Chemical bond by Newtonian trajectories in the $H_2^+$ ion

According to the correspondence principle, classical mechanics and quantum mechanics agree in the semiclassical limit, although presently it has become more and more clear how intriguing would be to try to fix a boundary between them. Here we give a significant example in which the agreement concerns Newtonian trajectories of an electron with initial data corresponding to a quantum ground state. The example is the simplest case in which a chemical bond occurs, i.e. the $H_2^+$ ion. By molecular dynamics simulations for the full system (two protons and one electron) we show that there exist initial data producing an ``effective potential'' among the protons, which superposes in a surprisingly good way the quantum one computed in the Born-Oppenheimer approximation (Fig~1). Preliminarily, following the perturbation procedure first exhibited by Born and Heisenberg in the year 1924, we recall why an effective potential should exist in a classical frame, and also describe the numerical procedure employed in computing it.