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The superconducting (SC) cuprate HgBa$_2$Ca$_2$Cu$_3$O$_8$ (Hg1223) has thehighest $T_{c}^{\rm opt}\simeq 138$ K (the experimental SC transitiontemperature at optimal hole doping) among cuprates at ambient pressure $P_{\rmamb}$. $T_{c}^{\rm opt}$ increases under pressure $P$ and reaches $164$ K at$P_{\rm opt}\simeq 30$ GPa, then decreases with increasing $P>P_{\rm opt}$. Tounderstand the microscopic origin of this dome-like $P$ dependence of$T_{c}^{\rm opt}$, we consider the $ab$ $initio$ low-energy effectiveHamiltonian (LEH) for the antibonding (AB) Cu$3d_{x^2-y^2}$/O$2p_{\sigma}$band. In the AB LEH for cuprates with $N_\ell \leq 2$ laminated CuO$_2$ planesbetween block layers, it was proposed that $T_{c}^{\rm opt}\simeq0.16|t_1|F_{\rm SC}$, where $t_1$ is the nearest neighbor hopping and the SCorder parameter $F_{\rm SC}$ at optimal hole doping mainly depends on$u=U/|t_1|$ ($U$ is the onsite effective Coulomb repulsion): $F_{\rm SC}$ ismaximal at $u_{\rm opt}\simeq 8.5$ and decreases sharply with decreasing $u$for $u<u_{\rm opt}$. In this paper, we show that (I) $|t_1|$ increases with$P$, whereas (II) $u$ decreases with $P$ in the AB LEH of Hg1223. Based on(I,II), the dome-like $P$ dependence of $T_{c}^{\rm opt}$ emerges if we assume(A) Hg1223 with $N_\ell =3$ obeys $T_{c}^{\rm opt}\simeq 0.16|t_1|F_{\rm SC}$,and (B) $u\gtrsim u_{\rm opt}$ at $P_{\rm amb}$ and $u\simeq u_{\rm opt}$ at$P_{\rm opt}$ by including expected corrections to our $ab$ $initio$calculation. The consequence of (A,B) is the following: At $P<P_{\rm opt}$, theincrease in $T_{c}^{\rm opt}$ is accounted for by (I) whereas $F_{\rm SC}$ isinsensitive to (II) and hence to $P$. At $P>P_{\rm opt}$, the decrease in$T_{c}^{\rm opt}$ is accounted for by (II): The rapid decrease in $F_{\rm SC}$with decreasing $u$ for $u<u_{\rm opt}$ dominates over (I). We support (A,B)based on studies of other cuprates.

Dome structure in pressure dependence of superconducting transition  temperature for HgBa$_2$Ca$_2$Cu$_3$O$_8$ -- Studies by $ab$ $initio$  low-energy effective Hamiltonian