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We review harmonic oscillator theory for closed, stable quantum systems. The H₂ potential energy curve (PEC) of Mexican hat-type, calculated with a confined Kratzer oscillator, is better than the Rydberg–Klein–Rees (RKR) H₂ PEC. Compared with QM, the theory of chemical bonding is simplified, since a confined Kratzer oscillator can also lead to the long sought for universal function, once called the Holy Grail of Molecular Spectroscopy. This is validated by reducing PECs for different bonds H₂, HF, I₂, N₂ and O₂ to a single one. The equal probability for H₂, originating either from $H_A+H_B$ or $H_B+H_A$, is quantified with a Gauss probability density function. At the Bohr scale, a confined harmonic oscillator behaves properly at the extremes of bound two-nucleon quantum systems.

The symmetry breaking due to a magnetic field applied on a hydrogen molecule H₂ generates an electric polarization. This magnetoelectric effect occurs for electrons in a triplet state provided the magnetic induction field is not aligned with the symmetry axis of the molecule.

We theoretically investigate the quantum correlations (in terms of concurrence of indistinguishable electrons) in a prototype molecular system (hydrogen molecule). With the assistance of the standard approximations of the linear combination of atomic orbitals and the configuration interaction methods we describe the electronic wavefunction of the ground state of the H₂ molecule. Moreover, we managed to find a rather simple analytic expression for the concurrence (the most used measure of quantum entanglement) of the two electrons when the molecule is in its lowest energy. We have found that concurrence does not really show any relation to the construction of the chemical bond.