Section 1: Theoretical Foundations of Intermolecular Interactions1: Bonds and Intermolecular Interactions – The Return of Cohesion to Chemistry

One of the fundamental territories of chemistry is the chemical bond, the glue from which an entire chemical universe is constructed, resulting in a rich and enticing variety of species held together by stabilizing interactions. Let me first define the interactions and the manner in which they will be classified in this introductory chapter.

There are interactions that pair-up or delocalize electrons over a two or more atoms, and thereby give rise to entities we call molecules. This chapter refers to these interactions as chemical bonds. The chemical bond defines a chemical identity. It also accounts for the ‘magic’ of chemistry, whereby one molecule disappears and a new one appears. Take for example the soft gray solid, sodium, (Na)_s, and mix it with the yellow-greenish gas, chlorine (Cl₂). And within seconds, lo and behold, you get a white solid, (Na⁺Cl⁻)_s. All three species have distinct identities, being defined by their chemical bonds, and as such the magic is a manifestation of breaking bonds and making new ones. Clearly, then, the term chemical bond defines the molecule, and hence also the chemical identity. Usage of this term is also a tribute to Gilbert Newton Lewis who invented it 100 years ago¹  as a quantum unit of bonding in molecules. By so doing, Lewis has ushered “the electronic-structure revolution in chemistry”.² 

There are however, stabilizing interactions also between and among molecules with saturated valences, such as hydrogen bonds, dipole–dipole and dispersion interactions, and so on. These interactions are responsible for another type of ‘magic’- the magic of aggregation, crystallization, gelation, coagulation, micelle formation, self-assembly, and evolution of large mesoscopic bodies. Consider for example discrete water molecules in the gas phase (vapor). As you lower the temperature, gradually you see water appearing as a liquid, and as temperature is further lowered, lo and behold, the liquid becomes a beautiful translucent solid, ice. Thus, the interactions between the discrete molecules bring about the formation of mesoscopic and macroscopic matter, and as such, we refer to these cohesive forces as intermolecular interactions.

Despite the identity of the forces involved in these two kinds of interactions and the fact that their energy representations are entirely identical (Figure 1.1), the distinction of bonds and intermolecular forces is not artificial. Thus, chemical bonding obeys magic numbers (electron pairs, octets, Hückel's 4n+2 numbers, etc.), and defines chemical identity, while the intermolecular interaction, as a creator of aggregates, is an extensive property that does not obey magic numbers and has much lesser selectivity. In other words, bonds are formed to satisfy the atomic valences, while intermolecular interactions are residual interactions formed between species, which have satisfied their formal valences. Consequently, diatomic bonds are generally stronger than the respective diatomic intermolecular interactions, and as such, ‘bonds’ make molecules, while ‘intermolecular interactions’ create aggregates of molecules.

It is deemed necessary therefore to conserve this distinction in view of the current vogues of calling every stabilizing interaction, even as in He₂, a bond; a tendency that has already been criticized.³  This distinction is also didactic, playing an important role in the comprehension and teaching processes of the structure of chemistry. In fact, it is this distinction that has enabled the development of chemistry as a molecular science.⁴ 

In many ways, bonding and intermolecular interaction is a field in a midst of a great renaissance.^5,6  There are new interesting theoretical approaches to probe the origins of bonds, many novel bonding motifs,^7,8  and experimental studies that describe imaging of bonds being broken and remade using atomic force microscopy.⁹  The bond is becoming again a central intellectual arena, and one can even find allusions to the bond as an elementary particle of chemistry, the so-called “bondon”.¹⁰  Likewise, the interest in intermolecular interactions has moved to new frontiers with bonding motifs such as “halogen bonds” and CH⋯HC interactions, and methods to calculate and analyze them.^11–15  There exist also new chemical bonds that are supported by dispersion interaction, such as very long C–C bonds.¹⁶  There are even some claims of visualizing hydrogen bonds and of bond breakage and remaking. This “return of the bond and intermolecular interactions” to the central arena of chemistry marks the revitalization of chemistry as an intellectual science.

This chapter tries to convey the feeling of the author that the chemical bond and intermolecular interactions are now frontier domains in chemistry. And, by the nature of this goal, the chapter does not present an exhaustive treatise of bonding but rather a bird's eye view with examples, which reflect personal tastes of the author and some reliance of science writers like Ball and Ritter.^5,6  As such, the chapter adopts in places an essay-like style that addresses the target audience, rather than bringing a rigorous equations-laden exposition that would appear in source material.

The chapter starts with a short historical perspective of the evolution of the bond concept from the times of alchemy, through the constitutional revolution by Lavoisier, Cannizzaro, and Dalton, to the electronic structure revolution by Lewis, and the second revolution of quantum mechanics, where valence bond theory (VBT) and molecular orbital theory (MOT) were developed and imported to mainstream chemistry. It continues with the energy perspective of the origins of bonding, and then shows the unity of outlooks on bonding, emerging from VBT, MOT and density functional theory (DFT). The narrative goes on with very brief description of new bonding motifs, and then it switches to intermolecular interactions.

With the advent of quantum mechanics there is a growing tendency to annex the chemical bonding territory to physics. Let me start then by establishing a claim on the chemical bond as an original territory of chemistry. As Siegfried musingly defines, the task of the historian is to “play tricks on the dead” (p. 16).⁴  To avoid this risk, I shall simply summarize my personal impressions on the matter as shaped by necessarily selective reading.

Long before Christian era, during the 6th to 4th centuries BCE, the great Greek philosophers (Thales, Anaxagoras, Empedocles, Plato, Aristotle, etc.) were postulating the existence of building blocks of matter, calling them elements or atoms, which were drawn/repelled by cohesive/repulsive (sometimes defining these forces as ‘love’ and ‘hate’). However, the Greek philosophers did not engage in experiments to prove or disprove their ideas.

The first experimental chemists were the alchemists (sometimes called the Neoplatonic philosophers) who succeeded the classical philosophers, at the end of the 4th century BCE during the post-Alexander the Great era in the Greek empire. The alchemists viewed their goal as the attempt to emulate the harmonious changes of Nature and its perfection. In this regard, the idea of “bonding” has alchemical origins, where the conjuctio or union of the opposites was considered to be the ultimate synthesis necessary to drive the change of lower matter to gold (including the improvement of the soul).¹⁷  When modern chemistry succeeded alchemy, chemists were trying to grapple with the ‘magic of chemistry’,⁴  the formation of substances and their transmutation. These efforts led eventually to the formulation of theories in which affinities were considered to unite substances and to cause selectivity. Affinity and its practical definition are the roots of the modern concept of bonding and selectivity.

In 1675, Lemery published his book Course de Chymie¹⁸  and used elective affinity to describe the processes in which one metal (or metal ion) is selectively replaced by others in the chemistry of salts (pp. 79, 83, and 94 in ref. 4).⁴  In 1718, Etienne Francois Geoffroy systematized this affinity phenomenon in his table of rapports (pp. 76, 93–96 in ref. 4).⁴  This wonderful mnemonic was a precursor of Mendeleyev's periodic table. It allowed Geofrroy to generalize the selectivity relationships between acids, bases (including metals) and salts. The historian Ursula Klein¹⁹  credits Geoffroy as the first to generalize the basic concept of modern chemistry – that of the compound held together by chemical affinity between the constituents.

In 1744 another French chemist, Guillaume Francois Rouelle, differentiated the sciences of chemistry and physics referring to the force that combines chemical elements, and is very different than the physical forces of mixing or mechanical forces between bodies as in Newton's theory.²⁰  The status of chemistry was rather high in those days, judging by the fact that the Swedish chemist Torben Bergman, who assembled thousands of reactions into an elective attraction/affinity table, presented his table of affinities as a gift to the Duke of Parma. Thus, 18th century chemists were already aware of chemical identities of substances and distinguished these identities from physical mixtures where these identities are not lost.

At the same period, chemists used an eclectic mixture of terms to describe the formation of substances from simpler building blocks. There were abstract terms, which originated in Aristotle's theory of the four elements, like protyle, the hypothetical primitive substance from which all elements were made, and subsequently combined to generate matter. There was a term phlogiston, which was referred to as a principle that causes materials like coal, sulfur, etc., to be inflammable. There was a practical definition of a simple body, one that could not be further decomposed to simpler ones by the experimental means of the day. And even atoms were invoked to represent material building blocks. All these terms painted a very fuzzy abstract picture of elementary bodies that engaged in elective combinations.⁴ 

A dramatic change had to occur before the term elective affinity could be defined as the effective concept we now call the chemical bond. This was brought about by two consecutive revolutions. The first one was the compositional revolution. It started with Lavoisier, who established the material essence and chemical identity of oxygen and other gases (which until then were considered as celestial influences that trickled into earth from heavens). This went on⁴  with ideas on compositions of substances and the coining of the term stoichiometry by Richter, and proceeded to the great debate of Proust and Berthollet about whether a chemical substance has definite proportions of its constituent bodies (Proust), or indefinite-variable ones that depend on the relative masses used for the two bodies (Berthollet). Berthollet's idea, apparently influenced by Newton's Law of attraction, would be proved as a wrong one, but at the same time it was a seminal recognition of chemical equilibrium and the mass laws in chemistry. Chemists were groping for significance of their experiments, and eventually all this activity culminated in the atomic hypothesis of Dalton.

The Quantized Weight of Matter: Dalton identified the abstract term elementary bodies with the discrete atoms of Democritus. But he went a step further by suggesting that these atoms possess unique weights that could be determined in the laboratory. As such, Dalton showed that matter manifests in quantized weights (page 237 in ref. 4),⁴  which chemists could determine using the available techniques of the day. He augmented this hypothesis by suggesting compositional rules by which the atoms combine in simple definitive proportions (AB, A₂B, AB₂, etc.). The chemical bond became a LEGO of atoms.

Still, the LEGO game was somewhat ineffective because the connectivity of the atom was not yet defined, and as a result, the atomic weights seemed to fluctuate, depending on the assumption of the combined ratio of the atoms. In some cases, this indeterminism suggested molecules with fractions of atoms. Avogadro, a physics professor from Turin, tried to resolve these difficulties and postulated in 1814 the existence of molecules, including ones that are made from identical atoms, like H₂ and O₂, which he called molécules elemantaires (pp. 259–263, in ref. 4).⁴  This idea defied all the beliefs of chemists that the atomic combination must involve union of opposites, and the idea was met with strong objections, notably from Dalton himself. The delay in the acceptance of Avogadro's idea continued the misery of chemists, so much so that the great French chemist Dumas exclaimed in 1843, “If I were master I would efface the word atoms from science”. Nevertheless, despite being an objectionable ‘heresy’, some chemists were taken by the concept of the elementary molecules. Thus, Cannizzaro showed in 1860 during the Symposium of Organic Chemistry in Karlsruhe, that Avogadro's molecular hypothesis settled all the problems in the atomic weights and created perfect order (pp. 259–263 in ref. 4).⁴  Now, the weights of the building blocks, the atoms, became stable and did not fluctuate wildly as before. Once atomic weights were definite, valences could be defined from the combining weights, and one could understand the existence of multiple molecules made from the same atoms, e.g. H₂O and H₂O₂. Thus, Avogadro's notion and its implementation by Cannizzaro formed the basis for an eventual definition of the long sought for chemical identity.

With matter being quantized into atomic weights, what was still missing was the nature of the forces that caused the union of atoms into molecules, and the ground rules of this game of LEGO. Hence, bonding theories followed and gradually replaced the affinities or the elective forces.

The first was the dualistic electrical theory of Davy and Berzelius²¹  that sprung from electrochemistry, which postulated attraction between oppositely charged atoms or groups of atoms. A similar theory was proposed later by Thomson, the discoverer of the electron, in which bonding arose from attraction of oppositely charged ions after electron transfer between the atoms. Later, this theory created a vogue of ionic or the so-called electromer theory, especially amongst American scientists.²²  However, from their onset, these ionic type theories have been disqualified by structuralists, led by Frankland, Kekulé, Couper, Butlerov, and so on, who came up with the concept of valence to describe the ‘structure’ of organic molecules (the inverted commas are added because this was not at all a physical structure, but something abstract). In a nutshell, the notion of oppositely charge ions could not account for the nonpolar (nonionic) substances, which typify organic molecules. This dichotomy of two types of compounds was a call for generalization, which needs a hero who will ‘pick up the gauntlet’ and meet the challenge. This hero was found in the person of Gilbert Newton Lewis.

Heroes are seldom born as such. More commonly they seize the opportunity, when it presents itself. The end of the 19th century and the turn of the 20th century witnessed the rise of physical chemistry, ‘the brave new science’ in which chemists like Ostwald, van ’t Hoff, Nernst, and Arrhenius performed quantitative measurements of chemical substances in solution.^23,24  These substances were generally strong and weak ‘electrolytes’, such as NaCl and acetic acid. The conductivity measurements showed the presence of ions in solution, but these were dependent on the strength of the electrolytes. These were exciting days for chemistry where theories of 3-dimensional structure (by van ’t Hoff and Le Bel), equilibrium, rates, osmotic pressure, and electrochemistry were being developed. There were many discussions, which urged the community to generalize chemistry and create an Allgemeine Chemie (a unified chemistry, p. 4 in ref. 23).²³  This has eventually led to the second chemical revolution, the electronic structure revolution, which was ushered by Lewis's hypothesis of the electron-pair bond in 1916 that gave the clue to the LEGO rules of the atomic combinations.

