On the singularities of the Green's-function method for electronic structure of solids

The matrix elements appearing in the secular determinant of the Green's-function method for electronic structure have simple poles at the free-electron energies $E_n={(\overrightarrow{\mathrm{k}}+{\overrightarrow{\mathrm{K}}}_n)}^2$ with ${\overrightarrow{\mathrm{K}}}_n$ a reciprocal-lattice vector, which arise from the poles of the free-electron Green's function. The resultant singularities in the secular determinant cause difficulties in a practical application of the method, particularly when a band energy lies close to a free-electron energy. In this work we determine the order of the poles of the determinant for any state, since with this information it is a simple matter to eliminate the poles and the attendant difficulties. We show that for a state that transforms as a partner of the irreducible representation ${\Gamma }_j$ of the group of the wave vector $\overrightarrow{\mathrm{k}}$, the order of the pole is given by the number of times ${\Gamma }_j$ is contained in the representation formed by all plane waves with wave vectors $\overrightarrow{\mathrm{k}}+{\overrightarrow{\mathrm{K}}}_n$ corresponding to the same free-electron energy, including those which do not transform into each other under the operations of the group. For the special case of a state with a general $\overrightarrow{\mathrm{k}}$ and with no "accidental" free-electron degeneracy the pole is simple. The general result is obtained by an expansion of the determinant about the pole and also by a simpler means employing Lloyd's formula for the density of states.