Pseudopotential perturbation theory

W. Fickett in "Detonation Properties of Condensed Explosives Calculated with an Equation of State Based on Intermole-cular Potentials , LosAlamosScientific-LabRept LA-2712(1962), pp 38-42, reports that pseudopotential theories are obtd by an approach completely different from perturbation theories. The problem of defining a system of detonation products consisting of both solid carbon in some form and a fluid mixt of the remaining product species has been formally rearranged to a single fictitious substance with an extremely complicated compn- temp-dependent potential function , called the pseudopotential. The fictitious substance corresponding to this potential is clearly non-conformal with the components of the mixt... 

Perturbation theory (PT). eDirac-Hartree-Fock (DHF). f Pseudopotentials by Hay and Wadt90. 

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 remarkable conclusion of this argument is that though pseudopotentials can be used to describe semiconductors as well as metals, the pseudopotential perturbation theory which is the essence of the theory of metals is completely inappropriate in semiconductors. Pseudopotenlial perturbation theory is an expansion in which the ratio W/Ey of the pseudopotential to the kinetic energy is treated as small, whereas for covalent solids just the reverse quantity, Ey/W, should be treated as small. The distinction becomes /wimportant if we diagonalize the Hamiltonian matrix to obtain the bands since, for that, we do not need to know which terms are large. Thus the distinction was not essential to the first use of pseudopotentials in solids by Phillips and KIcinman (1959) nor in the more recent application of the Empirical Pseudopotenlial Method u.scd by M. L. Cohen and co-workers. Only in approximate theories, which are the principal subject of this text, must one put terms in the proper order. 

This view runs into difficulties that have only recently been completely resolved. The principal one is that the pseudopotential form factor happens to be very small for this particular diffraction. In Fig. 18-4 is sketched the pseudopotenlial form factor for silicon obtained from the Solid Stale Table the form factor that gives the [220] diffraction is indicated. Because it lies so close to the crossing, it is small and the diffraction is not expected to be strong. Heine and Jones (1969) noted, however, that a second-order diffraction can take an electron across the Jones Zone this could be a virtual diffraction by a lattice wave number of [1 ll]27t/fl followed by a virtual diffraction by [I lT]27c/a. (Virtual diffraction is an expression used to describe terms in perturbation theory it can be helpful but is not essential to the analysis here.) This second-order diffraction would involve the large matrix elements associated with the [11 l]27t/a lattice wave number indicated in Fig. 18-4, and Heine and Jones correctly indicated that these are the dominant matrix elements. 

The electronic structure is reformulated in terms of free electrons and a d resonance in order to relate the band width W, to the resonance width T, and is then reformulated again in terms of iransilion-metal pseudopotential theory, in which the hybridization between the frce-electron states and the d state is treated in perturbation theory, The pseudopotential theory provides both a definition of the d-state radius and a derivation of all interatomic matrix elements and the frce-electron effective mass in terms of it. Thus it provides all of the parameters for the L.CAO theory, as well as a means of direct calcidation of many properties, as was possible in the simple metals. ... 

The expansion in F/( , - tj.) would suggest the possibility of incorporating the effect of the resonance in perturbation theory and thus extending the pseudopotential perturbation theory of simple metals to transition metals. This has in fact been done (Harrison, 1969) and the pseudopotential approach has been extensively developed by Moriarty (1970, I972a,b,c), but the application.s have been largely restricted to the ends of the transition scries where the expansion is clearest. 

In examining how changes in the electron states caused by the pseudopotential change the total energy of the electron gas, it is best not to use the Fermi-Thomas approximation, used in Section 16-F, but to compute the energy of the electrons directly by applying perturbation theory. For a particular electron of wave number k, Eq. (1-14) directly gives... 

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