When a diatomic ion dissociates, how does the electron density on each atom change? Let's say, we have a fluorine molecule (F2), and we take 3 electrons from it, so now its bond starts to stretch, and at last the bond breaks and the two atoms are farther and farther away from one another. When they're far enough from one another, one of them must have one more electron than the other. But what does this electron do during the dissociation process - does it choose one atom and stay there, or does it go back and forth between the two atoms? Do the two atoms share the electrons equally during the early stage and then they start to compete for one electron, making each atom's electron density fluctuate? Or does one of the atom's electron density keeps going down while the other's electron density keeps going up during the whole process?
 A: Dissociation is a sudden phenomenon at a critical distance separating the nuclei of the dimer: at shorter distances, having two ions without occupation of bonding orbitals between the two of them is an energetically higher state than the ground state of the bonded dimer. At the critical distance of dissociation, the energies of these scenarios coincide.
The two nuclei set up a potential energy for the electron(s) that is symmetric wrt. exchange of the two nuclei. As long as the dimer is bound, there is one ground state which obeys this symmetry. For the dissociated situation, there are two degenerate ground states, one with an extra electron bound to one of the nuclei, and the other way around, respectively. Only degenerate ground states can break the symmetry of the Hamiltonian; a non-degenerate ground state obeys the symmetry of the Hamiltonian.
The above is based on assuming an adiabatically slow separation of the dimer. If time-dependence comes into play in an ultrafast separation, experiments on the dissociation of ${{\rm H}_2}^+$ show that after only $15\cdot 10^{-15}$ seconds, the electron has "decided" which of the two protons to pick at a distance of 8 Bohr and cannot be driven force and back with a laser between the two anymore [Xu et al. Observing electron localization in a dissociating ${{\rm H}_2}^+$ molecule in real time. Nat. Commun. 8, 15849 (2017) (open access)].
