This work presents calculations of the parity non-conserving (PNC) electric dipole transitions 6s + 7s in Cs and of the 6p,,, + 7p,,, and 6p,,, + 6p,,, transitions in T1, made possible by the weak interaction between electrons and nucleons. In addition to effects that can be accounted for in an independent-particle description, we have calculated those second-order correlation effects that can be described as modifications of the valence orbitals to approximate Brueckner orbitals, including terms where the PNC correction occurs on intermediate orbitals, as well as on external orbitals. Similar calculations were performed for the electric dipole moment ( EDM) of the ground states of Cs and TI, expressed in terms of a possible electron EDM, which, if found, implies simultaneous violation of symmetry under parity and time reversal.