Despite outstanding advances throughout this century, we still lack final knowledge on the structure of the hydrogen molecule because of a number of insufficiencies in available models identified in the text. In this paper we use the recently achieved covering of quantum chemistry called hadronic chemistry in its iso-chemical branch, and introduce a new model of the hydrogen molecule characterized by a bond at short distances of the two valence electrons into a singlet quasi-particle state called iso-electronium. We study the iso-chemical model of the hydrogen molecule with a stable iso-electronium describing an oo-shaped orbit around the respective two nuclei, and another model with a weaker realization of the iso-electronium as a partially stable state. We show that the new model provides, apparently for the first time, an exact representation of the binding energy and other characteristics of the hydrogen molecules from axiomatic principles, without ad hoc modifications of theory. In subsequent papers we shall show that the new model permits apparently novel advances in the energy, liquefaction and other technological applications of the hydrogen.