  We study the parity and time-reversal ($\mathcal{PT}$) symmetric quantum
physics in a non-Hermitian non-relativistic hydrogen molecule with local
(Hubbard type) Coulomb interaction. We consider non-Hermiticity generated from
both kinetic and orbital energies of the atoms and encounter the existence of
two different types of exceptional points (EPs) in pairs. These two kinds of EP
are characteristically different and depend differently on the interaction
strength. Our discovery may open the gates of a rich physics emerging out of a
simple Hamiltonian resembling a two-site Hubbard model.
