Energy-band equation for a general periodic potential

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Abstract
A new approach—one not based on multiple scattering or the variational principle—to the problem of the band structure for a general (i.e., non-muffin-tin) periodic potential is presented. From the eigenvalue equation, the Bloch function is shown to be expandable as a multipole series around the origin and the coefficients are found to be given as functionals of the Bloch function and the cell potential. By introducing into this functional various representations of the Bloch function (as a superposition in which each individual term is not Bloch periodic), we obtain previously derived band-structure equations (some claimed to be exact). By deriving, however, a new representation for the Bloch function (as an on-shell superposition in which each term is Bloch periodic) and using this representation in the above functional, we obtain a new energy-band equation in which the potential can no longer be separated from the structure. When approximations are introduced, then Korringa-Kohn-Rostoker-type equations (in which the potential is separated from the structure) may be obtained. All the steps of this new approach are illustrated in the soluble case of a constant periodic potential. It is shown, in this case, that only the newly introduced band-structure equation is able to generate the correct eigenvalues.