Coulomb t-matrix effective interaction

One century ago Rutherford treated the Coulomb scattering classically and obtained the well known Rutherford scattering formula. Quantum mechanical derivation some time later obtained the Coulomb scattering amplitude giving rise to the same Rutherford scattering cross section. Surprising though the classical and quantum mechanical derivations agree remarkably. It is even more surprising that even after the lapse of the century there is no derivation of the t-matrix effective interaction for the Coulomb scattering. The wide range of applicability and utility of Coulomb scattering tC-matrix effective interaction needs no introduction. The Coulomb t-matrix using the well known quantum mechanical scattering amplitude, fc(θ) is derived where θ is the angle of scattering in the center of mass system