  Optical control of polyatomic molecules promises new opportunities in
precision metrology, fundamental chemistry, quantum information, and many-body
science. Contemporary experimental and theoretical efforts have mostly focused
on cycling photons via excitation of a single electron localized to an alkaline
earth (group 2)-like metal center. In this manuscript, we consider pathways
towards optical cycling in polyatomic molecules with multi-electron degrees of
freedom, which arise from two or more cycling electrons localized to $p$-block
post-transition metal and metalloid (group 13, 14, and 15) centers. We
characterize the electronic structure and rovibrational branching of several
prototypical candidates using ab initio quantum chemical methods. Despite
increased internal complexity and challenging design parameters, we find
several molecules possessing quasi-closed photon cycling schemes with highly
diagonal, visible and near-infrared transitions. Furthermore, we identify new
heuristics for engineering optically controllable and laser-coolable polyatomic
molecules with multi-electron cycling centers. Our results help elucidate the
interplay between hybridization, repulsion, and ionicity in optically active
species and provide a first step towards using polyatomic molecules with
complex electronic structure as a resource for quantum science and measurement.
