  We implement the Bogoliubov-de Gennes (BdG) equation in real-space using the
screened Korringa-Kohn-Rostoker (KKR) method. This allows us to solve,
self-consistently, the superconducting state for 3d crystals including
substitutional impurities with a full normal-state DFT band structure. We apply
the theoretical framework to bulk Nb with impurities. Without impurities, Nb
has an anisotropic gap structure with two distinct peaks around the Fermi
level. In the presence of non-magnetic impurities those peaks are broadened due
to the scattering between the two bulk superconducting gaps, however the peaks
remain separated. As a second example of self-consistent real-space solutions
of the BdG equations we examine superconducting clusters embedded within a
non-superconducting bulk metallic host. This allows us to estimate the
coherence length of the superconductor and we show that, within our framework,
the coherence length of the superconductor is related to the inverse of the gap
size, just as in bulk BCS theory.
