  New results recently obtained (\textit{Annals of Physics} (New York)
a.n.~168943) established some non-relativistic ground state solutions for
three-body molecules interacting through a Chern--Simons model. Within this
model, it was argued that Chern--Simons potential should not help improve the
fusion rates by replacing electrons with muons, in the case of particular
muonic molecules. This achievement motivated us to investigate quantitatively
whether or not the Maxwell--Chern--Simons electrodynamics could influence
positively, for example, the probability of having a muon-catalyzed fusion; its
contribution to electronic molecules is also considered in this letter. The
principal factors related to the probability of elementary nuclear fusion are
therefore numerically calculated and compared with their analogs admitting
other forms of interaction like $-1/\rho$ and $\ln (\rho)$. The analysis
carried on here confirms that one should not expect a significant improvement
in nuclear fusion rates in the case of muonic molecules, although,
surprisingly, the same is not true for electronic molecules, compared with
other theoretical predictions. Numerical predictions for the fusion rates for
$ppe$, $pp\mu$, $dde$ and $dd\mu$ molecules are given as well as the predicted
value for the tunneling rate for these molecules.
