any of the interactions that account for the association of atoms into molecules, ions, crystals, and other stable species that make up the familiar substances of the everyday world. When atoms approach one another, their nuclei and electrons interact and tend to distribute themselves in space in such a way that the total energy is lower than it would be in any alternative arrangement. If the total energy of a group of atoms is lower than the sum of the energies of the component atoms, they then bond together and the energy lowering is the bonding energy.

The ideas that helped to establish the nature of chemical bonding came to fruition during the early 20th century after the electron had been discovered and quantum mechanics had provided a language for the description of the behaviour of electrons in atoms. However, even though chemists need quantum mechanics to attain a detailed quantitative understanding of bond formation, much of their pragmatic understanding of bonds is expressed in simple, intuitive models. These models treat bonds as primarily of two kinds—namely, ionic and covalent. The type of bond that is most likely to occur between two atoms can be predicted on the basis of the location of the elements in the periodic table, and to some extent the properties of the substances so formed can be related to the type of bonding.

A key concept in a discussion of chemical bonding is that of the molecule. Molecules are the smallest units of compounds that can exist. One feature of molecules that can be predicted with reasonable success is their shape. Molecular shapes are of considerable importance for understanding the reactions that compounds can undergo, and so the link between chemical bonding and chemical reactivity is discussed briefly in this article.

Although simple models of bonding are useful as rules of thumb for rationalizing the existence of compounds and the physical and chemical properties and structures of molecules, they need to be justified by appealing to more sophisticated descriptions of bonding. Moreover, there are some aspects of molecular structure that are beyond the scope of the simple theories. To achieve this insight, it is necessary to resort to a fully quantum mechanical description. In practice, these descriptions entail heavy reliance on computers. Such numerical approaches to the chemical bond provide important information about bonding.

This article begins by describing the historical evolution of the current understanding of chemical bonding and then discusses how modern theories of the formation of chemical bonds have emerged and developed into a powerful description of the structure of matter. After the historical introduction, qualitative models of bonding are discussed, with particular attention given to the formation of ionic and covalent bonds and the correlation of the latter with molecular shapes. The more sophisticated quantum mechanical approaches to bond formation are then described, followed by a survey of a number of special cases that raise interesting problems or lead to important insights.

For a detailed discussion of the structure and properties of atoms, see the article atom. Chemical compounds are surveyed in the article chemical compound, and the elements are described in the article chemical element.

The early Greeks, most notably Democritus, argued that matter is composed of fundamental particles called atoms. The views of the atomists, however, lacked the authority that comes from experiment, and evidence of the existence of atoms was not forthcoming for two millennia until the emergence of quantitative, empirical science in the 18th century.

The crucial transformation of chemistry from a collection of vain hopes and alchemical meddlings to a corpus of reliable quantitative knowledge hinged on the contributions of the French aristocrat Antoine-Laurent Lavoisier (Lavoisier, Antoine-Laurent) (and his wife, Marie-Anne), shortly before he lost his head to the guillotine at the height of the Reign of Terror. Lavoisier opened the door to quantitative chemistry by establishing that the transformations of matter, which until his day had been described largely by a miasma of uncoordinated reports, could be investigated quantitatively by measuring the masses of substances consumed and produced in reactions. The most significant observation he made was that, even though one substance is transformed into another during the course of a reaction, the total mass of the products is the same as the total mass of the reactants. The implication of this observation is that, although the identity of the substances may change when a reaction occurs, something, at least, remains unchanged.

Lavoisier's experimentation inspired further studies that ultimately resulted in an overthrow of the view that matter is a structureless continuum. These observations culminated in the atomic (atomic theory) hypothesis developed by the English chemist John Dalton (Dalton, John), which states that matter is composed of indestructible particles which are unique to and characteristic of each element. Two major sets of observations helped to establish this view. First, it was found that compounds always have a fixed composition, regardless of their origin. Thus, it was determined that 18 grams of water always consists of 2 grams of hydrogen and 16 grams of oxygen, regardless of how the sample originated. Such observations overthrew, at least temporarily, the view held by the French chemist Claude-Louis Berthollet (Berthollet, Claude-Louis) that compounds have a variable composition. Modern research has shown, however, that there are in fact certain classes of compound in which the composition is variable. Nevertheless, they are a minority, and the law of constant composition is the rule rather than the exception.

The second step toward Dalton's synthesis was the recognition of the existence of related series of compounds formed by the same elements. It was established, for example, that, whereas 28 grams of carbon monoxide invariably consists of 12 grams of carbon and 16 grams of oxygen, carbon also forms the compound carbon dioxide, and 44 grams of this compound always consists of 12 grams of carbon and 32 grams of oxygen. In this example, the mass of oxygen that combines with a fixed mass of carbon to form carbon dioxide is exactly twice the quantity that combines to form carbon monoxide. Such observations strongly suggested that carbon dioxide contains exactly twice as many oxygen entities per carbon entity as carbon monoxide does. Dalton predicted that, when two elements combine in a series of compounds, the ratios of the masses of one element that combine with a fixed mass of the second are reducible to small whole numbers; this is now known as the law of multiple proportions.

Dalton brought these observations together and thereby established a language that, with minor emendation, has become universal in chemistry. He proposed that elements are composed of indestructible atoms, that each atom of an element is identical, that atoms of different elements differ in terms of mass, and that compounds consist of characteristic groupings of atoms. Because a compound is characterized by the grouping of atoms and each atom has a characteristic mass, it was at once easy to understand that compounds have a fixed composition by mass. Moreover, the existence of related families of compounds, which differ in an integral manner in their composition by mass, could immediately be explained by supposing that the various compounds differ in the number of atoms of one element that combine with one atom of a second element. Carbon monoxide, for instance, consists of one atom of carbon linked to one atom of oxygen, whereas carbon dioxide consists of one atom of carbon linked to two atoms of oxygen. Thus, in modern terms, carbon monoxide is denoted CO, whereas carbon dioxide is denoted CO₂.

The chemists of the 19th century established a large body of empirical information leading to the realization that patterns exist in the types of compounds that elements can form. The most useful rationalizing characteristic of an element is its valence, which was originally defined in terms of the maximum number of hydrogen atoms that could attach to an atom of the element. Hydrogen was selected as the probe of valence because investigators discovered that an atom of hydrogen is never found in combination with more than one other atom and thus regarded it as the most primitive of the elements. In this way it was established that oxygen (O) typically has a valence of 2 (as in water, H₂O), nitrogen (N) a valence of 3 (as in ammonia, NH₃), and chlorine (Cl) a valence of 1 (as in hydrogen chloride, HCl). Examining the patterns of bonding between elements made it possible to ascribe typical valences to all elements even though their compounds with hydrogen itself were unknown.

Although the concept of valence was highly suggestive of an intrinsic property of atoms, there were some puzzling aspects, such as the observation that some elements appear to have more than one common valence. The element carbon, for example, is found to have typical valences of 2 and 4.

A second general feature of bonding also became apparent in the early days of chemistry. It was found that there are two large classes of compound that can be distinguished by their behaviour when dissolved in water. One class consists of electrolytes: (electrolyte) these compounds are so called because they dissolve to give solutions that conduct electricity. Members of the other class, nonelectrolytes, dissolve to yield solutions that do not conduct electricity. The difference between the two classes gave rise to the view that there are two types of chemical bond. Electrolytes produce ions (ion) in solution; an ion is an electrically charged atom and transports its electric charge as it moves through a solution. Electrolytes therefore either consist of ions before they are dissolved or produce ions upon dissolving. Nonelectrolytes do not produce ions when they dissolve and do not consist of ions in their undissolved state.

It became apparent that some compounds are composed of ions, whereas others are composed of groups of atoms that are held together in a different manner. The latter compounds are termed covalent. In fact, it took a long time for the view to be confirmed that ions exist even before dissolution occurs, and only in the early 20th century was crucial evidence obtained that showed the presence of distinct entities, specifically sodium cations (positively charged atoms), Na⁺, and chloride anions (negatively charged atoms), Cl⁻, in solid sodium chloride (NaCl).

Unknown to Mendeleyev, and not discovered until the late 19th century and the beginning of the 20th, is another family of elements that were originally thought to be inert and hence were called the inert gases (noble gas). This family is headed by helium (He) and includes neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn). It was not until the 1960s that their chemical inertness was overcome, and some members of the family (essentially only krypton and xenon) were induced to form compounds. Accordingly, the name inert gas was replaced by the term noble gas, which reflects a chemical aloofness but not total inertness. This family of elements might at first have seemed irrelevant to an understanding of chemical bonds. However, the very fact that they do not readily form any bonds proved to be crucial to the development of modern theories of bond formation.

Until the early 20th century some regarded the atomic hypothesis as no more than an unsubstantiated hypothesis or a convenient accounting device. The reality of atoms and the molecules they formed was widely advocated but by no means universally accepted; however, opposition to the reality of atoms diminished as experimental evidence accumulated. Among such historically significant evidence were the quantitative measurements of the volumes of gases. Thus, it was noted that, when water is decomposed by electrolysis (i.e., by passing an electric current through it), the gases hydrogen and oxygen are produced in the ratio of 2:1 by volume. This observation led the Italian scientist Amedeo Avogadro to propose that equal volumes of gases (at the same temperature and pressure) contain equal numbers of molecules. The electrolysis of water was then seen to be consistent with a water molecule formed of two hydrogen atoms and one oxygen atom and hence consistent with the chemical formula H₂O. (It is now known that hydrogen gas consists of H₂ molecules and oxygen gas of O₂ molecules, but this important detail does not upset the interpretation.)

The measured volumes of gases supported the claims of the existence of atoms and molecules. The emergence of the science of mechanics furthered the understanding of atoms and molecules, as the properties of gases were predicted based on the assumption that they are composed of minute particles in ceaseless chaotic motion. From this kinetic model of gases (see gas: Kinetic theory of gases (gas)), it was possible to calculate the pressure exerted by a gas and the average speed of its molecules, and excellent agreement with observation was obtained.

The last opposition to the existence of atoms vanished in the early 20th century when techniques were developed that portrayed visual representations of atoms. The first such techniques made use of the diffraction of X rays, where the pattern of interference between rays that are reflected by a crystal can be interpreted in terms of the scattering from individual atoms. More images of atoms were produced in the 1960s by using methods that stripped electrons out of arrays of atoms at the surfaces of solids so that a map of the surface could be made, as well as by using improved techniques in electron microscopy that increased the resolving power of the microscope to nearly the point where individual atoms could be distinguished. The most recent and visually compelling evidence came in the 1980s with the development of scanning tunneling microscopy. In this technique a needle point sharpened to consist of a single atom is moved like a delicate plow just above the surface of a sample, and its position is monitored. The results appear in the form of a visual image of the sample's surface. The technique has been perfected to a point where it can be used to determine the locations of individual atoms. Of these techniques, electron microscopy comes the closest to an actual “sighting” of an atom, as the image requires the least construction. Images are obtained from X-ray diffraction data only after intense mathematical manipulation. Both field-emission and scanning tunneling microscopy give portrayals of the properties of a surface on an atomic scale and show atomlike features.

Most chemists were confident that atoms really exist long before these sophisticated techniques provided such irrefutable evidence. In the 19th century, when the compositions of countless compounds were being determined, it was found that in certain cases different compounds have the same chemical composition. Thus, the composition C₃H₄ was found for two entirely different organic compounds (as judged by both their physical and chemical properties)—namely, propyne and allene. Confident about their analyses, chemists were forced to the conclusion that the two compounds differ in the manner in which their constituent atoms are linked together. In modern terms, the compounds are represented, respectively, as:

(The nature of the links between atoms is the major topic of this article and is discussed in detail below.) Thus, the sense of molecular structure (i.e., the arrangement of atoms in space) entered chemistry and, by implication, supported the view that atoms are real.

About the same time (in the 1860s), a more subtle aspect of structure became apparent—that of the three-dimensional spatial disposition of atoms in molecules. The concept of molecular structure began with the realization that atoms have different neighbours in different compounds even though their overall chemical compositions might be the same (as in the two structures corresponding to the formula C₃H₄). This is a topological distinction, meaning that the distinction is based on which atom is linked to which atom. The additional distinction introduced is geometric, referring to the spatial disposition of atoms relative to one another. As an example of this kind of distinction, the compound dichloromethane (CH₂Cl₂) can be considered. The topological structure of this molecule is:

with the hydrogen and chlorine atoms linked to a central carbon atom. It was observed that there is only one such compound. The significance of this is that the molecule cannot be planar, because, if that were the case, two different molecules of formula CH₂Cl₂ would be found:

The fact that there is only one dichloromethane suggests that its molecules are tetrahedral, for then, in whichever arrangement the four hydrogen and chlorine atoms are linked to the central carbon atom, the molecule is identical (apart from its orientation in space, which is irrelevant):

With observations such as this, the sense of molecular shape entered chemistry and since then has assumed a central and fundamental position.

Nineteenth-century chemists had to infer the shapes of molecules from clever but indirect experimentation. The modern understanding of molecular shape is more direct (if one discounts the computing that intervenes between observation and representation). In particular, X-ray diffraction has provided incomparably detailed images of molecules even as large as those of proteins, which contain thousands of atoms. Scanning tunneling microscopy, although much more recent in inception, has provided realistic images that confirm beyond doubt the essential features of molecular geometry.