Lewis seized the opportunity and led the chemical community to the unified land. Initially, Lewis was seeking an understanding of the behavior of strong and weak electrolytes in solution (p. 135 in ref. 23).²³  But like the voyagers to America who sailed seeking the Spice Islands and discovered a new continent, so did Lewis, who was seeking to understand the behavior of electrolytes in solution, find instead the concept of the electron pair bond as an intrinsic property of the molecule that stretches between the covalent and ionic situations.¹ 

To appreciate the nature of the discovery, let us recall that Thomson discovered the electron in the early years of the first decade of the 20th century, and he showed that this was an elementary particle of matter. Nevertheless, physicists were relatively latecomers in theorizing an effective chemical bonding mechanism. In fact, as late as 1923, Born writes to Einstein about his perplexing attempts to understand the “homopolar bonding forces” such as those holding H₂, and adds: “Unfortunately, every attempt to clarify the concept fails”.²⁵ 

But, Lewis did not try to clarify the forces, nor did he let the awe of the Laws of Classical Physics inhibit his quest. Instead, he let himself be guided by his chemical overview (from the many, many molecules he went through) to hypothesize the electron pairing as a quantized unit of bonding which gave the clue to the nature of the atomic combination to molecules.²  Thus, Lewis was one of the first chemists to realize that the electron, which he referred to as ‘the atom of electricity’, is an essential building block of the chemical bond. Already in his 1913 paper,²⁶  Lewis used the electron to re-define the term valence numbers, which were addressed by Abbeg who showed in 1904 that the sum of the maximum positive and negative valences for an atom was eight. Thus, Lewis made the distinction between the number of bonds (or coordination number) and the oxidation state of the atom, by using an electron-based language. He then revealed his early insight about the nature of the chemical bond by illustrating the two bonding types in a salt like potassium chloride vs. a paraffin hydrocarbon such as methane. As such, Lewis was already then looking at chemistry coming out in two branches separated by a rift, inorganic chemistry and organic chemistry, and trying to unify them in terms of bonding types, which are the ionic and covalent bonds in today's language. In the end of the 1913 article, Lewis added the metallic bond as a third bond-type, in which “the electron is free to move even outside of the molecule”.²⁶ 

In his groundbreaking 1916 JACS article, “The Atom and the Molecule”,¹  Lewis made a major leap and in a stroke of genius formulated the concept of the electron pair bond. In this paper, he initially discusses the separation of the electrons in an atom to core (kernel) and valence shell, which tends to hold eight electrons, which are normally arranged symmetrically at the eight corners of the cube. This is an iconic representation of the octet rule (which he still calls “the group of eight”). Since in his view the atomic shells are “mutually penetrable” (a term showing he was intuitively cognizant of the modern term overlap), he tries to bind atoms in a manner that satisfies the octet rule, by letting the cubes share corners or edges as shown in Figure 1.2.

Lewis calls these shared edges or corners by the name “shared bonds” (later to be called “covalent bonds”). Note in Figure 1.2 that Lewis also has an ionic bond in A, a shared electron bond in C, and a shared one-electron bond B. As such, his bonding theory stretches over a gamut of bonding situations.

It is clear that Lewis was aware that a shared bond violates the laws of physics. As he was grappling with the difficulty, he postulated (his 6th postulate)¹  that in the small spaces, as at the size of the atom, the classical laws of physics may not work anymore (he then toys with possible mechanisms of electron pairing like Parson's magnetic model, but he leaves the question open without a commitment). This reflects on Lewis's personality; he is not afraid to abandon a useful law when faced with a situation that demands it. This is the type of courage that was displayed for example by Niels Bohr a few years earlier in suggesting his quantum atom that defied the laws of physics. This is the courage of reformers.

Typically, at some point in the paper, Lewis informs his readers that new measurements by Moseley show “that helium has a total not of eight but of either four or two electrons”, and that “from the resemblance of this element to the other inert gases”, one may conclude that “here [in Helium] the pair of electrons plays the same role as the group of eight in the heavier elements, and that in the row … comprising hydrogen and helium we have in place of the rule of eight the rule of two”. Later he writes: “and we may question whether in general the pair rather than the group of eight should not be regarded as the fundamental unit”. He accordingly draws Cl₂ and Cl:Cl, as represented by the beautiful cartoon of Bill Jensen, in Figure 1.3.

With this single concise pictogram, Lewis uses the Berzelius language of representing molecules (e.g., H₂O) and the structural diagrams of the structuralists, to which he adds the notion of the electron-pair bond; what an ingeniously portable device! He uses this symbol further to depict the H₂ molecule as H:H and the somewhat polar molecule HCl as H:Cl, where the electron pair is displaced towards Cl. Using the octet rule and the colon symbol, he describes ammonium as a species with four N:H bonds (previously it was represented as ammonia complexed to H⁺). The perchlorate ion is drawn in a very modern way (no extension of octet is allowed, as most theoreticians agree…), with four Cl:O bonds with all atoms obeying the octet and having formal charges.

This electron-pair-based pictogram allows Lewis to represent double and triple bonds in compact manners, as H₂C::CH₂ and HC:::CH, to arrange electron pairs in space and to anticipate thereby the VSPER rules of 3-dimensional structure. He anticipates resonance theory as well, by shifting electron pairs and forming new structures, which he calls electronic tautomerism and uses to explain color due to movement of loose electrons between the two extreme forms, with sufficiently low characteristic frequency to produce color. Here we can see clearly how he linked spectroscopy to electronic structure changes.

Furthermore, Lewis considers in his paper also reactivity of molecules as a reorganization of electron pairs. Thus, in a bond with high polarity, the pair will move together with the more electronegative atom, while in a homopolar bond, the pair can move with one atom or the other, depending on the conditions. It is clear that Lewis views the bond as a dynamic entity, and naturally he considers reactivity of molecules as a reorganization of these electron pairs. This is clearly a precursor of the “curved arrow” mnemonics of Robinson and Ingold,²  and of the subsequent formulation of heterolytic mechanisms like S_N1 by the Ingold–Hughes school. As such, his intellectual impact on chemistry became significant within a few years of the seminal publication. This can be witnessed from the summary of the Faraday Discussion in 1923,²⁷  where the following introductory statement was made: “It is to Professor G. N. Lewis that we are indebted for a very valuable conception in that he has given us a visual picture of a mode of union between the atoms alternative to … the electron type suggested by Sir J. J. Thomson… with the aid of two electrons held in common”. Lewis has thus mapped chemistry as a science of electronic structure and reorganization.

Lewis's work eventually had its greatest impact in chemistry through the work of Irving Langmuir,²⁸  who very ably articulated the Lewis concept, coining new and catchy terms, and applying the concept to many branches of chemistry including transition metal complexes. At last, the meaning of the mythical terms elective affinity and elective forces had an operational and an effective definition in terms of a quantized unit of bonding that allowed constructing a chemical universe.² 

Despite the great advantage of the Lewis model it still lacked the basic physics of pairing and the quantitative aspect of bonding. This was provided by the quantum mechanical (QM) theory of matter, which sprung up during the early decades of the 20th century, when in 1926–7 both Schrödinger and Heisenberg, published their QM theories of atoms. Even though the two theories looked very different, they were shown later to be different representations of identical physics.

The Schrödinger equation trickled fast into chemistry, soon providing the quantitative and physical basis for the Lewis bond, and eventually it formed the basis for what we call quantum chemistry. Quantum chemistry (QC) is now the second power of chemistry alongside experiment. This is how far the revolution has spread.

The Schrödinger equation describes the electron as both a particle and a wave, and hence it was called also wave mechanics. Its applications in chemistry have led to three theoretical models, which now encompass virtually all the chemical phenomena. This theoretical trinity is shown in Scheme 1.1.

The first two theories, which were developed in the 1920s, were valence bond theory (VBT) and molecular orbital theory (MOT).²⁹  The third one is density functional theory (DFT), which was developed in the 1960s, and entered chemistry quite a bit later. I shall briefly describe the historical background of only VBT and MOT, and will restrict this to the development of the notions of the chemical bond. Later on, I shall make comments about the origins of chemical bonding using these three theoretical frames.

Initially, the Schrödinger equation was applied to the hydrogen atom but encountered difficulties going even to helium, and it certainly did not show any clear prospects of solving the nature of the chemical bond. At the same time, the overwhelming support of Lewis's idea in chemistry presented an exciting agenda for research directed at understanding the mechanism by which an electron pair could constitute a bond.

The mechanism of electron pairing remained, however, a mystery until 1927 when Walter Heitler and Fritz London went to Zurich to work with Schrödinger. Schrödinger was not interested in bonding, but they were. In the summer of the same year they published their seminal paper, “Interaction Between Neutral Atoms and Homopolar Binding”,³⁰  in which they showed that the bonding in H₂ originates in the quantum mechanical “resonance” interaction which is contributed as the two electrons are allowed to exchange their positions between the two atoms. This wave function and the notion of resonance were based on the seminal work of Heisenberg,³¹  who showed earlier that, since electrons are indistinguishable particles, then for a two electron systems, with two quantum numbers n and m, there exist two wave functions, [φ_n(1)φ_m(2)] and [φ_n(2)φ_m(1)], whose positive and negative linear combinations, Ψ_A and Ψ_B, are split in energy by a new energy term which he called resonance using a classical analogy of two oscillators that, by virtue of possessing the same frequency, form a resonating situation with characteristic exchange energy.

In modern terms (including the electron's spin), the Heitler–London (HL) bonding in H₂ can be accounted for by the wave function drawn in 1, in Scheme 1.2. This wave function is a superposition of two covalent situations in which, in the left-hand form one electron has a spin-up (α spin) while the other spin-down (β spin), and vice versa in the right-hand form. Thus, the bonding in H₂ arises due to the quantum mechanical resonance interaction between the two patterns of spin arrangement that are required in order to form a singlet electron pair. This interaction accounts for about 75% of the total bonding of the molecule, and thereby projects that the wave function in 1, which is referred to as the HL- or covalent-wave function, can describe the chemical bonding in a satisfactory manner. This origin of the bonding was a remarkable feat of the new quantum theory, since until then it was not obvious how two neutral species could be at all bonded (see above, Born's statement to Einstein from 1923, and recall the objection to Avogadro's hypothesis). The QM phenomenon of resonance or interference provided the new ground rules for interaction of negatively charged particles (electrons) in microscopic systems and in the field of oppositely charged ones (nuclei).

In the winter of 1928, London extended the HL-wave function and drew the general principles of covalent bonding, for which he considered also ionic structures for homopolar bonds, but discarded their mixing as being too small. In essence, the HL theory was a quantum mechanical version of Lewis's electron-pair theory. Thus, in his letter to Lewis,²  and in his landmark paper,³²  Pauling pointed out that the HL and the London treatments are “entirely equivalent to G. N. Lewis's successful theory of shared electron pair…”. Although the final formulation of the chemical bond has a physicists’ dress, the origin is clearly the chemical theory of Lewis. His concept of electron pairing turned out to be the resonance or interference of two electronic structure forms in which the indistinguishable electrons exchange sites and lower the energy.

The HL-wave function formed the basis for the version of VB theory that became very popular later. In 1929 Slater presented his determinant-based method³³  and in 1931 he generalized the HL model to n-electrons by expressing the total wave function as a product of n/2 bond wave functions of the HL type.³⁴  In 1932 Rumer³⁵  showed how to write down all the possible bond pairing schemes for n-electrons and avoid linear dependencies between the forms, in order to obtain canonical structures. Further refinements of the new theory of bonding between 1928 and 1933 were mostly quantitative, focusing on improvement of the exponents of the atomic orbitals by Wang,³⁶  and on the inclusion of polarization function and ionic terms by Rosen³⁷  and Weinbaum.³⁸ 

The success of the HL model for H₂ and its relation to Lewis's model posed a wonderful opportunity for the construction of a general quantum chemical theory for polyatomic molecules. Pauling and Slater achieved this import of the HL theory to chemistry. In the same year, 1931, they both published a few seminal papers in which they developed the notion of hybridization, the covalent-ionic superposition, and the resonating benzene picture.^34,39–42  Especially effective were those of Pauling's papers that linked the new theory to the chemical theory of Lewis, and rested on an encyclopedic command of chemical facts, much like the knowledge applied by Lewis to find his ingenious concept 15 years before.¹  In the first paper,⁴¹  Pauling presented the electron pair bond as a superposition of the covalent HL form and the two possible ionic forms of the bond, as shown in 2 in Scheme 1.2, and discussed the transition from a covalent to ionic bonding. He then developed the notion of hybridization and discussed molecular geometries and bond angles in a variety of molecules, ranging from organic to transition metal compounds. In the following paper,⁴²  Pauling addressed bonding in molecules like diborane, and odd-electron bonds as in the ion molecule H₂⁺ and in dioxygen, O₂, which he represented as having two three-electron bonds, 3 in Scheme 1.2. These papers were followed by a stream of five papers, published during 1931–1933 in Journal of the American Chemical Society, and entitled “The Nature of the Chemical Bond”. This series of papers enabled the description of any bond in any molecule, and culminated in the famous monograph in which all the structural chemistry of the time was treated in terms of the covalent-ionic superposition theory, resonance theory and hybridization theory.⁴³  The book which was published in 1939 is dedicated to G. N. Lewis, and the 1916 paper of Lewis is the only reference cited in the preface to the first edition. VB theory in Pauling's view is a quantum chemical version of Lewis's theory of valence. As noted by Hager,⁴⁴  Pauling's biographer, Pauling discovered Lewis’ 1916 paper¹  by reading Langmuir's²⁸  1919 paper. Until reading these two papers in 1920, Pauling had been teaching a chemistry course at Oregon Agricultural College in which he used the image of a chemical bond as consisting of hooks and eyes, e.g., with the sodium atom having an eye and chorine having a hook. Thus, the birth of VB theory in chemistry was an ingenious quantum chemical dressing of the Lewis seminal idea by Heitler and London, and then Pauling. At last there was a basis for the unification of chemistry under a single electronic structure theory, as professed by Ostwald, the father of physical chemistry.²³ 

At about the same time that Pauling and Slater were developing their VB theory, Mulliken^45–48  and Hund^49,50  were developing an alternative approach called molecular orbital theory (MOT) that has a spectroscopic origin. The term MOT appears only in 1932, but the roots of the method can be traced back to earlier papers from 1928, in which both Hund and Mulliken made spectral and quantum number assignments of electrons in molecules, based on correlation diagrams tracing the energies from separated to united atoms.

