Physical chemistry textbooks devote several paragraphs to electrostatics, whereas textbooks on electromagnetism devote several chapters. These sections present a subset of electrostatics that are particularly useful in chemistry.

Source of the dielectric effect

The dielectric effect is the macroscopic manifestation of polarization; polarization stemming from both electronic polarization and from the rotation of polar molecules. This polarization produces induced charge. Experimental analysis shows that for most systems, the net effect can be described with functions of the initial charge and an experimentally measured constant property of each dielectric material called the relative permittivity (ε_r). ε_r is related to the polarizability of the dielectric material as examplified in the Clausius Mosotti equation. In the vast majority of cases, the dielectric effect is linear[1]. For each system, the geometry of the charge distributions (initial and induced) can be analyzed to ascertain the equations that describe that unique system.

Coulomb's Law describes the force between charges in vacuum[2]. When the charges are in dielectric matrix, the surrounding induced charge moderates this force. The electrostatic equations can then be found by determining the magnitude and geometry of the induced charge and using the total charge to build equations that apply for the specific system under study. For the example of a point charge[2], the total charge, initial plus induced, is equal to the initial charge divided by the permittivity.

There are multiple methods to analyze a system. For example, there are differences between macroscopic and microscopic analysis. When there is a discrepancy between the techniques, the explicit induced charge analysis is considered to provide the correct result[3].

A material’s permittivity (ability to form induced charge) is due to induced electronic polarization of molecules, plus induced rotation of molecules having permanent dipole moments, plus internal rotations and vibrations which produce net dipoles, all of which oppose and reduce the electrostatic field. These sources can be divided into two general sets, permittivity stemming from movement of electrons and intra-molecular vibrations, ε_∞, (ε_∞ is predominantly due to electronic polarization) and permittivity stemming from rotations of dipoles, ε_ν.   ε_∞ and ε_ν can be distinguished experimentally as reviewed in Gilson and Honig[1].

The relationship between microscopic induced polarization and macroscopic permittivity was investigated by Debye, Onslager, and Kirkwood among others[4]. Permittivity was calculated from the response of polar molecules to an electrostatic field and their interactions with surrounding molecules. In these calculations, vacuum permittivity (ε_r = 1) was used to describe the microscopic interactions since polarization (and induced charge) is handled explicitly. The effect of electronic polarization was considered as well as the field of the surrounding molecules. The resulting equations build a link between microscopic and macroscopic properties by quantifying dependence of the dielectric effect on induced polarization.

Many systems in chemistry can be analyzed using spherical symmetry; either they are spherically symmetric (e.g. Cl^-), they can be approximated as spherically symmetric (e.g. NH₄⁺), or they can be approximated as a sum of spherically symmetric terms (e.g. a sum of atoms compose a molecule). As a result, in chemistry a large portion of electrostatic terms can be approximated with a function based on spherical symmetry. A collection of solved examples are provided below.

1. Landau LD, Lifshi*t*s EM, Pitaevski*i LP (1984) Electrodynamics of continuous media. Course of theoretical physics / L.D. Landau & E.M. Lifshitz, vol 8, 2nd edn. Pergamon, Oxford Oxfordshire ; New York

2. Lorrain P, Corson DR, Lorrain F (2000) Fundamentals of electromagnetic phenomena. W.H. Freeman, New York

3. Lorrain P, Corson DR (1970) Electromagnetic fields and waves. 2d edn. W. H. Freeman, San Francisco

4. Hill NE, Vaughan WE, Price AH, Davies M (1969) Dielectric Properties and Molecular Behavior. The Van Nostrand Series in Physical Chemistry. Van Nostrand Reinhold Company, London

The dielectric effect in molecular modeling

The charge distribution of solute in media can be modeled with a quantum mechanics calculation. In practice charges are modeled with static charges where an electrostatic function is used which reflects multiple polarization effects.

