  We present a computational study of PbCoO$_3$ at ambient and elevated
pressure. We employ the static and dynamic treatment of local correlation in
form of density functional theory + $U$ (DFT+$U$) and + dynamical mean-field
theory (DFT+DMFT). Our results capture the experimentally observed crystal
structures and identify the unsaturated Pb $6s$ - O $2p$ bonds as the driving
force beyond the complex physics of PbCoO$_3$. We provide a geometrical
analysis of the structural distortions and we discuss their implications, in
particular, the internal doping, which triggers transition between phases with
and without local moments and a site selective Mott transition in the
low-pressure phase.