In MO theory, the electrons occupy delocalized orbitals made from linear combination of atomic orbitals. Drawing 4, Scheme 1.2, shows the molecular orbital of the H₂ molecule, and the delocalized σ_g MO can be contrasted with the localized HL description in 1. While the VB and MO wave functions look very different, in both the origins of the electron-pair-bonding, as in H₂, is the same. In HL theory, it is due to the resonance of the forms, which allows the electrons to exchange positions while in MO theory it is the ability of the two electrons to delocalize over the two centers. Both describe electron sharing à la Lewis.

Eventually, it would be the work of Hückel that would usher MO theory into the mainstream organic chemistry. After a chilly reception,⁵¹  Hückel's theory ultimately gave MO theory an impetus by being a successful and widely applicable tool. In 1930, Hückel used Lennard-Jones's MO ideas⁵²  on O₂, applied them to CX (X=C, N, O) double bonds and suggested the σ–π separation.⁵³  Using this novel treatment Hückel ascribed the restricted rotation in ethylene to the π-type orbital. Equipped with this facility of σ–π separability, Hückel turned to solve the electronic structure of benzene using his new Hückel-MO (HMO) approach.⁵⁴  The π-MO picture in the left hand drawing in Scheme 1.3 viewed the molecule as a whole, with a σ-frame dressed by π-electrons that occupy three completely delocalized π-orbitals. The HMO picture revealed a closed-shell π-component with energy being lower relative to that of three isolated π-bonds as in ethylene. In the same paper, Hückel treated the ion molecules of C₅H₅ and C₇H₇ as well as the molecules C₄H₄ (CBD) and C₈H₈ (COT). This allowed him to understand why molecules with six π-electrons had special stability, and why molecules like COT or CBD either did not possess this stability (i.e. COT) or had not yet been made (i.e. CBD) at his time. Already in this paper and in a subsequent one,⁵⁵  Hückel had built the foundations for what would become later known as the Hückel-Rule of aromaticity.²⁹  This rule, its extension to antiaromaticity, and its articulation by organic chemists in the 1950s–1970s constituted major causes for the acceptance of MO theory.²⁹ 

The description of benzene in terms of a superposition (resonance) of two Kekulé structures appeared for the first time in the work of Slater,³⁹  as a case belonging to a class of species in which each atom possesses more neighbors than electrons it can share with them, much like in metals. Two years later, Pauling and Wheland⁵⁶  applied the HL VB theory to benzene, using the five canonical structures in the right hand drawing in Scheme 1.3, and approximating matrix elements between the structures by retaining only close neighbor interactions. Their approach allowed them to extend the treatment to naphthalene and to a great variety of other species. Thus, in the HL VB approach, benzene is described as a resonance hybrid of the two Kekulé structures and the three Dewar structures;⁵⁶  and the latter had already appeared before in Ingold's idea of mesomerism which itself is rooted in Lewis's concept of electronic tautomerism.^1,2 

It is important to stress again that despite the different appearances of the description of e.g., benzene by the two theories, the fact is that in both cases the bonding energy of the six electrons originates from the ability of the electrons to delocalize over six centers, either via delocalized multi-center MOs or via electron-shifting among the VB structures in Scheme 1.3. Nevertheless, these two seemingly different treatments of benzene faced the chemical community with two alternative descriptions of one of its molecular icons, and this began the VB–MO rivalry.⁵⁷  Interestingly, already back in the 1930s, Slater⁴⁰  and van Vleck and Sherman⁵⁸  stated that since the two methods ultimately converge, it is senseless to quibble on the issue of which one is the better theory. However, this rational attitude does not seem to have made much of an impression on this religious war-like rivalry, which involved most of the prominent chemists in the Pauling–Mulliken generation and beyond.

Although the rivalry makes an interesting story, it has been told before, and will not be repeated herein, as it can be found in a few sources.^29,57,59  What has eventually overshadowed the old conceptual tensions is the development of very effective computational packages. The current arsenal of computational quantum chemistry involves a variety of methods based on MOT, VBT and DFT,^2,7,8,60  and which have brought about discoveries of new bonding motifs (see later).^5–8  Additionally, new techniques have been developed which enable one to ‘locate’ bonds within the total electron density of the molecule (e.g. atoms in molecules (AIM), electron localization function (ELF),⁷  etc.), and procedures that analyze chemical bonding in terms of energy components (e.g. Morokuma analysis, energy decomposition analysis (EDA), block-localized wave function (BLW), natural orbital for chemical valence (NOCV), interacting quantum atoms (IQA), etc.). These powerful quantitative possibilities have taken chemistry into the computational paradise, and have led to a renaissance in the chemical bond.^7,8 

Before discussing the new bonding motifs, it is important first to briefly comment on the origins of the chemical bond, in a manner that does not depend on the bond type or the choice of the representation of the approximate wave function.

It might be tempting to analyze the chemical bond in terms of forces acting on the atoms and bringing them to an equilibrium distance, where the forces vanish.⁶¹  This pristine idea, correct as it may be, does not provide chemical insight, because it does not differentiate different bond types. A more productive approach is the virial theorem,^61–67  which deals with the changes of the potential and kinetic energies that bring about the formation of a bond in equilibrium.

Let us digress for a moment, and define some necessary quantities before discussing the theorem. Scheme 1.4 shows the energy terms in the Hamiltonian. There are terms that depend on the coordinates of a single electron, H(1e), terms which depend on the coordinates of two electrons, H(2e), and terms that depend only on the nuclei, H(N,N). In a given geometry, the last term is a constant, and can be excluded from further discussion.

H(1e) is a sum of mono electronic kinetic- and potential-energy terms, T(1e) and V(1e), respectively. T(1e) corresponds to the sum of kinetic energies of the electrons moving in the field of the nuclei, while V(1e) accounts for the sum of attractive terms of the electrons in the field of all the positive nuclei. Finally, H(2e) accounts for the potential energy due to the pairwise electron–electron repulsion in the molecule, V(2e). In an atom there are only one center atomic terms; the expectation values of the kinetic energy term, 〈T(1e)〉 is positive, the potential energy 〈V(1e)〉 is negative, and 〈V(2e)〉 is positive. 〈V(1e)〉 dominates the energy balance on the atom, and generates a stable atom. However, in a molecule there are interference or resonance terms, which refer to the expectation values of T and V when an electron is allowed to ‘travel’ between two sites. Consider for example the simplest case of H₂⁺ that represents a one electron bond, spanned by the two atomic orbitals (AOs), 〈a| and 〈b|, such that in addition to the on-site terms, we have a mono-electronic resonance-interference term which is commonly labeled as β, where β is the reduced resonance integral, which is always negative and stabilizing:

In turn, β is a sum of kinetic energy and potential energy terms:

Kutzelnigg showed⁶⁷  that β_V is small and slightly repulsive, whereas β_T is dominant and is stabilizing. It may seem counterintuitive that β_V is positive, since it represents the attraction of the overlap population to the two nuclei. However, recalling that the overlap population accumulates by depleting the atomic population, it is clear that the reduced β_V, which is a balance between the very attractive on-site interaction vs. the interaction of the overlap population with the nucleus and the smaller inter-site attraction (eqn (1.1)), can be positive. Be that as it may, β_T dominates the resonance term.

Eqn (1.3) shows the expression of the dominant part of the resonance interaction,⁶⁷  and its dependence on the interatomic distance R and the orbital exponent ζ:

The orbital exponent gauges the compactness of the orbital. The larger the exponent ζ, the more compact is the orbital, the higher is the resonance interaction, and the stronger is the interaction that brings about bonding. Generally speaking, all the interference/resonance terms in MOT, VBT or DFT scale with β, and hence with β_T. In MOT and DFT,⁶⁰  these terms are simply the common mono-electron resonance-integral terms of the orbitals. Similarly, in VBT all the mixing terms between VB structures scale with β and hence are dominated by β_T.

Another important effect that scales with β_T is the Pauli repulsion between electrons having identical spins, e.g. as we encounter when two closed shell orbitals on two centers (A: :B) overlap, or when three electrons are localized on two centers (A: ˙B), etc. This well-known overlap repulsion is given by:

Here n is the number of identical-spin-electrons on the two centers, β the reduced resonance interaction between the two orbitals, and S the corresponding overlap. Since β is dominated by β_T and the latter is negative, Pauli repulsion raises the kinetic energy of the electronic system. This effect is again common to MOT, VBT, and DFT.

The virial theorem^61,62  for two atoms/fragments is expressed in eqn (1.5),

where E is the total energy, V and T are respectively the potential and kinetic components, and R is the interatomic distance between the two atoms/fragments.

The term on the left of eqn (1.5) is the force acting on the molecule times the respective length, R (the product is then the ‘work’ required for stretching or squeezing the distance away from the equilibrium value R_eq). At equilibrium, i.e. for R=R_eq the force −dE/dR is zero, which means that any properly optimized wave function at its own equilibrium distance R_eq must obey the following virial ratio of the kinetic to the potential energy:

Achieving this ratio will result in energy lowering, namely in bonding, as expressed in eqn (1.7):

But, how does bonding actually manifest? When atoms form a covalent bond they shrink by comparison to the free atoms. One can verify this atomic shrinkage by comparing in the periodic table the atomic radius vs. the covalent radius; the latter is invariably smaller. To understand this phenomenon, let us imagine two atoms/fragments, which are infinitely apart, where each is in equilibrium and obeying the virial ratio T/(−V)=½. Let us now shrink their valence orbitals. As a result, the potential energy of the valence electrons in the atom will become more negative (due to larger electron-nuclear attraction). However, the kinetic energy will increase (due to confinement of the electron to a smaller space). The increase of the kinetic energy is steeper because it increases in proportion to the square of the orbital exponent ζ, while the potential energy decreases only in a direct proportion to ζ. Consequently, the ratio T/(−V) of the two shrunk atoms/fragments will become higher than ½, due to the excessive T. As such, the two species will be driven to get closer and overlap their valence orbitals, such that the resulting resonance/interference term β, which as we recall scales as by β_T, will lower their kinetic energy, and restore the virial ratio, giving rise to a bonded species at equilibrium. Thus, the resonance term allows the atoms to shrink, such that in the net balance, the potential energy of the atoms/fragments can decrease and lower the total energy (eqn (1.7)).

Whenever the atoms/fragments possess also lone pairs (or other bond pairs), the shrinkage of the fragments will result in a much higher kinetic energy pressure, which will be augmented by the Pauli repulsions of the spectator lone pairs with the bond pair and among themselves. Therefore, bonds like F–F, O–O, Cl–Cl, S–S, etc., will need a much higher resonance/interference term β to restore the virial ratio and achieve a bond at equilibrium. The manner by which such bonds acquire large resonance energy terms is most apparent by considering the bonding using VBT. But, let us show that VBT, MOT and DFT have the same physical content of bonding.

We limit the following discussion in this section to two-center bonds that occur by sharing of either electron pairs or odd numbers of electrons.

Let us start with the simplest bond in H₂. The energy of the bond is the expectation value of the wave function for H₂ with respect to the Hamiltonian of the molecule, which as we just elaborated has two parts; one is the kinetic energy of the electrons and the potential energy due to their attraction to the nuclei, and the second is the electron–electron repulsion.

Figure 1.4 shows how MOT and VBT describe this bond. Thus, as shown in Figure 1.4a, in MOT the bond is described by the electron pair occupying σ_g MO (g is the orbital symmetry with respect to the inversion point in the mid-bond). To the right of the MO diagram we show shorthand notations of the corresponding Slater determinant Ψ_MO with two spin-paired electrons in σ_g, where spin down is indicated by the bar over the orbital while the absence of a bar indicates spin up. The σ_g MO is a bonding orbital, in the sense that its energy is below the AOs of the constituent H atoms, due to the delocalization of the electron over the two centers. This delocalization effect is imparted by the mono-electronic resonance part of the Hamiltonian (the term β in eqn (1.1)). So, MOT shows intuitively and pictorially that having an electron pair in the σ_g² configuration binds H₂.

