Dispersion forces

The understanding of cohesion has two main strands; first, what are the forces between the constituent particles of matter and, second, how does the operation of these forces give rise to the transformation of gases into liquids, liquids into solids, and to all other manifestations of cohesion, of which the elasticity of solids and the surface tension of liquids have, throughout the years, been the two that have attracted most attention. We have seen that in the 18th century there were some interesting speculations about the form of the forces, in particular that they fell off with r, the separation of the particles, as r-n, where n is greater than 2, its value for the law of gravitation. The second strand received some attention at this time but little progress was made. The situation was reversed by Laplace who found that he had to dismiss speculation about the nature or form of the forces with the dictum that all we could know of them was that they were ‘insensible at sensible distances’. He made, however, a substantial contribution to the second strand of the problem with his theory of capillarity and, in the hands of his followers, his ideas proved fruitful, if controversial, in the interpretation of the elastic properties of solids. No further progress could be made until the kinetic theory and the laws of thermodynamics had been established.

The aim and scope of this work are set out in the first chapter. Here I explain the conventions that I have used and thank those who have been kind enough to criticize my efforts.

The work is based on primary printed sources. A few letters and other informal documents have been used but only if they have already been printed. Secondary sources are given when they refer directly to the matter in hand or when they seem to be particularly useful. No attempt has been made, however, to cite everything that is relevant to the background of the subject since this would have led to the inflation of an already long bibliography. This policy has led to a fuller coverage of the 18th century than of the 19th where the secondary literature is potentially vast. In contrast, there are almost no directly useful secondary sources for the 20th century, but here the number of primary sources is impossibly large. It would have been easy to have given ten or more times the number listed. The choice is inevitably biased by the recent aspects of the subject upon which I have chosen to concentrate; others might have made other choices, but no one could give a comprehensive coverage of the last century.

Newton's legacy

The natural philosophers of the eighteenth century knew Newton's work [1] through his two books, the Principia mathematica of 1687 [2] and the Opticks of 1704 [3]. His belief in a corpuscular philosophy is clear in both, and is particularly prominent in the later editions of the Opticks, but the cohesive forces between the particles of matter are not the prime subject of either book. Together, however, they contain enough for his views on cohesion to be made clear. We, who are now privy to many of his unpublished writings, know how much more he might have said, or said earlier in his life, had he not been so fearful of committing himself in public on so controversial a topic. He was not the first to speculate in this field but his views were better articulated than those of his predecessors [4] and, what is perhaps more important, they carried in the 18th century the force of his ever-increasing authority. It was his vision that was transmitted to the physicists of the early 19th century, and we examine first the legacy that he left to his philosophical heirs. The account is restricted to the subject in hand; that is, how does matter stick together, and wider aspects of Newton's thought remain untouched.

In the Preface to the Principia he describes the success of his treatment of mechanics and gravitation, and then continues:

Laplace in 1805

In the field of capillarity it is usual to consider together the work of Young and Laplace, and it is true that they both obtained some of the same important results within a year of each other. Their aims and methods were, however, quite different. In reading Young we are reading 18th century natural philosophy; in reading Laplace we are reading 19th century theoretical physics [1]. This ‘sea-change’ in the early years of the new century is as dramatic as that of the ‘scientific revolution’ of the 17th century, and was due to the efforts of the great French school of mathematical physics of that time [2]. This is not the place to discuss the origin of this second revolution but to concentrate only on how it led to a revival of the subject of cohesion and to a second period of advance. The man responsible was Laplace [3].

The prevailing opinion in France at the end of the 18th century was that of Buffon and his followers; the cohesive forces were probably gravitational in origin and so followed the inverse-square law at large distances but departed from that law at short distances where the shapes of the particles affected the interaction. In 1796 Laplace discussed this view in the first edition of his Exposition du système du monde, noting, however, that the particles of matter would have to be of an inconceivably high density and extremely widely spaced if matter was to have its observed degree of cohesion and its known density [4].

1820–1870

The half-century that followed the decline of Laplace's influence in the 1820s was an exciting if confusing time for both physicists and chemists. Laplace and his contemporaries had created many of the mathematical tools that would be needed by the rising generation of theoretical physicists but these tools were to be used in decidedly non-Laplacian ways in the flourishing fields of thermodynamics, optics, electricity and magnetism. The men who were responsible for these developments were mainly German and British; French influence declined rapidly from about 1830. An important early figure was Franz Neumann but it was the brilliant generation that followed who were to lead these fields – Stokes (b.1819), Helmholtz (1821) [1], Clausius (1822), William Thomson (1824), Kirchhoff (1824) [2], and Maxwell (1831) [3]. Some of the views that they were to articulate were held instinctively by Faraday [4], the modest but acknowledged leader of the experimental scientists. The physicists often maintained that every theory should ultimately be reducible to mechanics but they nevertheless created theoretical structures that did not lend themselves to such a reduction. The fertility of field theories led, in Britain at least, to a disparagement of theories based on action at a distance, but in Germany matters were less polarised. The influence of Kant's philosophy led Helmholtz in particular to retain this concept, and Clausius and Boltzmann were later to be equally happy with it, at least as a pragmatic basis for molecular modelling.

