where, ρ ij (t) = a * i (t)a j (t) are the adiabatic electronic density matrix elements. Because of the classical path approximation, one can ignore the back reaction of the electronic-non-adiabatic transition on the nuclear degrees of freedom. Hence, we do not re-scale the velocity, which is one of the key ingredients in the original FSSH algorithm. Instead, following earlier work, the transition probabilities are rescaled to preserve the energy conservation with the following expression: