X-ray Diffraction and the Electrostatic Potential

The distribution of positive and negative charge in a crystal fully defines physical properties like the electrostatic potential and its derivatives, the electric field, and the gradient of the electric field. The electrostatic potential at a point in space, defined as the energy required to bring a positive unit of charge from infinite distance to that point, is an important function in the study of chemical reactivity. As electrostatic forces are relatively long-range forces, they determine the path along which an approaching reactant will travel towards a molecule. A nucleophilic reagent will first be attracted to the regions where the potential is positive, while an electrophilic reagent will approach the negative regions of the molecule. As the electrostatic potential is of importance in the study of intermolecular interactions, it has received considerable attention during the past two decades (see, e.g., articles on the molecular potential of biomolecules in Politzer and Truhlar 1981). It plays a key role in the process of molecular recognition, including drug-receptor interactions, and is an important function in the evaluation of the lattice energy, not only of ionic crystals. This chapter deals with the evaluation of the electrostatic potential and its derivatives by X-ray diffraction. This may be achieved either directly from the structure factors, or indirectly from the experimental electron density as described by the multipole formalism. The former method evaluates the properties in the crystal as a whole, while the latter gives the values for a molecule or fragment “lifted” out of the crystal. Like other properties derived from the charge distribution, the experimental electrostatic potential will be affected by the finite resolution of the experimental data set. But as the contribution of a structure factor F(H) to the potential is proportional to H−2, as shown below, convergence is readily achieved. A summary of the dependence of electrostatic properties of the magnitude of the scattering vector H is given in Table 8.1, which shows that the electrostatic potential is among the most accessible of the properties listed.