Parity-mixed coupled-cluster formalism for computing parity-violating amplitudes

We formulate a parity-mixed coupled-cluster (PM-CC) approach for high-precision calculations of parity non-conserving amplitudes in mono-valent atoms. Compared to the conventional formalism which uses parity-proper (PP) one-electron orbitals, the PM-CC method is built using parity-mixed (PM) orbitals. The PM orbitals are obtained by solving the Dirac-Hartree-Fock equation with the electron-nucleus electroweak interaction included (PM-DHF). There are several advantages to such a PM-CC formulation: (i) reduced role of correlations, as for the most experimentally-accurate to date ${}^{133}{\rm Cs}\,$ $6S_{1/2}-7S_{1/2}$ transition, the PM-DHF result is only 3% away from the accurate many-body value, while the conventional DHF result is off by 18%; (ii) avoidance of directly summing over intermediate states in expressions for parity non-conserving amplitudes which reduces theoretical uncertainties associated with highly-excited and core-excited intermediate states, and (iii) relatively straightforward upgrade of existing and well-tested large-scale PP-CC codes. We reformulate the CC method in terms of the PM-DHF basis and demonstrate that the cluster amplitudes are complex numbers with opposite parity real and imaginary parts. We then use this fact to map out a strategy through which the new PM-CC scheme may be implemented.