  High-precision tests of fundamental symmetries are looking for the parity-
(P), time-reversal- (T) violating electric dipole moment of the electron (eEDM)
as proof of physics beyond the Standard Model. Particularly, in polyatomic
molecules, the complex vibrational and rotational structure gives the
possibility to reach high enhancement of the P, T-odd effects in moderate
electric fields. Additionally, it is possible to increase the statistical
sensitivity by using laser cooling. In this work, we calculate the P, T-odd
electronic structure parameters $W_\mathrm{d}$ and $W_\mathrm{s}$ for the
promising candidates BaCH$_3$ and YbCH$_3$ for the interpretation of future
experiments. We employ high-accuracy relativistic coupled cluster methods and
systematically evaluate the uncertainties of our computational approach.
Compared to other Ba- and Yb-containing molecules, BaCH$_3$ and YbCH$_3$
exhibit larger $W_\mathrm{d}$ and $W_\mathrm{s}$ associated to increased
covalent character of the M--C bond. The calculated values are $3.22\pm 0.11
\times 10^{24}\frac{h\text{Hz}}{e\text{cm}}$ and $13.80\pm 0.35 \times
10^{24}\frac{h\text{Hz}}{e\text{cm}}$ for $W_\mathrm{d}$, and
$8.42\pm0.29$~$h$kHz and $45.35\pm1.15$~$h$kHz for $W_\mathrm{s}$, in BaCH$_3$
and YbCH$_3$, respectively. The robust, accurate, and cost-effective
computational scheme reported in this work makes our results suitable for
extracting the relevant fundamental properties from future measurements and
also can be used to explore other polyatomic molecules sensitive to various
violations of fundamental symmetries.
