Spectroscopy roadmap for Hg-based diatomic and polyatomic molecules toward NSD-PV

Mercury-based polyatomic molecules offer fundamental advantages for symmetry tests: multiple rotational modes and vibrational structure that can be exploited to reduce systematics while maintaining large internal electric fields and strong sensitivity to nuclear-spin-dependent parity violation (NSD-PV) due to the unique nuclear structure of Hg. A key bottleneck is the incomplete excited-state spectroscopy for heavy Hg-containing species, which limits transition selection for state preparation, readout, and (where feasible) optical cycling. We describe our ongoing spectroscopy and modeling across HgF, HgOH, HgNH2, and HgOCH3 that combines broadband searches with high-resolution laser spectroscopy and relativistic electronic-structure calculations to generate an experiment-facing set of level assignments and transition predictions. We highlight candidate electronic manifolds and vibrational modes that optimize photon-scattering efficiency, and we outline how the resulting molecular parameters feed the NSD-PV/anapole interpretation pipeline and motivate future statistical systematics once multiple nuclei are measured.