H. H. J o f f i University
of
Cincinnati
Ohio
I
I
A classical electrostatic view of
Chemical Forcer
In the present paper, we shall attempt to review the different types of forces involved in the formation of chemical compounds, solids and liquids. In spite of the need of non-classical, quantum mechanical methods to achieve anything more accurate than the crudest qualitative calculations of the energies arising from the operation of these forces, it will he shown that, basically, they are all of a classical, electrostatic nature, and that the salient features of their operation can he understood from simple electrostatic considerations, even though these are utterly incauahle of uroviding- a satisfactory . quantitative . treaiment. These forces must explain the existence of the following types of matter: I n the solid state, crystals may be classified into the following four extreme classifications, with various intermediates: (1) Ionic crystals, consisting of an ordered lattice of ions. (2) Molecular crystals, consisting of an orderly arrangement of neutral molecules. (3) Extended lattice crystals, such as SiOz, in which neither free ions, nor individual molecules can be identified. (4) Metallic crystals, in which a n orderly array of neutral atoms of one kind is held toeether in a manner approximating none of the other three classes. In the liquid state,we must take account of: (1) The cohesion of neutral molecules to form the disordered condensed phase. (2) The various interactions between solvent and solute, as well as the interactions between like and dislike species in liquid mixtures. I n the gas phase, we note the existence of molecules, i.e., groups of atoms combined into and acting as a unit. The same problem arises for the molecules of the molecular crystal, for the entire extended lattice crystal, and for the molecules in the liquid state.
Simple Electrostatic Forces
We shall start with the interaction of two ions, of charge el and ez a t a distance r. The attractive Coulomhic force between two such ions is f =edr3 and depends on the distance through the inverse second power. This type of force leads to typical ionic bonds, as encountered in ionic crystals.
-
-
Figure 1.
lon-dipole interaction.
Many neutral molecules are dipoles. I n the homogeneous field of an ion, such dipoles will he oriented in such a way that an attractive force results. Perfect orientation is counteracted by thermal motion, so that the attractive force varies with the degree of alignment. If the dipole (Fig. 1) is represented as consisting of two charges of magnitude *el separated by a distance 1, relatively small compared to the distance r from the ion of charge er, the force is f = -(2e,*l
coa O,)/ra
where el is the angle between the direction of the dipole and the ion-dipole axis. The dependence on r in this case is the inverse third power, and thus ion-dipole forces fall off more rapidly with distance than ion-ion forces. An important example of iondipole forces is the solvation of ions by polar molecules if no specific solvato complexes are formed.
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Next, there must also be attractive forces between pairs of dipoles. The field of one dipole tends to orient the direction of another, again subject to counteracting thermal motion. If one dipole consists of two charges +el separated by a distance 11, the other of charges *e2 separated by lz, and the two dipoles are a t a distance r (large compared to 21 and 12) and form angles nl and 0%with the line joining them, (Fig. 2), the attractive force is
ion-induced dipole forces are weaker than ion-dipole forces, hut are important in the solvation of ions by non-polar molecules. Just as the electrical field of an ion is capable of inducing a dipole in a non-polar molecule, the field of a permanent dipole has a similar effect. Again, the orientation of the induced dipole is such that it points toward the permanent dipole, (Fig. 4),andJeads to an attractive force, which is given by 30le,~l,~ f = -( 3 cos' 8, '9
These forces are inversely proportional to the fourth power of r, and so fall off more rapidly with increasing r than those discussed previously; they are the pre-
Figure 3.
Ion-induced dipole interadion.
Figure 4.
Dipole-induced dipole interaction.
dominant forces responsible for intermolecular attraction, and hence for condensation to liquid and for freezing of polar substances, such as chloroform. With the exception of forces involving quadrupoles and higher poles, the forces listed are the only ones conceivable for particles made up from a rigid distribution of charges. Atoms and molecules, however, are not rigid. The motions of positive nuclei and negative electrons are separate and distinct, and application of an electric field to an atom or non-polar molecule will induce a dipole moment. Thus an ion induces a dipole in any near atom or molecule, and since electronic motion is very rapid compared to atomic motion, the dipole is always in the direction of the line joining the atom or molecule, and always so oriented that the net force is attractive (Fig. 3). The magnitude of this force is where a! is the polarizability of the atom or molecule, el the charge of the ion, and r their separation. The dependence on r is of the inverse fifth power. These 650
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+ 1)
where the permanent dipole consists of charge *el separated by a distance 11 and inclined a t an angle el to the line joining it to the atom or molecule of polarizability a a t a distance r. Since the electrons in atoms or molecules are in a constant motion the angular distribution of which is random, any atom or molecule has associated with it a dipole of constantly changing magnitude and direction. Maybe the easiest way of visualizing such a dipole is by considering the motion of one electron in the field of a singly charged nucleus, i.e., the hydrogen atom. It is immediately apparent that a t any one moment this pair of particles represents a dipole moment-provided we make the classical assumption that a t a given moment the electron may be localized a t a particular point. Visualization of a momentary dipole for a many-electron atom is less straightforward; but there is no reason that, momentarily, the electrons should be symmetrically distributed about the nucleusalthough in the time average they are--and hence atoms represent momentary dipoles. The same is true of molecules. These changing dipoles of neighboring molecules are randomly oriented, and thus lead to no net force. But the existence of these temporary dipoles leads to the induction of added induced dipoles in neighboring molecules, and these always give an attractive force. These forces are the van der Waals or London dispersion forces, and as such are responsible, among other things, for the condensation of nonpolar substances, e.g., rare gases and methane to liquids and solids.
