  The strain derivatives of $T_c$ along the $a$ and $c$ axes have been
determined for HgBa$_2$CuO$_{4+\delta}$ (Hg1201), the simplest monolayer
cuprate with the highest $T_c$ of all monolayer cuprates ($T_c$ = 97 K at
optimal doping). The underdoped compound with the initial $T_c$ of 65 K has
been studied as a function of pressure up to 20 GPa by magnetic susceptibility
and X-ray diffraction (XRD). The observed linear increase in $T_c$ with
pressure is the same as previously been found for the optimally-doped compound.
The above results have enabled the investigation of the origins of the
significantly different $T_c$ values of optimally doped Hg1201 and the
well-studied compound La$_{2-x}$Sr$_{x}$CuO$_{4}$ (LSCO), the latter value of
$T_c$ = 40 K being only about 40% of the former. Hg1201 can have almost
identical CuO$_6$ octahedra as LSCO if specifically strained. When the apical
and in-plane CuO$_2$ distances are the same for the two compounds, a large
discrepancy in their $T_c$ remains. Differences in crystal structures and
interactions involving the Hg-O charge reservoir layers of Hg1201 may be
responsible for the different $T_c$ values exhibited by the two compounds.
