Self-Consistent Band-Structure Problem

We now specify the external potential included by the last term of (7.6) to be the Coulomb attraction 

In the atomic-sphere approximation the electron density is spherically symmetric inside spheres of radius S, giving 

the integrals extend over the atomic sphere of radius S, and we defined the Hartree potential vH(r) by 

The minimalisation of (7.26) results in a one-electron Schrodinger equation of the form (7.10), valid inside each atomic sphere and with an effective one-electron potential given by 

Matching the solutions of (7.10,28) from sphere to sphere finally gives the electronic energy-band structure of the crystal in question. 

---

This volume proposes to describe one particular method by which the self-consistent electronic-structure problem may be solved in a highly efficient manner. Although the technique under consideration, the Linear Muffin-Tin Orbital (LMTO) method, is quite general, we shall restrict ourselves to the case of crystalline solids. That is, it will be shown how one may perform self-consistent band-structure calculations for infinite crystals, and apply the results to estimate ground-state properties of real materials. 

Fourier Transform and Discrete Variational Method Approach to the Self-Consistent Solution of the Electronic Band Structure Problem within the Local Density Formalism. 

The Hartree approximation is summarized in Figure 2 in schematic form. The heavy arrow denotes the basic band structure problem as stated in the previous section. The p H arrow in the self-consistent loop denotes, in the unrefined Hartree method, Poisson s equation, by which the charge density p is converted into an electrostatic potential which is to be incorporated in the Hamiltonian. With... 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

Early band structure calculations for the actinide metals were made both with and without relativistic effects. As explained above, at least the mass velocity and Darwin shifts should be included to produce a relativistic band structure. For this reason we shall discuss only the relativistic calculations. There were some difficulties with the f-band structure in these studies caused by the f-asymptote problem , which have since been elegantly solved by linear methods . Nevertheless the non-self-consistent RAPW calculations for Th through Bk indicated some interesting trends that have also been found in more recent self-consistent calculations ... 

As for an appropriate band theory for the localized 4f-electron system, an attractive approach based on the p-f mixing model was proposed, and was plied to CeSb. A future problem is to refine the approach so as to carry out quantitative calculations in a self-consistent way. The anomalously large enhancement factors for the cyclotron effective masses and the y values observed in the Ce compounds cannot be explained by band structure alone. Quantitative analysis of the mass enhancement factor is a problem challenging to many-body theory. There is still much room for improvement for a complete understanding of the electronic structures of lanthanide compoimds. 

---