&=&
\hat h_{\bf S}({\bf S})+H^{\bf S}({\bf S})\;,
\label{eq:tot-hamS}\end{aligned}$$ describing a quantum subsystem in terms of the Hamiltonian $\hat H(\{\hat{\chi}\})$, depending on the operators $\{\hat\chi\}$, $V(\{\hat\chi\},{\bf S})$ is the subsystem-spin interaction potential, and the second line of the equation defines the adiabatic Hamiltonian $\hat h_{\bf S}$. The quantum-classical dynamics of the operator-valued quasi-probability function (defined in the spinor space of the total system), $\hat W^{\bf S}({\bf S},t)$, is dictated by the spin-bath QCLE [@sergi-spin1; @sergi-spin2] $$\begin{aligned}
\frac{\partial}{\partial t}\hat W^{\bf S}({\bf S},t)
&=&-\frac{i}{\hbar}\left[\begin{array}{cc} \hat{\cal H}({\bf S}) & 
\hat W^{\bf S}({\bf S},t)\end{array}\right]
\mbox{\boldmath$\cal D$}^{\rm S}
\left[\begin{array}{c}\hat{\cal H}({\bf S})\\ 
\hat W^{\bf S}({\bf S},t)\end{array}\right]
\;\nonumber\\
&=&-\frac{i}{\hbar}\left[\hat{\cal H}({\bf S}),\hat W^{\bf S}({\bf S},t)
\right]_{\mbox{\tiny \boldmath$\cal D$}^{\rm S}},
\label{eq:pWS}\end{aligned}$$ where $$\begin{aligned}
\mbox{\boldmath$\cal D$}^{\rm S}
&=&
\left[\begin{array}{cc} 0 & 1+\frac{i\hbar}{2}
\overleftarrow{\nabla}\mbox{\boldmath$\cal B$}^{\rm S}
\overrightarrow{\nabla}
\\
-1-\frac{i\hbar}{2}
\overleftarrow{\nabla}\mbox{\boldmath$\cal B$}^{\rm S}
\overrightarrow{\nabla}
& 0\end{array}\right]\;.\end{aligned}$$

We next set out to represent Equation (\[eq:pWS\]) in the adiabatic basis $|\alpha;{\bf S}\rangle$ defined by the eigenvalue equation $$\hat h_{\bf S}({\bf S})|\alpha;{\bf S}\rangle =E_{\alpha}({\bf S})|\alpha;{\bf S}\rangle\;.
\label{eq:adbasS}$$ It should be noted that, in contrast to the case of canonically conjugate phase space coordinates which depends only on the positions $Q$ and not on the conjugate momenta $P$, this adiabatic basis depends on all the non-canonical spin 