  We perform first-principles calculation to study the electronic structure of
HgBa$_{2}$Ca$_{n-1}$Cu$_{n}$O$_{2n+2+x}$ copper oxides up to $n = 6$ for the
undoped 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 calculations
predict an antiferromagnetic insulator ground state for the parent compounds
with an energy gap that decreases with the number of CuO$_{2}$ planes. We
report structural, electronic and magnetic order evolution with $x$ which agree
with 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 the
Sommerfeld parameter, which recently has been discussed as a possible signature
of quantum criticality generic to all cuprates.
