The superconducting (SC) cuprate HgBa2Ca2Cu3O8 (Hg1223) has the highest SC transition temperature Tc among cuprates at ambient pressure Pamb, namely, Tcoptâ138 K experimentally at the optimal hole doping concentration. Tcopt further increases under pressure P and reaches 164 K at optimal pressure Poptâ30 GPa, then Tcopt decreases with increasing P>Popt generating a dome structure [Gao , ]. This nontrivial and nonmonotonic P dependence of Tcopt calls for a theoretical understanding and mechanism. To answer this open question, we consider the low-energy effective Hamiltonian (LEH) for the antibonding (AB) Cu3dx2ây2/O2pÏ band derived generally for the cuprates. In the AB LEH for cuprates with Nââ€2 laminated CuO2 planes between block layers, it was proposed that Tcopt is determined by a universal scaling Tcoptâ0.16|t1|FSC [Schmid , ], where t1 is the nearest-neighbor hopping, and the SC order parameter at optimal hole doping FSC mainly depends on the ratio u=U/|t1| where U is the onsite effective Coulomb repulsion: The u dependence of FSC has a peak at uoptâ8.5 and a steep decrease with decreasing u in the region u<uopt irrespective of materials dependent on other parameters. In this paper, we show that |t1| increases with P, whereas u decreases with P in the Hamiltonian of Hg1223. Based on these facts, we show that the domelike P dependence of Tcopt can emerge at least qualitatively if we assume Hg1223 with Nâ=3 follows the same universal scaling for Tcopt, and Hg1223 is located at the slightly strong coupling region uâłuopt at Pamb and uâuopt at Popt by taking account of expected corrections to our calculation. The consequence of these assumptions is the following: With increasing P within the range P<Popt, the increase in Tcopt is accounted for by the increase in |t1|, whereas FSC is insensitive to the decrease in u around âuopt and hence to P as well. At P>Popt, the decrease in Tcopt is accounted for by the decrease in u below uopt, which causes a rapid decrease in FSC dominating over the increase in |t1|. We further argue the appropriateness of these assumptions based on the insight from studies on other cuprate compounds in the literature. In addition, we discuss the dependencies of u and |t1| on each crystal parameter (CP), which provides hints for design of even higher Tcopt materials. Published by the American Physical Society 2024