  A new {\it ab initio} theoretical formulation to calculate $Z_{eff}$ and
hence the positron annihilation rates is presented using the on-shell and
half-offshell T-matrix scattering amplitudes without any explicit use of the
scattering wave function. The formulation is exact and universal, as long as
the dynamics of the system can be described by a Lippmann-Schwinger type
equation. It could serve as an effective tool as all the $T-$, $K-$, and
$S-$matrix formulations, yield directly the scattering amplitudes; not the wave
function. We also present a basic numerical test of the formulation.