The importance of the determination (and understanding) of molecular structure cannot be overestimated. At the simplest level, the properties of small molecules (including the ubiquitous and important water molecule, H₂O) stem in large measure from their shapes and not merely from their atomic compositions. The oceans, for instance, might not exist if water molecules were linear rather than angular, for the interactions between H₂O molecules would not be as strong, and hence it is doubtful whether life would have emerged if water molecules were linear. At the most complex level, that of proteins, geometric structure is essential to biochemical function and thus has a critical role in all living systems.

The concept of atoms thus emerged from the meticulous measurement of mass and volume, which in the earliest days of chemistry were the only quantitative probes of matter available. The reality of atoms was established by the explanatory power of the model on the one hand and by ever more direct images of microscopic entities on the other. As the atomic model of matter became more firmly established, attention turned to the existence of molecules, which are specific assemblages of atoms. As molecules were examined, it was discovered that they have characteristic links between atoms and that the atoms are positioned in three-dimensional arrangements that are characteristic of the compound and of the constituent atoms.

The questions raised by this fund of knowledge remained unanswered until the internal structure of atoms began to be unraveled at the end of the 19th century. The classic view, proposed by Dalton, that atoms are irreducible, unchangeable entities virtually eliminated the prospect of understanding their properties, for it implied the absence of internal structure. The mutability of atoms, and hence the first glimmerings of an understanding of their constitution and their properties, came with the discovery of the electron as a universal constituent of matter. The electron was the first subatomic particle to be discovered and in due course proved to be the most important one for the explanation of the chemical bond. This importance stems in large part from the ease with which electrons can be removed from one atom and transferred to another. As will be seen below, this transferability of electrons is the key to bond formation, and all theories of the chemical bond focus on the redistribution of an atom's electrons when it links to another atom.

More will be said about the essential features of the arrangement of electrons in atoms in the following section. The key to understanding the structure of the periodic table and hence the pattern of bonding between atoms was the realization that electrons are arranged in shells that surround a central positively charged nucleus. Each shell can contain a characteristic maximum number of electrons. The outermost shell contains the electrons that are involved in bond formation, for they are the least tightly bound to the nucleus and thus can be removed most readily. This shell is called the valence shell. The most important feature of the valence shell is that for the noble gases it is complete (in the sense explained below) with its full complement of electrons (i.e., eight, excepting the case of helium). Thus, the formation of chemical bonds appears to be related to the incompleteness of the valence shell.

The role of the valence shell in bond formation was expounded by the American chemist Gilbert N. Lewis about 1916. Important independent studies were made by Walther Kossel, and later contributions followed from Irving Langmuir. First, Lewis proposed that ionic bonds (ionic bond) are formed by the complete transfer of electrons from the valence shell of one atom into the valence shell of another atom and that the transfer proceeds until the valence shells of both have reached the electronic composition characteristic of the nearest noble gas atom in the periodic table. Thus, sodium has one electron in its valence shell, and its loss results in a singly charged cation, Na⁺, with a neonlike arrangement of electrons. Chlorine, on the other hand, has a valence shell that needs one more electron to achieve the closed shell characteristic of its noble gas neighbour, argon, and so readily forms the singly charged anion Cl⁻. Thus, it is easy to comprehend the formation of sodium chloride as a collection of Na⁺ ions and Cl⁻ ions.

Lewis proposed that a covalent bond consists of two electrons that are shared between atoms rather than being fully donated by one atom to another. He had no means of knowing why a pair of electrons should be so important (that understanding would come only later with the introduction of quantum mechanics), but his insight rationalized a great body of chemical facts. As in the formation of ionic bonds, Lewis emphasized the importance of the nearest-noble-gas valence shell and proposed that, as in the formation of ionic bonds, electron sharing continues until each atom possesses a noble gas configuration.

In summary, Lewis' ideas are expressed by his celebrated octet rule, which states that electron transfer or electron sharing proceeds until an atom has acquired an octet of electrons (i.e., the eight electrons characteristic of the valence shell of a noble gas atom). When complete transfer occurs, the bonding is ionic. When electrons are merely shared, the bonding is covalent, and each shared electron pair constitutes one chemical bond.

Such is the basis of the theory of chemical bonding that is still widely held. There is much to explain and more to understand, however, and there are many important exceptions to Lewis' ideas, which cannot as a consequence provide a complete explanation of bonding. The following sections step back from this historical account and put Lewis' important ideas in a broader context that will show more of their power. At the same time the more advanced treatment of bonding will transcend Lewis' ideas and account for features of bonding that his views could not embrace.

To understand bond formation, it is necessary to know the general features of the electronic structure of atoms—that is, the arrangement of electrons around the central nucleus. For background information about this subject and further details, see the article atom.

The modern version of atomic structure begins with Ernest (later Lord) Rutherford's (Rutherford atomic model) recognition that an atom consists of a single, central, massive, positively charged nucleus surrounded by electrons. The number of protons in the nucleus is the atomic number, Z, of the element. (For hydrogen Z = 1, and for carbon Z = 6.) A proton is positively charged, and an electron carries an equal but opposite negative charge. For an atom to be electrically neutral, it must contain the same number of extranuclear electrons as there are protons in the nucleus. Hence, the number of electrons in a neutral atom of atomic number Z is also Z. A hydrogen atom consequently has one electron, and a carbon atom has six electrons.

The first attempt to introduce quantum theory to account for the structure of atoms was made by the Danish physicist Niels Bohr in 1913. He asserted that the electron in a hydrogen atom occupies one of an array of discrete (but infinite in number) orbits, each orbit being progressively farther from the nucleus and labeled with an integer n = 1, 2, . . . . This integer is an example of a quantum number, which in general is an integer (in some cases, a half-integer) that labels the state of a system and which, through an appropriate formula, determines the values of certain physical properties of the system. By matching the centrifugal effect of the electron's motion in its orbit to the electrostatic attraction of the nucleus for the electron, Bohr was able to find a relation between the energy of the electron and the quantum number of its orbit. The result he obtained was in almost perfect agreement with the observed values of the energy levels of a hydrogen atom that had previously been obtained by spectroscopic methods.

Bohr's triumph was the first apparently successful incorporation of quantum theoretical ideas into the description of a mechanical system. The numerical success of the model has turned out to be coincidental, however, and Bohr's model is now regarded as no more than a historically important step in the evolution of quantum mechanics. The cracks in its validity were noted quite soon after its introduction. Thus, it was remarked that Bohr had not really derived the existence of discrete orbits from more fundamental principles but had merely imposed them on the model. Furthermore, all attempts to extend his theory to atoms that consisted of more than one electron (helium, with two electrons, for instance) utterly failed. Although the model was augmented by more elaborate specifications of the orbits (most notably, first, by allowing for elliptical orbits and introducing a second quantum number to specify the elongation of the ellipse and, second, by allowing for the effects of relativity), the failure to generalize to many-electron atoms remained a fatal flaw.

Current understanding of atomic structure had to await the introduction of quantum mechanics by the scientists Werner Heisenberg of Germany and Erwin Schrödinger (Schrödinger, Erwin) of Austria in the mid-1920s. Indeed, the structure of the hydrogen atom that is still employed today was developed by Schrödinger in the four papers with which he introduced his version of quantum mechanics—wave mechanics—to the world. The quantum mechanical model of the hydrogen atom has the same numerical agreement with experiment that proved so coincidental with the Bohr model, but it is more fundamentally founded (i.e., the discreteness of the allowed energy states emerges from more general aspects and is not imposed), and the model can be extended (albeit with difficulty) to many-electron atoms. Moreover, unlike Bohr's theory, it is consistent with the fundamental principles of quantum mechanics—specifically the wave character of the electron and the requirements of the uncertainty principle, which states that the position and momentum (mass times velocity) of a particle cannot be specified simultaneously.

In the quantum mechanical model of the hydrogen atom, the location of the electron is expressed in terms of a probability distribution, so one speaks of the probability that an electron will be found at a particular location near a nucleus. The probability distribution, in turn, is determined by a mathematical function known as a wavefunction (wave function), denoted ψ. Wavefunctions for the distribution of particles are a general feature of quantum mechanics, and for electrons in atoms they are known as atomic orbitals (orbital). The name orbital is intended to express a distribution that is less precise than the explicit orbits of the Bohr model. The probability of finding an electron at a specified location is proportional to the square of the amplitude of the wavefunction at that point. Hence, the sign (positive or negative) of the orbital is not relevant to the location of the electron, because taking the square of ψ eliminates any negative sign it may have. However, as explained below in The quantum mechanics of bonding: Molecular orbital theory, (chemical bonding) the sign is of crucial importance in the discussion of bonding between atoms and so cannot be ignored.

Three quantum numbers are needed to specify each orbital in an atom, the most important of these being the principal quantum number, n, the same quantum number that Bohr introduced. The principal quantum number specifies the energy of the electron in the orbital, and, as n increases from its lowest value 1 through its allowed values 2, 3, . . . , the energies of the corresponding orbitals increase. The ground state, or lowest energy state of the hydrogen atom, is the state in which it is normally found and has n = 1, it consists of a single electron in the orbital closest to the nucleus. As n increases, so does the average distance of the electron from the nucleus, and, as n approaches infinity, the average distance also approaches infinity. The energy required to elevate the electron from the orbital with n = 1 to the orbital with n = ∞ is called the ionization energy of the hydrogen atom; this is the energy required to remove the electron completely from the atom.

The quantum number n labels the shell of the atom. Each shell consists of n² individual orbitals with the same principal quantum number and hence (in the hydrogen atom) the same energy. Broadly speaking, each shell consists of orbitals that lie at approximately the same distance from the nucleus. The shells resemble the layers of an onion, with successive shells surrounding the inner shells.

The next quantum number needed to specify an orbital is denoted l and called the orbital angular momentum quantum number. This quantum number has no role in determining the energy in a hydrogen atom. It represents the magnitude of the orbital angular momentum of the electron around the nucleus. In classical terms, as l increases, the rate at which the electron circulates around the nucleus increases. The values of l in a shell of principal quantum number n are limited to the n values 0, 1, 2, . . . , n − 1, and the value of l of an orbital in a given shell determines the subshell to which that orbital belongs. It follows from the allowed values of l that there are n subshells in a shell of principal quantum number n. As will be explained, there are 2l + 1 orbitals in a given subshell.

Although subshells are uniquely specified by the values of n and l, it is conventional to label them in a slightly different manner. A subshell with l = 0 is called an s subshell, one with l = 1 is called a p subshell, and one with l = 2 is called a d subshell. Other subshells are encountered, but these three are the only ones that need to be considered here. The three subshells of the shell with n = 3, for example, are called the 3s, 3p, and 3d subshells.

As noted above, a subshell with quantum number l consists of 2l + 1 individual orbitals. Thus, an s subshell (l = 0) consists of a single orbital, which is called an s orbital; a p subshell (l = 1) consists of three orbitals, called p orbitals; and a d subshell (l = 2) consists of five orbitals, called d orbitals. The individual orbitals are labeled with the magnetic quantum number, m_l, which can take the 2l + 1 values l, l − 1, . . . , −l. The orbital occupied in the lowest energy state of the hydrogen atom is called a 1s orbital, signifying that it belongs to (and is in fact the only member of) the shell with n = 1 and subshell with l = 0.

The atomic orbitals differ in shape. That is, the electrons they describe have different probability distributions around the nucleus. Indeed, a part of the reason why orbitals differ in energy is that the electrons that occupy them are likely to be found in different regions around the parent nucleus and hence experience the latter's attraction with different strengths. The fact that all orbitals of a given shell in the hydrogen atom have the same energy despite having different shapes is surprising and is associated with a cancellation of different contributions to the energy. (This so-called degeneracy, the possession of the same energy by different wavefunctions, is also associated with the coincidental numerical agreement of Bohr's model with experiment.) As soon as a second electron is present, however, the degeneracy is lost.

When an electron is described by the wavefunction corresponding to a particular orbital, the electron is said to occupy that orbital. In the ground state of a hydrogen atom, the electron occupies the 1s orbital, while in an excited state it occupies one of the other orbitals to which it has moved. A unique feature of an s orbital is that an electron that occupies it may be found right at the nucleus. All other orbitals have zero amplitude at the nucleus, and an electron that occupies one of them has zero probability of being found there. This apparently slight detail has remarkable consequences: it is largely responsible, for instance, for the structure of the periodic table and hence for the pattern of the compounds that the elements can form and for the properties of the substances that make up the tangible world. Several apparently trivial differences of this kind are responsible for the richly varied properties of matter.

In the building-up principle, Z electrons (for a neutral atom of an element of atomic number Z) are placed in succession into an array of hydrogen-like atomic orbitals in such a way as to achieve the lowest possible total energy. Thus, to account for the structure of a helium atom (for which Z = 2), one electron is allowed to occupy a hydrogen-like 1s orbital, and then a second electron is allowed to join it, giving the electron configuration 1s² (which is read “one-s-two”).

To produce the ground-state electron configuration of the next element, lithium (Z = 3), one more electron is added. However, that electron cannot occupy the 1s orbital, for it has a property known as spin, which is fundamental to its behaviour. Spin is an intrinsic property of an electron, like its mass or charge. In elementary treatments, spin is often visualized as an actual spinning motion. However, it is a quantum mechanical property without a classical counterpart, and to picture spin in this way can be misleading. Nevertheless, for the present discussion, such a picture is useful. An electron has a fixed amount of spin, in the sense that every electron in the universe is continually spinning at exactly the same rate. Although the spin of an electron is constant, the orientation of the axis of spin is variable, but quantum mechanics restricts that orientation to only two possibilities. The two possible spin states of an electron are represented by the arrows ↑ and ↓ and are distinguished by the spin magnetic quantum number, m_s, which takes the values +¹/₂ (for the ↑ spin) or −¹/₂ (for the ↓ spin).

