The moments of a charge distribution provide a concise summary of the nature of that distribution. They are suitable for quantitative comparison of experimental charge densities with theoretical results. As many of the moments can be obtained by spectroscopic and dielectric methods, the comparison between techniques can serve as a calibration of experimental and theoretical charge densities. Conversely, since the full charge density is not accessible by the other experimental methods, the comparison provides an interpretation of the results of the complementary physical techniques. The electrostatic moments are of practical importance, as they occur in the expressions for intermolecular interactions and the lattice energies of crystals. The first electrostatic moment from X-rays was obtained by Stewart (1970), who calculated the dipole moment of uracil from the least-squares valence-shell populations of each of the constituent atoms of the molecule. Stewart’s value of 4.0 ± 1.3 D had a large experimental uncertainty, but is nevertheless close to the later result of 4.16 ± 0.4 D (Kulakowska et al. 1974), obtained from capacitance measurements of a solution in dioxane. The diffraction method has the advantage that it gives not only the magnitude but also the direction of the dipole moment. Gas-phase microwave measurements are also capable of providing all three components of the dipole moment, but only the magnitude is obtained from dielectric solution measurements. We will use an example as illustration. The dipole moment vector for formamide has been determined both by diffraction and microwave spectroscopy. As the diffraction experiment measures a continuous charge distribution, the moments derived are defined in terms of the method used for space partitioning, and are not necessarily equal. Nevertheless, the results from different techniques agree quite well. A comprehensive review on molecular electric moments from X-ray diffraction data has been published by Spackman (1992). Spackman points out that despite a large number of determinations of molecular dipole moments and a few determinations of molecular quadrupole moments, it is not yet widely accepted that diffraction methods lead to valid experimental values of the electrostatic moments.
Lorentz transformation of a charge-current density and “relativistic polarization” of a moving current loop