Two daunting questions face the authors of prospective textbooks. (1) For whom is the book intended? (2) What makes it different from other books intended for the same audience? We first address these questions.

One might think from counting the mathematical equations that the book is intended for a theoretical physicist. This is partially true, for indeed we hope the subject is presented in a way that will satisfy a rigorously inclined mathematical physicist that “valency and bonding” is not just murky chemical voodoo, but authentic science grounded in the deepest tenets of theoretical physics.

Beyond Chapter 1 the reader may be relieved to find few if any equations that would challenge even a moderately gifted high-school student. The emphasis on orbital diagrams and “doing quantum mechanics with pictures” might then suggest that the book is intended for undergraduate chemistry students. This is also partially true. For example, we believe that our treatment of homonuclear diatomic molecules (Section 3.2.9) should be accessible to undergraduates who commonly encounter the topic in introductory chemistry courses.

Our principal goal has been to translate the deepest truths of the Schrödinger equation into a visualizable, intuitive form that “makes sense” even for beginning students, and can help chemistry teachers to present bonding and valency concepts in a manner more consistent with modern chemical research. Chemistry teachers will find here a rather wide selection of elementary topics discussed from a high-level viewpoint.

A quantum-mechanical calculational model can be uniquely specified by identifying (i) the method of approximation and (ii) the orbital basis set that underlies the model. The model is conventionally specified by a keyword label of the form method/basis, where method and basis are suitable identifying abbreviations or acronyms. Simple examples are “RHF/6-31G*” (for the RHF restricted Hartree–Fock method and the 6-31G* basis set) or “B3LYP/6-311 + +G**” (for the B3LYP hybrid density functional method and the 6-311 + +G** basis set). In this appendix we briefly describe the principal method and basis types that are now well established in the literature, particularly as implemented in the Gaussian program. Consult notes 1 and 2 for background information and original references.

Methods

Quantum-mechanical approximation methods can be classified into three generic types: (1) variational, (2) perturbative, and (3) density functional. The first two can be systematically improved toward exactness, but a systematic correction procedure is generally lacking in the third case.

Variational methods

Variational approximation methods are identified by the form of the variational trial function, particularly by the number and types of Slater determinants.

The simplest approximation corresponds to a single-determinant wavefunction. The best possible approximation of this type is the Hartree–Fock (HF) molecular-orbital determinant. The HF wavefunction is constructed from the minimal number of occupied MOs (i.e., N/2 for an N-electron closed-shell system), each approximated as a variational linear combination of the chosen set of basis functions (vide infra).

Introduction

Given the power of the concept of covalency and the deeper electronic implications that were realized by G. N. Lewis's octet-rule and shared-electron-pair concepts, it is natural to wonder whether these advances are limited to s/p-block elements or apply to the entire periodic table.

Soon after the quantum revolution of the mid 1920s, Linus Pauling and John C. Slater expanded Lewis's localized electronic-structural concepts with the introduction of directed covalency in which bond directionality was achieved by the hybridization of atomic orbitals. For normal and hypovalent molecules, Pauling and Slater proposed that spn hybrid orbitals are involved in forming shared-electron-pair bonds. Time has proven this proposal to be remarkably robust, as has been demonstrated by many examples in Chapter 3.

Hypervalent main-group compounds and most transition-metal complexes do not conform to Lewis's octet rule, because they exceed the ideal eight-electron count. Pauling first addressed this issue for main-group compounds by expanding the available valency of second- and higher-row elements by the addition of valence d orbitals to the set of orbitals that are active in bond formation. By using spn dm hybrid orbitals to form directed covalent bonds and by considering the strong role of resonance among ionic normal-valent structures, Pauling was able to rationalize a large number of hypervalent structures from the p block of the periodic table. However, as we have seen in Section 3.5, modern quantum-mechanical analyses of electron-density distributions downplay the importance of spn dm hybridization in hypervalent p-block compounds, emphasizing instead the importance of ionic resonance and 3c/4e bonding interactions.

