  We use density functional theory methods to study the electronic structures
of a series of $s-p$ cubic perovskites $ABX_{3}$: the experimentally available
SrBiO$_{3}$, BaBiO$_{3}$, BaSbO$_3$, CsTlF$_{3}$, and CsTlCl$_{3}$, as well as
the hypothetical MgPO$_{3}$, CaAsO$_{3}$, SrSbO$_{3}$, and RaMcO$_3$. We use
tight-binding modeling to calculate the interatomic hopping integrals
$t_{sp\sigma}$ between the $B-s$ and $X-p$ atomic orbitals and charge-transfer
energies $\Delta$, which are the two most important parameters that determine
the low-energy electron and hole states of these systems. Our calculations
elucidate several trends in $t_{sp\sigma}$ and $\Delta$ as one moves across the
periodic table, such as the relativistic energy lowering of the $B-s$ orbital
in heavy $B$ cations leading to strongly negative $\Delta$ values. Our results
are discussed in connection with the general phase diagram for $s - p$ cubic
perovskites proposed in Ref. 26, where the parent superconductors SrBiO$_{3}$
and BaBiO$_{3}$ are found to be in the regime of negative $\Delta$ and large
$t_{sp\sigma}$. Here, we explore this further and search for new materials with
similar parameters, which could lead to the discovery of new superconductors.
Also, some considerations are offered regarding a possible relation between the
physical properties of a given $s - p$ compound (such as its tendency to bond
disproportionate and the maximal achievable superconducting transition
temperature) and its electronic structure.
