An electron electric dipole moment (eEDM) search using laser-coolable triatomics such as YbOH is one of the most sensitive probes for physics beyond the standard model. The eEDM-induced energy shift is proportional to the polarization ($P$) of the molecule. Similarly to diatomics with an $\mathrm{\Omega }$-doubling structure, it was assumed that for triatomics with an $l$-doubling structure, related to the vibrational angular momentum, $P$ can be easily saturated to almost 100% with a moderate external electric field. We developed a method for the calculation of the properties of triatomic molecules and applied it to a calculation of $P$ of ${}^{174}\mathrm{YbOH}$ in the first excited $v=1$ bending mode. Calculations showed that most of the levels reach less than 50% efficiency. We showed that this fact is related to the Hund's case $b$ coupling scheme of YbOH. As the coupling scheme (for $\mathrm{\Omega }=1/2$ molecules) approaches the $a$ (or $c$) case, polarization increases up to 100%. The results of our calculations should be used for the correct extracting of the eEDM value from YbOH experiments and similar calculations are required for other molecules.

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• Fine & hyperfine structure
• Relativistic & quantum electrodynamic effects in atoms, molecules,& ions
• Molecules
• Quantum correlations, foundations & formalism