A new ab initio method of calculating $Z_{eff}$ and positron   annihilation rates using coupled-channel T-matrix amplitudes
  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. 
P K Biswas