Because of its spin, an electron must obey a fundamental requirement known as the Pauli exclusion principle. This principle (which is a consequence of the more fundamental Pauli principle) states that no more than two electrons may occupy a given orbital and, if two electrons do occupy one orbital, their spins must be paired (denoted ↑ ↓; that is, one electron must be ↑ and the other must be ↓). The Pauli exclusion principle is responsible for the importance of the electron pair in the formation of covalent bonds. It is also, on a more cosmic scale, the reason why matter has bulk; that is to say, all electrons cannot occupy the orbitals of lowest energy but are instead located in the many shells that are centred on the nucleus. Also owing to the existence of spin, two objects do not simply blend into one another when they are in contact; the electrons of adjacent atoms cannot occupy the same space, thereby prohibiting the combining of two atoms into one. Here again is an example of a seemingly trivial property, in this case spin, having consequences of profound and macroscopic importance. In this instance, the spin of the electron is responsible for the existence of identifiable forms of matter.

With the Pauli exclusion principle in mind, one can see that in helium the 1s orbital (and hence the entire n = 1 shell, for that shell consists of only a single orbital) is full. The helium atom is said to be a closed-shell species. There is an obvious connection between the remarks made earlier concerning the inertness of helium and the fact that its valence shell is complete. The details of this connection will be considered below. With the n = 1 shell complete, the third electron of lithium must enter an orbital of the next higher shell, that with n = 2. This shell consists of two subshells, which are composed of the single 2s orbital and the three 2p orbitals, respectively.

The next problem that must be addressed is the experimental (i.e., spectroscopic) fact that the third electron occupies the 2s orbital rather than any of the three 2p orbitals to give the configuration 1s²2s¹. In a hydrogen atom all the orbitals of a shell are degenerate. That is not the case, however, in atoms where more than one electron is present; in such instances, within a given shell the s subshell lies at lower energy than the p subshell. The lower energy of an ns orbital relative to that of an np orbital arises from the ability of an s electron to be found extremely close to the nucleus.

If the electrons in ns and np orbitals were distributed equally outside the closed shells that constitute the helium-like core of the atom, then they would be equally repelled by the two core electrons. As a result, they would experience a lower effective nuclear charge, the difference between the true charge of the nucleus and the net charge experienced after allowing for the repulsion of any electrons present. The reduction of the actual nuclear charge by the effect of the other electrons in the atom is referred to as the shielding of the nuclear charge. Next, it is necessary to note that a 2s electron can penetrate through the core (that is, have nonzero probability of being found closer to the nucleus than the bulk of the core electron density). If penetration occurs, the electron experiences the full nuclear charge and hence has a lower energy than an electron in an orbital that cannot penetrate through the shielding core. It is this combination of the effects of penetration and shielding that results in an ns orbital having a slightly lower energy than an np orbital, for the latter has zero amplitude at the nucleus.

It follows from this discussion that, for a lithium atom to achieve the lowest possible energy, the third electron should occupy the 2s orbital, in accord with spectroscopic evidence. Successive elements complete first the 2s subshell (at beryllium, Be; Z = 4) and then begin the 2p subshell. The three orbitals of the 2p subshell are completed after the addition of six more electrons, which occurs at neon (Ne; Z = 10).

Another aspect of the building-up principle needs to be mentioned at this point, although its significance will not become fully apparent until later. When there are several orbitals of the same energy available for occupation, the electron configurations observed in atoms are found to be reproduced if Hund's rule is adopted. This rule states that, if more than one orbital is available for occupation by the electrons currently being accommodated, then those electrons occupy separate orbitals and do so with parallel spins (both ↑, for instance, which would be denoted ↑↑). The occupation of separate orbitals minimizes the repulsion energy between the electrons and hence leads to a lower energy than if they were confined to the same region of space. The requirement of Hund's rule that the electrons have parallel spins is more subtle. When electrons have parallel spins, they are constrained by quantum mechanics to stay apart from one another; as a result the atom can shrink slightly and hence improve the energy of attraction between its electrons and nucleus.

At neon the entire n = 2 shell is complete. At this point it should be noticed that the second noble gas, neon, has a closed-shell electron configuration, as does the first noble gas, helium. Note also that eight electrons are needed to pass from helium to neon, that eight is the maximum number of electrons that the n = 2 shell can accommodate, and that there are eight columns of elements in the main part of the periodic table. Thus, a combination of the Pauli exclusion principle and the effects of penetration and shielding has explained the essential structure of this table.

The element that follows neon in the periodic table is sodium (Na), with Z = 11. Its additional electron is excluded by the Pauli principle from neon's closed shell and must enter the next higher energy shell, in which n = 3. This shell contains three subshells, 3s, 3p, and 3d, and, as a result of the effects of penetration and shielding, the energies of these subshells lie in the order 3s \\< 3p \\< 3d. It follows that the incoming electron enters the 3s orbital, resulting in the ground-state electron configuration of a sodium atom being 【Ne】3s¹, where 【Ne】 represents the neon-like 1s²2s²2p⁶ closed shell. It is a striking feature of this discussion that the electron configuration of sodium is the exact analogue of the electron configuration of lithium (Li), 【He】2s¹, with its helium-like closed-shell core. Moreover, sodium belongs to the same family as lithium and has strikingly similar chemical properties, including the ability to form ionic compounds that contain singly-charged cations, namely Na⁺ and Li⁺, respectively.

The third row of the periodic table (sodium through argon) is in fact a replication of the second row (lithium through neon), the only difference being that a more distant shell of s and p orbitals (the shell with n = 3) is being occupied. The elements of this row bear a strong family resemblance, particularly in terms of their valences, to the elements directly above them in the second row. Moreover, after eight members, the row terminates at the noble gas argon, with a closed set of 3s and 3p subshells.

Chemistry, though, is a subtle subject, and its variety depends on that subtlety. The detail needed at this point (but which will not be unduly dwelt upon) is that the effects of penetration and shielding are so pronounced that the 4s orbital is so substantially lowered in energy by its ability to penetrate close to the nucleus that it lies lower than the 3d orbitals, even though those orbitals belong to a shell of lower principal quantum number. Thus, after argon, the next electron enters the 4s orbital, not a 3d orbital, giving the configuration 【Ar】4s¹ for potassium, where 【Ar】 represents the configuration of argon. Indeed, potassium is similar in chemical properties to sodium, which is consistent with its analogous electron configuration.

The pattern suggested by this discussion now continues as electrons are added, and the next row of the table replicates the electron configurations of the fourth row. The general pattern of the periodic table is now established.

The columns of the periodic table, which contain elements that show a family resemblance, are called groups (group). All members of a particular group have analogous outermost (valence) electron configurations, suggesting that all members of a group should show a family relationship in the types and numbers of the chemical bonds that they are able to form. The horizontal rows of the periodic table are called periods. Each period corresponds to the successive occupation of the orbitals in a valence shell of the atom, with the long periods corresponding to the occupation of the orbitals of a d subshell. Successive periods down the table correspond to successively higher values of n for the valence shell. The first period (consisting of only hydrogen and helium) corresponds to n = 1, the second period (from lithium to neon) to n = 2, and so on. These successive periods correspond to atoms in which the valence shell is outside a more electron-rich core of inner completed shells. Each of the first six periods terminates at a noble gas, with a closed-shell electron configuration. The replication of analogous electron configurations that characterizes the periodic table is an example of the periodicity of the elements and is responsible for the overall pattern of the elements when arranged as Mendeleyev, with chemical insight and without the benefit of quantum mechanics, had originally proposed.

The elements show a rich variety of periodicities. Emphasis will be placed on the periodicity of the properties that are of direct relevance to the formation of chemical bonds. These properties are essentially the size of atoms and the energy required to remove electrons from or attach them to neutral atoms.

Broadly speaking, the radii of atoms increase from the top to the bottom of the periodic table and decrease from left to right. Hence, the largest atoms are found at the lower left of the table, and the smallest ones are found at the upper right. The increase in radius down each group stems from the fact that in successive periods one more layer of the atomic “onion” is being formed; that is, electrons are being added to a new shell outside a closed-shell core of the atom. Thus, lithium consists of one electron outside a compact, helium-like core, sodium consists of a single electron outside a neon-like core (which itself has a helium-like core deep within its structure), and so on down the group.

The decrease in atomic radius from left to right across a period is perhaps more surprising, for a contraction in size occurs despite the presence of more electrons in each successive element. Thus, lithium has three electrons, and beryllium (Be) has four, but beryllium is slightly smaller than lithium. Fluorine, with nine electrons, might be expected to be a significantly larger atom than lithium, but the opposite is true. The explanation of this seemingly counterintuitive trend is that, although successive elements have a larger number of electrons, they also have a higher nuclear charge because of the increasing number of protons. That positive charge draws in the surrounding electrons to make the atom more compact. The inner-shell, or core, electrons, which do not increase upon going across a period, effectively shield the outer-shell electrons from the positive charge of the nucleus. The outer-shell electrons that are added upon going across a period, however, do not shield other valence electrons from the increasing charge of the nucleus as well as the core electrons do. Thus, the outer-shell electrons are pulled in more closely by the greater charge of the nucleus. There is clearly competition (as is so often the case in chemistry) between the inflating effects of the presence of more electrons and the contracting effects of the stronger nuclear charge. With a few exceptions, the latter influence dominates slightly, and successive atoms are smaller on moving across a period.

Ions (ion), both cations (cation) and anions, show a similar variation in size with the position of their parent elements in the periodic table. However, there are two gross differences. First, cations (which are formed by the loss of electrons from the valence shell of the parent atom) are invariably smaller than their parent atoms. In some cases the difference can be considerable (more than 50 percent). In effect, the outer layer of the atomic “onion” is discarded when the valence electrons are lost, so the radius of the cation is that of the compact atomic core.

Anions (anion), which are formed by the gain of electrons by an atom—most commonly into the incomplete valence shell—are invariably larger than the parent atoms. In this case, the additional electrons repel the electrons that are already present, and the entire atom inflates.

Next in order of importance for determining the number and type of chemical bonds that an atom may form is the ionization energy of the element. It is the minimum energy needed to remove an electron from an atom of the element. The energy is required because all the electrons of an atom are attracted by the positive charge of the nucleus, and work must be done to drag the electron off the atom to produce a cation. Chemical bond formation stems from the transfer or sharing of electrons, and so the energy required to remove an electron is a crucial criterion in the ability of an atom to form a bond.

In broad terms, the variation of ionization energies throughout the periodic table mirrors the variation in atomic radii, with small atoms typically having high ionization energies and large atoms usually having small ones. Thus, the elements with the lowest ionization energies (and hence from which an electron is most readily removed) are found at the lower left of the periodic table, near cesium and francium, and elements with the highest ionization energies are found at the upper right of the table, close to fluorine and helium. The variation in ionization energy correlates with the variation in atomic radius because a valence electron in a bulky atom is on average far from the nucleus and therefore experiences only a weak attraction to it. On the other hand, a valence electron in a small atom is close to its parent nucleus and is subject to a strong attractive force.

At this point the relative inertness of the noble gases (noble gas) can be in part explained. They lie on the right of the periodic table, and the members of the family that are closest to helium (namely, neon and argon) have among the highest ionization energies of all the elements. Thus, their electrons are not readily available for bond formation. Only lower in the group, at krypton and xenon, do the ionization energies become comparable to those of other elements, and these elements can be coaxed into compound formation by sufficiently aggressive reagents (most notably by fluorine).

An important feature of the ionization energy is that the energy required to remove a second electron from an atom is always higher than the energy needed to remove the first electron. Once an electron has been removed, there are fewer electrons to repel one another in the cation, so more work must be done to drag the next electron away from the nucleus. The same is true of the third electron, which is even less available than the second electron. However, an important point is that, if an electron needs to be removed from the core of the atom (as is the case for a second electron removed from sodium), then the ionization energy may be exceedingly high and not attainable in the course of a typical chemical reaction (as will be justified below). The reason for the high ionization energies of core electrons is largely that these electrons lie much closer to the nucleus than do the valence electrons, and thus they are gripped by it much more strongly.

It is a general rule that for elements on the left in the periodic table, which have one, two, or three electrons in their valence shells, sufficient energy is attainable in chemical reactions for their removal, but not enough energy is available for removing any electrons from inner shells. Hence, sodium can form Na⁺ ions, magnesium can form Mg²⁺ ions, and aluminum can form Al³⁺ ions.

One reason for the importance of noble gas configurations in chemical bond formation now becomes apparent. Once a noble gas, closed-shell configuration is obtained, the ready removal of electrons to form cations ceases (as does the opportunity for the partial removal of electrons for the sharing required in the formation of covalent bonds, as discussed below). A large energy barrier is encountered when going beyond the removal of the valence electrons of an atom.

Ionization energies do not correlate with atomic radii exactly, because there are other influences beyond the distance of the electron from the nucleus that determine the energy needed to remove an electron. These influences include the details of the occupation of the orbitals in the valence shell. Once again, the origin of a further possibility for competition becomes apparent, in this case between effects that stem from size alone and those that are determined by the energy requirements for ionization.

