On calculating the Berry curvature of Bloch electrons using the KKR   method
  We propose and implement a particularly effective method for calculating the Berry curvature arising from adiabatic evolution of Bloch states in wave vector k space. The method exploits a unique feature of the Korringa-Kohn-Rostoker (KKR) approach to solve the Schr\"odinger or Dirac equations. Namely, it is based on the observation that in the KKR method k enters the calculation via the structure constants which depend only on the geometry of the lattice but not the crystal potential. For both the Abelian and non-Abelian Berry curvature we derive an analytic formula whose evaluation does not require any numerical differentiation with respect to k. We present explicit calculations for Al, Cu, Au, and Pt bulk crystals. 
M. Gradhand, D.V. Fedorov, F. Pientka, P. Zahn, I. Mertig, and B.L.   Gy\"orffy
cond-mat.mes-hall cond-mat.mtrl-sci