Extraction of nuclear induced Lambda-doubling effects from velocity-modulation spectroscopy of HfF⁺

This study analyzes high-resolution velocity-modulation spectroscopic data from the JILA eEDM group to examine nuclear-induced Λ-doubling effects in the ¹Π₁ electronic state of ¹⁷⁷HfF⁺ and ¹⁷⁹HfF⁺ ions. In contrast to spin-rotation-induced Λ-doubling, which enables molecular polarization under relatively small external electric fields (<10 V/cm), nuclear-induced effects in heavy-metal molecules with nonzero nuclear spin may introduce complexities that could hinder effective polarization. This analysis provides a comparison with quantum chemistry calculations, suggesting Λ-doubling in the ¹Π₁ state to be on the order of several hundred MHz. Although the Λ-doublet splitting in the EDM-sensitive ³Δ₁ state, predicted to be a few MHz, remains difficult to probe, this work provides a useful benchmark for theoretical models of the ¹Π₁ state, offering insights into their reliability for the ³Δ₁ state. The methods developed here may be applied to other molecules, such as ²²⁷ThF⁺, ²²⁹ThF⁺, and ¹⁸¹TaO⁺, which are of interest for studying CP-violating effects through nuclear Schiff moments or magnetic quadrupole moments.