Third in importance for bond formation after size and ionization energy is the energy change accompanying the attachment of electrons to a neutral atom. This energy is expressed as the electron affinity, which is the energy released when an electron is attached to an atom of the element. In many cases, the electron affinity is positive, signifying that energy is indeed released when an electron attaches to an atom. Such is the case when the incoming electron enters a vacancy in the valence shell of the atom. Although it is repelled by the electrons already present, it is sufficiently close to the nucleus for there to be a net attraction. Hence, the energy of the electron is lower when it is a part of the atom than when it is not. However, if the incoming electron has to start a new shell because the orbitals of the neutral atom are full, then it remains so far from the nucleus and so strongly repelled by the electrons already present that there is a net repulsion, and energy must be supplied to attach the electron to form an anion. In such cases, the electron affinity is negative.

Here lies the second part of the overall reason why a noble gas configuration is the end of the road for the formation of ions (ion)—in this case anions. Once the noble gas configuration has been attained, there may be serious energy disadvantages in the attachment of additional electrons. Thus, a chlorine atom can accept one electron to complete its valence shell, and Cl⁻ is a common species. An oxygen atom can accept two electrons to complete its shell, and O²⁻ is also common. These remarks conceal certain difficulties, but they are broadly true and account for the formation of the anions characteristic of the elements located on the right in the periodic table.

Electron affinities vary through the periodic table, and their periodicity is more complex than that of ionization energies. Broadly speaking, however, electron affinities are largest close to the upper right of the periodic table near fluorine. (As indicated above, the closed-shell noble gases have lower electron affinities.)

In summary, the low ionization energies and low electron affinities of the elements on the lower left of the periodic table account for the readiness of their atoms to form cations. They also correlate, as discussed below, with the fact that these elements are metallic, for that property depends on the ready loss of electrons. On the other hand, the high ionization energies and high electron affinities of elements on the upper right of the periodic table (with the exception of the noble gases) account for their ready formation of anions (and for the fact that they are generally nonmetals, since that property is associated with the difficulty of removing electrons from atoms).

This synoptic view of ion formation is summarized by the concept of electronegativity, χ. There are numerous definitions of electronegativity. Qualitatively, the electronegativity of an element is the ability of one of its atoms to attract electrons toward itself when it is part of a compound (this definition was originally proposed by the American chemist Linus Pauling). Such an ability is high if the ionization energy of the element is high (so that the atom is reluctant to give up electrons) and if its electron affinity is also high (for then it is energetically favourable for it to acquire electrons). It follows that atoms with high electronegativities are those in the upper right-hand corner of the periodic table, close to fluorine (but excluding the noble gases). Such elements are likely to form anions when they form compounds. Elements with low ionization energies (so that they readily give up electrons) and low electron affinities (so that they have little tendency to acquire electrons) have low electronegativities (i.e., they are electropositive) and occur at the lower left of the periodic table. Such elements are likely to form cations during compound formation. (The effect of electronegativity on the polarity of a bond is discussed below in the section The polarity of molecules (chemical bonding).)

Emphasis has been placed on ion formation in this section, and hence it may appear that covalence was unduly neglected. However, the scene is now set for an introduction to the whole range of bonding types, and it will be explained how the atomic (atom) property of electronegativity helps to unify the discussion.

It has been shown that, for reasons related to the energy requirements for electron removal or addition, only the electrons in valence shells play a significant role in the formation of bonds between atoms. Henceforth this article will concentrate on these electrons alone. Lewis introduced the conventions of representing valence electrons by dots arranged around the chemical symbol of the element, as in H·, Na·, and .Cl:.... , and of discussing bond formation as the transfer of dots from one symbol to another. This seemingly simplistic device turns out to be very useful for establishing the characteristics of chemical bonds and will be examined in this section.

In Lewis terms, the formation of an ionic bond stems from the transfer of electrons from one atom to another. When such a transfer occurs, all the valence electrons on the more electropositive element (from one of the first three groups on the left in the periodic table) are removed to expose the core of the atom. The electrons so released are accepted into the empty orbitals of the valence shell of the more electronegative atom (typically from the groups immediately to the left of the noble gases); the valence shell is thereby filled. Thus, the formation of the ionic compound sodium chloride (salt) can be represented by the following process:

The formation of aluminum oxide ( alumina) involves selecting enough aluminum and oxygen atoms to ensure that all the electrons released by the aluminum atoms (three from each one) are accommodated by oxygen atoms (each of which can accept two electrons):

(The numbers of atoms required to balance the electrons donated and accepted is indicated by the chemical formula Al₂O₃ for aluminum oxide.)

That the transfer of electrons represented by these diagrams leads to a lowering of energy can be checked by assessing the energies associated with them. There is more to the process than a straightforward consideration of ionization energy and electron affinity. The ionization energy of sodium is larger than the electron affinity of chlorine, so energy is required to remove an electron from a sodium atom and attach it to a chlorine atom. That is, at first sight it appears that the total energy of a Na⁺ ion and a Cl⁻ ion is greater than that of a sodium atom and a chlorine atom. If that were the case, then it would be hard to understand how sodium chloride could be a stable species relative to a gas of sodium and chlorine atoms.

There are in fact two errors in such a simple approach. First, the argument has ignored the favourable energy of interaction between the cation and the anion. The net energy of formation of a Na⁺ ion and a Cl⁻ ion is the sum of three terms. The first is the energy investment needed to ionize a sodium atom. The second is a somewhat smaller energy that is released when the electron from the sodium atom attaches to a chlorine atom. At this stage, the net energy change is positive, indicating a higher energy than for the two atoms. However, because there is an attraction between opposite charges, there is a further release of energy as a result of the interaction of the two ions. This additional favourable contribution to the energy varies with the separation of the ions and strengthens as the two ions approach one another. Thus, at large separations the neutral atoms have the lowest energy, but as the two atoms are brought together a point is reached at which the lowest total energy is obtained if an electron transfers from the sodium atom to the chlorine atom. At this distance, and at shorter distances, Na⁺Cl⁻ is the lower-energy species.

The second feature omitted from the argument is that an ionic compound does not consist of an isolated cation and anion. An ionic compound is typically a solid formed from an array of alternating cations and anions. The packing of ions together and their electrostatic interactions with one another account for the typical features of ionic compounds—namely, their brittleness and high melting points. Moreover, when studying the stability of such compounds, one should more appropriately consider the energy changes associated with their formation from the elements in their standard state (such as solid metallic sodium and gaseous chlorine molecules) than from a gas of atoms of the elements.

At this stage, the ions are allowed to come together to form a crystalline (crystal) array. This step releases a large quantity of energy called the lattice energy of the compound. Energy is released in the process of crystal formation because first a cation becomes surrounded by anions, then that cluster of anions becomes surrounded by cations, and so on. As a result of this packing, every cation has anions as neighbours, and every anion has cations around it, and there is a strong overall attractive interaction among the many ions of opposite charge in the crystal. For sodium chloride, the lattice energy is so great that more energy is released in this step than is required for all the preceding steps combined, and solid sodium chloride therefore has a lower energy than sodium metal and chlorine gas. It is for this reason that, when sodium reacts with chlorine, a large quantity of heat is released.

A Born-Haber cycle gives an indication of the factors that favour ionic bonding. Overall, the lattice energy must be great enough to overcome the energy required for ion formation. It follows that only elements with reasonably low ionization energies can contribute, as cations, to ionic materials, for too large an ionization energy could not be recovered from the resulting lattice energy. In practice, this criterion means that only metallic (metal) elements are likely to form cations, and two elements are unlikely to form an ionic compound unless one of them is a metal. Moreover, the steep increase in ionization energy required to break into a closed shell precludes the loss of all but the valence electrons. Furthermore, no more than about three electrons per atom can be lost before the increase in ionization energy becomes prohibitive.

It can also be seen from the Born-Haber cycle that elements will contribute anions to an ionic compound only if their electron affinity is positive or, at least, not too strongly negative. Elements with positive electron affinities are likely to form anions (as long as a metal is present). A negative electron affinity can be tolerated provided it is not too great and the additional energy investment can be recovered from a greater lattice energy. That is the reason why ionic oxides (oxide) are so common: although energy is required to push the second electron on an oxygen atom to make an O²⁻ ion, the resulting ion produces such a high lattice energy that the energy investment is overcome.

Ionic bonding is likely to occur if the lattice energy of the compound is large, for a large lattice energy can compensate for some strongly demanding energy requirements, most notably for cation formation, earlier in the cycle. High lattice energies are achieved if the ions that form the lattice are small and highly charged, for small ions can pack together closely and interact strongly with one another. The O²⁻ ion of oxides is small (oxygen lies well to the right in the periodic table) and is reasonably highly charged (it has two negative charges; three negative charges is about the limit for monatomic anions). As a result, ionic oxides are widely formed by metallic elements. Although it is conceivable that O³⁻ ions could be formed if enough energy were provided to overcome the repulsion from the many electrons present in O²⁻, the necessary energy would not be recovered from the lattice energy, for O³⁻ anions are so large that the lattice energy of any compound they would form would be small. Once again, the termination of electron gain at a noble gas configuration is not so much a sign of some magic stability of such a species but rather a consequence of the fact that after such a configuration has been attained there is insufficient opportunity for achieving a lower energy.

The actual pattern in which cations and anions pack together is the one that results in the greatest lattice energy (that is, the greatest lowering of energy of the ions relative to the gas of ions). For further details on crystal arrangements, see crystal.

When none of the elements in a compound is a metal, no atoms in the compound have an ionization energy low enough for electron loss to be likely. In such a case, covalence prevails. As a general rule, covalent bonds are formed between elements lying toward the right in the periodic table (i.e., the nonmetals). Molecules of identical atoms, such as H₂ and buckminsterfullerene (C₆₀), are also held together by covalent bonds.

In Lewis terms a covalent bond is a shared electron pair. The bond between a hydrogen atom and a chlorine atom in hydrogen chloride is formulated as follows:

In a Lewis structure of a covalent compound, the shared electron pair is represented by a line, so the Lewis structure of hydrogen chloride is denoted HCl:.... . The electron pair represented by the line is called a bonding pair; the three other pairs of electrons on the chlorine atom are called lone pairs and play no direct role in holding the two atoms together.

Each atom in the hydrogen chloride molecule attains a closed-shell octet of electrons by sharing and hence achieves a maximum lowering of energy. In general, an incomplete shell means that some attracting power of a nucleus may be wasted, and adding electrons beyond a closed shell would entail the energetic disadvantage of beginning the next shell of the atom concerned. Lewis' octet rule is again applicable and is seen to represent the extreme means of achieving lower energy rather than being a goal in itself.

A covalent bond forms if the bonded atoms have a lower total energy than the widely separated atoms. The simplest interpretation of the decrease in energy that occurs when electrons are shared is that both electrons lie between two attracting centres (the nuclei of the two atoms linked by the bond) and hence lie lower in energy than when they experience the attraction of a single centre. This explanation, however, requires considerable modification to capture the full truth about bonding, and it will be discussed further below when bonding is considered in terms of quantum mechanics.

Lewis structures of more complex molecules can be constructed quite simply by extending the process that has been described for hydrogen chloride. First, the valence electrons that are available for bonding are counted (2 × 1 + 6 = 8 in O (water), for example, and 4 + 4 × 7 = 32 in carbon tetrachloride, CCl₄), and the chemical symbols for the elements are placed in the arrangement that reflects which are neighbours:

Next, one bonding pair is added between each linked pair of atoms:

The remaining electrons are then added to the atoms in such a way that each atom has a share in an octet of electrons (this is the octet-rule part of the procedure):

Finally, each bonding pair is represented by a dash:

(Note that Lewis structures do not necessarily show the actual shape of the molecule, only the topological pattern of their bonds.)

In some older formulations of Lewis structures, a distinction was made between bonds formed by electrons that have been supplied by both atoms (as in H−Cl, where one shared electron can be regarded as supplied by the hydrogen atom and the other by the chlorine atom) and covalent bonds formed when both electrons can be regarded as supplied by one atom, as in the formation of OH⁻ from O²⁻ and H⁺. Such a bond was called a coordinate covalent bond or a dative bond and symbolized O → H⁻. However, the difficulties encountered in the attempt to keep track of the origin of bonding electrons and the suggestion that a coordinate covalent bond differs somehow from a covalent bond (it does not) have led to this usage falling into disfavour.

The Lewis structures illustrated so far have been selected for their simplicity. A number of elaborations are given below.

First, an atom may complete its octet by sharing more than one pair of electrons with a bonded neighbour. Two shared pairs of electrons, represented by a double dash (=), form a double bond. Double bonds are found in numerous compounds, including carbon dioxide: (carbon dioxide)

Three shared pairs of electrons are represented by a triple dash (≡) and form a triple bond. Triple bonds are found in, for example, carbon monoxide, nitrogen molecules, and acetylene, shown, respectively, as:

A double bond is stronger than a single bond, and a triple bond is stronger than a double bond. However, a double bond is not necessarily twice as strong as a single bond, nor is a triple bond necessarily three times as strong. Quadruple bonds, which contain four shared pairs of electrons, are rare but have been identified in some compounds in which two metal atoms are bonded directly together.

There is sometimes an ambiguity in the location of double bonds. This ambiguity is illustrated by the Lewis structure for ozone (O₃). The following are two possible structures:

In such cases, the actual Lewis structure is regarded as a blend of these contributions and is written:

The blending together of these structures is actually a quantum mechanical phenomenon called resonance, which will be considered in more detail below. At this stage, resonance can be regarded as a blending process that spreads double-bond character evenly over the atoms that participate in it. In ozone, for instance, each oxygen-oxygen bond is rendered equivalent by resonance, and each one has a mixture of single-bond and double-bond character (as indicated by its length and strength).

