Dissociation of diatomic molecules
Where do their electrons go?


IIRC, in the gas phase, the energy required for heterolysis is usually much greater than that required for homolysis. Consequently, usually, each of the atoms of a diatomic molecule retains one of the originally bonded electrons on dissociation.


  Which quantum mechanical technique can decide the question -
  Hartree-Fock?


In principle, one could scan the potential energy surface1) by elongating the bond using some quantum chemical method to look on what actually  happens. Beaware though, that the most used variant of the Hartree-Fock (HF) method, the so-called restricted Hartree-Fock (RHF) method is know to be ill-suited for such calculations: it is known to fail completely when the homolytical dissociation of a closed-shell molecule into two open-shell fragments is considered.

Such behaviour is usually refered to as the Hartree-Fock "dissociation catastrophe", although, strictly speaking, only the restricted variant of the Hartree-Fock method is a subject to it. The unrestricted Hartree-Fock (UHF) method does not suffer from the "dissociation catastrophe" and tends to give at least qualitatively correct dissociation limit. So, use UHF at least. Even better, use some correlated method.



1) Simply a curve in this case, since there is only single degree of freedom (bond length).