d_{\alpha\beta})^2}
+
\delta_{\alpha\beta}
\frac{P}{M}\cdot d_{\alpha'\beta'}^*
e^{(E_\alpha'-E_\beta')M\partial/\partial(P\cdot\hat d_{\alpha'\beta'}^*)^2}
\label{eq:T_aa'bb'_MJ}
\;,\end{aligned}$$ where $\hat d_{\alpha\beta}$ is the normalized coupling vector. Within the momentum-jump approximation [@simu; @sergi-theor], the action of the transition operator on the bath momenta can be easily obtained in closed form: $$\begin{aligned}
e^{(E_\alpha-E_\beta)M\partial/\partial(P\cdot\hat d_{\alpha\beta})^2} P
&=&
P - P\left(P\cdot \hat d_{\alpha\beta}\right)
+\hat d_{\alpha\beta}
\sqrt{\left(P\cdot\hat d_{\alpha\beta}\right)^2 
+ M\left(E_\alpha-E_\beta\right)}
\;.\end{aligned}$$

Considering Equations (\[eq:TrW\]) and (\[eq:prop\_sstp\_W\]), together with its SSTP implementation just described, one can see that the solution of the QCLE can be obtained from an ensemble of classical-like trajectories, where each trajectory (whose initial conditions arise from a Monte Carlo sampling [@frenkel] of the $X$’s), involves deterministic evolution segments on a given adiabatic energy surfaces interspersed with stochastic quantum transitions, caused by the momentum-jump operator in Equation (\[eq:T\_aa’bb’\_MJ\]).

The SSTP algorithm [@sergi-theor; @sstp] maps the calculation of averages through the QCLE (\[eq:qcle\]) onto a stochastic process. It is a hybrid Molecular Dynamics/Monte Carlo procedure suffering from two main problems. The first is given by the momentum-jump approximation, which is not valid in general. One can avoid this approximation by devising different integration schemes, but usually at the expense of other approximations [@jlgh]. The second problem is not just associated with the SSTP 