Some problems have always been with us. No one knows when man first asked ‘What is the origin of our world?’ or ‘What is life?’, and progress towards satisfactory answers has been slow and exceedingly difficult. One aim of this study is to take such a perennial theme, although one narrower than either of these two problems, and see how it has been tackled in the Western world in the last three hundred years. The topic is that of cohesion – why does matter stick together? Why do gases condense to liquids, liquids freeze to solids or, as it has been put more vividly, why, when we lift one end of a stick, does the other end come up too? Such questions make sense at all times and the attempts to answer them have an intrinsic interest, for the subject of cohesion has at many times in the last three centuries been an important component of the physical science of the day. It has attracted the attention of some of the leading scientists of each era, as well as a wide range of the less well known. It is a part of our history that is worth setting out in some detail, a task that I think has not yet been attempted.

In this chapter we examine some results for four model systems consisting of rings of H atoms. These calculations show how the number of atoms in a complex reaction may influence rates of reaction, particularly through the activation energy. The systems are as follows.

• Four H atoms in a rectangular geometry of D2h symmetry. The rectangle is characterized by two distances, RA and RB. We map out a region of the ground state energy for this four-electron system as a function of the two distances.

• Six H atoms in a hexagonal geometry of D3h symmetry. This is not a regular hexagon, in general, but, like the system of four H atoms, is characterized by two distances we also label RA and RB. These two distances alternate around the ring. We also calculate the map of the ground state energy for this six-electron system.

• Eight H atoms in an octagonal geometry of D4h symmetry and the specific shape characterized by the RA and RB variables as above. For this larger system we only determine the saddle point with respect to the same sort of variables.

• Ten H atoms in a decagonal geometry of D5h symmetry and the RA and RB variables. Again, we determine only the saddle point.

Since the geometries of these systems are in most regions not regular polygons, we will symbolize them as (H2)n, emphasizing the number of H2 molecules rather than the total number of atoms.

As was seen in the last chapter, the effect of permutations on portions of the wave function is important in enforcing their correct character. The permutations of n entities form a group in the mathematical sense that is said to be one of the symmetric groups. In particular, when we have all of the permutations of n entities the group is symbolized Sn. In this chapter we give, using the theory of the symmetric groups, a generalization of the special treatment of three electrons discussed above.

There are several more or less equivalent methods for dealing with the twin problems of constructing antisymmetric functions that are also eigenfunctions of the spin. Where orbitals are orthogonal the graphical unitary group approach (GUGA), based upon the symmetric group and unitary group representations, is popular today. With VB functions, which perforce have nonorthogonal orbitals, a significant problem centers around devising algorithms for calculating matrix elements of the Hamiltonian that are efficient enough to be useful. In the past symmetric group methods have been criticized as being overcomplicated. Nevertheless, the present author knows of no other techniques for obtaining what appears to be the optimal algorithm for these calculations.

This chapter is the most complicated and formal in the book. Looking back to Chapter 4 we can obtain an idea of what is needed in general.

Benzene is the archetypal aromatic hydrocarbon and its study has been central to the understanding of aromaticity and resonance from the early times. In addition, it has the physical property of having its π electrons reasonably independent from those in σ bonds, leading early quantum mechanics workers to treat the π electrons alone. Since benzene is a ring and the rules for forming Rumer diagrams have one draw noncrossing lines between orbital symbols written in a circle, the Rumer diagrams correspond to the classical Kekulé and Dewar bond schemes that chemists had postulated far earlier than the VB treatments occurred. This parallel has intrigued people since its first observation and led to many discussions concerning its significance. It has also led to considerable work in more qualitative “graphical methods” for which the reader is directed to the literature. (See, inter alia, Randić[59].)

We will examine benzene with different bases and also discuss some of the ideas that consideration of this molecule has led to, such as resonance and resonance energy.

We show again the traditional five covalent Rumer diagrams for six electrons and six orbitals in a singlet coupling and emphasize that the similarity between the ring of orbitals and the shape of the molecule considerably simplifies the understanding of the symmetry for benzene.