General principles of band theory

The simple hard-wall boundary condition, eqn (5.1), does yield the correct ordering of the energy levels as shown in Fig. 5.1. The only exceptions are the 3s and lh states, which have their sequence reversed compared to that of the self-consistent jellium predictions. Experimentally, the most frequently occurring sodium clusters are indeed Na8 and Na20 as expected from their special stability in Fig. 5.3. 

The variation in equilibrium bulk properties between one sp-valent metal and the next cannot be understood within the jellium model, since it has obscured the chemical behaviour of the elements by smearing out the ion 

We note that in one dimension na is a direct lattice vector, whereas m(2n/a) is a reciprocal lattice vector. Their product is an integral multiple of 2n. 

These four equations, eqs (5.20)-(5.23), have a solution only if the determinant of the coefficients of , , C, and D vanishes. After non-trivial determinantal manipulation we find 

This equation may be simplified by considering the limit in which the barrier thickness becomes increasingly thin (i.e. t 0) but the barrier height becomes increasingly high (i.e. V0 - oo) in such a way that the area under the barrier remains constant, that is 

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The theoretical prediction of the optical absorption profile of a solid using first-principles methods has produced results in reasonable agreement with experiment for a variety of systems [2-4], For example, several ionic crystals were studied extensively, generally using the Hartree-Fock one-electron approximation [5], through the extreme-ultraviolet. Lithium fluoride was the focus of a particularly detailed comparison [6-8], providing excellent confirmation of the applicability of the band theory of solids for optical absorption. 

Althou, in principle, the general theory is superior to the band theory, the appropriate techniques for its application are not yet developed sufficiently well and a unified approach to a quantitative description of the structures and the physical properties of crystals is still lacking. The less generally valid band theory can at present give clearer and more convincing explanations of changes in the physical properties of crystals caused by variations in the temperature, pressure, magnetic and electric fields intensities, impurity concentrations, etc. However, many problems encoimtered in the study of chemical bonds in crystals cannot be considered within the framework of the standard band theory. They include, for example, determination of the elastic, thermal, and thermodynamic properties of solids, as well as the structure and properties of liquid and amorphous semiconductors. 

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