  Ca$_2$FeMnO$_6$ (CFMO) double perovskite was studied using first principles
density functional theory and tight-binding (TB) Hamiltonian modeling using
extended Hubbard model. We have shown by electronic structure analysis that
charge- and magnetic-ordering are driven by charge disproportionation at low
temperature caused by partial localization of O-$2p$ ligand holes at alternate
Fe sites that creates Jahn-Teller distortion, which leads to metal to insulator
transition (MIT) in CFMO. Our results suggests MIT was triggered by negative
charge-transfer energy of self-hole doping, responsible for symmetry lowering
transitions. Notably, the band-gap was found to fundamentally controlled by the
strength of the charge-transfer energy, and not by the Mott-Hubbard
interactions, which can be modeled by composition, pressure or stoichiometry
modulations. The fundamental insights presented in this work will help
understand similar physics and mechanisms in other class of perovskites and
correlated metals.
