  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.
