Bonding in Solids and Electronic Properties

Traditionally, bonding in metals has been approached through the idea of free electrons, a sort of electron gas. 

The free electron model regards a metal as a box in which electrons are free to roam, unaffected by the atomic nuclei or by each other. The nearest approximation to this model is provided by metals on the far left of the Periodic Table—Group 1 (Na, K, etc.), Group 2 (Mg, Ca, etc.)—and aluminium. These metals are often referred to as simple metals. 

The theory assumes that the nuclei stay fixed on their lattice sites surrounded by the inner or core electrons whilst the outer or valence electrons travel freely through the solid. If we ignore the cores then the quantum mechanical description of the outer electrons becomes very simple. Taking just one of these electrons the problem becomes the well-known one of the particle in a box. We start by considering an electron in a one-dimensional solid. 

The electron is confined to a line of length a (the length of the solid), which we shall call the x-axis. Because we are ignoring the cores, there is nothing for the electron to 

The electron is not allowed outside the box and to ensure this we put the potential to infinity outside the box. Since the electron cannot have infinite energy, the wave function must be zero outside the box and since it cannot be discontinuous, it must be zero at the boundaries of the box. If we take the sine wave solution, then this is zero at =0. To be zero at x=a as well, there must be a whole number of half waves in the box. Sine functions have a value of zero at angles of nn radians where n is an integer and so 

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Because a book of this size could not cover all topics in solid state chemistry, we have chosen to concentrate on structures and bonding in solids, and on the interplay between crystal and electronic stracture in determirring their properties. Examples of sohd state devices are used throughout the book to show how the choice of a partictrlar solid for a particular device is determined by the properties of that solid. 

In recent times, the bond indicators , which are the ground state properties of the solid related to its cohesion (metaUic radii, cohesive energy, bulk moduli), have been interpreted in the light of band calculations. The bond in metals and in compounds has been described by an easily understandable and convincing thermodynamic formalism, which we shall illustrate in this chapter. Essentially, narrow bands, as the 5 f electrons form, are considered to be resonant with the wider (spd) conduction band. The 5 f electronic population is seen as a fluid the partial (bonding) pressure of which assists in cohesion along with the partial pressure of another fluid constituted by the conduction electrons of (s and d) character. ... 

Figure 28.1 The electronic structure of a solid can be described in terms of a band model in which bonding electrons are primarily found in a low-energy valence band, while conduction is typically associated with antibonding or nonbonding high-energy orbitals known as the conduction band. In the case of a semiconductor (left), these two bands are separated by a quantum-mechanical forbidden zone, the band gap. Excitation of electrons from the valence band to the conduction band gives rise to the bulk optical and electronic properties of the semiconductor. In the case of a metal (right), the conduction band and valence band overlap, giving rise to a continuum of states.

Currently, it is fair to say that asymmetric synthesis using chiral quaternary ammonium fluorides remains an underdeveloped field, and the various useful stereoselective carbon-carbon bond-forming processes described in this chapter are only the start of an exploration of the vast synthetic potential of this process, particularly when combined with current knowledge of organosilicon chemistry. It seems that the key issue to be addressed is the rational molecular design of chiral quaternary ammonium cations with appropriate steric and electronic properties. These are expected to be readily tunable to impart not only a sufficient reactivity but also an ideal chiral environment to the requisite nucleophile involved in a desired chemical transformation. Clearly, the continuous accumulation of information related to the structure of fluoride salts and their reactivity and selectivity should create a solid basis for this field, offering - in time - a unique yet reliable tool for sophisticated bond construction events with rigorous stereocontrol, under mild conditions. 

A satisfactory theory of metallic bonding must account for the characteristic properties of high electrical and thermal conductivity, metallic lustre, ductility and the complex magnetic properties of metals which imply the presence of unpaired electrons. The theory should also rationalise the enthalpies of atomisation A/f tom of metallic elemental substances. A/f tom is a measure of the cohesive energy within the solid, and we saw in Chapter 5 how it plays an important part in the thermochemistry of ions in solids and solutions. The atomisation enthalpies of elemental substances (metallic and nonmetallic) are collected in Table 7.1. There is a fair correlation between A/Z tom an(J physical properties such as hardness and melting/boiling points. 

Quantum-chemical ab initio calculations have become an alternative to experiments for determining accurately structures, vibrational frequencies and electronic properties as well as intermolecular forces and molecular reactivity.28-31 Two specific approximations were developed to solve the problems of surface chemistry periodic approximation, where quantum-chemical method employs a periodic structure of the calculated system and cluster approximation, where a model of solid phase of finite size is created as a cutoff from the system of solid phase (it produces unsaturated dangling bonds at the border of the cluster). Cluster approximation has been widely used for studying interactions of molecules with all types of solids and their surfaces.32 This approach is powerful in calculating the systems with deviations from the ideal periodic structure like doping and defects. 

IN THIS PART of the book, we shall attempt to describe solids in the simplest meaningful framework. Chapter 1 contains a simple, brief statement of the quantum-mechanical framework needed for all subsequent discussions. Prior knowledge of quantum mechanics is desirable. However, for review, the premises upon which we will proceed are outlined here. This is followed by a brief description of electronic structure and bonding in atoms and small molecules, which includes only those aspects that will be directly relevant to discussions of solids. Chapter 2 treats the electronic structure of solids by extending the framework established in Chapter 1. At the end of Chapter 2, values for the interatomic matrix elements and term values are introduced. These appear also in a Solid State Table of the Elements at the back of the book. These will be used extensively to calculate properties of covalent and ionic solids. 

Bond orbitals are constructed ft om s/r hybrids for the simple covalent tetrahedral structure energies are written in terms of a eovalent energy V2 and a polar energy K3. There are matrix elements between bond orbitals that broaden the electron levels into bands. In a preliminary study of the bands for perfect crystals, the energies for all bands at k = 0 arc written in terms of matrix elements from the Solid State Tabic. For calculation of other properties, a Bond Orbital Approximation eliminates the need to find the bands themselves and permits the description of bonds in imperfect and noncrystalline solids. Errors in the Bond Orbital Approximation can be corrected by using perturbation theory to construct extended bond orbitals. Two major trends in covalent bonds over the periodic table, polarity and metallicity, arc both defined in terms of parameters from the Solid State Table. This representation of the electronic structure extends to covalent planar and filamentary structures. 

Metallic solids Recall from Chapter 8 that metallic solids consist of positive metal ions surrounded by a sea of mobile electrons. The strength of the metallic bonds between cations and electrons varies among metals and accounts for their wide range of physical properties. For example, tin melts at 232°C, but nickel melts at 1455°C. The mobile electrons make metals malleable—easily hammered into shapes—and ductile—easily drawn into wires. When force is applied to a metal, the electrons shift and thereby keep the metal ions bonded in their new positions. Read Everyday Chemistry at the end of the chapter to learn about shape-memory metals. Mobile electrons make metals good conductors of heat and electricity. Power lines carry electricity from power plants to homes and businesses and to the electric train shown in Figure 13-21a. 

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