  We study beryllium dihydride (BeH$_2$) and acetylene (C$_2$H$_2$) molecules
using real-space diffusion Monte Carlo (DMC) method. The molecules serve as
perhaps the simplest prototypes that illustrate the difficulties with biases in
the fixed-node DMC calculations that might appear with the use of effective
core potentials (ECPs) or other nonlocal operators. This is especially relevant
for the recently introduced correlation consistent ECPs (ccECPs) for $2s2p$
elements. Corresponding ccECPs exhibit deeper potential functions due to higher
fidelity to all-electron counterparts, which could lead to larger local energy
fluctuations. We point out that the difficulties stem from issues that are
straightforward to address by upgrades of basis sets, use of T-moves for
nonlocal terms, inclusion of a few configurations into the trial function and
similar. The resulting accuracy corresponds to the ccECP target (chemical
accuracy) and it is in consistent agreement with independent correlated
calculations. Further possibilities for upgrading the reliability of the DMC
algorithm and considerations for better adapted and more robust Jastrow factors
are discussed as well.