At the same time, we can see that the bonding can be improved if we enable the two electrons in σ_g to get away from each other and lower their Coulomb repulsion. This is achieved by usage of configuration interaction (CI). CI causes the σ_u² configuration to mix with σ_g², thus generating the MO-CI wave function (Ψ_MO-CI) in Figure 1.4b, where c is the mixing coefficient. The mixing is enabled by the bi-electronic part of the Hamiltonian, H(2e), which lowers the electron–electron repulsion in the electron pair, and energetically improves the electron-pair-bonding of H₂.^29,60 

This looks somewhat like a paradox, that adding a high-energy configuration lowers the energy. But it can be immediately elucidated by mapping the Ψ_MO and Ψ_MO-CI wave functions to VB ones.²⁹  This can be done easily, by expressing σ_g as a sum a+b of the corresponding AOs, and σ_u as their difference, a−b. Plugging these expressions into the Slater determinants of Ψ_MO and Ψ_MO-CI enables one to expand these determinants into AO-based ones.²⁹  Thus as shown in Figure 1.4a, the Ψ_MO gives rise to a mixture of 50% covalent HL VB structure, Φ_cov, and 50% ionic structures, Φ_ion(1) and Φ_ion(2), which localize the electron pair once on H_a and the other time on H_b. Repeating the procedure for Ψ_MO-CI (Figure 1.4b) shows that the CI augments the contribution of the covalent structure, which becomes 1+c, while diminishing the ionic ones, which becomes 1−c. Clearly therefore, the CI brings about correlation between the two electrons, such that they are mostly far apart in the covalent structure and close together in the ionic structures. This is called static correlation or Coulomb correlation (or still, left–right correlation, or Coulomb hole). It is important to note that Ψ_MO-CI describes the electron pair bond as a covalent-ionic superposition, á la Lewis–Pauling.

One could do everything in the reverse order: start now from the VB configurations and derive the same wave function as obtained in the MO-CI procedure.^29,60  Thus, as shown in Figure 1.4c, initially we utilize the two determinants needed for constructing Φ_cov, which is stabilized by the resonance between the two spin arrangement patterns. This resonance (achieved via the mono-electronic part H(1e) of the Hamiltonian) causes the two electrons to become spin-paired to a singlet. The covalent structure, with the electrons being one on each center, exhibits 100% correlation, and this is why the covalent structure is the lowest in energy compared with the ionic structures that keep the electrons together on a single site. We can now bring in the ionic wave function, Φ_ion(1)+Φ_ion(2), and let it mix into Φ_cov. This mixture of the covalent and ionic structures will describe the electron-pair-bond in the spirit of Lewis and Pauling. The electron pair is now less correlated compared to the pure Φ_cov, but the bond has gained some covalent-ionic resonance energy (RE_cov-ion), which over-compensates for the small loss of correlation. This full circle shows that there is no difference between MO-CI and VB when both methods are carried out to their fullest. In fact, one can start from MO-CI and recover both the Φ_cov and the RE_cov-ion quantity, which are the elements of VB theory.⁶⁸ 

When the bond in question is in a more complex molecule e.g., F₂, the bond energy can be further improved by CI that allows the spectator electron pairs (lone pairs), as well as the doubly occupied orbitals in the ionic structures, to adapt themselves to the changes in the static correlation caused in the bonding electrons and the local charges of the ionic structures. This ‘grooming’ of the electron pairs surrounding the bond in question is called dynamic correlation. Dynamic correlation is rather pictorial in VB, showing how the lone pairs of F–F adapt their shapes and sizes to the local charges on the two atoms, and how they contribute to enhance the bond energy. This differential dynamic correlation can be retrieved in MO-based CI.⁶⁸  Thus we can go back and forth and cross the mirrors between the MOT and the VBT worlds, and even retrieve the key concepts of bonding. This realization is helpful, because it allows us to use for any given problem the theory that provides the most lucid insight.

DFT as implemented in computational quantum chemistry, uses Kohn–Sham orbitals,⁶⁰  which are pictorially identical to those used in MOT, and differ only in that the KS orbitals include already some dynamic correlation effects due to the exchange–correlation terms. For larger molecules DFT uses the NOCV technique¹⁵  to define the bond orbitals, from fragment orbitals that look very much as in MOT, and in a manner akin to VBT. Thus, the description of the electron-pair-bond in DFT is in no way different than in MOT and VBT; it originates from the one-electronic part of the Hamiltonian augmented by the correlation effect. The terminology is different but the physical essence is the same.⁶⁰ 

Odd-electron bonds are quite common now in the chemistry of main group elements,^69,70  and have been known since the beginning of the 20th century when Lennard-Jones described the O₂ molecule⁵²  using MOT and subsequently Pauling did the same using VBT (2 in Scheme 1.2). Figure 1.5 shows the description of simple odd-electron bonds by MOT and VBT (DFT is skipped since it suffers from a pathological self-interaction error for these species). Figure 1.5a shows the one-electron bond, H₂⁺. Using MOT, the bond is described by the σ_g¹ configuration, which maps directly to the VB description as a resonating mixture of two one-electron forms. In both cases, the bond is due to “sharing”, i.e. delocalization of the single electron over the two centers (the β term). Unlike the electron-pair bond, there is no static/Coulomb correlation to take care of, and the MOT and VBT wave functions are identical.

A prototypical 3e-bond is He₂⁺, which is shown in Figure 1.5b. Here too the MOT and VBT wave functions are identical and the bonding is due either to an excess electron in the bonding σ_g orbital, or to the resonance between the two 3e-forms; both are β terms. In more complex species, like F₂⁻, dynamic correlation improves the bonding, by allowing the orbitals to respond to the instantaneous charges of the atoms. As such, once again we can see that MOT and VBT describe the same bonding mechanism for archetypical odd-electron bonds: odd electron binding is due to the resonance/interference term β, which scales by β_T, and lowers the kinetic energy at the bonding region, thus restoring the virial ratio.

In discussing the emergence of new bonding motifs one is limited to a very small selection of topics. As such, I followed with some modifications a recent selection of highlighted topics.^5,71  I duly apologize for any inadvertent omissions in the following discussion.

VBT has given rise to the emergence of new bonding motifs in two-center bonds. Let us therefore start with VBT, while being cognizant that the emerging features are in principle identical to those in MO-CI⁶⁸  or DFT, but in the latter approaches these features are not immediately apparent.

Figure 1.6 depicts the essential elements for discussing the emergence of new bonding features using VBT. Figure 1.6a shows the three VB structures needed to describe an electron pair bond between two fragments A and B (where A and B are identical or different). For classical covalent bonds, the covalent structure Φ_cov is generally stabilized relative to the dissociation limit (A˙+˙B) by the covalent spin pairing energy D_cov. We now bring in two ionic structures, Φ_ion(1) and Φ_ion(2), and mix them with the covalent structure. The VB mixing diagram shows that the consequence is the further stabilization of the Lewis bond state (Ψ_bond(A–B)) by the resonance energy quantity, RE_cov-ion, due to the covalent-ionic mixing. Since this mixing is common also to homonuclear bonds, we refer to this quantity as the charge-shift resonance energy, RE_CS.⁷²  Of course, when one of the ionic structures is the lowest in energy, the final bond is ionic, which may in principle enjoy some RE_CS stabilization too.

The above analysis shows that the electron-pair bond in VBT can be described by three independent quantities, which are indicated in the triangle in Figure 1.6b. Two corners of the triangle are occupied by the covalent and ionic VB structures, while the third corner is the RE_CS quantity, due to their mixing. In principle therefore, we might expect three distinct families of electron-pair bonds that emerge from these independent quantities. One family is dominated by the covalent structure and its bond energy is primarily D_cov, with a minor contribution of RE_CS. The second family is dominated by one of the ionic structures, Φ_ion, and its bonding energy is by and large the Coulomb attraction of the opposite charges, with a minor contribution of RE_CS. These are the two classical Pauling families of covalent and ionic bonds. Alongside them we expect in principle to find a third family of bonds where most, if not all, of the entire bond energy is provided by the RE_CS quantity. This is the charge-shift bonding (CSB) family.^72,73 

Figure 1.7 shows the energy curves of a few bonds plotted against the internuclear distances. For each case one can find the curve for the dominant VB structure in blue, and the one for the ‘exact’ state of the bond in red. The three bond families manifest nicely in these plots.

Thus, for H₂ (Figure 1.7a) the covalent curve dominates the wave function as well as the bonding energy; the exact curve is very close to it. H–H is a classical covalent bond. By contrast, for F–F (Figure 1.7b) the covalent structure is repulsive throughout the range of internuclear distances. Here, the bond and the bonding energy arise entirely from the RE_CS due to the mixing of the ionic structures into the covalent structure. F–F is therefore a CSB. The second panel, in Figure 1.7c and d, shows two heteronuclear bonds B–H and F–H. It is seen that B–H is a classical covalent bond, whereas F–H is a CSB. Finally, the third panel in Figure 1.7 shows the Na–F and Na–Cl bonds. In both of these cases, the dominant VB structure is ionic, and the full exact curve is very close to the ionic curve (RE_CS is small). These two bonds belong to the classical family of ionic bonds.

It is important to recognize that CSBs may appear as classical covalent bonds when one examines the weights of the covalent and ionic structures. Thus, for example, if we compare F₂ and H₂ we find that both wave functions are dominated by the covalent structure that has virtually identical weights of 0.76 and 0.73, respectively. This further shows that the bonding in F₂ is not associated with either the covalent or the ionic structures, and is exclusively determined by the RE_CS feature. The CSB family is unique.

There are many other bonds which belong to the CSB family: Cl–Cl, Br–Br, O–O, S–S, N–N, N–F, N–O, C–F, Si–Cl, etc. All of these bonds are typified by repulsive or weakly bonded covalent structures, whereas the bonding energy is provided exclusively or almost so by RE_CS. It is possible to show that the electronegativity scale is proportional to RE_CS. In fact, the origin of the scale is arguably rooted in RE_CS.⁷³ 

Recall that the analysis of the virial theorem led us to conclude that unlike in H₂, the bonding electrons in F₂ suffer severe Pauli repulsion exerted by the lone pairs. As such, the covalent structure in F₂ is destabilized by the Pauli repulsion of the lone pairs with the bond pair. This kinetic energy pressure together with the shrinkage of the valence orbitals raise dramatically the kinetic energy of the bond and tip the virial ratio off balance. The requisite kinetic energy lowering and restoration of the virial ratio is achieved by increasing the RE_CS, through the covalent-ionic mixing.

As we mentioned before, there are electron density methods that enable one to define and identify bonds. The ELF (electron localization function)⁷⁴  method uses a local function related to the Pauli repulsion to find the domains of different spin-paired electron pairs in the molecule. The integration of the electron density of these domains gives the population N̄ of these domains.

Another electron density based method is AIM (atoms in molecules).⁷⁵  In AIM a bond is characterized by a bond path, which defines a maximum density path connecting the bonded atoms. The point along the path at which the density is at minimum is called the bond critical point (BCP), r_c. One then characterizes the bond type by determining the electron density ρ(r_c) at the critical point, and the Laplacian of the density at the critical point. The Laplacian is defined in eqn (1.8):

It is composed of two terms; G(r_c), which is the kinetic energy density, and V(r_c) which is the potential energy density; G(r_c)>0 and V(r_c)<0. According to AIM, a significant density and a negative Laplacian at the critical point typify a covalent bond with a shared density. When the Laplacian is negative, this means that the potential energy density dominates it. On the other hand, a positive Laplacian corresponds to a closed shell interaction, which typifies either ionic bonds or cases with closed-shell Pauli repulsion such as He₂. However, as seen from Figure 1.8b, a positive Laplacian occurs also in CSBs.

Figure 1.8 shows the differences in ELF and AIM qualifiers for the covalent bond C–C, and the CSB F–F. Thus, Figure 1.8a is the C–C bond in ethane, which is characterized as a classical covalent bond in VBT. It is seen that ELF finds a bond region (the green cylinder) between the carbon atoms, and the integrated electron density in this region N̄ amounts to 1.81e, close to an electron pair. The AIM analysis shows that C–C has a significant critical point density and a largely negative Laplacian as expected from a covalent bond with a shared density. Thus, similar to VBT, both ELF and AIM characterize C–C as a classical covalent bond.

In a striking contrast, the F–F bond in Figure 1.8b has a severely depleted ELF basin with N̄=0.44e, and the basin looks split with most of the electron density being outside of the bond region. Similarly, in AIM, while F–F has a significant critical point density, its Laplacian is positive, indicating repulsive Pauli interactions and excess of kinetic energy. Thus, much like VBT, both ELF and AIM characterizes F–F as CSB. As shown recently,⁶⁸  the Φ_cov wave function, the RE_CS quantity, and the contrasting appearance of the H₂ vs. F₂ bonding in Figure 1.8, could all be extracted as well from MO-CI theory.