Introduction

The close relationship between chemistry and electricity was surmised long before the discovery of quantum mechanics. The inverse-square law for the force between electrically charged objects (“Coulomb's law”) was established as early as 1766 by the American chemist Joseph Priestley, on the basis of a method suggested by Benjamin Franklin. Galvani's discoveries of 1780 and the subsequent development of the voltaic pile sparked intensive investigations on the chemical effects of electricity, culminating in Humphry Davy's electrolytic discoveries of alkali elements and general recognition of the “natural electrical energies of the elements” by about 1806. Within a decade, J. J. Berzelius had formulated his influential “dualistic” electrochemical theory, asserting that “in every chemical combination there is a neutralization of opposite electricities.”

Studies of electrolysis and electromagnetic induction were greatly advanced by Davy's assistant, Michael Faraday, who also introduced the modern terminology of “cation” and “anion.” Long before the 1897 “discovery” of the electron by J. J. Thomson, strong chemical evidence pointed to the existence of such “corpuscles of electricity,” as could be inferred particularly from Faraday's studies. Indeed, the word “electron” was coined in 1874 by the chemist J. Stoney, who also estimated the electronic charge from the value of Faraday's constant. Thus, the essential components of a simple ionic bonding picture, much as presented in current textbooks, were available at least a half-century before the discovery of Schrödinger's equation and the modern quantal view of electronic behavior.

Although the Berzelius ionic theory achieved successes in interpreting inorganic compounds, it met persistent difficulties in the emerging domain of organic chemistry.

Introduction

“Covalency” refers to the partial sharing of electrons that leads to mutual attraction of atoms, the profound and characteristic mystery of chemical bonding. The concept of covalency largely evolved from classical organic chemistry (the study of compounds of carbon, hydrogen, and nearby main-group elements) and achieved considerable maturity in the mid nineteenth century, long before Rutherford's model of atomic structure made it possible to consider the deeper electronic implications of this concept. The latter step was achieved most notably by G. N. Lewis, who showed how the octet rule and shared-electron-pair concepts could provide a comprehensive rationalization of structural bonding principles. The elementary Lewis-structure picture of nonbonding (one-center) and bonding (two-center) valence electron pairs forms the starting point for practically all aspects of molecular bonding in s- and p-block elements, the subject of the present chapter. Extension of covalency concepts to d-block elements will be considered in Chapter 4.

Let us briefly outline the topics to be addressed, which span a rather wide range of covalent and noncovalent “effects” in the bonding of s/p-block elements. The localized Lewis-structure picture of covalent and polar covalent bonding is described in Section 3.2, starting from the simplest diatomic species (e.g., H2 +, H2, and dialkali analogs) and proceeding through singly and multiply bonded polyatomic species. Inadequacies of the single-Lewis-structure picture (“delocalization effects”) are described in Section 3.3, including conjugative and aromatic “resonance” effects, which are conventionally represented as contributions from additional Lewis structures. Weaker “hyperconjugative” delocalizations are described in Section 3.4, including those responsible for internal rotation barriers, anomeric effects, and other stereoelectronic phenomena.

Introduction

Figure B.1 depicts the periodic table of the elements in the current IUPAC-approved form, with US-style assignments of group numbers as used throughout this book. This standard textbook table (STT) provides an accepted common vocabulary for the “row” (period) and “column” (group) assignments of each chemical element, and is thus an asset to clear pedagogical communication.

However, the STT somewhat muddles the natural groupings into s block, p block, d block, and f block of filling valence shells, which, from a fundamental perspective, is the most important feature of elemental electron configuration. The STT evolved from Mendeleev's original conception of serially ordering the elements in periodic tabular repeat patterns to reflect conspicuous chemical groupings (such as maximum oxidation number, physical state under standard-state conditions, and the like), long before the discovery of quantum theory and realization of the close associations between chemical and spectroscopic behavior. The STT chemical groupings reflect subtle biases toward ambient thermal conditions or the ambiguities arising from selected chemical criteria or the current extent of chemical knowledge. For theoretical purposes, certain alternative representations of chemical periodicity may therefore be advantageous in depicting the fundamental valence-electron occupancy patterns that underlie both chemistry and spectroscopy.

