Charge densities come of age.

In 1915, just three years after the discovery of X-ray diffraction by Von Laue, Peter Debye noted that “the scattering from light atoms should get more attention, since along this way it should be possible to determine the arrangement of electrons in crystals”. Debye!s statement preceded the development of quantum mechanics and Born!s definition of electron probability distribution, but correctly assumed that the electron distribution was an observable that had become accessible. Interestingly, it took the better part of a century for this vision to be realized and for X-ray charge-density analysis to become a true analytical technique that was capable of providing quantitative insight into controversial issues and sufficiently rapid to be applicable to a series of related problems. In 1990 we wrote, “At present, charge-density analysis is far from a routine technique”, pointing out the need for time-consuming collection of large data sets and the limitations in accuracy of the experimental measurements. These limitations have now to a large extent been overcome as a result of the development of more-intense X-ray sources, sensitive area detectors that allow rapid (and redundant) data collection, much improved cryogenic techniques, and last, but not least, the dramatic increase in computing power. As a result, not only has the accuracy improved but the analysis can be fast and precise, as demonstrated by Koritsanszky et al. in 1998 with the determination of accurate experimental electronic properties of dl-proline monohydrate within 1 day. It illustrates the disappearance of time limitations, making the time required for experimental charge-density analysis comparable with that for theoretical calculations. Of course, the experiment yields the charge density for the molecule in the solid state rather than the isolated molecule or complex, thus incorporating in the case of molecular crystals the small but subtle effects of the molecular environment. It must also be kept in mind that the charge distribution and not the wave function is accessible, a crucial distinction, though recently wave functions derived from experiment have been obtained by constraining Hartree–Fock variational calculations to fit the experimental structure factor amplitudes. However, the theory of “atoms in molecules”, as pioneered by Richard Bader, which provides a quantitative link between the total electron density and the allimportant physical properties of a molecule, bypasses the wave function in the analysis. Topological analysis of the total density has been exploited to obtain net atomic moments, including charges, and to infer the nature of chemical bonding directly from the electron-density distribution. The chemical bond analysis derives much of its power from the characteristics of the topological bond path between atoms, including the density (1) at the bond critical point (BCP) and the Laplacian of the electron density at the BCP and in other regions around the atoms. At the BCP, 1 is a minimum along the bond path but is a maximum along the perpendicular directions. The results recently reported by Luger and co-workers on the bonding in a [1,1,1]propellane with its inverted carbon atoms are a culmination of a series of careful studies of charge density on strained hydrocarbons and other small molecules from Luger!s laboratory. The work on the propellane addresses the important issue of the nature of this relatively short (< 1.6 ?) C–C interaction, which has been the subject of earlier theoretical and experimental studies. The analysis showed the presence of a bond path between the two inverted carbon atoms with a significant electron density at the BCP, corresponding to a bond order of 0.71, as derived with Bader!s empirical relationship, in very good agreement with theory. The agreement is less close for the value of the Laplacian (51(rBCP) at the BCP, which is much more positive according to the experiment, and indicates a closed-shell rather than a shared interaction. To assess this discrepancy, it must be realized that the experimental Laplacian, being a second derivative, is quite sensitive to the functions used in both the experimental and theoretical analyses. To obtain precise experimental information on a second-derivative function such as the Laplacian, very high order X-ray data would be needed. These are weak or even absent as the Xray scattering falls off with scattering angle as a result of interference and the effects of thermal motion. The experimental Laplacian may therefore be rather dependent on the functions used in the fitting of the experimental observations. Similarly, the theoretical Laplacian may vary with the nature of the basis set and its completeness. In this context it is relevant that the X-ray refinement on which the static density is based is performed with Slater-type functions, while Gaussian functions are used in the theoretical computation with which comparison is made. So the dis[*] Prof. P. Coppens Department of Chemistry State University of New York at Buffalo Buffalo, New York, 14260-3000 (USA) Fax: (+1)716-645-6948 E-mail: [email protected] Highlights