  We present a systematic study of the interplay between lattice parameters and
the energy of the optical phonons as well as the antiferromagnetic coupling
strength, $J$, in the high-$T_{\text{c}}$ superconducting cuprate
$R$Ba$_2$Cu$_3$O$_{6+\delta}$ (\mbox{R-123}, $R =$ Y, Dy, Gd, Sm, Nd) with
hole-doping $p$ ($0.00<p \lesssim 0.04$). The energy of the $B_{1g}$-mode at
$\nu_{B1g}\approx~335$ cm$^{-1}$ has been found to relate systematically to the
inverse of the lattice parameter $a$. Our results confirm the temperature
dependent phonon splitting for Nd-123 at low doping, which has been reported
for optimally-doped Nd-123. Surprisingly, $J$ is independent of $a$ for the
first four $R$ families, and a general consistency between
$T_{\text{c}}^{\text{max}}$ and $J$, as suggested in a previous investigation,
could not be confirmed.
