  In previous works, we introduced a geometric route to define our Ehrenfest
Statistical Dynamics (ESD) and we proved that, for a simple toy-model, the
resulting ESD does not preserve purity. We now take a step further: we
investigate decoherence and pointer basis in the ESD model by considering some
uncertainty in the degrees of freedom of a simple but realistic molecular
model, consisting of two classical cores and one quantum electron. The
Ehrenfest model is sometimes discarded as a valid approximation to
non-adiabatic coupled quantum-classical dynamics because it does not describe
the decoherence in the quantum subsystem. However, any rigorous statistical
analysis of the Ehrenfest dynamics, such as the described ESD formalism, proves
that decoherence exists. In this article, decoherence in ESD is studied by
measuring the change in the quantum subsystem purity and by analysing the
appearance of the pointer basis to which the system decoheres, which for our
example is composed by the eigenstates of the electronic Hamiltonian.
