  Construction of astrophysically realistic initial data remains a central
problem when modelling the merger and eventual coalescence of binary black
holes in numerical relativity. The objective of this paper is to provide
astrophysically realistic freely specifiable initial data for binary black hole
systems in numerical relativity, which are in agreement with post-Newtonian
results. Following the approach taken by Blanchet, we propose a particular
solution to the time-asymmetric constraint equations, which represent a system
of two moving black holes, in the form of the standard conformal decomposition
of the spatial metric and the extrinsic curvature. The solution for the spatial
metric is given in symmetric tracefree form, as well as in Dirac coordinates.
We show that the solution differs from the usual post-Newtonian metric up to
the 2PN order by a coordinate transformation. In addition, the solutions,
defined at every point of space, differ at second post-Newtonian order from the
exact, conformally flat, Bowen-York solution of the constraints.
