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.
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