Lewis structures and the octet rule jointly offer a succinct indication of the type of bonding that occurs in molecules and show the pattern of single and multiple bonds between the atoms. There are many compounds, however, that do not conform to the octet rule. The most common exceptions to the octet rule are the so-called hypervalent compounds. These are species in which there are more atoms attached to a central atom than can be accommodated by an octet of electrons. An example is sulfur hexafluoride, SF₆, for which writing a Lewis structure with six S−F bonds requires that at least 12 electrons be present around the sulfur atom:

(Only the bonding electrons are shown here.) In Lewis terms, hypervalence requires the expansion of the octet to 10, 12, and even in some cases 16 electrons. Hypervalent compounds are very common and in general are no less stable than compounds that conform to the octet rule.

The existence of hypervalent compounds would appear to deal a severe blow to the validity of the octet rule and Lewis' approach to covalent bonding if the expansion of the octet could not be rationalized or its occurrence predicted. Fortunately, it can be rationalized, and the occurrence of hypervalence can be anticipated. In simple terms, experience has shown that hypervalence is rare in periods 1 and 2 of the periodic table (through neon) but is common in and after period 3. Thus, the octet rule can be used with confidence for carbon, nitrogen, oxygen, and fluorine, but hypervalence must be anticipated thereafter. The conventional explanation of this distinction takes note of the fact that, in period-3 elements, the valence shell has n = 3, and this is the first shell in which orbitals are available. (As noted above, these orbitals are occupied after the 4s orbitals have been filled and account for the occurrence of the transition metals in period 4.) It is therefore argued that atoms of this and subsequent periods can utilize the empty d orbitals to accommodate electrons beyond an octet and hence permit the formation of hypervalent species.

In chemistry, however, it is important not to allow mere correlations to masquerade as explanations. Although it is true that d orbitals are energetically accessible in elements that display hypervalence, it does not follow that they are responsible for it. Indeed, quantum mechanical theories of the chemical bond do not need to invoke d-orbital involvement. These theories suggest that hypervalence is probably no more than a consequence of the greater radii of the atoms of period-3 elements compared with those of period 2, with the result that a central atom can pack more atoms around itself. Thus, hypervalence is more a steric (geometric) problem than an outcome of d-orbital availability. How six atoms can be bonded to a central atom by fewer than six pairs of electrons is discussed below.

Less common than hypervalent compounds, but by no means rare, are species in which an atom does not achieve an octet of electrons. Such compounds are called incomplete-octet compounds. An example is the compound boron trifluoride, BF₃, which is used as an industrial catalyst. The boron (B) atom supplies three valence electrons, and a representation of the compound's structure is:

The boron atom has a share in only six valence electrons. It is possible to write Lewis structures that do satisfy the octet rule.

However, whereas in the incomplete octet structure the fluorine atoms have three lone pairs, in these resonance structures one fluorine atom has only two lone pairs, so it has partly surrendered an electron to the boron atom. This is energetically disadvantageous for such an electronegative element as fluorine (which is in fact the most electronegative element), and the three octet structures turn out to have a higher energy than the incomplete-octet structure. The latter is therefore a better representation of the actual structure of the molecule. Indeed, it is exactly because the BF₃ molecule has an incomplete-octet structure that it is so widely employed as a catalyst, for it can use the vacancies in the valence shell of the boron atom to form bonds to other atoms and thereby facilitate certain chemical reactions.

Another type of exception to the Lewis approach to bonding is the existence of compounds that possess too few electrons for a Lewis structure to be written. Such compounds are called electron-deficient compounds. A prime example of an electron-deficient compound is diborane (borane), B₂H₆. This compound requires at least seven bonds to link its eight atoms together, but it has only 2 × 3 + 6 × 1 = 12 valence electrons, which is enough to form only six covalent bonds. Once again, it appears that, as in hypervalent compounds, the existence of electron-deficient compounds signifies that a pair of electrons can bond together more than two atoms. The discussion of the quantum mechanical theory of bonding below shows that this is indeed the case.

A number of exceptions to Lewis' theory of bonding have been catalogued here. It has further deficiencies. For example, the theory is not quantitative and gives no clue to how the strengths of bonds or their lengths can be assessed. In the form in which it has been presented, it also fails to suggest the shapes of molecules. Furthermore, the theory offers no justification for regarding an electron pair as the central feature of a covalent bond. Indeed, there are species that possess bonds that rely on the presence of a single electron. (The one-electron transient species H₂⁺ is an example.) Nevertheless, in spite of these difficulties, Lewis' approach to bonding has proved exceptionally useful. It predicts when the octet rule is likely to be valid and when hypervalence can be anticipated, and the occurrence of multiple bonds and the presence of lone pairs of electrons correlate with the chemical properties of a wide variety of species. Lewis' approach is still widely used as a rule of thumb for assessing the structures and properties of covalent species, and modern quantum mechanical theories echo its general content.

The following sections discuss how the limitations of Lewis' approach can be overcome, first by extending the theory to account for molecular shapes and then by developing more thorough quantum mechanical theories of the chemical bond.

There is a sharp distinction between ionic (ionic bond) and covalent bonds (covalent bond) when the geometric arrangements of atoms in compounds are considered. In essence, ionic bonding is nondirectional, whereas covalent bonding is directional. That is, in ionic compounds there is no intrinsically preferred direction in which a neighbour should lie for the strength of bonding to be maximized. In contrast, in a covalently bonded compound, the atoms adopt specific locations relative to one another, as in the tetrahedral arrangement of hydrogen atoms around the central carbon atom in methane, CH₄, or the angular arrangement of atoms in H₂O.

The rationalization of the structures adopted by purely ionic solids is essentially a straightforward exercise in the analysis of electrostatic interactions between ions. The problem of the structures of covalent compounds, both individual molecules, such as methane, and covalently bonded solids, such as diamond, is much more subtle, for it involves delving into the characteristics of the electron arrangements in individual atoms. Thus, if the formation of a covalent bond is regarded as corresponding to the accumulation of electrons in a particular region of an atom, then, to form a second bond, electrons can be accumulated into only certain parts of the atom relative to that first region of enhanced electron density. As a result, the bonds will lie in a geometric array that is characteristic of the atom. The remainder of this section focuses on this problem, but a detailed quantum mechanical analysis is required for a full understanding of the matter.

The theory of molecular shape known as valence-shell electron-pair repulsion (VSEPR) theory grew out of Lewis' theory, and, like that approach to bonding, VSEPR focuses on the role of electron pairs. It stems from the work of the British chemists H.M. Powell and Nevil V. Sidgwick in the 1940s and was extensively developed by R.J. Gillespie in Canada and Ronald S. Nyholm in London during the 1960s. As such, it postdates quantum mechanical theories of bonding and shape but should be seen (as is so common a motivation in chemistry) as an attempt to identify the essential features of a problem and to formulate them into a simple qualitative procedure for rationalization and prediction.

A Lewis structure, as shown above, is a topological portrayal of bonding in a molecule. It ascribes bonding influences to electron pairs that lie between atoms and acknowledges the existence of lone pairs of electrons that do not participate directly in the bonding. The VSEPR theory supposes that all electron pairs, both bonding pairs and lone pairs, repel each other—particularly if they are close—and that the molecular shape is such as to minimize these repulsions. The approach is commonly applied to species in which there is an identifiable central atom (the oxygen atom in H₂O, for instance), but it is straightforward to extend it to discussions of the local shape at any given atom in a polyatomic species.

When applying VSEPR theory, attention is first focused on the electron pairs of the central atom, disregarding the distinction between bonding pairs and lone pairs. These pairs are then allowed to move around the central atom (at a constant distance) and to take up positions that maximize their mutual separations. As in the methane molecule, four pairs adopt a tetrahedral disposition. The arrangements adopted by two through six pairs are summarized in the table. At this stage, the atoms that are attached by the bonding pairs are introduced, and the shape of the molecule is reported on the basis of the arrangement of these atoms.

The angle between electron pairs in a tetrahedral arrangement is 109.5°. However, although H₂O is indeed angular and NH₃ is trigonal pyramidal, the angles between the bonds are 104° and 107°, respectively. In a sense, such close agreement is quite satisfactory for so simple an approach, but clearly there is more to explain. To account for variations in bond angle, it is supposed that electron pair repulsions are greatest between lone pairs, less between lone pairs and bonding pairs, and least between bonding pairs. The justification of this ordering has proved somewhat elusive; qualitatively it is presumed that lone pairs, being attached only to a single centre, spread over a greater volume than bonding pairs, which are pinned between two attracting centres. Whatever the reason may be, the order correlates quite well with observation. Thus, in H₂O, the two lone pairs move apart a little, and the two bonding pairs move away from them by closing the angle between one another. Likewise, in NH₃ the three bonding pairs move back from the single lone pair to minimize their interaction with it. As a result, the H−N−H bond angle decreases slightly. In each case, the predicted angle is less than the tetrahedral angle, as is observed experimentally.

VSEPR theory is quite successful at predicting (or, at least, rationalizing) the overall shapes of molecules. Thus, the hypervalent species SF₆ (sulfur hexafluoride), with six bonding pairs, is predicted and found to be a regular octahedron, and PCl₅ (phosphorus pentachloride), with five bonding pairs, is predicted and found to be a trigonal bipyramid. The XeF₄ (xenon tetrafluoride) molecule is hypervalent with six electron pairs around the central xenon (Xe) atom. These pairs adopt an octahedral arrangement. Four of the pairs are bonding pairs, and two are lone pairs. According to VSEPR theory, the repulsion between the lone pairs is minimized if they lie on opposite sides of the xenon atom, leaving the four equatorial pairs as bonding pairs.

This analysis suggests that XeF₄ should be a planar species, which is found to be the case.

can be treated as having the equivalent of four pairs (two ordinary pairs and two superpairs) around the sulfur atom in a tetrahedral arrangement. All four pairs are bonding, so the ion is predicted to be a regular tetrahedron, which it indeed is. The same conclusion about the shape of the molecule would be drawn from another possible Lewis structure, in which each bond is single:

The actual molecule is a resonance hybrid of these and related structures; but, as each one corresponds to the same geometry, no particular Lewis structure need be selected before one can make a prediction based on VSEPR theory. In other words, resonance (resonance, theory of) does not affect the shapes of molecules.

Examples of the manner in which VSEPR theory is applied to species in which there is no central atom are provided by ethane (C₂H₆), ethylene (C₂H₄), and acetylene (C₂H₂), the Lewis structures for which are, respectively, the following:

The VSEPR theory is simple yet powerful. Nevertheless, like any simplified model, it has its limitations. First, although it predicts that the bond angle in H₂O is less than the tetrahedral angle, it does not make any attempt to predict the magnitude of the decrease. Second, the theory makes no predictions about the lengths of the bonds, which is another aspect of the shape of a molecule. Third, it ascribes the entire criterion of shape to electrostatic repulsions between bonding pairs, when in fact there are numerous contributions to the total energy of a molecule, and electrostatic effects are not necessarily the dominant ones. Fourth, the theory relies on some vague concepts, such as the difference in repelling effects of lone pairs and bonding pairs. There also are some species for which VSEPR theory fails. Nevertheless, despite these limitations and uncertainties, VSEPR theory is a useful rule of thumb and can be used with reasonable confidence for numerous species.

There are three main properties of chemical bonds that must be considered—namely, their strength, length, and polarity. The polarity of a bond is the distribution of electrical charge over the atoms joined by the bond. Specifically, it is found that, while bonds between identical atoms (as in H₂) are electrically uniform in the sense that both hydrogen atoms are electrically neutral, bonds between atoms of different elements are electrically inequivalent. In hydrogen chloride, for example, the hydrogen atom is slightly positively charged whereas the chlorine atom is slightly negatively charged. The slight electrical charges on dissimilar atoms are called partial charges, and the presence of partial charges signifies the occurrence of a polar bond.

The polarity of a bond arises from the relative electronegativities of the elements. electronegativity, it will be recalled, is the power of an atom of an element to attract electrons toward itself when it is part of a compound. Thus, although a bond in a compound may consist of a shared pair of electrons, the atom of the more electronegative element will draw the shared pair toward itself and thereby acquire a partial negative charge. The atom that has lost its equal share in the bonding electron pair acquires a partial positive charge because its nuclear charge is no longer fully canceled by its electrons.

The existence of equal but opposite partial charges on the atoms at each end of a heteronuclear bond (i.e., a bond between atoms of different elements) gives rise to an electric dipole. The magnitude of this dipole is expressed by the value of its dipole moment, μ, which is defined as the product of the magnitude of the partial charges times their separation (essentially, the length of the bond). The dipole moment of a heteronuclear bond can be estimated from the electronegativities of the atoms A and B, χ_A and χ_B, respectively, by using the simple relation

where D denotes the unit debye, which is used for reporting molecular dipole moments (1 D = 3.34 × 10⁻³⁰ coulomb·metre). Moreover, the negative end of the dipole lies on the more electronegative atom. If the two bonded atoms are identical, it follows that the dipole moment is zero and the bond is nonpolar.

As the difference in electronegativity between two covalently bonded atoms increases, the dipolar character of the bond increases as the partial charges increase. When the electronegativities of the atoms are very different, the attraction of the more electronegative atom for the shared electron pair is so great that it effectively exercises complete control over them. That is, it has gained possession of the pair, and the bond is best regarded as ionic (ionic bond). Ionic and covalent bonding (covalent bond) therefore can be regarded as constituting a continuum rather than as alternatives. This continuum can be expressed in terms of resonance by regarding a bond between atoms A and B as a resonance between a purely covalent form, in which the electrons are shared equally, and a purely ionic form, in which the more electronegative atom (B) has total control over the electrons:

As the electronegativity difference increases, the resonance lies increasingly in favour of the ionic contribution. When the electronegativity difference is very large, as between an electropositive atom like sodium and an electronegative atom like fluorine, the ionic structure dominates the resonance, and the bonding can be regarded as ionic. Thus, as the electronegativity difference of the two bonded elements increases, a nonpolar bond gives way to a polar bond, which in turn becomes an ionic bond. There are, in fact, no purely ionic bonds, just as there are no purely covalent bonds; bonding is a continuum of types.

