Energy-Band Methods

As shown by Ruderman and Kittel (77) and Bloembergen and Rowland (78), Aij in a solid is dependent on the nature of the energy bands in the solid. For metals A is proportional to the product of the square of the electron density of Fermi surface electrons at the nucleus and the effective mass, and decreases as the inverse cube of the internuclear distance. Insulators have been treated by the energy band method (78) and by a molecular method (79) where each atom is considered to be bonded to its nearest neighbors. Unfortunately, both of these methods involve approximations in the evaluation of An which are quite crude at present. 

Kasowski, R. V. 1982. Self-consistent extended-muffin-tin orbital energy-band method Application to LiC6.Phys. Rev. B 25 4189-4195. 

Other methods for detennining the energy band structure include cellular methods. Green fiinction approaches and augmented plane waves [2, 3]. The choice of which method to use is often dictated by die particular system of interest. Details in applying these methods to condensed matter phases can be found elsewhere (see section B3.2). 

Many phenomena in solid-state physics can be understood by resort to energy band calculations. Conductivity trends, photoemission spectra, and optical properties can all be understood by examining the quantum states or energy bands of solids. In addition, electronic structure methods can be used to extract a wide variety of properties such as structural energies, mechanical properties and thennodynamic properties. 

It is possible to identify particular spectral features in the modulated reflectivity spectra to band structure features. For example, in a direct band gap the joint density of states must resemble that of critical point. One of the first applications of the empirical pseudopotential method was to calculate reflectivity spectra for a given energy band. Differences between the calculated and measured reflectivity spectra could be assigned to errors in the energy band... 

Wang C S and Callaway J 1978 BNDPKG. A package of programs for the calculation of electronic energy bands by the LCGO method Comput. Phys. Commun. 14 327... 

Marous P M 1967 Variationai methods in the oomputation of energy bands Int. J. Quantum Chem. 1 S 567... 

Simply doing electronic structure computations at the M, K, X, and T points in the Brillouin zone is not necessarily sufficient to yield a band gap. This is because the minimum and maximum energies reached by any given energy band sometimes fall between these points. Such limited calculations are sometimes done when the computational method is very CPU-intensive. For example, this type of spot check might be done at a high level of theory to determine whether complete calculations are necessary at that level. 

A unique feature of our method is that we fit the above Hamiltonian simultaneously to the energy bands and the total energy of a given matei.u.1. We write the total energy as follows ... 

Figure 9.11 Several minimum energy paths for O2 dissociation on PtsCo) 11) (labeled by respective initial states), generating ML of atomic O, compared with equilibrium and 2% compressed Pt(lll). The points on each path are the images or states used to discretize the path with the climbing-image nudged elastic band method. The zero of the energy axis corresponds to an O2 molecule and the respective clean surfaces at infinite separation. The points located on the right vertical axis represent atomic O at ML. (Reproduced with permission from Xu et al. [2004].)...

Henkelman G, Jdnsson H. 2000. Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. J Chem Phys 113 9978-9985. 

Henkelman G, Uberuaga BP, Jonsson H. 2000. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J Chem Phys 113 9901-9904. 

Hamada, N. and Ohnishi, S. (1986) Self-interaction correction to the local-density approximation in the calculation of the energy band gaps of semiconductors based on the full-potential linearized augmented-plane-wave method, Phys. Rev., B34,9042-9044. 

The empirical approach [7] was by far the most fruitful first attempt. The idea was to fit a few Fourier coefficients or form factors of the potential. This approach assumed that the pseudopotential could be represented accurately with around three Fourier form factors for each element and that the potential contained both the electron-core and electron-electron interactions. The form factors were generally fit to optical properties. This approach, called the Empirical Pseudopotential Method (EPM), gave [7] extremely accurate energy band structures and wave functions, and applications were made to a large number of solids, especially semiconductors. [8] In fact, it is probably fair to say that the electronic band structure problem and optical properties in the visible and UV for the standard semiconductors was solved in the 1960s and 1970s by the EPM. Before the EPM, even the electronic structure of Si, which was and is the prototype semiconductor, was only partially known. 

This book systematically summarizes the researches on electrochemistry of sulphide flotation in our group. The various electrochemical measurements, especially electrochemical corrosive method, electrochemical equilibrium calculations, surface analysis and semiconductor energy band theory, practically, molecular orbital theory, have been used in our studies and introduced in this book. The collectorless and collector-induced flotation behavior of sulphide minerals and the mechanism in various flotation systems have been discussed. The electrochemical corrosive mechanism, mechano-electrochemical behavior and the molecular orbital approach of flotation of sulphide minerals will provide much new information to the researchers in this area. The example of electrochemical flotation separation of sulphide ores listed in this book will demonstrate the good future of flotation electrochemistry of sulphide minerals in industrial applications. 

Figure 6.7 Typical outcome if the images from Fig. 6.6 are adjusted using the elastic band method with spring constants that are too small. The curve shows the energy profile for the true MEP between the two energy minima.

In the photoemission spectroscopic method, it is customary to distinguish between a core level region, which is probed in XPS (see Fig. 2), and which contains the response coming from the bound or core levels, and a valence band region, which is explored by both XPS and UPS, and which contains the response coming from the outer electrons in a solid, those of the ground state energy bands. 

In most atomic programs (5) is actually solved self-consistently either in a local potential or by the relativistic Hartree-Fock method. There is, however, an important time-saving device that is often used in energy band calculations for actinides where the same radial Eq. (5) must be solved If (5.a) is substituted into (5.b) a single second order differential equation for the major component is obtained... 

Self-consistent energy band calculations have now been made through the LMTO method for all of the NaCl-type actinide pnictides and chalcogenides . The equation of state is derived quite naturally from these calculations through the pressure formula extended to the case of compounds . The theoretical lattice parameter is then given by the condition of zero pressure. 

H. Jonsson, G. Mills and K. W. Jaeobsen, Nudged Elastic Band Method for Finding Minimum Energy Paths of Transitions , in Classical and Quantum Dynamics in Condensed Phase Simulations , ed. B. J. Berne. G. Ciccotti andD. F. Coker, page 385 (World Scientific, 1998). 

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