It is apparent from the above analyses that homonuclear CSBs have very different features than classical covalent ones, even though they may appear covalent by just looking at weights of the covalent structures in the VB wave functions, or by naively considering that a homonuclear bond like F–F must be covalent. In addition to the above computational features, there are quite a few experimental manifestations of CSBs, which set these bonds apart from classical covalent bonds. For example, the RE_CS quantity for the dihalogen molecules, X₂, can be quantified based on the difference of the barriers for halogen atom transfer vs. H atom transfer, RE_CS=4[ΔE^‡_HXH−ΔE^‡_XHX].⁷³  Other manifestations concerning the expression of CSB in experimental electron density of bonds in propellanes, and in chemical reactivity have been discussed elsewhere.⁷³ 

CSBs are formed by electronegative and lone-pair rich atoms, or bonds that suffer from Pauli repulsion pressure, such as the internal bond in [1.1.1]propellane.⁷³  Dative bonds, e.g., the bond between BH₃ and an amine, carbocations with amines or water molecules, etc., are CSBs.^76,77  All hypercoordinated species, e.g., XeF₂, PCl₅, etc. are CSBs.^73,78  Odd-electron bonds, such as F₂⁻, Cl₂⁻, etc. are CSBs.⁷³  CSB is a new family of bonding alongside the traditional classical covalent and ionic bond families. The ubiquitous CSB family is one of the discoveries of CQC.

The Isolobal Analogy: Perhaps the most productive bonding concept in the past four decades has been the isolobal analogy, developed by Hoffmann.⁷⁹  Hoffmann derived this analogy by going back and forth between delocalized and localized MOs of transition metal complexes, in a manner that created a bridge between MOT and VBT, and allowed the predictions of many new organometallic complexes, which were not considered before.

The isolobal analogy implies that the bonding capabilities of transition metal complex fragments can be deduced by comparing their frontier orbital lobes and occupations to those of common organic fragments. Scheme 1.5 exemplifies the essence of the isolobal analogy between transition-metal complex fragments and small organic fragments. Thus for example, (CO)₅Mn is a fragment that possesses an odd electron in a σ-type hybrid on the Mn. As such, the fragment is isolobal to a methyl-radical fragment that has an analogous singly occupied σ-type hybrid orbital. Similarly, (CO)₄Fe has two lobes, one σ- and the other π type, and is therefore isolobal to the CH₂ fragment, and finally (CO)₃Co has one σ- and two π type lobes, all singly occupied, and is hence isolobal to a CH fragment.

Once the bonding capabilities are defined, one can think about new as well as old molecules made from these fragments. For example, Mn₂(CO)₁₀ is an analog of ethane having a σ(Mn–Mn) bond, (CO)₅Mn–Mn(CO)₅, analogous to the σ(C–C) bond in ethane. Similarly, by analogy to tetrahedrane which is made of four HC fragments, one can think about the transition-metal based tetrahedrane, Co₄(CO)₁₆, constructed from four (CO)₃Co fragments. Mixing of fragments, and devising new ones, gives rise to multitude of new molecules. The isolobal analogy is a bonding concept that emerged from computational quantum chemistry in the late 1970s using extended Hückel theory. Despite the simplicity of the method and the passage of time, the concept is as vibrant as ever, showing how productive theory of chemical bonding can be, when it bridges two branches of chemistry.

Dative Bonds and Bonding in Carbones: Another concept that was imported from coordination chemistry to organic chemistry is the idea of coordinative or dative bonds to carbon atoms. In 1973 Kaska et al.⁸⁰  described the compound hexaphenylcarbodiphosphorane by analogy to coordinative bonding in transition metal complexes, as being composed of the coordinatively unsaturated C(0) coordinated by two Ph₃P: molecules using dative bonds, which were represented by arrows from the Lewis donor phosphines to the Lewis acid C(0); Ph₃P:→C←:PPh₃. Subsequently, Frenking and coworkers⁸¹  used DFT tools and developed this description into a productive concept that involves the tetreles (Group 14 atoms, E=C, Si, Ge, Sn, Pb) and their isoelectronic analogues, e.g. N⁺. An example of such a compound is shown in Scheme 1.6a, in which the bare C in its ¹D state (e.g., with two lone pairs, 2s² and 2p_x²; x is in the molecular plane) acts as a bidentate Lewis acid accepting two dative bonds from appropriate electron-donor ligands, L:. At the same time, its two remaining lone pairs on C, which are the doubly occupied HOMO and HOMO−1 in Scheme 1.6b, can participate in π-back bonding to the ligands.

Unlike the normal bonding motif that predicts allenic type linear molecules as in Scheme 1.6c, carbones are strongly bent (Scheme 1.6a). Using the idea of dative L:→C bonds, Frenking et al. showed that in some cases, the central carbon in a carbone has two lone pairs which are largely the two filled 2p orbitals, shown in Scheme 1.6b. As such, the bonding in these compounds appears different than the classical electron-sharing Lewis bonding, which should result in allenic bonding type (as in Scheme 1.6c), with two π bonds rather than localized lone pairs. The dative bonding picture led to syntheses of new molecules.⁸¹ 

The binding model of carbones was analyzed using DFT computations and EDA analysis, coupled with the NOCV technique,¹⁵  which enables extraction of the energy due to orbital mixing within DFT. It turns out that the orbital mixing term is the dominant one in these bonds, and the σ-orbital mixing is much more significant than the π-back bonding.

The Bonds in Carbones Are CSBs: The usage of arrows, to indicate dative bonds, initiated a dispute regarding the meaning of these arrows vis-à-vis normal Lewis bonds. On the one side was Krossing et al., who argued for a traditional Lewis model with covalent and ionic structures,⁸²  and on the other, Frenking,⁸³  who insisted that the arrow icon describes the bonding as donor–acceptor bonds that are different to Lewis bonds. In the spirit of building bridges, the author of this chapter agrees with both sides. But how so?

Recall that dative bonds have been predicted to be CSBs,⁷⁶  and then shown as such by VB computations.⁷⁷  A CSB from an L: donor to an acceptor like C(0) involves mixing of a no-bond structure, Φ_NB, with a covalent structure, Φ_cov, as shown in the VB mixing diagram in Scheme 1.7, which describes the σ type bonding in a generic carbone. There are other structures wherein the lone pairs of C(0) donate electrons to vacant π type orbitals on L:, but because the σ bonds were shown^81,83  to be dominant, we omit the π back-bonding for the sake of simplicity. Since Φ_NB is either nonbonding or repulsive because of Pauli repulsion, most if not all the bonding in carbones will arise from the VB mixing. As such the L:→C(0) bond is a CSB, which does not arise from the stabilization of any one of its constituent VB structures, but rather from their VB mixing. Indeed, the orbital mixing reported by Frenking et al. is very large and is the origin of the L:→C(0) bond. Furthermore, as the VB mixing is proportional to β we see once again that the driving force for the L:→C(0) CSB is the lowering of the kinetic energy in the bonding region.

When L: is a very good donor (better than L:=R₃N:), the Φ_cov structure will be stabilized and intensify the VB mixing and the RE_CS. At some critical donor capability, Φ_cov, as will be the lowest structure, and the bond will be a CSB similar to other electron-pair bonds like F–F, H–F, etc. Thus, as commented above, the present author seems to agree with both sides of the Krossing–Frenking dispute. The L:→C(0) bond is a multi-structure VB bond á la Lewis–Pauling, but at the same time, the case drawn in Scheme 1.7 is a CSB, owing its origins to the VB mixing between the structures rather than to either one of them.

Supra-multiple Bonds: By supra-multiple bonds I refer to bond multiplicity that exceeds a triple bond. This glass ceiling has been broken ever since Cotton described the Re–Re quadruple bonding in [Re₂Cl₈]²⁻.⁸⁴  Since then many other transition metal and lanthanide/actinide complexes were found to exceed the glass ceiling as well. Thus, based on CASSCF and CASPT2 methods, it has been revealed that Cr₂, W₂, and U₂ complexes and dimers^85,86  reach quintuple and even sextuple bonding. However, usage of effective bond orders (EBO) reduced significantly all these supra-multiple bonds since the formally antibonding orbitals, which are the counterparts of the bonding orbitals, are significantly populated. The EBO index has gained much popularity as the means to gauge the number of bonds in the molecule.

Recently, this author and his coworkers proposed,⁸⁷  based on both full-CI and VB calculations, that C₂ and its isoelectronic species involve quadruple bonding, with three internal strong bonds (one σ and two π) and a fourth inverted σ bond which is weaker (Scheme 1.8). For C₂, the π bonds and the inverted σ were further shown to have propensity towards CSB.

This idea about C₂ was cordially debated,⁸⁸  and recently criticized,^89–91  while the criticism was also responded.⁹²  One of the lessons from this dispute is that there is a disparity between the number of electron-pair bonds that emerge from direct VB calculations or from localization of a full-valence CASSCF wave function, and the bond indices such as the EBO for CASSCF. Thus, in C₂, VB and the transformed full-valence CASSCF wave functions show the existence of four bonding pairs,⁹³  whereas the EBO index is only 2.15–2.3 or so. This disparity may well be common to other supra-multiple bonds, many of which have strong multi-reference character.

One might argue that in principle, the EBO has at least two basic blind spots: one has to do with the 2σ_u orbital of C₂, which is formally antibonding (thus reducing the EBO) but actually this orbital is nonbonding, and hence it should not contribute to the reduction of the EBO. The other blind spot is general and it may trickle into all bonds. Thus, consider the covalent wave function of H₂, which has by itself a strong bonding interaction (95.8 kcal mol⁻¹).⁷³  The covalent wave function can be expressed in terms of two MO configurations with equal weights (when overlap is neglected; otherwise non-equal weights⁶⁸ ), one with σ_g² configuration, the other with σ_u² configuration.⁶⁸  As such, the covalent structure of H₂ has by definition EBO=0, despite its very strong bonding interaction. Consequently, the EBO reduction in molecules with strong multi-reference character will be more prominent than in molecules having dominant single-reference character.

The description of supra-multiple bonds is a new frontier area that will continue evolving, as the dispute on the number of bonds eventually settles.

A recent exciting bonding motif was found by Schreiner and his group,¹⁶  who synthesized molecules with very long C–C bonds, 1.647–1.704 Å. Two of these are shown in Figure 1.9. Thus the molecule in Figure 1.9a, made of diamantine–diamantine coupling, has a C–C bond 1.647 Å long. Similarly, the molecule in Figure 1.9b, the hexa-(3,5-di-tert-butylphenyl)-ethane exists as a stable molecule with a long C–C bond of 1.67 Å. Despite the long bonds, these molecules are stable up to temperatures higher than 200 °C and their bond dissociation energies (BDEs) are close to 60 kcal mol⁻¹. Schreiner et al.,¹⁶  and subsequently, Grimme and Schreirner⁹⁴  showed that approximately 40–50% of the BDEs of these long C–C bonds are contributed by the dispersion interactions due to short and sticky H⋯H contacts (1.94 and 2.28 Å), between the CH⋯HC faces of the diamondoid moieties or of the tertiary-butyl groups of the diarylethane. Grimme¹²  and Schriener and Wagner⁹⁵  reviewed the many other molecules with bonds supported by dispersion. This new bonding motif was discovered by the interplay of experiment and computational quantum chemistry.

Are the Long C–C Bonds CSBs? These molecules have not yet been computed by higher-level theories like VBT or MO-CI. Looking at these molecules in Figure 1.9, one can see that many lone pairs line up the space of the C–C bond, and it suffers most likely from severe Pauli repulsion. As such, one might speculate that the long C–C bond in the Schreiner compounds is a CSB, and that the dispersion may augment the charge-shift resonance energy to sustain these bonds. This is another frontier.

Multi-centered bonds have been known to occur in organic conductors made from conjugated molecules, such as tetrathiafulvalene (TTF) and tetracyanoquinodimethane (TCNQ), which after undergoing oxidation or reduction form stacks that conduct electricity.⁹⁶  For example, reducing TCNQ by alkali metals or by copper, generates solids with stacks of TCNQ˙⁻ radical anions, which form dimers at low temperatures. Similarly, oxidation of TTF generates stacks of partially oxidized TTF, which correspond approximately to oxidation of about 50% of the TTF in the stack. Analogous donors form low energy organic superconductors.

The ceramic-based high T superconductors eventually eclipsed the field of organic conductors. However, the interest in the π-overlapping stacks was rekindled,⁹⁷  when these species were analyzed as models of multi-centered long bonds, known also as “pancake bonds”.⁹⁸  What enables the eventual cohesion of the two anions are the counter-ions, which counteract the anion–anion repulsion and hold the “pancake”.^97–100  The dimer of tetracyanoethene radical anion (TCNE˙⁻)₂ was analyzed (Scheme 1.9a) by several groups.

