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We perform first-principles calculation to study the electronic structure ofHgBa$_{2}$Ca$_{n-1}$Cu$_{n}$O$_{2n+2+x}$ copper oxides up to $n = 6$ for theundoped parent compound $(x = 0)$ and up to $n = 3$ for the doped compound $(x> 0)$ by means of the SCAN meta-GGA density functional. Our calculationspredict an antiferromagnetic insulator ground state for the parent compoundswith an energy gap that decreases with the number of CuO$_{2}$ planes. Wereport structural, electronic and magnetic order evolution with $x$ which agreewith experiments. We find an enhanced density of states at Fermi level at $x\approx 0.25$ for the single-layered compound manifesting in a peak of theSommerfeld parameter, which recently has been discussed as a possible signatureof quantum criticality generic to all cuprates.

First-principles electronic structure investigation of HgBa$_{2}$Ca$_{n-1}$Cu$_{n}$O$_{2n+2+x}$ with the SCAN density functional