Electronic trajectories in atomic physics: The chemical bond in the H₂ ⁺ ion

The H₂ ⁺ ion is the simplest example in which a chemical bond exists, created by one electron between two protons. As all chemical bonds, it is usually considered inexplicable in a classical frame. Here, in view of the extremely large velocities attained by the electron near the protons, we consider a relativistic extension of the standard classical three-body model. This has a great impact since the reference unperturbed system (clamped protons) is no more integrable, and indeed by molecular dynamics simulations, we find that the modification entails the existence of a large region of strongly chaotic motions for the unperturbed system, which lead, for the full system, to a collapse of the molecule. For motions of generic type, with the electron bouncing between the protons, there exists an open region of motions regular enough for producing a bond. Such a region is characterized by the property that the electron's trajectories have an angular momentum p_φ along the inter-nuclear axis of the order of the reduced Planck's constant ℏ. Moreover, special initial data exist for which the experimental bond length and oscillation frequency of the protons (but not the dissociation energy) are well reproduced. Also, well reproduced is the quantum potential, albeit only in an extended interval about the minimum.