One way to understand this dimer is to regard it as being bound by an electron-pair multi-center bond made from the two overlapping singly occupied π* orbitals, as shown by the MO in Scheme 1.9b. Using a two configuration CASSCF, which describes this bonding, led to small binding energy, which had to be augmented by multi-reference perturbation theory (MRPT2), which indicated that the bonding must be dominated by dispersion.⁹⁹  Tian and Kertesz⁹⁸  used MCSCF calculations (including both π and π* orbitals) augmented by AIM characterization of the Laplacian along the bond paths. They concluded that the electron pair bond between the two TCNE˙⁻ radical anion is a CSB, in line with its positive Laplacian, as shown in Scheme 1.9c. More recently, Braida et al. used VBSCF calculations¹⁰⁰  augmented by quantum Monte Carlo methods (to retrieve missing correlation). Their conclusion is depicted in Scheme 1.9d, showing that the (TCNE˙⁻)₂ dimer is bonded by pericyclic 3-electron bonding, which is CSB. These unusual species define also a new frontier for the chemical bond.

Unusual bonding motifs expand the frontiers of the chemical bond. This is the case for bonds based on bound triplets. Bound triplet pairs are uncommon because they violate the Pauli exclusion rule. Nevertheless, VB calculations, which were carried out in 1999,¹⁰¹  on the weakly bonded ³Li₂ (³Σ$u+$) dimer, provided the necessary insight, showing how the Pauli repulsion in the triplet pair is transformed into bonding.¹⁰²  The VB structures, which participate in the state-wave-function of ³Li₂ (the z-axis is along the Li⋯Li distance), and the respective orbital representations are summarized in Figure 1.10a–d. Figure 1.10a shows that in addition to the fundamental structure, ³Φ_SS, there are two ionic structures indicated as ³Φ_SZ(ion), which are generated by transferring an electron from the 2s AO of one Li atom to 2p_z of the other, and a covalent configuration, ³Φ_ZZ(cov), in which the electrons are excited from 2s to 2p_z on the two atoms (2p_x,y are unimportant and can be neglected). The mixing of the latter three configurations into the fundamental and repulsive ³Φ_SS configuration generates some bonding.

The corresponding orbitals of this state are illustrated in Figure 1.10b–d. Thus, by mixing ³Φ_ZZ(cov) into ³Φ_SS (in Figure 1.10b) we get a two VB-configuration wave function. We can transform the orbitals of these two configurations without any change of the energy, and we obtain the hybrid orbitals depicted in Figure 1.10c; one of the hybrid orbitals points in and the other is pointing out. As such, the overlap of these hybrids is reduced compared to the 2s–2s overlap in Figure 1.10b, and hence, populating the triplet electrons, one in each of these two hybrids, reduces the triplet Pauli repulsion. At the same time, the resonance interaction between the two structures, (in/out)⇔(out/in), further stabilizes the triplet pair. Finally, mixing the ionic structure endows each one of these hybrids with a tail on the other atom (Figure 1.10d), and thereby it augments the stabilization of the triplet state, leading to a “bound triplet pair”.

What Happens as the Metal Cluster Grows? As these clusters grow, each Li atom maintains with its close neighbors ionic and covalent configurations, and the total number of VB structures that stabilize the triplet pair increases steeply. Consequently, the bonding energy per single Li atom in the cluster increases also dramatically. The bond energy per one Li atom converges for a cluster of 12 atoms. As shown in Figure 1.11, even without a single electron pair bond, the bond dissociation energy increases from mere 1.7 kcal mol⁻¹ for ³Li₂ to 145 kcal mol⁻¹ for ¹³Li₁₂! Thus, the bond energy per atom starts at 0.85 kcal mol⁻¹ and converges to about 12 kcal mol⁻¹. This means that the bonding of ³Li₂ embedded within ¹³Li₁₂ reaches 24 kcal mol⁻¹, which is approximately the bond dissociation energy of a singlet Li–Li bond. For the ⁿ⁺¹Au_n and ⁿ⁺¹Cu_n clusters the pair-bonding energy reaches 33 and 39 kcal mol⁻¹, respectively.

Since the bonding of the dimers is weak, there has been a tendency to refer to these clusters as van der Waals clusters. However, the bonding energies for all the ⁿ⁺¹M_n clusters were calculated with standard DFT methods that lack dispersion interactions. Adding dispersion exaggerates the depth of the minima of the dimers compared with CCSD(T) data and with experiment, but it does not affect the binding energies for larger clusters. Indeed, as was pointed out,^102–104  the bonding in these clusters is highly sensitive to orbital hybridization and orbital mixing effects. VBT provides a clear description of the orbital-related effects and their impact on the bonding energy of these clusters of triplet bound pairs, and the theory further predicts their unique structures. To suit their uniqueness, the bonding in these clusters was termed no-pair ferromagnetic (NPFM) bonding.¹⁰²  NPFM bonded clusters can be made to have even stronger bonding and to have chiral structures; chirality and magneticity in the same molecule. Two of these chiral molecules, which were predicted by CQC, are shown in Figure 1.12.

NPFM clusters are related to Bose-Einstein condensates and Fermi-gas condensates, and can be synthesized using helium-droplet isolation.¹⁰⁵  They can be detected by Stern–Gerlach techniques and spectroscopy.¹⁰⁵  NPFM bonding may become a frontier area in the map of chemical bonding.

The chemical bond is alive and is teeming with novelties. A glimpse into that was allowed above. Many more novelties can be found in the recent perspective, “Pushing the Limits of the Chemical Bonding”, written by Ritter,⁶  discussing novel aspects of bonding in clusters e.g., TaB₁₀⁻, B₁₆²⁻ , Au₂₀, etc., with inorganic aromaticity,¹⁰⁶  and the Jemmis Rules which extend the Wade–Mingos rules to boron-rich clusters.¹⁰⁷  Chemical bonding on surfaces and catalysis,¹⁰⁸  bonding in heavy main group elements¹⁰⁹  and the role of relativistic effects on bond energies and bonding.¹¹⁰  All these novelties push the frontiers of the chemical bond into the central intellectual arena of chemistry.

Up to the 19th century there is no distinction between the historical developments of the concepts of bonding and intermolecular interactions. Both started with the question, why does matter stick together? This question can be traced to the Greeks who were the first to ask it in a manner that did not involve the influence of deities.^111,112  The philosophical discussions of the Greeks eventually led to theories of matter in terms of elements or atoms. In the 18th century chemists were seeking what determined chemical identity, and this quest eventually led them at the end of the 19th century to the recognition of molecules made from atoms (the constitutional revolution). This recognition differentiated the sciences of chemistry and physics, leading to a revolution of chemistry, which has become a molecular science with clear definitions of bond and structure.

At the same time, this autonomy of chemistry relegated to physics the topics of cohesion between molecules to form macroscopic matter, and related problems like wetting, capillarity, surface tension, micelles, etc. It is from those problems that the notion of intermolecular interactions evolved and matured into the science we see now. This is a rich science that involves dispersion interaction, dipole–dipole interactions, hydrogen bonds, and many other interactions that are crowned by the name Z-bonds; e.g., halogen-bonds, pnictogen-bonds, tetrel-bonds, chalcogen-bonds, etc. How did this rich culture evolve?

Here, the historian Rowlinson (chapters 2–4)¹¹¹  distinguishes three periods, which he names after Newton, Laplace, and van der Waals and beyond. But eventually, and especially due to the effort to understand water and the discovery of the ubiquitous hydrogen-bonding (H-bonding), the problem returned into the hands of physical chemists, crystallographers, quantum chemists, and it eventually became widespread in mainstream chemistry. To condense the following historic discussion, I shall consider only dispersion and H-bonding since they also represent the major interaction that shape mesoscopic and macroscopic matter in nature.

Newton himself was an avid alchemist who carried out many experiments including on mercury and its oxide (which according to speculation inflicted him for a period of time with the Mad Hatter Syndrome). Newton believed that both light and matter were particulate. Gravity suggested that particles should attract one another. In contrast, Boyle's gas law suggested that particles of gas repelled each other. Since gases condense to liquids and then to solids, Newton insisted on attraction, but since the solid did not collapse into itself, he concluded that there must also be repulsion at shorter range. He formulated the force between particles as composed of a gravitational attraction given by the inverse-square law (1/r²) and a repulsion given by 1/r. However, the inverse-square law diverged when integrated over many particles and long distance, and in any event, Newton did not provide a cause for these forces. Furthermore, the action from a distance required an instantaneous force, which was conceived as being supernatural.

Experiments on the nature of capillarity by contemporaries of Newton (e.g., Newton's demonstrator Hauksbee) led to the recognition that the interparticle forces must be very short ranged, but since liquids rise even in macroscopic tubes, this suggested that the forces are also long-ranged. Despite these conflicting conclusions, the experiments led eventually to the understanding of surface wetting and tension, which later influenced Laplace's theory of capillarity.

The Newtonian legacy continued to the 18th century, but for many reasons, the progress was painfully slow. The experimental infrastructure was too crude to deal with these weak forces. In addition, most natural philosophers of the era had inadequate knowledge of the Newton–Leibnitz calculus and of mechanics to make headways. Importantly, they had difficulties in accepting action at a distance. And finally, the talents of the science preferred the fields of Astronomy, Electricity and so on. Cohesion was not fashionable. Consequently, the Newtonian heirs believed in attraction and repulsion but were unable to define them in a manner that made sense to their contemporaries; to use d'Alembert's words, “an explanation so vague condemns itself”.

Nevertheless, at the end of the 18th century, action at a distance started to be tacitly accepted, and physicists like John Leslie and Thomas Young established (1802–1804) good intuitive understanding of capillarity. For example, Leslie explained capillarity as attractive forces between water and glass, and the movement of water particles to be in places close to the glass. Young explained capillarity in terms of surface tension, which manifests as a balance of forces of attraction and repulsion.

At about the same time, the Bernoullis, Euler, d'Alembert, Lagrange and Laplace put the Newtonian mechanics in the form familiar now. And it became evident that objecting to action at a distance was not scientifically productive. These changes caused a transition that was led by French scientists, notably Laplace, who described, in mathematical forms, capillarity, surface tension, and interaction of matter particles.

Laplace was a close friend of Berthollet, who strongly believed that chemical affinity was proportional to mass and hence, was gravitational and related to capillarity and surface tension. But, by 1805, Laplace settled on a view that the attractive forces were short-ranged and pairwise, but of unknown nature. He further hypothesized that caloric (the particle of heat that surrounded the particles of matter) was the agent of repulsion that stopped matter from collapsing in by keeping particles apart. These opposite forces served Laplace to develop the mathematical tools for discussing the nature of gases, liquids and solids.

Laplace's model was essentially static. But by the mid-19th century, the new fields of physics, light, electromagnetism, and heat, where field and wave theories reigned, eclipsed this view. Corpuscular/molecular theory and hence also cohesion fell out of vogue! Positivism dominated physics, and all corpuscular phenomena had to be understood in macroscopic visible/measureable terms. The Laplacian physics of interparticle forces lost the battle, but the ideas remained with a few followers like Poisson, and the ideas returned later in the century when van der Waals and others fruitfully united them with a kinetic view of matter.

The new way to analyze intermolecular interactions was provided from the concepts of energy and dynamics, which removed the obstacles created by Laplace's static particles and incorrect description of heat (as caloric).¹¹¹  Newton wrote, “energy is motion”, and although he did not necessarily mean the kinetic energy of gases, what he said inspired others. Already in the 18th century Bernoulli suggested that gases contained particles in rapid movement. Within 1845–1855, two Englishmen, Herapath, a teacher turned journalist, and Waterston, an engineer, published their kinetic theories, disregarding Bernoulli's. This was not the right time, because of the dominant positivism in Physics. Nevertheless, these attempts did not stop, and later Joule and Krönig (from a technical college in Berlin) published in 1856 nonmathematical and intuitive kinetic theories. Krönig's paper induced Clausius and Maxwell into action and they published their own versions of the topic, and within the treatise of energy, they alluded to the interactions between molecules and the behavior of gases and liquids (Chapter 3 in ref. 111).¹¹¹ 

Thermodynamics was developed in the 19th century also by non-mainstream individuals (Chapter 4 in ref. 111).¹¹¹  This started with Carnot's book in 1824, in which he regarded heat as the quantity being conserved. The experiments of Joule in the conversion of heat to mechanical energy, and the calculations of Mayer, both convinced physicists that it was energy not heat that was conserved. This was followed by the derivation of the 1st and 2nd laws of thermodynamics by Clausius and Thomson (William). Helmholtz's publication, On the Conservation of Force, was published in 1847 and it sealed the acceptance of the conservation of energy as a law of nature. He postulated that this conservation meant that all forces in nature are attractive or repulsive and acting along lines joining particles of matter. He also introduced the notion of potential energy. Joule, Clausius, Maxwell and Helmholz held that energy originated from motion.

In the meantime, chemists contributed to the corpuscular theory of matter through Dalton's work (itself influenced by the attraction-repulsion mechanism in Laplace's theory of cohesion) and later through the theory of Davy and Berzelius on the nature of the chemical bond as an electrostatic attraction. Dumas presented, in his course in 1836, a theory of attractive forces between particles of matter, in which he made a clear distinction between the ‘cohesion of the physicists’ and the ‘forces of affinity’ which lead to ‘formation of chemical compounds’. There were a few more attempts, but chemists were by and large out of the field of cohesion in the 19th century.

