  Small polarons (SPs) are known to exist in most important transition metal
oxides (TMOs); however, the nature of small polaron formation remains
enigmatic, and a fundamental understanding of how SPs impact the intrinsic
electronic structure and optical properties of these materials is largely
lacking. In this work, we employ first-principles calculations to investigate
SP formation in Co$_{3}$O$_{4}$, a highly promising material for a wide range
of emerging energy applications, and we resolve the conflicting findings that
have been reported on the electronic structure of the system. We confirm that
the intrinsic band gap of Co$_{3}$O$_{4}$ is 1.6 eV, and we show that the
formation of hole small polarons significantly influences the optical
absorption spectra, leading to a 0.8 eV transition that is often misinterpreted
as the band edge that defines the fundamental gap. In addition, we discuss how
uniaxial strain can be utilized to probe the Jahn-Teller distortion of SP
states and in turn, effect their optical transitions. Beyond Co$_{3}$O$_{4}$,
our study suggests a general roadmap for establishing a first-principles
computational approach that can simultaneously achieve an accurate description
of SP states, electronic band structure and optical transitions of polaronic
magnetic oxides.
