  We present a formulation of the so-called Fermi sea contribution to the
conductivity tensor of spin-polarized random alloys within the fully
relativistic tight-binding linear muffin-tin-orbital (TB-LMTO) method and the
coherent potential approximation (CPA). We show that the configuration
averaging of this contribution leads to the CPA-vertex corrections that are
solely due to the energy dependence of the average single-particle propagators.
Moreover, we prove that this contribution is indispensable for the invariance
of the anomalous Hall conductivities with respect to the particular LMTO
representation used in numerical implementation. Ab initio calculations for
cubic ferromagnetic 3d transition metals (Fe, Co, Ni) and their random binary
alloys (Ni-Fe, Fe-Si) indicate that the Fermi sea term is small against the
dominating Fermi surface term. However, for more complicated structures and
systems, such as hexagonal cobalt and selected ordered and disordered Co-based
Heusler alloys, the Fermi sea term plays a significant role in the quantitative
theory of the anomalous Hall effect.