At the end of the 19th century Boyle's Law was extended by Charles, Gay-Lussac, and Dalton, and become the ideal gas law, connecting pressure, volume and temperature:

Here T is a temperature on a scale whose zero was found to be at approximately −270 °C (where the pressure p is zero). This equation was in fact a statement that the kinetic energy (cT and later RT) determines the volume and pressure of the gas. For years it was known that the behavior of real gases was non-ideal and the deviation reflected molecular properties. Joule and Thomson explored systematically the deviations and revised eqn (1.9) to (1.10):

B(T) is known today as the second virial coefficient and is one of such coefficients associated with intermolecular forces.

For liquids there were known some basic facts about the vapor pressure as a function of temperature, and the change from liquid to vapor involving a large intake of heat (the latent heat of evaporation). The melting of solids involved less intake of heat. The exceptional behavior of ice/water between 0 and 4 °C was also known, but no explanation existed. Heavier vapors such as chlorine, CO₂, and H₂S could be liquefied. Faraday showed (1844) that a variety of gases could be liquefied by combination of pressure and cooling. Andrews (from Queen's College) discussed in 1869 the critical point where liquid and gas coexist, and showed the existence of a continuum in the passage vapor→liquid. His interpretation was something close to a hypothesis of the action of ‘a molecular force of great attractive power’. Other ideas and developments included Earnshaw's theorem that no static system with inverse-square power law could be in equilibrium, Herapath's derivation of the mean velocity of molecules of a gas from the state equation (eqn (1.9)), and the derivation by Young, Dupré and Waterston, of the range of intermeolcular distance as 1 Å or so; much too short but of the right order of magnitude. In the 1850s and 1870s, Clausius and Maxwell published their new kinetic theories of gases based on molecular motion, rotation and vibration, mean sizes and mean paths, and how those lead to the existence of matter in gaseous and condensed phases. In 1870 Clausius derived the virial theorem, in which positions and speeds were linked. One could then derive for an ideal gas (where all the pairwise intermolecular forces are zero) that the kinetic energy of all the particles was equal to 3pV, which is equal to 3RT. In 1868–1871, Boltzmann derived the kinetic energy of molecules as 3RT/2 using his statistical theory. The corpuscular-molecular view was back in focus.

Van de Waals, a schoolmaster at The Hague, started his PhD in Leiden in 1871 and defended it in 1873 (Chapter 4 in ref. 111).¹¹¹  He was awarded the Nobel Prize for his work in 1910. His conviction of the existence of molecules led him to address the pressure and volume modification in the equation of state (eqn (1.9)) by the intermolecular interaction as well as by the molecular size, by reliance on the virial theorem. He assumed that the temperature gauges the kinetic energy of the molecules, that the effect of intermolecular forces is expressed through the pressure, and that the molecules were hard objects. Thus, he surmised that molecules at the surface of a fluid were pulled inward and thereby affect the pressure in proportion to the number of molecules being pulled per unit volume and to the number doing the pulling in the interior. This correction was a term proportional to the square of the molecular density. The free space available for motion was corrected by taking into account the effective volume (co-volume) occupied by the molecules. His new state equation was given then as follows:

where the constants, a and b, are related to molecular properties and interactions. He then showed that the new equation applies to gas and liquid, through the critical point where gas and liquid are coexistent, and that this application enabled him to quantify the molecular diameter and the range of intermolecular forces, e.g., 4.0 and 2.9 Å, respectively, for dimethyl ether. Van der Waals still referred to the forces as “Newtonian”, not in a gravitational sense, but rather as acting at a distance. In 1896 Boltzmann referred to these as van der Waals cohesive forces – a name that has stuck to this day.

Molecules became associated with electricity through the works of Faraday, Berzelius and Davy, then by the young school of physical chemistry of Ostwald, van ’t Hoff and Arrhenius, and subsequently and mostly so by the work of J. J. Thomson. At the same time, it was clear that the van der Waals equation of state needed many corrections and could not be derived in a closed form, and these corrections defined an agenda for research that tried to derive the intermolecular interaction potential from electrostatic theory.

A first attempt by J. D. van der Waals (the son) to derive the intermolecular interaction in the van der Waals equation of sate by looking at molecules as electrical dipoles and estimating their interaction, failed. In the meantime, Bohr described the atom as a quantal object, with spherical orbits, and the success of this model in predicting the spectrum of the hydrogen atom implied that atoms have no permanent dipoles. This conclusion was further confirmed from experimental evidence of diatomic molecules, such as H₂ and O₂, in electric fields, which showed that these molecules had no dipoles.

The rescue came eventually through considerations of the polarizability of atoms and molecules in electric fields. The behavior of matter in electric fields has been actively studied since the days of Faraday, and the various experiments allowed relating the polarizability of a molecule to the refractive index of the material (or the permittivity of matter vs. vacuum) by Clausius and Mossotti and by Maxwell. Then Debye included permanent dipoles, and eventually also quadrupoles and beyond. However, the attempts to relate these electrical properties to the virial coefficients in the van der Waals equation were not entirely successful.

Furthermore, in the first decades of the 20th century, it became apparent that even noble gases had significant interatomic interaction that caused them to liquefy. It was also apparent that classical electrostatics failed to account for these attractive interactions, and the various treatments led to potentials with inverse powers of 1,2,3,4,5,7 and 8, but not 6. In 1927 in the Faraday Discussion meeting on “cohesion and related problems” it was clear that there was little progress in the understanding of these intermolecular potentials, and one of the participants (presumably Lennard-Jones) raised a hope that the new quantum mechanical theory (QMT) might solve the problems.

The rise of the new QMT in the years 1926–1929, and especially of the Schrödinger equation, revolutionized physics and enabled treatment of molecules and hence also intermolecular interactions, using the Born–Oppenheimer approximation (Chapter 5 in ref. 111).¹¹¹ 

An initial treatment of the interaction between hydrogen atoms by Wang in 1927, who was convinced by Debye to try, led to the following expression for the attractive potential at a long distance:

The importance of this result was its demonstration that two neutral atoms without permanent dipoles/multipoles could maintain an attractive interaction according to the new QMT.

At the same time, Heitler and London came up with their dramatic discovery of the short-range interaction that leads to the bond between two hydrogen atoms. They also found two new aspects: (i) at very short distances the potential was repulsive due the repulsion between the nuclei, which were not shielded anymore by the electrons, and (ii) when the electrons’ spins in the H⋯H system were parallel there was repulsion throughout. They also showed that the He⋯He interaction was repulsive, and these electron-electron repulsive terms were soon associated with the Pauli exclusion rule. It became clear therefore that the interaction between molecules must involve an attractive energy varying as r⁻⁶ and a repulsive energy term that dies faster at long distances. Later, London verified Wang's expression using triplet H⋯H, and this was followed by other verifications. Then Pauling and Beach derived the value C₆=6.499 03 au.¹¹¹  The attractive interaction was interpreted to originate from the creation of instantaneous dipoles of the two H atoms, and since the induction of a dipole by one atom on the other is proportional to r⁻⁶, r being the interatomic distance, then the attractive potential is proportional to r⁻⁶.

In 1930 London published a simple derivation of this attractive potential for two oscillating dipoles. Since the frequency of the oscillating dipole is related to the dispersion of light by matter, London dubbed the attractive interaction as ‘dispersion energy’, which became known also as ‘London energy’. As shown by London, the energy associated with the oscillation frequency, hν₀, reflects the looseness of binding of the valence electrons, and hence it can be replaced by the ionization energy (I) of the molecule and the molecular polarizability, U(r)=−0.75 Iα²/r⁶. Slater and Kirkwood further modified the Iα² term to N^1/2Iα^3/2, where N is the number of valence electrons. Others modified the potential to include the effect of higher multipoles. Later on, between 1931 and 1938, the potential was modified by Lennard-Jones and later by Buckingham to a 12,6-potential. The efforts after these years were enormous, they included physical chemists, who focused on the Ar⋯Ar problem, using a variety of methods, including simulations, inversion from spectroscopy to potential, and from virial coefficients to pair potentials.

QM calculations were too demanding for this problem, until recently. This fact, and the extreme weakness of the corresponding pairwise interaction energy have held back the acceptance of the van der Waals/Dispersion/London interactions into mainstream chemistry. The weakness of the interaction is not a problem, because the dispersion is cumulative, and if a molecular fragment experiences many such interactions, the total dispersion energy can amount to tens of kcal per mol. Currently there is a good arsenal to calculate van der Waals interactions in ‘real’ molecules. Quantum chemical methods like CCSD(T) enable quantification of these interactions and have already revealed exciting findings, such as the proposal¹¹³  that the DNA double helix is held mostly by dispersion interactions between the stacked bases. There are also DFT functionals that include dispersion from first principles, as well as empirically corrected functionals of various types.^12,114  The major ‘entrance’ of dispersion into mainstream chemistry really occurred in the 21st century, much due to the work of Grimme and others, who have shown how ubiquitous dispersion can be, and how significant in fact it is. Grimme's dispersion correction (D3) involves pairwise 6,8 terms and a three body term with 1/r⁹ dependence. The usage of dispersion is now widespread and its impact is found in structure, bonding, and reactivity, in all branches of chemistry. However, understanding of dispersion is still lacking.

Pairwise hydrogen bonding (H-bonding; a term which appeared for the first time in 1923 in Lewis's book, ‘Valence’) interactions are not as weak as van der Waals interactions. An average H-bond is a few kcal per mol strong. But, in some cases, as in (FHF)⁻, the interaction becomes a normal covalent bond. The notion of H-bonding emerged in the late 19th century and early 20th century, from the interest in aqueous solutions and the structure of water. The anomalies of water have been known for a long time, and have puzzled the community of physicists and chemists alike (Chapter 5 in ref. 111).¹¹¹  In the second decade of the 20th century, it was established that the water molecule has a permanent dipole moment. In 1922 Bragg senior¹¹¹  solved the structure of ice and interpreted it á la NaCl as a lattice of ions. Chemists on the other hand, described the water molecule with two covalent-polar bonds as Lewis depicted in his 1916 JACS paper.¹  Lewis also mentioned aggregation of water molecules due to polarity of the bonds.

Apparently, the first paper to mention the impact of H-bonding interaction was a 1912 paper by Moore and Winmill,¹¹⁵  who described the weak basicity of Me₃N as a weak interaction between the base and water, Me₃N⋯HOH. Eight years later, Latimer and Rodebush¹¹⁶  discussed the importance of this interaction in highly associated liquids, such as water and HF, the formation of dimers by acetic acid, and many other phenomena. In 1933 Bernal and Fowler published a landmark paper¹¹⁷  on the structure of water, and described ice to be composed of tetrahedral structure involving a central H₂O interacting with four others via O–H⋯O interactions, wherein the three atoms are collinear.

This architectural element of the hydrogen bond (H-bond), its ubiquity in nature, its importance in creating beautiful structures like the α-helix in proteins, its role in the genetic machinery, in crystal engineering and in enzyme catalysis, caused chemists to embrace this intermolecular interaction rather quickly, and very soon it became one of the hottest and most visited topics in chemistry.¹¹⁸ 

The architecture and character of macroscopic-, mesoscopic- and nano-materials are fashioned by-and-large by intermolecular interactions, made from pairwise weak interactions like dispersion and H-bonding, which accumulate and become hugely forceful (e.g., the Gecko phenomenon). It was immediately recognized that the architectural elements in these interactions and their potential strength make them ideal for design and engineering of new materials. As such, the major interest in these interactions has migrated to chemistry. Chemists like to attach to each interaction a name that serves as a chemical qualifier. To economize the myriad of names, consider the generic symbol, X–Z⋯Y, to represent a generic ‘Z-bond’, where Z can be hydrogen, halogen, chalcogen, pnicogen, tetrel, etc., and X and Y are molecular fragments that ‘donate’ and ‘accept’ Z, respectively.¹  When both Z and Y are hydrogen, the respective interactions are referred to as dihydrogen bonds, as e.g., the CH⋯HC interactions in hydrocarbons. Additionally, Y can be an anion or a cation, and X–Z an aromatic ring, or an olefin, and then we talk about anion–π and cation–π interactions, and so on.

The term ‘Z-bonds’ generally refers to the stabilizing interaction that occurs between atoms or groups, which in the conventional chemical sense have satisfied their formal valence. Chemists, commonly analyze a given interaction using energy decomposition analyses (EDA) of various types (see above). These interactions involve a few common energy terms, which refer to dispersion, electrostatic, promotion and deformation energies, Pauli repulsions, and bonding terms (arising from delocalization and charge transfer). Some of the methods use perturbation theory to calculate the various interactions, and other methods extract these quantities from energy difference calculations of the aggregate vs. the separate molecules/species and then extract the separate interaction terms. There are differences and similarities among the various methods, which differ in the manner by which they take the reference state, but there is no general consensus on the “best” method.¹⁴ 

Since ‘Z-bonds’ are formed between molecules/fragments, which have satisfied their valence, one might have thought that the intermolecular interactions should not involve any covalency. This turns out to be a simplistic approach and recent studies have shown that some of these ‘Z-bonds’ may involve considerable covalency. This issue tends also to be controversial and it is therefore interesting to focus on it. In so doing, I will limit myself to H-bonds (Z=H, X, Y – electron rich centers), Hal-bonds (Z=halogen/halide, X, Y – electron rich), and diH-bonds (CH⋯HC).