(J Figure 5.
Quadrupoler
In addition to the various forces listed, it must be realized that all molecules represent quadrupoles (and higher electric poles); two kinds of quadrupoles which are of different orders of magnitude should be distinguished. If bonds in a molecule are polar and
the lack of dipole moment in the molecule results from the molecular symmetry (CClh COX), rather large quadmpole moments can be expected (Fig. 5a). I n molecules without polar bonds (e.g., homonuclear diatomic molecules), however, the high concentration of electrons in the bond leads to a quadrnpole moment of smaller magnitude (Fig. 5h). The Coulombic forces between quadrnpoles on the one hand, and ions, dipoles, and quadrupoles on the other can readily he calculated. Their magnitude, however, is generally so small that they make little or no contribution. Finally, it must be pointed out that the equations given above break down when the separation of charges becomes of the same order of magnitude as the separation of particles. These equations give, however, an excellent qualitative picture of the forces involved and may well be used for calculations of orders of magnitude. We have thus accounted for the forces involved in four items of our original listing: the ionic forces of an ionic crystal, the van der Waals and London forces of the molecular crystal, the cohesive forces of liquids, and the forces leading to solvation of ions and molecules. The interactions leading to molecule formation and to the formation of extended lattices are covalent bonding forces, and the metallic crystal may be envisaged in terms of the same covalent bonding forces, or by other quantum mechanical models, all of which will be discussed in the next section. Covalent Bonds
Aside from this array of direct electrostatic interactions between ions, dipoles, induced dipoles, and higher poles, we must consider covalent or electron bond forces. Even though these forces are often specifically distinguished from electrostatic forces, we will show that they also are of electrostatic origin. However, their understanding requires the methods of quantum mechanics. Quantum mechanics can be considered as based on the Schrodinger equation H*
=
W*
as a fundamental postulate, which cannot be proved a pm'om', but which is justified a posteriori by comparison of the results derived with experiments. In this equation H is the Hamiltonian operator; Y,a function of all the space and spin coordinates of all particles under consideration, and of time, is called a wave function, or eigenfunction; and W is a differential operator involving time. For most molecular problems it is practical to remove from Y a factor containing the time dependence, in which case W simply becomes the total energy of the system. Without going into detail, we must take a closer look a t the Hamiltonian operator H. It consists of two parts: the first of these involves the partial differentiation of Y with respect to space coordinates, and represents the kinetic energy of the particles. The second part of the Hamiltonian operator represents potential energy of the particles. When written down completely, it is an infinite series, involving interaction terms for charges, dipoles, multipoles, both of electrostatic and magnetic nature. It is, however, a common experience of people working in quantum mechanics that, for calculation of energy
quantities, including the binding energy of molecules, only the terms involving attraction and repulsion of charged particles are of importance. Hence a single term erej/r2, representing the mutual electrostatic interaction of the particles i and j, is needed for each pair of particles. Additional terms, of course, must be included for any externally applied forces, but such forces are rarely of concern in discussions of bond energies. In the treatment of molecules, the particles which are considered are the individual electrons (particularly valence electrons) and the nuclei (or kernels, i.e., nuclei including inner shells of electrons). I n this case, the second part of the Hamiltonian represents the electrostatic attraction between electrons and nuclei (or kernels), and the electrostatic repulsions between pairs of electrons and between pairs of nuclei (or kernels). Unfortunately it is impossible to solve the Schrodinger equation rigorously, for more than two particles, and consequently approximation methods must be used. Two main methods are available, and if carried to completion, give identical results. Usually however, it is impractical to carry either method to completion, and very approximate results are generally accepted. The two methods are the valence bond (VB) method (resonance method) due largely to Heitler, London, Slater, and Pauling, and the molecular orbital (MO) method, developed mostly by Hund, Mulliken, and Hiickel. Both methods are, in principle, capable of explaining the existence of molecules and chemical bonds. I n practice, the calculations are so cumbersome that few treatments even approaching completeness have been reported save for some of the simplest moleculesmostly diatomic ones, linear triatomic ones, and hydrides. However, these calculations indicate that the existence of molecules, and of chemical bonds such as are responsible for extended lattices, are tractable by arguments-although quatum mechanical onesbased only on the electrostatic interactions of nuclei and electrons. The same types of calculations can he applied to metallic crystals, although they become extremely cumbersome, involving, for example, the resonance of a single covalent bond between large numbers of nearest neighbors. A more effective argument is the treatment of the metal as a network of positive nuclei (or kernels) imbedded in a sea of electrons. This model can also be treated rather successfully quantum mechanically if only the electrostatic terms are included in the potential energy portion of the Hamiltonian operator. Bibliography RICE, 0. K., "Electronic Structure and Chemical Binding," McGraw-Hill Book Co., New York, 1940. PAULING, L., "Nature of the Chemical Bond," McGraw-Hill Book Co., New Yark, 1935. PITZER, K. S., "Quantum Chemistry," Prcntice-Hall, Inc., New York. 1953. SLATER; J. C., "Quantum Theory of Matter," McGraw-Hill Book Co., New York, 1951. STREITWEISER, A., "Molecular O~bital Theory for Organic Chemists," John Wiley and Sons, New York, 1961. KAUZMAN, W., "Quantum Chemistry," Academic Press Inc., New York, 1957. Volume 40, Number 12, December 1963
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