Even a homonuclear bond, which is a bond between atoms of the same element (as in 2 (chlorine)), is not purely covalent, because a more accurate description would be in terms of ionic-covalent resonance:

That the species is nonpolar despite the occurrence of ionic contributions stems from the equal contributions of the ionic structures Cl⁻Cl⁺ and Cl⁺Cl⁻ and their canceling dipoles. That Cl₂ is commonly regarded as a covalently bonded species stems from the dominant contribution of the structure Cl−Cl to this resonance mixture. In contrast, the valence bond theory wavefunction (see below The quantum mechanics of bonding: Valence bond theory (chemical bonding)) of hydrogen chloride would be expressed as the resonance hybrid

In this case, the two ionic structures would contribute different amounts (because the elements have different electronegativities), and the larger contribution of H⁺Cl⁻ is responsible for the presence of partial charges on the atoms and the polarity of the molecule.

A polyatomic molecule will have polar bonds if its atoms are not identical. However, whether or not the molecule as a whole is polar (i.e., has a nonzero electric dipole moment) depends on the shape of the molecule. For example, the carbon-oxygen bonds in carbon dioxide are both polar, with the partial positive charge on the carbon atom and the partial negative charge on the more electronegative oxygen atom. The molecule as a whole is nonpolar, however, because the dipole moment of one carbon-oxygen bond cancels the dipole moment of the other, for the two bond dipole moments point in opposite directions in this linear molecule. In contrast, the water molecule is polar. Each oxygen-hydrogen bond is polar, with the oxygen atom bearing the partial negative charge and the hydrogen atom the partial positive charge. Because the molecule is angular rather than linear, the bond dipole moments do not cancel, and the molecule has a nonzero dipole moment.

The polarity of H₂O is of profound importance for the properties of water. It is partly responsible for the existence of water as a liquid at room temperature and for the ability of water to act as a solvent for many ionic compounds. The latter ability stems from the fact that the partial negative charge on the oxygen atom can emulate the negative charge of anions that surround each cation in the solid and thus help minimize the energy difference when the crystal dissolves. The partial positive charge on the hydrogen atoms can likewise emulate that of the cations surrounding the anions in the solid.

The preceding discussion has outlined the general approach to covalent bonding and has shown how it is still widely employed for a qualitative understanding of molecules. It is incomplete in many respects, however. First, the role of the electron pair remains unexplained but appears to be the hinge of both Lewis' theory and the VSEPR theory. Second, there is evidence that suggests that Lewis' theory overemphasizes the role of electron pairs. More fundamentally, little has been said about the distribution of bonding electrons in terms of orbitals, although it has been shown that in atoms the distributions of electrons are described by wavefunctions. Finally, the models that have been described have little quantitative content: they do not lead to bond lengths or precise bond angles, nor do they give much information about the strengths of bonds.

A full theory of the chemical bond needs to return to the roots of the behaviour of electrons in molecules. That is, the role of the electron pair and the quantitative description of bonding must be based on the Schrödinger equation and the Pauli exclusion principle. This section describes the general features of such an approach. Once again, the discussion will be largely qualitative and conceptual rather than mathematical and numerical. However, the character of the presentation here should not be taken to imply that the current understanding of molecules is not rigorous, quantitative, and precise.

Several difficulties are encountered at the outset of the application of the Schrödinger equation to molecules. Even the simplest molecules consist of two nuclei and several electrons, and interesting molecules may contain a thousand atoms and tens of thousands of electrons. So that any progress of a generally applicable kind can be made, approximations are necessary.

One approximation is common to all discussions of molecules. The Born-Oppenheimer approximation, which was introduced by Max Born and J. Robert Oppenheimer in 1927, separates the motion of the electrons in a molecule from the motion of the nuclei. The separation is based on the fact that the nuclei are much heavier than the electrons and move more slowly. Hence, even though nuclei do move, the electrons can respond to their new positions almost instantaneously. That being the case, it is permissible to consider the nuclei as stationary in a given arrangement and then to solve the Schrödinger equation for the electrons in that stationary framework of nuclei. In order to explore how the energy of the molecule changes as the nuclei change their positions, a series of static nuclear arrangements can be selected, and the Schrödinger equation solved for the electrons in each stationary arrangement.

Even the Born-Oppenheimer approximation is only one of the approximations needed for the study of the molecule. It separates out the nuclear motion and leaves untouched the need to solve the Schrödinger equation for several (and perhaps tens of thousands) of electrons. Two major alternative approximations beyond the Born-Oppenheimer approach have been devised to tackle this aspect of the problem. The first to be proposed (by Walter Heitler and Fritz London in 1927 and substantially developed by John Slater and Linus Pauling in the 1930s) is valence bond (VB) theory. This theory introduced language into chemistry that is still widely used, particularly in the discussion of organic compounds, but it has been somewhat overshadowed in quantitative investigations by its rival. The latter, molecular orbital (MO) theory, was introduced in 1927 by Robert S. Mulliken and Friedrich Hund. It has undergone considerable development and is the principal model for the calculation of molecular properties and for general discussions of compounds.

The basis of VB theory is the Lewis concept of the electron-pair bond. Broadly speaking, in VB theory a bond between atoms A and B is formed when two atomic orbitals, one from each atom, merge with one another (the technical term is overlap), and the electrons they contain pair up (so that their spins are ↓↑). The merging of orbitals gives rise to constructive interference—i.e., an enhancement of amplitude—between the wavefunctions in the areas where they overlap, and hence an enhanced amplitude results in the internuclear region. As a consequence of the formation of this region of heightened amplitude, there is an increased probability of finding he electrons in the internuclear region (so echoing Lewis' conception of the bond) and, by implication, a lowering of the energy of the molecule.

The VB theory can be put in the broader context of quantum mechanics by drawing on the superposition principle and the Pauli exclusion principle (see quantum mechanics). The two principles establish more precisely the type of orbital merging that is required and also show that, to achieve that merging, the two electrons must pair their spins. The technical justification will not be presented here.

As an illustration of the VB procedure, consider the structure of H (water)₂O. First, note that the valence-shell electron configuration of an oxygen atom is 2s²2p_x²2p_y¹2p_z¹, with an unpaired electron in each of two 2p orbitals, and

is the Lewis diagram for the atom. Each hydrogen atom has an unpaired 1s electron (H·) that can pair with one of the unpaired oxygen 2p electrons. Hence, a bond can form by the pairing of each hydrogen electron with an oxygen electron and the overlap of the orbitals they occupy. The electron distribution arising from each overlap is cylindrically symmetrical around the respective O−H axis and is called a σ bond. The VB description of H₂O is therefore that each hydrogen atom is linked to the oxygen atom by a σ bond formed by pairing of a hydrogen 1s electron and an oxygen 2p electron. Because a wavefunction can be written for this structure, an energy can be calculated by solving the Schrödinger equation, and a bond length can be determined by varying the nuclear separation and identifying the separation that results in the minimum energy.

The term σ bond is widely used in chemistry to denote an electron distribution like that in an oxygen-hydrogen bond, specifically one that has cylindrical symmetry about the line between the two bonded atoms. It is not the only type of bond, however, as can be appreciated by considering the structure of a nitrogen molecule, N₂. Each nitrogen atom has the valence-shell electron configuration 2s²2p_x¹2p_y¹2p_z¹. If the z direction is taken to lie along the internuclear axis of the molecule, then the electrons in the two 2p_z orbitals can pair and overlap to form a σ bond. However, the 2p_x orbitals now lie in the wrong orientation for head-to-head overlap, and they overlap side-to-side instead. The resulting electron distribution is called a π bond. A π bond also helps to hold the two atoms together, but, because the region of maximum electron density produced by the overlap is off the line of the internuclear axis, it does not do so with the same strength as a σ bond. The 2p_y electrons can pair and overlap in the same way and give rise to a second π bond. Therefore, the structure of an N₂ molecule consists of one σ bond and two π bonds. Note how this corresponds to and refines the Lewis description of the :N≡N: molecule.

In summary, a single bond in a Lewis structure corresponds to a σ bond of VB theory. A double bond corresponds to a σ bond plus a π bond, and a triple bond corresponds to a σ bond plus two π bonds.

Valence bond theory runs into an apparent difficulty with CH (methane)₄. The valence-shell electron configuration of carbon is 2s²2p_x¹2p_y¹, which suggests that it can form only two bonds to hydrogen atoms, in which case carbon would have a valence of 2. The normal valence of carbon is 4, however. This difficulty is resolved by noting that only the overall energy of a molecule is important, and, as long as a process leads to a lowering of energy, it can contribute even if an initial investment of energy is required. In this case, VB theory allows promotion to occur, in which an electron is elevated to a higher orbital. Thus, a carbon atom is envisaged as undergoing promotion to the valence configuration 2s¹2p_x¹2p_y¹2p_z¹ as a CH₄ molecule is formed. Although promotion requires energy, it enables the formation of four bonds, and overall there is a lowering of energy. Carbon is particularly suited to this promotion because the energy involved is not very great; hence the formation of tetravalent carbon compounds is the rule rather than the exception.

The discussion is not yet complete, however. If this description of carbon were taken at face value, it would appear that, whereas three of the CH bonds in methane are formed from carbon 2p orbitals, one is formed from a carbon 2s orbital. It is well established experimentally, however, that all four bonds in methane are identical.

Quantum mechanical considerations resolve this dilemma by invoking hybridization. Hybridization is the mixing of atomic orbitals on the same atom. When the 2s and three 2p orbitals of a carbon atom are hybridized, they give rise to four lobelike sp³ hybrid orbitals that are equivalent to one another apart from their orientations, which are toward the four corners of a regular tetrahedron. Each hybrid orbital contains an unpaired electron and can form a σ bond by pairing with a 1s electron of a hydrogen atom. Hence, the VB structure of methane is described as consisting of four equivalent σ bonds formed by overlap of the s orbitals of the hydrogen atoms with sp³ hybrid orbitals of the carbon atom.

This description conforms to the Lewis description. It also explains naturally why ethylene is a planar molecule, because twisting one end of the molecule relative to the other reduces the overlap between the 2p orbitals and hence weakens the π bond. All double bonds confer a torsional rigidity (a resistance to twisting) to the parts of molecules where they lie.

There is a VB wavefunction for each of these so-called Kekulé structures. (They are so called after Friedrich August Kekulé, who is commonly credited with having first proposed the hexagonal structure for benzene in 1865; however, a cyclic structure had already been proposed by Joseph Loschmidt four years earlier.) The actual structure is a superposition (sum) of the two wavefunctions: in VB terms, the structure of benzene is a resonance hybrid of the two canonical structures. In quantum mechanical terms, the blending effect of resonance in the Lewis approach to bonding is the superposition of wavefunctions for each contributing canonical structure. The effect of resonance is the sharing of the double-bond character around the ring, so that each carbon-carbon bond has a mixed single- and double-bond character. Resonance also (for quantum mechanical reasons) lowers the energy of the molecule relative to either contributing canonical structure. Indeed, benzene is a molecule that is surprisingly resistant to chemical attack (double bonds, rather than being a source of molecular strength and stability, are usually the targets of chemical attack) and is more stable than its structure suggests.

One of the difficulties that has rendered VB computationally unattractive is the large number of canonical structures, both covalent and ionic, that must be used in order to achieve quantitatively reliable results; in some cases tens of thousands of structures must be employed. Nevertheless, VB theory has influenced the language of chemistry profoundly, and the concepts of σ and π bonds, hybridization, and resonance are a part of the everyday vocabulary of the subject.

The alternative quantum mechanical theory of the electronic structures of molecules is MO theory. This approach was introduced about the same time as VB theory but has proved more amenable to quantitative implementation on computers. It is now virtually the only technique employed in the computational investigation of molecules. Like VB theory, it has introduced a language that is widely used in chemistry, and many chemists discuss chemical bonds in terms that combine both theories.

Just as an atomic orbital is a wavefunction that describes the distribution of an electron around the nucleus of an atom, so a molecular orbital (an MO) is a wavefunction that describes the distribution of an electron over all the nuclei of a molecule. If the amplitude of the MO wavefunction is large in the vicinity of a particular atom, then the electron has a high probability of being found there. If the MO wavefunction is zero in a particular region, then the electron will not be found there.

Although an MO can in principle be determined by solving the Schrödinger equation for an electron in the electrostatic field of an array of nuclei, in practice an approximation is always adopted. In this approximation, which is known as the linear combination of atomic orbitals (LCAO) approximation, each MO is constructed from a superposition of atomic orbitals belonging to the atoms in the molecule. The size of the contribution of an orbital from a particular atom indicates the probability that the electron will be found on that atom. The actual shape of the molecular orbital (and indirectly its energy) is a reflection of the extent to which the individual atomic orbitals interfere with one another either constructively or destructively.

