Eigenvalue equation for a general periodic potential and its multipole expansion solution

It is shown that the Bloch function, as an element of the Hilbert space spanned by Bloch-periodic planes waves, can be also represented as an on-shell superposition of Bloch-periodic orbitals which, however, exist in a distribution sense. By using this result and the multipole expansion of the Bloch function at the origin, it is further shown that the integral eigenvalue equation of the Bloch function for a general periodic potential is equivalent (both necessary and sufficient) to an algebraic system of homogeneous linear equations for the coefficients of the multipole expansion of the Bloch function at the origin, akin to the much simpler case of a finite-range potential. In contrast to the Korringa-Kohn-Rostoker (KKR) equation, the contribution of the cell potential (whether of the muffin-tin or general form) introduces a supplementary structure dependence. However, the separation between structure and potential, typical for the KKR equation, can be restored by introducing various approximations.