Electron iso-density surfaces provide a thermodynamically consistent representation of atomic and molecular surfaces

Abstract

The surface area of atoms and molecules plays a crucial role in shaping many physiochemical properties of materials. Despite its fundamental importance, precisely defining atomic and molecular surfaces has long been a puzzle. Among the available definitions, a straightforward and elegant approach by Bader describes a molecular surface as an iso-density surface beyond which the electron density drops below a certain cut-off. However, so far neither this theory nor a decisive value for the density cut-off have been amenable to experimental verification due to the limitations of conventional experimental methods. In the present study, we employ a state-of-the-art experimental method based on the recently developed concept of thermodynamically effective (TE) surfaces to tackle this longstanding problem. By studying a set of 104 molecules, a close to perfect agreement between quantum chemical evaluations of iso-density surfaces contoured at a cut-off density of 0.0016 a.u. and experimental results obtained via thermodynamic phase change data is demonstrated, with a mean unsigned percentage deviation of 1.6% and a correlation coefficient of 0.995. Accordingly, we suggest the iso-density surface contoured at an electron density value of 0.0016 a.u. as a representation of the surface of atoms and molecules.

Fig. 1. Agreement of iso-density surfaces from quantum chemical calculations with experimental results.

The mean unsigned percentage error (MUPE) is plotted as a function of the electron iso-density cut-off. Source data are provided as a Source Data file.

Fig. 2. Comparison of iso-density surfaces (in Ų) from quantum chemical calculations with experimental surfaces.

a, b Cut-off density 0.0016 a.u. c, d 0.002 a.u. e, f 0.001 a.u. The residuals are plotted at the right. Source data are provided as a Source Data file.

Fig. 3. Iso-density surfaces of two low-energy conformers of 1-pentanethiol contoured at cut-off density of 0.0016 a.u.

The surface areas of 159.8 Ų (left) and 149.2 Ų (right) are significantly different from the experimentally determined surface (156.5 Ų). The Boltzmann averaged iso-density surface for multiple conformers is 157.2 Ų, in agreement with the experimental value.