  While the intrinsic anomalous Hall conductivity is normally written in terms
of an integral of the electronic Berry curvature over the occupied portions of
the Brillouin zone, Haldane has recently pointed out that this quantity (or
more precisely, its ``non-quantized part'') may alternatively be expressed as a
Fermi-surface property. Here we present an {\it ab-initio} approach for
computing the anomalous Hall conductivity that takes advantage of this
observation by converting the integral over the Fermi sea into a more efficient
integral on the Fermi surface only. First, a conventional electronic-structure
calculation is performed with spin-orbit interaction included.
Maximally-localized Wannier functions are then constructed by a post-processing
step in order to convert the {\it ab-initio} electronic structure around the
Fermi level into a tight-binding-like form. Working in the Wannier
representation, the Brillouin zone is sampled on a large number of equally
spaced parallel slices oriented normal to the total magnetization. On each
slice, we find the intersections of the Fermi-surface sheets with the slice by
standard contour methods, organize these into a set of closed loops, and
compute the Berry phases of the Bloch states as they are transported around
these loops. The anomalous Hall conductivity is proportional to the sum of the
Berry phases of all the loops on all the slices. Illustrative calculations are
performed for Fe, Co and Ni.