The procedure can be introduced by considering the H₂ molecule. Its molecular orbitals are constructed from the valence-shell orbitals of each hydrogen atom, which are the 1s orbitals of the atoms. Two superpositions of these two orbitals can be formed, one by summing the orbitals and the other by taking their difference. In the former, the amplitudes of the two atomic orbitals interfere constructively with one another, and there is consequently an enhanced amplitude between the two nuclei. As a result, any electron that occupies this molecular orbital has a high probability of being found between the two nuclei, and its energy is lower than when it is confined to either atomic orbital alone. This combination of atomic orbitals is therefore called a bonding orbital. Moreover, because it has cylindrical symmetry about the internuclear axis, it is designated a σ orbital and labeled 1σ.

The MO formed by taking the difference of the two 1s orbitals also has cylindrical symmetry and hence is also a σ orbital. Taking the difference of the two atomic orbitals, however, results in destructive interference in the internuclear region where the amplitude of one orbital is subtracted from the other. This destructive interference is complete on a plane midway between the nuclei, and hence there is a nodal plane—i.e., a plane of zero amplitude—between the nuclei. Any electron that occupies this orbital is excluded from the internuclear region, and its energy is higher than it would be if it occupied either atomic orbital. The orbital arising in this way is therefore called an antibonding orbital; it is often denoted σ* (and referred to as “sigma star”) or, because it is the second of the two σ orbitals, 2σ.

The central importance of the electron pair for bonding arises naturally in MO theory via the Pauli exclusion principle. A single electron pair is the maximum number that can occupy a bonding orbital and hence give the greatest lowering of energy. However, MO theory goes beyond Lewis' approach by not ascribing bonding to electron pairing; some lowering of energy is also achieved if only one electron occupies a bonding orbital, and so the fact that H₂⁺ exists (with the electron configuration 1σ¹) is no longer puzzling.

The molecular orbitals of other species are constructed in an analogous way. In general, the orbitals in the valence shells of each atom are considered first (not, initially, the electrons those orbitals contain). Then the sets of these orbitals that have the same symmetry with respect to the internuclear axis are selected. (This point is illustrated below.) Bonding and antibonding combinations of each set are then formed, and from n atomic orbitals n such molecular orbitals are formed. The molecular orbital energy- level diagram that results is constructed by putting the molecular orbitals in order of increasing number of internuclear nodal planes, the orbital with no such nodal plane lying at lowest energy and the orbital with nodal planes between all the atoms lying at highest energy. At this stage, the valence electrons (valence electron) provided by the atoms are allowed to occupy the available orbitals in accord with the general rules of the building-up principle, with no more than two electrons in each orbital and in accord with Hund's rule if more than one orbital is available for occupation.

To see how the MO approach transcends the Lewis approach (and, in this instance, the VB approach as well), consider the electronic configuration of O₂. The same MO energy-level diagram (with changes of detail) can be used because the oxygen atoms provide the same set of atomic orbitals. Now, however, there are 2 × 6 = 12 valence orbitals to accommodate. The first 10 electrons reproduce the configuration of N₂. The last two enter the 2π* antibonding orbital, thereby reducing the net configuration to one σ bond and one π bond. That is, O₂ is a doubly-bonded species, in accord with the Lewis structure O=O. However, because there are two 2π orbitals and only two electrons to occupy them, the two electrons occupy different orbitals with parallel spins (recall Hund's rule). Therefore, the magnetic fields produced by the two electrons do not cancel, and O₂ is predicted to be a paramagnetic species. That is in fact the case. Such a property was completely outside the competence of Lewis' theory to predict and must be contrived in VB theory. It was an early major triumph of MO theory.

The principal qualitative difference between MO theory and VB theory becomes obvious when the objects of study are polyatomic, rather than diatomic, species. The benzene molecule is considered again but in this case from the viewpoint of its molecular orbitals. The atomic orbitals that provide the so-called basis set for the molecular orbitals (i.e., the ones from which the MOs are constructed) are the carbon 2s and 2p orbitals and the hydrogen 1s orbitals. All these orbitals except one 2p orbital on each carbon atom lie in the plane of the molecule, so they naturally form two sets that are distinguished by their symmetries. This discussion concentrates on the molecular orbitals that are constructed from the six perpendicular 2p orbitals, which form the π orbitals of the molecule; the remaining orbitals form a framework of σ orbitals.

Each carbon atom supplies one electron to the π system (the other 24 valence electrons have occupied the 12 low-energy σ orbitals that are not directly of interest here). These six electrons occupy the three lowest-energy molecular orbitals. Notice that none of the net antibonding orbitals is occupied; this is a part of the explanation of the considerable stability of the benzene molecule.

In the VB description of the benzene molecule, each double bond is localized between a particular pair of atoms, but resonance spreads that character around the ring. In MO theory, there are three occupied π orbitals, and hence three contributions to double-bond character, but each electron pair is spread around the ring and helps to draw either all the atoms together (the 1a orbital) or several of the atoms together (the two 1e orbitals). Thus, delocalization distributes the bonding effect of an electron pair over the atoms of the molecule, and hence one electron pair can contribute to the bonding of more than two atoms.

Several problems that remained unsolved in the earlier discussion of Lewis structures can be unraveled. It has already been shown that one electron can contribute to bonding if it occupies a bonding orbital; therefore the problem of the existence of one-electron species is resolved.

Hypervalence is taken care of, without having to invoke octet expansion, by the distributed bonding effect of delocalized electrons. Consider SF₆, which according to Lewis' theory needs to use two of its 3d orbitals in addition to its four 3s and 3p orbitals to accommodate six pairs of bonding electrons. In MO theory, the four 3s and 3p orbitals of sulfur and one 2p orbital of each fluorine atom are used to build 1 + 3 + 6 = 10 molecular orbitals. These 10 MOs are delocalized to varying degrees over the seven atoms of the molecule. Half of them have a net bonding character and half of them a net antibonding character between the sulfur and fluorine atoms. There are 6 sulfur valence electrons to accommodate and 6 × 1 = 6 fluorine electrons for a total of 12. The first 10 of these electrons occupy the net bonding orbitals; the remaining two occupy the lowest-energy antibonding orbital. In fact, this orbital is so weakly antibonding that it is best to regard it as nonbonding and as having little effect on the stability of the molecule. In any event, its weakly antibonding character is distributed over all six fluorine atoms, just as the other five pairs of electrons help to bind all six fluorine atoms to the central sulfur atom. The net effect of the 12 electrons is therefore bonding, and delocalization eliminates the need to invoke any role for d orbitals. The quantitative description of the forms and energies of the molecular orbitals is improved by the inclusion of 3d orbitals in the basis set, but only a small admixture is needed. There is certainly no need to invoke 3d orbitals as a necessary component of the description of bonding and no need to regard this hypervalent molecule as an example of a species with an expanded octet. Octet expansion is a rule of thumb, a correlation of an observation with the presence of available d orbitals, and not a valid explanation.

The other remaining outstanding problem is that of electron-deficient compounds, as typified by B (borane)₂H₆. Such molecules are classified as electron deficient because, in Lewis terms, there are fewer than two electrons available per bond. However, a consequence of delocalization is that the bonding influence of an electron pair is distributed over all the atoms in a molecule. Hence, it is easy to construct molecular orbitals that can achieve the binding of eight atoms by six electron pairs. The question to consider is not why electron-deficient compounds exist but why they are so rare. The answer lies in the smallness of the boron and hydrogen atoms, which allows them to get so close to one another that the cluster can be held together efficiently by a few delocalized pairs of electrons. Lewis was fortunate because the rules he adduced were generally applicable to larger atoms; there are more large atoms in the periodic table than there are atoms that are small enough for electron pair delocalization to be a dominant feature of their structures.

The language that molecular orbital theory brings to chemistry is that of bonding and antibonding orbitals and delocalization of electrons. The theory is presented here as an alternative to valence bond theory, and the formulation of the theory is quite different. However, both theories involve approximations to the actual electronic structures of molecules, and both can be improved. Valence bond theory is improved by incorporating extensive ionic-covalent resonance; molecular orbital theory is enhanced by allowing for a variety of occupation schemes for molecular orbitals (the procedure of configuration interaction). As these two improvement schemes are pursued, the wavefunctions generated by the two approaches converge on one another and the electron distributions they predict become identical.

Valence bond theory is widely used when the molecular property of interest is identifiable with the properties of individual bonds. It is therefore commonly employed in organic chemistry, where the reactions of molecules are often discussed in terms of the properties of their functional groups. The latter are small localized regions of a molecule (such as a double bond) or particular clusters of atoms (such as an OH group). Molecular orbital theory is widely used to describe properties that are most naturally discussed in terms of delocalization. Such properties include the spectroscopic properties of molecules, in which electromagnetic radiation is used to excite an electron from one molecular orbital to another and all the atoms contribute to the shift in electron density that accompanies the excitation.

Molecules cohere even though their ability to form chemical bonds has been satisfied. The evidence for the existence of these weak intermolecular forces is the fact that gases can be liquefied, that ordinary liquids exist and need a considerable input of energy for vaporization to a gas of independent molecules, and that many molecular compounds occur as solids. The role of weak intermolecular forces in the properties of gases was first examined theoretically by the Dutch scientist Johannes van der Waals, and the term van der Waals forces is used synonymously with intermolecular forces. Under certain conditions, weakly bonded clusters of molecules (such as an argon atom in association with a hydrogen chloride molecule) can exist; such delicately bonded species are called van der Waals molecules.

The first of the four bonding interactions discussed here is the dipole–dipole interaction between polar molecules. It will be recalled that a polar molecule has an electric dipole moment by virtue of the existence of partial charges on its atoms. Opposite partial charges attract one another, and, if two polar molecules are orientated so that the opposite partial charges on the molecules are closer together than their like charges, then there will be a net attraction between the two molecules. This type of intermolecular force contributes to the condensation of hydrogen chloride to a liquid at low temperatures. The dipole–dipole interaction also contributes to the weak interaction between molecules in gases, because, although molecules rotate, they tend to linger in relative orientations in which they have low energy—namely, the mutual orientation with opposite partial charges close to one another.

The second type of attractive interaction, the dipole–induced-dipole interaction, also depends on the presence of a polar molecule. The second participating molecule need not be polar; but, if it is polar, then this interaction augments the dipole–dipole interaction described above. In the dipole–induced-dipole interaction, the presence of the partial charges of the polar molecule causes a polarization, or distortion, of the electron distribution of the other molecule. As a result of this distortion, the second molecule acquires regions of partial positive and negative charge, and thus it becomes polar. The partial charges so formed behave just like those of a permanently polar molecule and interact favourably with their counterparts in the polar molecule that originally induced them. Hence, the two molecules cohere. This interaction also contributes to the intermolecular forces that are responsible for the condensation of hydrogen chloride gas.

The third type of interaction acts between all types of molecule, polar or not. It is also somewhat stronger than the two attractive interactions discussed thus far and is the principal force responsible for the existence of the condensed phases of certain molecular substances, such as benzene, other hydrocarbons, bromine, and the solid elements phosphorus (which consists of tetrahedral P₄ molecules) and sulfur (which consists of crown-shaped S₈ molecules). The interaction is called the dispersion interaction or, less commonly but more revealingly, the induced-dipole–induced-dipole interaction. Consider two nonpolar molecules near each other. Although there are no permanent partial charges on either molecule, the electron density can be thought of as ceaselessly fluctuating. As a result of these fluctuations, regions of equal and opposite partial charge arise in one of the molecules and give rise to a transient dipole. This transient dipole can induce a dipole in the neighbouring molecule, which then interacts with the original transient dipole. Although the latter continuously flickers from one direction to another (with an average of zero dipole overall), the induced dipole follows it, and the two correlated dipoles interact favourably with one another and cohere.

The interactions described so far are not limited to molecules of any specific composition. However, there is one important intermolecular interaction specific to molecules containing an oxygen, nitrogen, or fluorine atom that is attached to a hydrogen atom. This interaction is the hydrogen bond, an interaction of the form A−H···B, where A and B are atoms of any of the three elements mentioned above and the hydrogen atom lies on a straight line between the nuclei of A and B. A hydrogen bond is about 10 times as strong as the other interactions described above, and when present it dominates all other types of intermolecular interaction. It is responsible, for example, for the existence of water as a liquid at normal temperatures; because of its low molar mass, water would be expected to be a gas. The hydrogen bond is also responsible for the existence as solids of many organic molecules containing hydroxyl groups (−OH); the sugars glucose and sucrose are examples.

Many interpretations of the hydrogen bond have been proposed. One that fits into the general scheme of this article is to think of the A−H unit as being composed of an A atomic orbital and a hydrogen 1s orbital and to consider a lone pair of electrons on B as occupying a B orbital. When the three atoms are aligned, these three orbitals can form three molecular orbitals: one bonding, one largely nonbonding, and one antibonding. There are four electrons to accommodate (two from the original A−H bond and two from the lone pair). They occupy the bonding and nonbonding orbitals, leaving the antibonding orbital vacant. Hence, the net effect is to lower the energy of the AHB grouping and thus to constitute an intermolecular bond. Once again, on encountering the hydrogen bond, one encounters a twist in the conventional attitude; the question raised by this interpretation is not why such a bond occurs but why it does not occur more generally. The explanation lies in the small size of the hydrogen atom, which enables the balance of energies in the molecular orbital scheme to be favourable to bonding.

Hydrogen bonding occurs to atoms other than nitrogen, oxygen, and fluorine if they carry a negative charge and hence are rich in readily available electrons. Thus, hydrogen bonding is one of the principal mechanisms of hydration of anions in aqueous solution (the bonding of H₂O molecules to the solute species) and hence contributes to the ability of water to act as a good solvent for ionic compounds. It also contributes to the hydration of organic compounds containing oxygen or nitrogen atoms and thus accounts for the much greater aqueous solubility of alcohols than hydrocarbons.