The electrostatic term used in most force fields models is a macroscopic term applicable at large distances. For short distances, alternate methods may be beneficial. Consider, for example, modeling the water bridge where the strength of the bond can change markedly with rotation of the water molecule. The macroscopic (pairwise) dielectric function does not accommodate the rotation of solvent molecules and is not suitable for a water bridge. Simulations with polarizable atoms can be used to model rotations and subsequent induced charge, with consequent increase in computational resources. In practice, macroscopic equations are employed, often with modification of the permittivity (dielectric constant) to maximize correlation with experiment.

Molecular mechanics calculations use partial-charge models where charges located on atom centers represent approximate molecular charge distributions. (When a charge distribution is spherically symmetric, the charge behaves as though it were located at the center[1]) These charge distributions are then used to calculate interactions within and between molecules. The equation for electrostatic force between point charges is Coulomb's law, V=1/4πε₀ q_iq_j/r²_ij. Partial charge models, with accompanying force field models, have been developed so that they agree with experimental values of binding energies, solvation energies, heats of vaporization, and other calculated or experimentally determined properties. The common practice of modifying the permittivity used during model development presents additional considerations.

A variety of methods are used to model the dielectric effect on the molecular scale, resulting in a wide range of numerical results, especially regarding binding affinities and relative strengths of near versus far interactions (below). These quantities have important practical applications in determining specificity in drug-protein interactions, modeling materials, and predicting packing in crystals.

Effect of modifying the dielectric constant used during force field development

In chemistry, equations span the range from atomic scale to macroscopic. In force field models, when close to a charge (a couple of Angstroms), polarization is only partially realized. For example, it can be shown that at vdW radius from a charged atom, the electrostatic field is the field calculated using the vacuum dielectric constant, D=1.0 (see Example 2) and D=1.0 is used during charge model development. When the models are later used, ‘D’ is frequently set between 2 and 4, D=2-4, as indicated for proteins where many of the distances are greater than 10 Angstroms.

When vacuum permittivity is used during force field model development, but a larger permittivity is used when the force field model is later used, it affects the results. For example, for an experimental value of the RCOOH - RCOOH binding affinity of 6 kcal/mole, imagine a force field model which employed a vacuum permittivity of ε₀ =1.0 to yield binding affinity of 6 kcal/mole. The same model would instead yield binding affinities of 3 or 1.5 kcal/mole (van der Waals attraction being comparatively small) if permittivity values of 2 or 4, respectively, were used. This introduces difficulty when designing a small pharmaceutical molecule intended to bind to a protein where variations of 1.0 kcal/mole in binding affinity strongly influence compound efficacy.

This aspect is elucidated in the microscopic calculation of a charge in solution. The electrostatic field is calculated as a function of distance from the charge. Analysis of the simulation results [2],Fig 3 produces a screening function that is somewhat larger than one for nearest neighbors, that has complex behavior at short distances and becomes smooth at longer distances. This effective dielectric analysis is valid for one charge. However when there are multiple charges and the media is simultaneously polarized by multiple charges and analysis is more complex.

It would be useful to have self-consistent, inexpensive methods for handling the dielectric effect at molecular scales. In particular, it would be helpful to decompose electronic polarization potential energy into terms that depend on individual charges, permitting the separation of pairwise terms. Fortunately, this effort is greatly aided by the relative homogeneity of permittivity due to electronic polarization, whereby polarization terms have approximate radial symmetry. Permittivity due to electronic polarization varies between 1.5 and 2.5[3] for most materials and an electronic polarizability permittivity of ε_r=2 is commonly seen in the literature. The remainder of the dielectric effect, which is due to rotation of permanent and induced dipoles, can be determined independently of electronic polarization by simulating the movement of atoms.

Implicit macroscopic stabilization vs. explicit microscopic terms

The water bridge demonstrates a difference between molecular scale analysis (used in force field models) and classical analysis.

Consider a water bridge between two molecules. Force field models detail the interactions in the system. However, if the water molecule is represented as dielectric material and the macroscopic equation is used, then the permittivity of water lowers the calculated binding affinity by a factor of 80 even though empirically the magnitude is similar to the interaction without the water.

