The Electrostatic Moments of a Charge Distribution

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 

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 (Fig. 7.1) agree quite well. 

A comprehensive review on molecular electric moments from X-ray diffraction 

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The Electrostatic Potential Outside a Charge Distribution in Terms of the Multipole Moments... 

As described above (see section on Polarizability), fluctuating charges represent another type of model that can yield self-consistent results for the electrostatic moments of a molecule. Here the idea is to replace induction sites with some sort of movable or fluctuating charge distribution that responds to the field of the other molecules.Many variants of this approach have been reported, several of which we comment on. 

Robinson (1967) has used the Unsold approximation for the energy levels to express the polarizabilities in terms of the electrostatic moments of the ground-state electron distribution. The expressions have been applied to X-ray charge densities by Zyss, Baert, and coworkers (Fkyerat et al. 1995 F. Hamzaoui, F. Baert and J. Zyss, private communication). A detailed description of the derivation and the approximations involved is beyond the scope of this treatise. However, it should be mentioned that the severe approximations are made that all excited-state energy levels are equal, and that the exciting light frequency is equal to zero. 

The elimination of the dipole moment is not mathematical chicanery, but stems from the need to describe different physics for ions and polar molecules. The electrostatic interactions of a polar molecule are dominated by the asymmetry of its charge distribution, which is described in a series expansion as a gradient. By contrast, an ion represents a maximum or minimum (depending on its charge) in the electrostatic potential. For an optimal description, the gradient (dipole) term should be eliminated. 

This way, the electrostatic potential of a given charge distribution is decomposed into contributions due to the various multipole moments of the distribution. 

In a complete description of a molecule s charge distribution, the dipole moment is just one term in a series monopole, dipole, quadrupole, octupole, hexadecapole, etc. A monopole is just a point charge—the dominant term for ions. For neutral molecules organic chemists usually truncate the series after the dipole. However, the quadrupole moment of a molecule can often be quite important. As such, we take a moment here to remind you aboutsome basic electrostatics. 

Consider the interaction of a neutral, dipolar molecule A with a neutral, S-state atom B. There are no electrostatic interactions because all the miiltipole moments of the atom are zero. However, the electric field of A distorts the charge distribution of B and induces miiltipole moments in B. The leading induction tenn is the interaction between the pennanent dipole moment of A and the dipole moment induced in B. The latter can be expressed in tenns of the polarizability of B, see equation (Al.S.g). and the dipole-mduced-dipole interaction is given by... 

The Self-Consistent Reaction Field (SCRF) model considers the solvent as a uniform polarizable medium with a dielectric constant of s, with the solute M placed in a suitable shaped hole in the medium. Creation of a cavity in the medium costs energy, i.e. this is a destabilization, while dispersion interactions between the solvent and solute add a stabilization (this is roughly the van der Waals energy between solvent and solute). The electric charge distribution of M will furthermore polarize the medium (induce charge moments), which in turn acts back on the molecule, thereby producing an electrostatic stabilization. The solvation (free) energy may thus be written as... 

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