Nearly-Free-Electron Perturbation Theory

Here kp denotes a plane wave with wave vector of magnitude kp, and and are contributions to the effective mass from energy dependence and nonlocality. Such effective masses have been calculated by Weaire and others (Ref. 51, pp. 64-69 Ref 75). The total effective mass m is almost always close to unity, which to some extent justifies simple local pseudopotential theories—on the other hand, any property which involves and in some other combination than i V iiot be accurately treated without their inclusion in the theory (Ref 51, pp. 64-69). 

We have already discussed the next order of perturbation theory for local potentials (Section 3.5). For the complications of nonlocality and energy dependence here see Heine and Weaire (Ref 11, pp. 319-331) and Harrison (Ref 39, pp. 259-294). 

Some further remarks are in order on the subject of truncation of the pseudopotential. Most current semiempirical studies involve quite large secular determinants (say 50 x 50) but set f(g) equal to zero for g 2kp. However, a somewhat cruder procedure, that of truncating the basis set at g = 2kp, resulting in a smaller secular determinant, has also been widely used. This procedure may be put on a formal basis by the use of Lowdin perturbation theory, by which a larger secular determinant is reexpressed as a smaller one, with correction terms. For a local pseudopotential the correction terms are given by (Ref 51, pp. 78-83) 

The corrections are indeed small for typical potentials. It is unfortunate that there does not seem to be any such formal procedure for discussion of the errors implicit in the other type of truncation mentioned above. 

The basic idea is to use the known solution of (32) for some k = k as a first approximation for other values of k. 

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How to proceed with these matrix elements will depend upon which property one wishes to estimate. Let us begin by discussing the effect of the pseudopotential as a cause of diffraction by the electrons this leads to the nearly-free-electron approximation. The relation of this description to the description of the electronic structure used for other systems will be seen. We shall then compute the screening of the pseudopotential, which is necessary to obtain correct magnitudes for the form factors, and then use quantum-mechanical perturbation theory to calculate electron scattering by defects and the changes in energy that accompany distortion of the lattice. 

The first solution to this problem was produced phenomenologically by Mooser and Pearson. The solution for A B compounds is reproduced in Figure 9. Similar solutions apply not only to A"B semiconductors and insulators, but also to many intermetallic compounds including transition metals. This work provides the first step toward explaining structural and phase transitions in chemically homologous families of binary crystals. It has made the question of the proper treatment of chemical bonding in crystals susceptible to theoretical analysis, whereas formerly work based on mechanical models (ionic compounds) or quantum mechanical perturbation theory (nearly-free-electron metals) made the same problem appear insoluble. 

The simple metals, whose conduction bands correspond to s and p shells in isolated atoms, include the alkali metals, the divalent metals Be, Mg, Zn, Cd, and Hg, the trivalent metals Al, Ga, In, and Tl, and the tetravalent metals (white) Sn and Pb. Almost all of their properties which are related to electronic band structure are explicable by nearly-free-electron theory using pseudopotentials (Sections 3.5 and 3.6). The extent to which they conform in detail to this generalization varies from one case to another. For all the metals cited simple pseudopotential theory is fairly successful in predicting or fitting Fermi surface properties. This will be evident from a consideration of the comparisons of theoretical and fitted pseudopotential parameters already shown in Figure 12. However, the use of perturbation theory is not very critical in this context [i.e., the contribution of screening to the values of v q) which are of interest is not large]. In other contexts the validity of perturbation theory is more critical, and indeed the use of pseudopotential-perturbation theory is then not always so successful. An example is the calculation of phonon dispersion relations by such methods, which has enjoyed remarkable success for Na, Mg, and j(i2i,i22) jjjjQ difficulties for the heavier metals and those... 

The energy perturbation given in Equation 2.10 technically applies only at the boundary, ki=k2. As one moves away from the boundary, the magnitude of this interaction decreases quadratically with k. This is the same behavior as at the bottom of the free-electron curve. Another way to view this behavior is that near any extremum (maximum or minimum) of an arbitrary function, a power-law expansion of that function is always quadratic. Thus, near enough to any local maximum or minimum of an E(k) diagram, the behavior of an electron will always appear free-electron like. This provides a partial, if circular, justification of the approximations made above. A rigorous justification is provided by quantum mechanical perturbation theory and may be found in most quantum mechanics texts. 

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