A polarizable atom can also form a bridge. Microscopically, the more polarizable the atom the stronger the bridge, and the stronger the net bonding. Macroscopically, more polarizable material reduces net bonding.

Of course, the dielectric effect is a macroscopic phenomenon and it is not appropriate to use the macroscopic equation. Yet this does not explain all of the differences between microscopic and macroscopic analysis.

The difference lies in the question asked. In classical electrostatics the question is: given a charge and its surrounding induced charge, what force is exerted on a distant charge? In the classical analysis the attraction between the charge and its induced charge is not considered. However, in the microscopic perspective, the binding energy between the bridge water and its surrounding molecules is a considerable portion of the net binding energy. The water bridge is an exaggerated example. A more subtle, yet more pervasive and equally important concern is the potential energy between charges and the charges they induce.

The magnitude of the induced charge due to electronic polarization is significant. The total induced charge for an ion is half the magnitude of the ionic charge, q_induced=-q/2. Moreover, the induced charge is close in position to the ionic charge, resulting in an important attraction term.

As the configurations of the molecules change, the potential energy between the charges and their induced charges change. The system is complex because each partial charge interacts with the induced charge of all other partial charges. It is the potential energy between all partial charges and all induced charges is needed.

Macroscopic vs. microscopic electrostatic equations in EE

Classical electromagnetism is macroscopic analysis where the induced charge layer is assumed to be infinitely thin. Molecular scale classical equations do not use permittivity. Instead they use total charge density which includes induced charge as well as initial charge.

A classic text in optics by Frederick Wooten [4] provides a very good overview of microscopic vs. macroscopic electrostatic equations, including the approximations made. To guide you through the text: Maxwell’s 1st equation is essentially a mathematical transformation of Coulombs Law, pg 16 has the microscopic version of Maxwell’s equation, pg17 has the macroscopic version, pg 25 has the approximations used for isotropic media, and pg 26 has the approximate macroscopic version of Maxwell’s equations.

Solved examples using spherical symmetry

In the classical examples below, Gauss's Law is used, which is of particular interest to chemists since Gauss's Law does not involve macroscopic approximations and is exact at the molecular scale. Gauss's Law relates the electrostatic field at the surface of a volume with the total charge (initial plus induced charge) inside the volume.

Gauss's Law is the standard used when other methods of analyses are in question [1]. Gauss's Law is particularly simple to apply when the volume is a sphere and the charge is at the center of the sphere or is spherically symmetric about the center. In that case, the magnitude of the electrostatic field is equal at all points on the surface of the sphere. The surface area of the sphere is 4pr2 and the field is, E=1/4p ε₀ q_totalr/r² where r is a unit vector. The below examples are provided in Lorrain and Corson, pg 111-113.

Example 1: Consider a point charge in an infinite dielectric. The charge induces a charge about it. The magnitude of the induced charge can be determined with Gauss's Law and the classical macroscopic relation D= ε₀ ε_r E where D can be thought of as the field that would be present if there were no dielectric. A sphere is drawn around the assembly of the charge and its induced charge and the electrostatic field flux is integrated over the surface of the volume. This provides the total charge of q_total=q+q_induced=q/ ε_r.

Example 2: We move now to a charge in the center of a sphere of dielectric. Again an induced charge develops immediately surrounding the charge, moderating the electrostatic field (E) in the interior of the dielectric by a factor of 1/ ε_r, that is E_interior=E_vac/ ε_r. Since the net induced charge must be zero, there is an equal and opposite induced surface charge on the exterior surface of the sphere. Analysis shows that the field outside the sphere is the same field that is generated when the charge is in vacuum with no dielectric present. E_outside=E_vac (q) .

References

1. Lorrain P, Corson DR (1970) Electromagnetic fields and waves. 2d edn. W. H. Freeman, San Francisco

2. Russell, A. W. a. S. T. Quarterly Review of Biophysics 1984, 17(3), 283-422.

3. Gilson MK, Honig BH (1986) The dielectric constant of a folded protein. Biopolymers 25 (11):2097-2119

4. Wooten F (1972) Optical properties of solids. Academic Press, New York