Electron-band theory

Switendick was the first to apply modem electronic band theory to metal hydrides [5]. He compared the measured density of electronic states with theoretical results derived from energy band calculations in binary and pseudo-binary systems. Recently, the band structures of intermetallic hydrides including LaNi5Ht and FeTiH v have been summarized in a review article by Gupta and Schlapbach [6], All exhibit certain common features upon the absorption of hydrogen and formation of a distinct hydride phase. They are ... 

Chemical bonds are defined by their frontier orbitals. That is, by the highest molecular orbital that is occupied by electrons (HOMO), and the lowest unoccupied molecular orbital (LUMO). These are analogous with the top of the valence band and the bottom of the conduction band in electron band theory. However, since kinks are localized and non-periodic, band theory is not appropriate for this discussion. 

Insulators are not amenable to treatment of catalytic activity in terms of electronic band theory. 

Both the LCAO and NFE methods are complementary approaches to one-electron band theory, in which electrons are allowed to move independently of one another, through an averaged potential generated by all the other electrons. The true Hamiltonian is a function of the position of all the electrons in the solid and contains terms for all the interactions between these electrons, that is, all of the electron-electron Coulombic repulsions. Electronic motion is correlated the electrons tend to stay away from one another because of Coulombic repulsion. 

The archetypal examples are the 3d transition metal monoxides NiO, CoO, FeO, MnO, VO, and TiO. AU of these oxides possess the rock-salt stmcture (which makes both cation-cation and cation-anion-cation overlap important). One-electron band theory correctly predicts the metallic behavior observed in TiO, which is expected of a... 

CO. For that matter, in regards to predicting the type of electrical behavior, one has to be careful not to place excessive credence on actual electronic structure calculations that invoke the independent electron approximation. One-electron band theory predicts metallic behavior in all of the transition metal monoxides, although it is only observed in the case of TiO The other oxides, NiO, CoO, MnO, FeO, and VO, are aU insulating, despite the fact that the Fermi level falls in a partially hUed band. In the insulating phases, the Coulomb interaction energy is over 4 eV whereas the bandwidths have been found to be approximately 3 eV, that is, U > W. 

Mott originally considered an array of monovalent metal ions on a lattice, in which the interatomic distance, d, may be varied. Very small interatomic separations correspond to the condensed crystalline phase. Because the free-electron-Uke bands are half-filled in the case of ions with a single valence electron, one-electron band theory predicts metallic behavior. However, it predicts that the array will be metallic, regardless of the interatomic separation. Clearly, this can t be true given that, in the opposite extreme, isolated atoms are electrically insulating. 

A metallic state predicted by one-electron band theory is not stable when its Fermi surface is nested, and becomes susceptible to a metal-to-insulator transition under a suitable perturbation. We now examine the nature of the nonmetallic states that are derived from a normal metallic state upon mixing its occupied and unoccupied band levels. For simplicity, consider the 2D representation of the nested Fermi surface shown in (100), where the vector q is one of many possible nesting vectors. The occupied and unoccupied wave vectors are denoted by k and k, respectively. Each unit cell will be assumed to contain one AO x Suppose we choose the k and k values to satisfy the relationship... 

Fig. 22. L3 palladium edge of Pd metal (dotted line) compared with one-electron band theory (solid line) taking account of the partial (1 = 2) local density of states, of the inelastic mean free path and of the core-hole lifetime. The dashed line shows the total density of states of palladium metal, which is quite different from the absorption spectrum. The zero of the energy scale is fixed at the Fermi energy...

We now need to know how the probability of finding an electron of specified energy varies across the permitted band. In the first and simplest version of the Electron Band Theory, electrons were assumed to move in a field of uniform positive potential (i.e. ion cores were neglected), and mutual electrostatic repulsion was ignored. Application of the Schrodinger equation and Fermi-Dirac statistics leads to the conclusion that a collection of N electrons at the absolute zero occupies the N/2 lowest levels, those at the maximum being said to be at the Fermi surface Ef. 

It is seen that only copper has a complete d band, and it may be that this is the reason for the catalytic inactivity of very pure copper, and for the very marked promotion by such small traces of nickel. Data on the catalytic activity of pure copper in a variety of other reactions is required before this point can be decided. It is also noted that cobalt, which on the electron band theory might be expected to be the most active metal in fact is (see Table III). ... 

The recent experimental confirmation of the existence of one-dimensional metallic systems has led to a rapid increase in the experimental and theoretical study of these conducting systems. The objective of this section is to acquaint the reader with the physical basis of the concepts currently being used to explain the experimental results. Emphasis is given to the development of one electron band theory because of its central importance in the description of metals and understanding the effects of lattice distortion (Peierls transition), electron correlation, disorder potentials, and interruptions in the strands. It... 

Fig. 16. The Mott metal-insulator transition as a function of separation between lattice sites, a. Curve A is the conductivity versus the inverse of the lattice spacing predicted by Mott. Curve B is conductivity versus the inverse of the lattice spacing predicted by one electron band theory, assuming a finite mean free path for electrons in the metallic phase.

According to the developed classification (see Chap. 1), this film system belongs to the category of the ion-semiconductor ones with very low total conductivity— that is, to dielectric type of systems. Its low conductivity constitutes of electronic (semiconductive) and ionic (migrative) parts. The electronic band theory prescribes the exponential temperature dependence of the electronic conductivity. Then, the electronic current can be written as... 

Leon Nicolas Brillouin (1889-1969), French physicist, professor at the Sortionne and College de France in Paris, after 1941 in the USA at the University of Madison, Columbia University, Harvard University. His contributions included quantum mechanics and solid stEite theory (he is one of the fbuitders of electronic band theory). 

For a meaningful discussion of electronic factors in catalysis it is necessary to briefly review the nature of chemisorption bonds. Two theories of the metallic state have been accepted, the electron band theory and the valence bond theory. Both theories recognize the existence of two separate functions for valence electrons in metals one function is to bind the atoms together and the other is to account for magnetic and conductive properties. In the electron band theory, as particularly applied to the transition metals, the s-electron energy band is broad with a low maximum... 

Let the local functions Wj f r) = W<(r — q), (i = 1, 2,..., n ) be centered at point q of the direct lattice and span the space of the irrep / of the site-symmetry group Gq C G with matrices d > gq) and characters X H9q) 9q Gq) The nature of these functions depends on the physical problem under consideration. In the electron-band theory of crystals lF) (r — qA) are atomic functions of atom A. In phonon spectroscopy applications — qA) mean the components of atomic displacements of... 

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