Diatomic molecules with an energetically low-lying ${}^3\mathrm{\Delta }_1$ state are attractive platforms to detect new physics beyond the Standard Model, such as parity- and time-reversal-violating phenomena. One of the advantages of using a ${}^3\mathrm{\Delta }_1$ state is its tiny $\mathrm{\Lambda }$ splitting due to the coupling between the electronic and rotational angular momenta, which facilitates polarizing the molecules in small external electric fields. Theoretical estimation of the magnitude of the $\mathrm{\Lambda }$ splitting is helpful for planning new experiments. In this study, we present a theoretical model to calculate the $\mathrm{\Lambda }$ splitting. Our model integrates the relativistic four-component wave function and the traditional rotational Hamiltonian based on Hund's case (a). The multireference character of the wave function is taken into account. Our calculations for PtH and ${\mathrm{ThF}}^{+}$ molecules qualitatively agree with experiment. The $\mathrm{\Lambda }$ splitting of ${\mathrm{TaO}}^{+}$ for the rotational ground state is predicted to be around 9 kHz. This tiny splitting can reduce the systematic uncertainty, but in a practical experiment, it may cause depolarization during rotation ramp-up.

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• Molecular spectra
• Molecule trapping & guiding
• Rotational states
• Spin-orbit coupling
• Molecules