  We present a detailed investigation of the electronic structure and bonding
characteristics of hydrogen-based molecular systems (\ch{H2+}, \ch{H2},
\ch{H2-}) using the Exact Diagonalization Ab Initio (EDABI) approach within the
framework of combined first- and second-quantization. By analyzing the relative
contributions of kinetic exchange and effective Coulomb interactions, we
provide a comprehensive understanding of covalency, atomicity, and ionicity as
a function of interatomic distances. Our approach leverages exact solutions of
the extended Heitler-London model to quantify these interactions, extending the
analysis to the discussion of properties of excited states and the dissociation
limit to these molecules. The findings reveal significant differences in
bonding characteristics, particularly highlighting the stability and bonding
nature of the neutral \ch{H2} molecule compared to its ionic counterparts. This
study not only enhances an understanding of molecular interactions in hydrogen
systems but also demonstrates the potential of the EDABI approach in developing
more accurate computational models in quantum chemistry.
