Non-spherical potentials in the band theory of solids

This study is concerned with the problem of calculating electronic band structures of solids from a priori potentials, in particular potentials which are not spherically symmetric about the centre of the Wigner-Seitz cell. Band structure calculations are performed using the Cellular Method, and the adaptation of this method to deal with non-spherical potentials is a major concern of this work.
A review is presented of the work so far published on non-spherical potentials in the OPW, APW and KKR methods, and the ideas to be built on in the present work are considered more carefully. The details of the Cellular Method are expounded, paying particular attention to the stages which require modification in the non-spherical treatment.
The superposition method introduced by Mattheiss, and by which many of the potentials used in band structure calculations are constructed from the appropriate atomic potentials, is extended to include non-spherical terms, and a general expression for the potential is obtained for the f.c.c. lattice. This expression takes the form of an expansion in spherical harmonics correctly orientated with respect to the crystal axes.
We then proceed to consider the non-spherical corrections introduced when the potential used in the Cellular Method is the true potential within the whole circumscribing sphere of the Wigner-Seitz cell, which overlaps the neighbouring cells. These non-spherical corrections to the potential are non-zero in the region between the inscribed and circumscribing spheres of the cell, and therefore affect the radial wave functions over this region - the very region over which their values are required in order to perform the boundary condition fitting of the Cellular Method.
Methods are developed for the treatment of general non-spherical potentials which can be written in terms of a reasonably convergent expansion in spherical harmonics. First of all a rigorous treatment, in v;hich it is found necessary to label the radial functions with two sets of angular momentum quantum numbers is developed. This treatment involves the calculation of many more radial functions than the usual spherical treatment and the form of trial function produced necessitates certain changes and generalisations of the Cellular Method, It is, therefore, important to look for possible approximations which might be employed to facilitate a quicker, and more simple solution to the problem Two such methods are explored.
A suite of computer programs is set up to deal with the calculations necessary to employ the methods developed, and these programs are described in some detail. The methods are then applied to the non-spherical correction arising from the strict division of the potential field into cells mentioned above, using the potential of Chodorow for f.c.c. copper as a basis. Various tests of the methods developed are undertaken. An iterative approach to the problem, much quicker to compute than the rigorous treatment, is shown to give results in excellent agreement with those of the rigorous" treatment for the case considered, and is certainly worthy of further investigation in cases where the non-spherical corrections are larger. Both of these treatments are shown to retain adequate convergence of the cellular expansion.
The corrections due to the newly proposed potential scheme are shown to be often negligible, and never more than 0,008 Rydbergs, for the lower lying band energies at the symmetry points in k-space calculated.