Fully relativistic ab initio calculations of the energies of chiral molecules including parity-violating weak interactions

The parity-odd perturbation operator for the inelastic electron-nucleon scattering by weak neutral currents (exchange of virtual $Z^0$ bosons) has been implemented into a fully relativistic four-component Dirac-Hartree-Fock scheme. Dirac-Hartree-Fock electronic structure calculations on ${\mathrm{H}}_2{\mathrm{O}}_2,$ ${\mathrm{H}}_2{\mathrm{S}}_2,$ ${\mathrm{H}}_2{\mathrm{Se}}_2,$ ${\mathrm{H}}_2{\mathrm{Te}}_2,$ and ${\mathrm{H}}_2{\mathrm{Po}}_2$ provides a demonstration of the higher than $Z^5$ scaling of the parity-violating energy shift $(Z$ is the nuclear charge) in chiral molecules. To our knowledge, the calculations for ${\mathrm{H}}_2{\mathrm{Te}}_2$ and ${\mathrm{H}}_2{\mathrm{Po}}_2$ are the first for molecules containing heavy elements from period 5 or 6 of the Periodic Table, and the parity-violating energy shifts are some of the highest reported in any ab initio study. It has been shown that special care is needed in the basis set expansion of the wave function because of the coupling between the large and small components of the Dirac wave function through the ${\gamma }^5$ matrix. Estimates of the remaining errors in the calculations have been given. A comparison with the calculated parity-violating energy shift of ${\mathrm{H}}_2\mathrm{TeO}$ have confirmed the importance of the single-center theorem, which states that the parity-violating energy shift is suppressed in molecules containing only a single heavy atomic center. Due to the close correspondence between parity-violating energy shifts and observable parity-odd properties, our results have important consequences for the current search for an experimental confirmation of parity-odd effects in molecular physics: (i) The experiments should be performed on molecules containing heavy (period 5 or 6) elements. (ii) Molecules with more than one heavy atomic center will be extremely favorable due to the single-center theorem.