Hydrogen bonds are also responsible for the transmission of genetic information from one generation to another, for they are responsible for the specific keying together of cytosine with guanine and thymine with adenine moieties that characterizes the structure of the DNA double helix.

Chemical bonds and intermolecular forces are jointly responsible for the existence of the solid phases of matter. This section reviews some of the types of solid that are encountered and relates them to the topics discussed earlier.

The structures of ionic solids have already been described in some detail. They consist of individual ions that are stacked together in such a way that the assembly has the lowest possible energy. These ions may be monatomic (as in sodium chloride, which consists of Na⁺ and Cl⁻ ions) or the ions may themselves be covalently bonded polyatomic species. An example of the latter is ammonium nitrate, in which the cation is NH₄⁺ and the anion is NO₃⁻; the N−H and N−O bonds within the ions are covalent. Ionic compounds are generally hard and brittle and have high melting points.

The structures of molecular solids, which are solids composed of individual molecules, have also been touched on in the section on intermolecular forces. These molecules are held to one another by hydrogen bonds (if they can form them), dispersion forces, and other dipolar forces—in that order of decreasing importance—and the molecules stack together in a pattern that minimizes their total energy. Examples of such solids include ice, in which hydrogen bonding is of paramount importance, and polyethylene, in which dispersion forces are dominant. Unless hydrogen bonds are present (in which case molecular solids resemble ionic solids in brittleness), molecular solids are generally soft and have low melting points because the bonds between the molecules are easily overcome.

Some solids have a network character in certain directions and a more molecular character in other directions. Once again, carbon provides the paradigm example, for the form of carbon known as graphite consists of a stack of sheets of hexagonal rings of carbon atoms. In the plane of the sheets, the bonding is covalent (and resembles an extended version of the bonding in benzene). The sheets themselves are held together by binding that is so weak that it is sometimes referred to as a van der Waals interaction. The anisotropy of the structure of graphite accounts for the anisotropy of its electrical conductivity (which is higher in the plane of the sheets than perpendicular to them). The ability of graphite to shed sheets of carbon (a feature utilized in the manufacture of pencils) and to act as a high-temperature lubricant (because the sheets can slide over one another) appears to be consistent with this structure but in fact seems to depend on the presence of impurities between the sheets.

The remaining major type of solid is a metal. A metal is characterized by its lustre, the ease with which it may be deformed (rather than shattered) by hammering, and its high electrical and thermal conductivities. Metals also tend to have higher densities than other types of solid. The starting point for theories of the structures of metals is to regard them as consisting of cations of the metal atoms embedded in a sea formed by the discarded valence electrons. The mobility of these electrons accounts for the mechanical, optical, and electrical properties of metals. The spherical cations can pack closely together yet still give rise to locally neutral electrical assemblies. This is because of the ability of the electrons to spread between the cations and neutralize their charges regardless of how closely they are packed. The closeness of the packing of the atoms accounts for the high densities of metals.

In the context of theories of the chemical bond, a metal is one extremely large homonuclear molecule. (For an alternative point of view, see the article crystal) If a sample of sodium metal is thought of as consisting of n sodium atoms where each atom has a 3s orbital for use in the construction of molecular orbitals and each atom supplies one electron to a common pool, then from these n atomic orbitals n molecular orbitals can be constructed. Each orbital has a characteristic energy, and the range of energies spanned by the n orbitals is finite, however great the value of n. If n is very large, it follows that the energy separation between neighbouring molecular orbitals is very small and approaches zero as n approaches infinity. The molecular orbitals then form a band (band theory) of energies. Another similar band can be formed by the overlap of the 3p orbitals of the atoms, but there is a substantial band gap—i.e., a region of energy in which there are no molecular orbitals—between the two bands.

Although the 3s band is virtually continuous, it actually consists of n discrete molecular orbitals, each of which, by the Pauli exclusion principle, can contain two paired electrons. It follows that the 3s band of sodium, which is occupied by the pool of n electrons, is only half full. There are empty molecular orbitals immediately above the uppermost filled orbitals, and it is easy for a perturbation, such as an applied potential difference or an oscillating electromagnetic field of incident light, to move the electrons into these unoccupied levels. Hence, the electrons are very mobile and can conduct an electric current, reflect light, transmit energy, and rapidly migrate to new locations when the cations are moved by hammering.

The full theory of the structure of metals is a highly technical subject (as are the full theories of the other topics discussed here). This brief introduction has been intended only to show that the ideas of molecular orbital theory can be naturally extended to account for the general features of the structures and properties of solids.

This section treats several aspects of molecular structure that are of more specialized interest and shows how particular classes of compounds are described. Molecular orbital theory will be used as a framework for the discussion, but aspects of valence bond theory will be incorporated when it is natural (in the sense of being commonplace in chemistry) to use them.

A particular class of compounds that once gave rise to some difficulty in the explanation of the origin of their bonding are the complexes of transition metal (transition element) ions. There are numerous examples of such species; they have in common a structure in which a central metal ion is surrounded by a number of ions or molecules, called ligands (ligand), that can also exist separately. The most common complexes have six ligands arranged in an octahedron around the central ion. An example is 【Fe(H₂O)₆】²⁺, where Fe denotes iron. This species can essentially be regarded as an Fe²⁺ ion, with an electron configuration 【Ar】3d⁶, surrounded by six H₂O molecules linked to the metal ion through their oxygen atoms.

Complex formation is an example of a particular class of reactions known as Lewis acid-base reactions (acid–base reaction). The general form of Lewis acid-base reactions involves the formation of a covalent bond between a species that supplies an electron pair, which is called a Lewis base, and a species that can accept an electron pair, which is called a Lewis acid. In complexes of the formula 【M(H₂O)₆】ⁿ⁺, the central metal ion acts as the Lewis acid and the ligand molecules act as the Lewis bases by virtue of a lone pair of electrons on the oxygen atom (only one of the lone pairs is in a position to act in this way). In general, a Lewis acid-base reaction is represented by the scheme A + :B → A−B. Such reactions occur widely in chemistry, but the singular characteristic of metal ions is that they can act as acceptors to several ligands. The actual number of ligands that attach to a metal ion is in part controlled by the spatial problem of packing ligands together around a central ion.

Crystal field theory is an artificial parameterization of the bonding in complexes, for it models the actual bonding in terms of an array of point charges. A superior theory is a modification of crystal field theory known as ligand field theory, which is more securely based in MO theory and allows for a more appropriate degree of delocalization of electrons over the metal ion and the ligands.

In essence, in ligand field theory molecular orbitals of complexes of first-transition-series (i.e., period-4) metals are constructed from the five 3d orbitals of the central metal cation and one orbital from each of the six ligand atoms that are directly attached to the metal cation. It follows that in such an octahedral complex there are 5 + 6 = 11 molecular orbitals to accommodate the 3d electrons of an 【Ar】3dⁿ species and 12 electrons from the six ligand atoms, giving 12 + n electrons in all. The 11 MOs span a range of energies. Twelve of the electrons occupy the six lowest-energy MOs, which are largely ligand-atom in character. The remaining n electrons are to be accommodated in the e_g and t_2g sets of orbitals. The energy separation between these two sets of orbitals, the ligand-field splitting energy (LFSE) is the ligand field version of the CFSE in crystal field theory, and from this point on the construction of the lowest-energy electron configuration is much the same as in crystal field theory. However, ligand field theory is less artificial, allows for electron delocalization, and is more readily extended to more complex patterns of bonding between the central metal ion and the ligands (such as the incorporation of bonds with π symmetry).

A particular class of complexes consists of the organometallic compounds, in which there are bonds between a metal atom and a carbon atom. Among the most important of such compounds are the carbonyls (carbonyl group), which are complexes in which one or more of the ligands is a carbon monoxide molecule, CO, either linked to one atom or bridging two. Another interesting class of organometallic compounds is composed of the metallocenes, informally called “sandwich compounds,” in which the metal atom sits between two planar hydrocarbon rings, analogous to the meat in a sandwich. Of these, ferrocene 【Fe(C₅H₅)₂】 was among the first to be synthesized.

The stabilities of organometallic compounds follow certain empirical rules, among which the 18-electron rule is the analogue of the octet rule of main-group compounds. According to this rule, the most stable organometallic compounds are those having 18 electrons in the valence shell, a term in this context extended to include the outermost d orbitals. Nickel tetracarbonyl, Ni(CO)₄, a poisonous gas used in the refining of nickel, has 10 electrons provided by the neutral nickel atom and two from each of the four CO ligands, giving 18 electrons in all.

The electronic structures of organometallic compounds can be expressed most effectively in MO terms, and they can be regarded as no more than a special case of ligand field theory. There are certain details that make them particularly interesting, however. To give a sense of the detail in which the structure of a metal-carbon bond may be expressed, attention will be focused here on the link between a metal atom (M) and a carbonyl group (CO): M−CO.

A CO molecule has much the same electronic structure as an N₂ molecule, because it has the same number of electrons—that is, the two species are isoelectronic. There are two important features of this structure, which differ in detail from N₂ on account of the different electronegativities of the two elements in CO. One is that the highest occupied molecular orbital, the HOMO, is largely confined to the carbon atom and can be interpreted as being a lone pair occupying an orbital with σ symmetry. This lone pair enables CO to act as a Lewis base and to link to the metal atom by forming a σ bond to it by overlap with one of the lobes of a d orbital. However, the lowest unfilled molecular orbital, the LUMO, has π symmetry and can accept electrons from an appropriate d orbital of the metal, and thus it can help the ligand to act as a Lewis acid. It is this ability of the ligand to act as both a Lewis base and a Lewis acid that is responsible for the stability of metal carbonyls.

The usefulness of the concept of a 3c,2e bond stems from two observations. The first is that diborane is in fact only one of a large class of compounds of boron and hydrogen, the boranes and the borohydride anions, in which the same feature is found. The second observation is that a 3c,2e bond can be formed by three boron atoms. Intricate networks of atoms can be formed in this way—for example, some having the form of closed frameworks (the closo-boranes), some looking like untidy birds' nests (the nido-boranes), and some resembling spiderwebs (the arachno-boranes). Which type of structure is obtained correlates with the number of valence electrons in the molecule, and the correlation is expressed by Wade's rules. These rules are empirical, but they can be justified by a consideration of the numbers of 3c,2e and ordinary 2c,2e bonds that are needed in each type of structure. They constitute an excellent example of how chemists utilize the concept of bond formation and deploy a mixture of valence bond and molecular orbital concepts to establish or rationalize helpful correlations between the number of electrons present and the structure of the species.

One component in the structure of a quadruple bond is a σ (sigma bond) bond formed by the cylindrically symmetrical overlap of two d orbitals. There are also two π (pi bond) bonds formed by the overlap of two appropriately orientated d orbitals. The new feature is the δ bond, which is formed by the face-to-face overlap of two parallel d orbitals and has a distinctly different symmetry with respect to the internuclear axis than the other two types of bond. A quadruple bond therefore consists of a σ bond, two π bonds, and one δ bond. The reduction in bond strength that would occur if one d orbital were rotated away from its partner so that overlap is lessened accounts for the torsional rigidity of the bond and the observed shape of the species.

In conclusion, a brief introduction to the manner in which these qualitative ideas are implemented computationally follows. The computation of molecular structures by numerical solution of the Schrödinger equation is a highly developed discipline. The principal difficulty is the large number of interactions between electrons that must be taken into account; this fact makes computational quantum chemists some of the most demanding users of computers and, increasingly, of supercomputers.

There are two strands of approach to the computation of molecular structure. In the semiempirical approach, the calculation draws on a number of experimentally determined characteristics to help in the overall calculation. In the ab initio approach, the calculation proceeds from first principles (the Schrödinger equation) and makes no use of imported information. The former approach was dominant in the 1970s, but increases in computing power have led to an ascendancy of ab initio techniques since then. The latter are intrinsically more reliable because there can be no certainty that a quantity determined in one context is appropriate to a particular molecule.

The central aim of computations is to identify the lowest-energy arrangement of a given set of atoms and to identify that arrangement as the structure of the molecule. The calculational strategy adopted is to seek self-consistency in the calculation, and, for that reason, the computations are referred to as self-consistent field (SCF) procedures. Thus, a particular electronic distribution is proposed, and the distribution of the electrons is recalculated on the basis of this first approximation. The distribution is then calculated again on the basis of that improved description, and the process is continued until there is negligible change—i.e., until the electron distribution has achieved self-consistency.

The implementation of this basic strategy can take a number of forms, and rival techniques have given rise to a large number of acronyms, such as AM1 (Austin Method 1) and MINDO (Modified Intermediate Neglect of Differential Overlap), which are two popular semiempirical procedures.

With self-consistency established, the wavefunctions are available for detailed scrutiny. One illustration must suffice. There is certain evidence that carcinogenic or pharmacological activity correlates with certain aspects of the charge distribution in molecules. Instead of dealing with the primitive concept of partial charges, numerical wavefunctions can be used to map the details of the charge distribution and hence to screen molecules for possible activity. This approach is potentially of considerable utility for pharmaceutical products as it can help to reduce the amount of in vivo screening of novel products.

Computational procedures have advanced to the stage where the role of the environment (for example, the water around enzyme molecules) can be incorporated. They are also being applied to the demanding calculations that are needed to describe the replacement of one grouping of chemical bonds into another that takes place in the course of chemical reactions. Thus, as well as dealing with the static considerations of structure, modern treatments of the chemical bond are now confronting the dynamic problems of reactions.