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The current constraint on the electron electric dipole moment ($e$EDM),$|d_e|<4.1\times 10^{-30}$ ${e {\cdotp} {\rm cm}}$ (90\% confidence), wasrecently established using the trapped $^{180}$Hf$^{19}$F$^+$ molecular ions inthe $J=1$ rotational level of its $ ^3\Delta_1$ electronic state [T. S. Roussy,L. Caldwell, T. Wright, et al., arxiv:2212.11841]. The extensive experimentalstudy of the HfF$^+$ cation provides detailed spectroscopy of the$\Omega-$doublet levels in the external rotating electric and magnetic fields.We showed that previously developed theoretical approaches can fully reproducethe latest experimental data. Their justification from the first principles isvery important for the examination of both modern molecular theory and possiblesystematic uncertainties in the interpretation of the experimental dataobtained with high accuracy.

Revisited $\mathcal{T}$, $\mathcal{P}$-odd spin-rotational Hamiltonian of HfF$^+$ for precise $e$EDM measurements