How covalent are H-bonds? In his book (p. 452), Pauling⁴³  estimated that the covalency of the water dimer is 5–6%. In most cases indeed, the consensus is that the weak H-bond is mostly electrostatic.¹¹⁹  A recent paper by Zhang et al.,¹²⁰  reported that the H-bond in 8-hydroxyquinoline on a Cu(1.1.1) support could be visualized in real space using non-contact-atomic force microscopy (NC-AFM). In this technique the AFM tip is functionalized with CO that acts as a sensor of the forces exerted by the electron density of the adsorbed molecules. Not long after this publication, Swart et al.¹²¹  used a tetramer of bis(para-pyridyl)acetylene, which involves C–H⋯N H-bonds and nonbonded N⋯N moieties at 3 Å apart. They showed that the CO on the AFM tip bends by interacting with intermolecular potential of the molecules, leading to a semblance of electron density also between the nonbonded N⋯N atoms. Thus, generally the Zhang work is being dismissed. Still however, another recent AFM force measurement of the iron-sulfur cluster rubredoxin reveals that the strength of the NH⋯S H-bonding network affects markedly the strength of the Fe–S bond.¹²²  Moreover, Elgabarty el al.,¹²³  quantified the covalence of the H-bond in water by measuring the anisotropy of the proton magnetic shielding tensor. They reported that the O⋯H H-bond possesses a density of 10me⁻ and its strength due to this charge transfer is ca. 2.4–3.5 kcal mol⁻¹. The prospects of these experimental techniques are exciting despite the cautionary buzz around the Zhang imaging experiment.

How does theory respond to the opening question? In general, the answer remains controversial. Depending on the computational method, the covalence of H-bonds ranges between significant (from NEDA(NBO) analysis) and slight (from other methods, e.g., Morokuma, BLW, AIM, IQA, etc. see above). Nevertheless, all theoretical analyses suggest some covalency, due to charge transfer from the H-acceptor moiety (e.g., Y:⁻) to the H-donor (X–H).^124,125  Furthermore, the linear structure of the H-bond is considered to arise from this charge transfer/covalency due to the donation from a lone pair orbital on Y: to the σ*_XH orbital of the H-donor.¹²⁵ 

A recent series of papers^126,127  issued another cautionary alert on the practice of mining the H-bond distances from crystal structures and deducing their relative stability, based on a bond-length-bond-strength (BLBS) principle. For example, in polyprotic acids, such as K⁺[CO₃H]⁻ the H-bonds O–H⁻⋯O⁻ are actually repulsive and are supported as “short H-bonds” only because of the strong interactions of the anions to the cation. Moreover, using H-bond distances from neutron diffraction data and calculating the bond strengths using a perturbation method^127,128  shows a breakdown of the BLBS relationship due to the fact that the crystallographic data reflect the packing of ions and not necessarily the strength of any specific short distances. A subsequent study¹²⁹  showed that when the H-donor (X) and H-acceptor (Y) are ions of opposite charge, the so formed H-bonds are highly covalent with bonding energies between 80 and 210 kcal mol⁻¹.

A resonating VB model^{29,43,119,130,131}  appears to provide a productive outlook on the problem, by showing how covalency changes from being minimal in weak H-bonds to highly significant in strong H-bonds. Thus, if one views the H-bond as a species along the proton transfer coordinate from X to Y, one requires primarily three structures to model the process. As shown in Scheme 1.10a, these are the two covalent structures of the X–H and H–Y bonds, and a third protonic structure, ⁻X: H⁺ :Y⁻. Mixing of the two covalent structures in an H-bonding geometry, e.g. X–H⋯Y⁻, generates some covalency in the H-bond, while a very significant mixing of the protonic structure into both covalent structures contributes to covalency in both X–H and H–Y linkages in a given geometry. Scheme 1.10b shows the example of the formation of the symmetric (FHF)⁻ species (a “low barrier H-bond”) which resides in a deep minimum highly stabilized by the large covalent-ionic resonance energy of the covalent and protonic structures.^130,131  For other systems, where the protonic VB structure is not as low in energy, the H-bond is simply a precursor cluster en route of the proton transfer reaction, and as such will possess varying degrees of covalency due to resonance interaction of one covalent structure with the other. This is exemplified in Scheme 1.10c, which describes a proton transfer for X and Y groups that are not as electronegative as F. Here the H-bonded clusters are precursors of the transition state for proton transfer.^130,131 

In summary, therefore, despite the fact that the H-bond is now 104 years old, the field is still alive and full with surprises, controversies, and exciting developments.

One of the intriguing Z-bonds is the halogen bond (Hal-bond), which involves an interaction of an electron rich center Y: with an electronegative halogen substituent Z in an XZ molecule. In fact, the first Hal-bond was discovered back in 1863 when Guthrie made the ammonia–iodine complex H₃N:⋯I–I.¹³²  Subsequently, when Mulliken formulated his charge-transfer (CT) theory, which has provided a framework for the understanding of the unique spectroscopy of these complexes,¹³³  the Hal-bond was called by the generic name, a charge-transfer complex (CTC). The fascination with the Hal-bond arises from the facts that, (i) the interaction involves two electron rich centers, Y: and the electronegative halogen Z, and (ii) the Hal-bond is highly directional and its XZY angle is close to 180°, and as such it constitutes an architectural element, much like H-bonds. Interest in the subject has hugely surged because of the fact that this ‘bond’ turns out to be ubiquitous in biological materials¹³³  such as proteins, nucleic acids, and interactions of drugs with biological objects.

One of the useful theoretical concepts for comprehending the intriguing features of the Hal-bond is electrostatic in origin.¹¹  Thus, despite the partial negative charge of the halogen (Z), still it has a region of positive electrostatic potential at the head of its lone pair and in the opposite direction to the X–Z axis, called a ‘σ-hole’. As such, the σ-hole confers electrostatic and polarization interactions, which account for the ability of the electronegative halogens to accept an interaction from electron rich centers, as well as for the linear X–Z⋯Y angle. The current discussions of experimental and theoretical studies of Hal-bonds are virtually dominated by the ‘σ-hole’ notion and its electrostatic effects.

A recent computational study^134,135  showed however, that like the H-bond, here too there is significant covalency. Thus, VB calculations and BLW analysis of over 50 Hal-bonds, led to the results shown in Figure 1.13. Thus, for the great majority of Hal-bonds the major contribution to the bonding interaction (ΔE_b) is due to charge-transfer (CT)/covalency, as seen by the green columns. On average, the polarization (blue) is much smaller, while electrostatic interaction and long-range dispersion are negligible (not shown), and the Pauli repulsion is destabilizing (not shown). Furthermore, the CT interaction is almost twice as large as the bonding energy of the species (in brown), and one can say therefore that the Hal-bond originates in the CT interaction. Others reached a similar conclusion for other sets of Hal-bonds.¹³⁶  The conclusion is clear-cut, most of the Hal-bonds are held by charge transfer interactions as envisioned more than 60 years ago by Mulliken.

In a recent study,¹³⁵  it was demonstrated that the linear angle of the Hal-bond, X–Z⋯Y, originates also in the CT interaction due to the overlap of the lone-pair orbital on Y with the σ*_Z–X orbital. It is very clear that if at all possible, then real space imaging of Hal-bonds would be an exciting development.

As we showed in Section 1.5.4.c, the seemingly very week CH⋯HC dispersion interactions can accumulate and contribute greatly to the stabilization of long C–C bonds. Furthermore, this interaction controls the boiling point and melting points of hydrocarbons.¹³⁷  CH⋯HC is by and large a dispersive interaction, but how is it possible to conceptualize it in terms of a clear physical model?

Recalling that the original physical model derived by London involved oscillating dipoles, the appropriate way to conceptualize the effect is to carry our VB calculations of the CH⋯HC interaction in a series of alkanes and observe the change in the wave function.¹³⁷  Each C–H bond is described in VBT as a linear combination of one major covalent structure and two minor ionic structures. The ionic structures of each bond come in two opposing polarities, C⁺ :H⁻ and C:⁻ H⁺. At infinite intermolecular separation the weights of the two structures are nearly identical. However, as the two C–H bonds approach one another, one expects the mechanism of oscillating dipoles to be turned on, in order to stabilize the dimer. This should be apparent by looking at the weights of the VB structures with oscillating-opposing dipole combinations C⁺ :H⁻//H⁺ C:⁻ and C:⁻ H⁺//:H⁻ C⁺, wherein the interacting H's have opposite charges, compared with those where the two interacting H's have identical charge. Figure 1.14 shows the interaction of two C–H bonds of two methane molecules. It is seen that at the equilibrium distance (R_H⋯H=2.501 Å) the combined weights of the oscillating-opposing dipole combinations increases to ∼1.7 times the weights at infinite distance, while the combined weights of the structures with the identical charges on the interacting H's decreases to ∼0.34 of the value at infinite separation. This is a vivid demonstration of the oscillating dipoles mechanism.

In H₃CH⋯HCH₃, virtually all the stabilization energy originates from the oscillating dipoles mechanism. Namely, the methane dimer is stabilized by pure dispersion interaction. However, as the alkanes grew, this mechanism continued to flesh out, but it became less and less important. More significantly, the stabilization energy was brought about through some sticky covalent interactions between the two molecules, due to reorganization of the bonding electrons of the two interacting CH bonds via recoupling these electrons to H⋯H and C⋯C ‘bonds’, and charge transfer interactions between the two moieties that create long-range C⋯H bonds. Once again it is seen that the pristine mechanism, derived from the physics of interacting atoms, gets significantly more complex as the molecular species grows.

In fact, the size effect is spectacular.^137,138  Figure 1.15 displays two graphane planes, which like porcupines have axial C–H bonds basically pointing towards each other, and maintaining CH⋯HC interactions (each C–H in one graphane sheet experiences interaction with 3C–H's in the other sheet). The interaction energy is approximately three times the number of carbons in a graphane sheet, reaching 51.6 kcal mol⁻¹ for [73]graphane. BLW analysis shows that the major stabilizing factor is the long-range dispersion (presumably between the oscillating dipoles of the C–H bonds in the two sheets). However, there is an important two-way charge transfer energy due to the σ_CH→σ*_CH interactions of the CH⋯HC. As seen in Figure 1.15, the ΔE_CT term, which accounts for ∼15% of the total binding energy, results in the accumulation of electron density in the interface area between two layers. This accumulated electron density thus acts as an electronic “glue” for the graphane layers and constitutes an important driving force in the self-association and stability of graphane at ambient conditions.

The dispersion interactions then are more than the name suggests, and involve in addition orbital interactions of significant size. In relating the computed values to experiments, e.g., vaporization energies or heats of melting, one must account also for entropy. Also, recently measured dispersion energies of alkanes in solution appear to be an order of magnitude smaller than the values suggested by calculations and derived from vaporization enthalpies.¹³⁹  A possible cause for such a reduction is the dispersion interaction between the alkane and the surrounding solvent molecules, which causes the alkanes to turn on the dipole oscillating interaction with the solvent molecules, at the expense of diminishing the same alkane–alkane interaction due to mismatch of the respective oscillating dipoles. Again, all these uncertainies and complications are a sign of an alive science.

Intermolecular interactions have been for a long while a territory of physics, while chemists seemed content with their own ‘neverland’¹⁴⁰  of molecules and their chemical bonds. However, at turn of the 20th century, hydrogen bonding interactions were discovered and quickly revealed their omnipresence as a design force in matter. Probably this change has turned the attention of chemists to other intermolecular interactions and notably to dispersion/London interactions, halogen bonds, etc. The understanding of intermolecular interactions, and their usage as an element of design of nanomaterials and crystals, define a current central intellectual arena of chemistry.

I tried to describe in this chapter a very brief history of “cohesion” in chemistry by overviewing the conceptual evolution of the bond that glues atoms to molecules, and that of the intermolecular interactions that assemble molecules into larger aggregates of macroscopic matter. Subsequently, I reviewed novel bonding motifs and new aspects of intermolecular interactions, thus showing that the field of cohesion in chemistry is vibrant, exciting, and it still presents problems that await solution. The imaging of bonds, bond breaking and remaking, and perhaps also of hydrogen bonds and maybe in the future of halogen bonds too, mark a degree of hype and excitement in this field. In fact, as I completed this chapter, I noticed two studies which return to the basics, define a generalized electronegativity,¹⁴¹  and a new parameter that can distinguish bonds and weak interactions.¹⁴²  This is high time for bonding and cohesion in chemistry.

(a) The dative bonds between amines and carbocations were shown to be CSBs in a recent experimental study;¹⁴⁴  (b) the quadruple bonding in C₂ was reproduced in a CASSCF recent study;¹⁴⁵  (c) the inverted bond in C₂ was recently also found in B₂.¹⁴⁶  This calls for a new Aufbau principle for bonding in diatomic molecules.

I am thankful to all my friends who participated in the bonding march: W. Wu, C. Wang, Y. Mo, P. C. Hiberty, D. Danovich, B. Braida, P. Su, H. Rzepa, P. R. Schreiner, and S. Alvarez. My special thanks to David Danovich who has read the paper and checked the references and art items. The paper is dedicated to Santiago Alvarez whose gentle coercion brought me to look at dispersion using VB theory, and to Philippe Hiberty my longtime partner in VB theory.