, we define a molecular weight function $w\left(r\right)w\left(r\right)$:

$w\left(r\right)=\frac{{\rho }_{promolecule}}{{\rho }_{procrystal}}\phantom{\rule{0ex}{0ex}}w\left(r\right) = \frac\left\{\rho_\left\{promolecule\right\}\right\}\left\{\rho_\left\{procrystal\right\}\right\}\\$

$w\left(r\right)=\frac{\sum _{a\in molecule}{\rho }_{a}\left(r\right)}{\sum _{b\in crystal}{\rho }_{b}\left(r\right)}w\left(r\right) = \frac\left\{\sum\limits_\left\{a \in molecule\right\}\rho_a\left(r\right)\right\}\left\{\sum\limits_\left\{b \in crystal\right\}\rho_b\left(r\right)\right\}$

${\rho }_{a}\left(r\right)\rho_a\left(r\right)$ is a spherically-averaged atomic electron density centred on nucleus $aa$, and the promolecule and procrystal are sums over the atoms beloning to the molecule and to the crystal, respectively.

The Hirshfeld surface is then defined in a crystal as that region around a molecule where $w\left(r\right)\ge \frac{1}{2}w\left(r\right) \ge \frac\left\{1\right\}\left\{2\right\}$ That is, the region where the promolecule contribution to the procrystal electron density exceeds that from all other molecules in the crystal.

Note: Although the Hirshfeld surface is based on Hirshfeld's stockholder partitioning scheme, Fred Hirshfeld did not discover it, nor did he ever describe it. We named it in his honour in 1997.

## ¶ Attributes of the Hirshfeld Surface

• It is not a simple function of molecular geometry. The surface shape relies on the interactions between molecules in the crystal as well as between atoms in the molecule.
• Hirshfeld surfaces smoothly enclose almost all of the available space around molecules.
• Intermolecular voids, typically occupying less than 5% of the unit cell colume, exist where more than two molecules contribute significantly to the total local electron density.
• Molecular volumes defined in this manner interlock somewhat like molecular lego.
• Surface characteristics reflect the interplay between different atomic sizes, and intermolecular contact distances, and hence intermolecular interactions, in a very subtle way.
• Computationally straightforward and rapid.
• Various properties can be encoded on the surfaces.