The Electronic Structure of Solids

The values computed for from the experimental values of the quantities on the right of Eq. (28.10) agree with the values predicted by Eq. (28.6) to about 4 % for the alkali halides. 

The increase in cohesive energy in the 2-2 salts explains the generally lower solubility of these salts (for example, the sulfides, as compared with that of the alkali halides). The greater the cohesive energy, the more difficult it is for a solvent to break up the crystal. 

Consider metallic sodium. The sodium atom has eleven electrons in the configuration ls 2s 2p 3s. Bringing many sodium atoms together in the crystal scarcely affects the energies of the electrons in the Is, 2s, and 2p levels, since the electrons in these levels are screened f rom the influence of the other atoms by the valence electron the corresponding bands are filled. The levels in the valence shell are very much influenced by the presence of other atoms and split into bands as shown in Fig. 28.3(a). The 3s and 3p bands have been 

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As we have outlined, a very wide variety of methods are available to calculate the electronic structure of solids. Empirical TB methods (such as discussed in section B3.2.2) are the least expensive, affordmg the... 

Slater, J. C., Handbuch der Physik, Springer, Berlin, 1956, Band XIX, Elektrische Leitungsphanomene I, p. 1. The electronic structure of solids. ... 

One way of experimentally exploring the electronic structure of solids is by means of photoemission spectroscopies such as UPS and X-ray photoelectron spectroscopy (XPS), where photoexcited electrons are analyzed dispersively as a function of their kinetic energy. The electronic structure of the reference material TTF-TCNQ will be extensively discussed in Section 6.1. Figure 1.31 shows the XPS spectra of the S2p core line for (TMTTF)2PF6 (black dots) and BEDT-TTF (grey dots). 

The problem of first-principles calculations of the electronic structure of solid surface is usually formatted as a problem of slabs, that is, consisting of a few layers of atoms. The translational and two-dimensional point group symmetry further reduce the degrees of freedom. Using modern supercomputers, such first-principles calculations for the electronic structure of solid surfaces have produced remarkably reproducible and accurate results as compared with many experimental measurements, especially angle-resolved photoemission and inverse photoemission. 

The first successful first-principle theoretical studies of the electronic structure of solid surfaces were conducted by Appelbaum and Hamann on Na (1972) and A1 (1973). Within a few years, first-principles calculations for a number of important materials, from nearly free-electron metals to f-band metals and semiconductors, were published, as summarized in the first review article by Appelbaum and Hamann (1976). Extensive reviews of the first-principles calculations for metal surfaces (Inglesfeld, 1982) and semiconductors (Lieske, 1984) are published. A current interest is the reconstruction of surfaces. Because of the refinement of the calculation of total energy of surfaces, tiny differences of the energies of different reconstructions can be assessed accurately. As examples, there are the study of bonding and reconstruction of the W(OOl) surface by Singh and Krakauer (1988), and the study of the surface reconstruction of Ag(llO) by Fu and Ho (1989). 

Drickamer, H. G. (1965) The effect of high pressure on the electronic structure of solids. Solid State Phys., 17,1-133. 

In work on the electronic structure of solids, Lowdin[40] pointed out that if the Hamiltonian matrix for a system were a polynomial function of the overlap matrix of the basis, H and S would have the same eigenvectors and the energy eigenvalues would be polynomial functions of the eigenvalues of S. A number of consequences of this sort of relationship are known, but so far as the author is aware, no tests of such an idea have ever been made with realistic H and S matrices. This may be accomplished by examining the commutator, since if... 

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. 

Energy and charge transport in saturated and conjugated polymeric solids represent limiting cases in the applicability of the precepts of band theoretical descriptions of the electronic structure of solids. Discussions of the nature of intrinsic localized electronic states and their consequences to treatments of transport phenomena in such materials comprise an important section of these proceedings. 

In the present contribution, we will examine the fundamentals of such an approach. We first describe some basic notions of the tight-binding method to build the COs of an infinite periodic solid. Then we consider how to analyze the nature of these COs from the viewpoint of orbital interaction by using some one-dimensional (ID) examples. We then introduce the notion of density of states (DOS) and its chemical analysis, which is especially valuable in understanding the structure of complex 3D sohds or in studying surface related phenomena. Later, we introduce the concept of Fermi surface needed to examine the transport properties of metallic systems and consider the different electronic instabilities of metals. Finally, a brief consideration of the more frequently used computational approaches to the electronic structure of solids is presented. 

Density Functional Theory (DFT) has become the method of choice for the study of the electronic structure of solids. Advances in computer technology have made possible the development of DFT-based codes providing a detailed ab initio description of the electronic structme of complex materials. Following the two celebrated papers by Hohenberg and Kohn and Kohn and Sham, a wide variety of approaches have been developed and turned into very efficient computational tools. These approaches differ in the way they represent the density, potential, and Kohn-Sham orbitals. Essentially DFT approaches can be classified in two main groups all electron methods and pseudopotential methods. 

Manipulation of the raw data can provide accurate information on variation of optical constants (in particularly, k, the absorption coefficient) with wavelength. This yields information on the band edge and on the electronic structure of solids (particularly metals and semiconductors) in the valence region... 

Consider, for instance, the electron structure of solids (periodic or aperiodic) and the Schrddinger equation... 

Another standing topic during the last two decades has been to evaluate the electronic structure of solids, surfaces and adsorbates on surfaces. This can be done using standard band structure methods [107] or in more recent years slab codes for studies of surfaces. An alternative and very popular approach has been to model the infinite solid or surface with a finite cluster, where the choice of the form and size of the cluster has been determined by the local geometry. These clusters have in more advanced calculations been embedded in some type of external potential as discussed above. It should be noted that these types of cluster have in general quite different geometries compared with... 

The net interfacial electron flux is the result of many discrete tunneling events between electron levels in phases A and B. Usually, electron tunneling is an elastic process it occurs between an occupied and an empty electron level of the same energy. Hence, the probability of electron tunneling depends on the distribution of the electron levels at both sides of the interface, and on their occupancy. The electronic structure of solids is thus of primary importance for the kinetics of interfacial electron transfer. In analogy, electrochemical electron transfer can be regarded as the result of discrete tunneling events between electron levels in a solid (the elec-... 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

In addition to the thermal bath of nuclear motions, important groups of solids— metals and semiconductors provide continua of electronic states that can dominate the dynamical behavior of adsorbed molecules. For example, the primary relaxation route of an electronically excited molecule positioned near a metal surface is electron and/or energy transfer involving the electronic degrees of freedom in the metal. In this section we briefly outline some concepts from the electronic structure of solids that are needed to understand the interactions of molecules with such environments. 

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