The theory of periodicity in electronic Aufbau of atomic configurations

According to the general Sturm–Liouville theory, the eigenvalues of any one-dimensional Schrödinger-type eigenvalue equation can be monotonically ordered according to the number of nodes (oscillations between negative and positive values) in the corresponding eigenfunctions.

We examine in this chapter how the motion of the reactants as they approach each other governs chemical reactivity. This allows us to use a two-body point of view where the internal structure of the colliding species is not explicitly recognized. All that we can do therefore is lead the reactants up to a reaction. But we will not be able to describe the chemical rearrangement itself nor to address such questions as that of energy disposal in the products. Chapter 5 takes up these themes. On the other hand, without the approach of the reactants there cannot be a bimolecular reaction. The tools already at our disposal are sufficient to discuss this approach motion. As expected, the striking distance that we have called the impact parameter will be a key player. We do all of this in Section 3.2. What we will obtain is information about the dependence of the reaction cross-section on the collision energy.

In chemical kinetics one characterizes the role of energy in chemical reactivity by the temperature dependence of the reaction rate constant. In Section 3.1 we review the input from chemical kinetics – the Arrhenius representation of the rate constant – then go from the rate constant to the reaction cross-section. Next we go in the opposite direction, from the microscopic reaction cross-section to the macroscopic rate constant. What we obtain thereby is the Tolman interpretation of the activation energy as the (mean) excess energy of those collisions that lead to reaction.

This chapter explores how the directed nature of chemical bonding affects molecular collisions and chemical reactions. Whereas the concept of size or cross-section leads to numbers (scalar quantities), the concept of chemical shape leads to vectors (numbers tied to directions). Consequently, this topic is more mathematically challenging, but its study yields important insights into the nature of chemical transformations. As expected, just as the size of a molecule depends on the probe selected to measure this quantity, the chemical shape of a molecule also depends sensitively on the probe chosen for the measurement. It is important to stress that the physical shape of a molecule is most commonly determined from molecular spectroscopy; it is usually expressed in terms of bond angles and bond lengths. The chemical shape of a molecule refers to the apparent size and shape of a molecule as experienced by another atom or molecule that collides with it.

The steric factor and early history of stereodynamics

The notion of spatial requirements for a chemical reaction dates to the introduction of a steric factor p in simple versions of the collision theory of reactions, Section 3.2.5. This was forced upon the theory by the smaller than expected, or even much smaller, overall magnitude of the reaction cross-section. Unfortunately, the early theory gave no clue for determining the value of p, which has largely been treated as an adjustable parameter to make simple collision theory agree with observation.

The photoionization spectrum of diphenylamine provides an unusual and interesting illustration of the Franck–Condon principle. Diphenylamine (DPA), illustrated in Figure 17.1, is a relatively large molecule to study by gas phase spectroscopy and it might be thought that the vibrational structure in its electronic spectra would be highly congested and difficult to interpret. After all, this is a molecule with 66 vibrational modes! However, it was shown in Section 7.2.3 that only totally symmetric modes generally need to be considered in interpreting electronic spectra. Also, there is the further simplification that not all of the totally symmetric modes need be Franck–Condon active, i.e. will give a significant progression. DPA is an excellent example of this, with the main structure arising from a single vibrational mode.

Before spectra are considered, the experimental procedure, carried out by Boogaarts and co-workers [1], will be outlined. Mass-analysed threshold ionization (MATI) spectroscopy was employed. This technique, which was briefly described in Section 12.6, is essentially the same as ZEKE spectroscopy but employs ion rather than electron detection. It has the advantage over ZEKE spectroscopy in that ions can be separated according to their mass, which in most cases enables the spectral carrier to be determined with confidence. By analogy with ZEKE spectroscopy, a cation ← neutral molecule electronic absorption spectrum is effectively obtained.