= E_\alpha(Q)|\alpha;Q\rangle \;.
\label{eq:adia_states}$$ The representation of the QCLE in the adiabatic basis is sketched in Appendix \[sec:adbas\_rep\]. In the adiabatic basis, the QCLE is given by Equation (\[eq:qcle\_adbas\]) and the quantum-classical Liouville super-operator matrix elements are given in Equation (\[eq:liouville\]).

To derive the SSTP algorithm, we divide the time interval $t$ into $n$ equal small steps $\tau=t/n$. If one is able to calculate the propagation over a single $\tau$, the dynamics over the whole interval can be reconstructed by sequential iteration of the procedure. Let us then consider the quantum-classical propagator over a small step $\tau$ for the matrix elements of the operator-valued quasi-probability function $\hat W(X)$ in the adiabatic basis. Such a propagator is written as $$\left(e^{-i\tau{\cal L}}\right)_{\alpha\alpha,\beta\beta'}
\approx
\delta_{\alpha\beta} \delta_{\alpha'\beta'}
e^{-i\int_0^\tau ds \omega_{\alpha\alpha'}(s)}
e^{-i\tau L_{\alpha\alpha'}}
\left(1+\tau {\cal T}_{\alpha\alpha',\beta\beta'}\right)
\;.
\label{eq:prop_sstp}$$ On the right-hand side of Equation (\[eq:prop\_sstp\]), we have introduced $\omega_{\alpha\alpha'}$, the Bohr frequency defined in Equation (\[eq:Bohr-omega\]), $iL_{\alpha\alpha'}$ is a classical-like Liouville operator, defined in Equation (\[eq:classical-like\_L\]), and ${\cal T}_{\alpha\alpha',\beta\beta'}$ is the transition operator defined in Equation (\[eq:T\_aa’bb’\]). The SSTP dynamics of the matrix elements of $\hat W(X,t)$ is given by $$W_{\alpha\alpha}(X,\tau)
=
\sum_{\beta\beta'}
\delta_{\alpha\beta} \delta_{\alpha'\beta'}
e^{-i\int_0^\tau ds \omega_{\alpha\alpha'}(s)}
e^{-i\tau L_{\alpha\alpha'}}
\left(1+\tau {\cal T}_{\alpha\alpha',\beta\beta'}\right)
W_{\beta\beta}(X) \;.
