$\mathcal{CP}$-violation sensitivity of closed-shell radium-containing polyatomic molecular ions

Closed-shell atoms and molecules such as Hg or TlF provide some of the best low-energy tests of hadronic $\mathcal{CP}$-violation which is considered to be a necessary ingredient to explain the observed excess of matter over antimatter in our universe. $\mathcal{CP}$-violation is, however, expected to be strongly enhanced in octupole deformed nuclei such as $^{225}$Ra. Recently, closed-shell radium-containing symmetric-top molecular ions were cooled sympathetically in a Coulomb crystal [M. Fan et al., Phys. Rev. Lett. 126, 023002 (2021)] and shown to be well-suited for precision spectroscopy in the search for fundamental physics [P. Yu and N. R. Hutzler, Phys. Rev. Lett. 126, 023003 (2021)]. In closed-shell molecules hadronic $\mathcal{CP}$-violation contributes to a net electric dipole moment (EDM) that violates parity and time-reversal symmetry ($\mathcal{P,T}$), which is the target of measurements. To interpret experiments, it is indispensable to know the electronic structure enhancement parameters for the various sources of $\mathcal{P,T}$-violation which contribute to the net $\mathcal{P,T}$-odd EDM. In this paper we employ relativistic Hartree--Fock and density functional theory calculations to determine relevant parameters for interpretation of possible EDM measurements in RaOCH$_3^+$, RaSH$^+$, RaCH$_3^+$, RaCN$^+$, and RaNC$^+$ and perform accurate relativistic coupled cluster calculations of the Schiff moment enhancement in RaSH$^+$ to gauge the quality of the density functional theory approach. Finally, we project to bounds on various fundamental $\mathcal{P,T}$-odd parameters that could be achievable from an experiment with RaOCH$_3^+$ in the near future and asses the complementarity of this experiment to experiments with Hg and TlF.