  Using our new post-Newtonian (PN) smoothed particle hydrodynamics (SPH) code,
we have studied numerically the mergers of neutron star binaries with
irrotational initial configurations. Here we describe a new method for
constructing numerically accurate initial conditions for irrotational binary
systems with circular orbits in PN gravity. We then compute the 3D hydrodynamic
evolution of these systems until the two stars have completely merged, and we
determine the corresponding GW signals. We present results for systems with
different binary mass ratios, and for neutron stars represented by polytropes
with $\Gamma=2$ or $\Gamma=3$. Compared to mergers of corotating binaries, we
find that irrotational binary mergers produce similar peak GW luminosities, but
they shed almost no mass at all to large distances. The dependence of the GW
signal on numerical resolution for calculations performed with N>10^5 SPH
particles is extremely weak, and we find excellent agreement between runs
utilizing N=10^5 and N=10^6 SPH particles (the largest SPH calculation ever
performed to study such irrotational binary mergers). We also compute GW energy
spectra based on all calculations reported here and in our previous works. We
find that PN effects lead to clearly identifiable features in the GW energy
spectrum of binary neutron star mergers, which may yield important information
about the nuclear equation of state at extreme densities.
