coordinates $\bf S$. In this basis, Equation (\[eq:pWS\]) becomes $$\begin{aligned}
\frac{\partial}{\partial t}\hat W^{\bf S}_{\alpha\alpha'}
&=&
-i\omega_{\alpha\alpha'} W^{\bf S}_{\alpha\alpha'}
- H^{\rm S}\overleftarrow{\nabla}^{\bf S}
\mbox{\boldmath$\cal B$}^{\bf S}
\langle\alpha|\overrightarrow{\nabla}^{\bf S}\hat W^{\bf S}|\alpha'\rangle
\nonumber\\
&+&\frac{1}{2}
\langle\alpha|\hat h_{\bf S}\overleftarrow{\nabla}^{\bf S}
\mbox{\boldmath$\cal B$}^{\bf S}
\overrightarrow{\nabla}^{\bf S}\hat W^{\bf S}|\alpha'\rangle
-\frac{1}{2}
\langle\alpha|\hat W^{\bf S}\overleftarrow{\nabla}^{\rm S}
\mbox{\boldmath$\cal B$}^{\rm S}\overrightarrow{\nabla}^{\bf S}
\hat h_{\bf S}|\alpha'\rangle
\;,\end{aligned}$$ where $\omega_{\alpha\alpha'}=E_\alpha({\bf S}) - E_{\alpha'}({\bf S})/\hbar$ is the Bohr frequency. Defining the spin coupling vector $$\begin{aligned}
d^{\bf S}_{\alpha\alpha'}
&=&
\langle\alpha;{\bf S}|\overrightarrow{\nabla}^{\bf S}|\alpha';{\bf S}\rangle,
\;\end{aligned}$$ one finds the two identities $$\begin{aligned}
\langle\alpha;{\bf S}|\left(\overrightarrow{\nabla}^{\bf S}
\hat W^{\bf S}({\bf S})\right)|\alpha';{\bf S}\rangle
&=&
\overrightarrow{\nabla}^{\bf S}W^{\bf S}_{\alpha\alpha'}({\bf S})
+\sum_\beta d^{\bf S}_{\alpha\beta}W^{\bf S}_{\beta\alpha'}({\bf S})
-\sum_{\beta'}W^{\bf S}_{\alpha\beta'}({\bf S}) d^{\bf S}_{\beta'\alpha'}
\label{eq:drho}
\\
\langle\alpha;{\bf S}|\left(\overrightarrow{\nabla}^{\bf S}
\hat{h}_{\bf S}({\bf S})\right) |\alpha';{\bf S}\rangle
&=&
\overrightarrow{\nabla}^{\bf S} h_{\bf S}^{\alpha\alpha'}
-\Delta E_{\alpha\alpha'} d^{\bf S}_{\alpha\alpha'}
\label{eq:dh}\end{aligned}$$ where $\Delta E_{\alpha\alpha'}=E_{\alpha}-E_{\alpha'}$. Using Equations (\[eq:drho\]) and (\[eq:dh\]), the spin-bath QCLE may be rewritten as $$\begin{aligned}
\frac{\partial}{\partial t}W^{\rm S}_{\alpha\alpha'}({\bf S},t)
&=&-\sum_{\beta\beta'}\left(i\omega_{\alpha\alpha'}\delta_{\alpha\beta}\delta_{\alpha\alpha'}
+iL_{\alpha\alpha'}\delta_{\alpha\beta}\delta_{\alpha\alpha'}
+{\cal T}^{\rm S}_{\alpha\alpha',\beta\beta'}
+{\cal S}_{\alpha\alpha',\beta\beta'}\right)
W^{\rm S}_{\beta\beta'}({\rm S},t)\;,
\label{eq:qcsd-dyna}\end{aligned}$$ where we have defined the classical-like spin-Liouville operator $$\begin{aligned}
iL_{\alpha\alpha'}&=&
H_{\rm S}
\overleftarrow{\nabla}^{\rm S}
\mbox{\boldmath$\cal B$}^{\rm S}
\overrightarrow{\nabla}^{\rm S}
+
\frac{1}{2} E_{\alpha'} \overleftarrow{\nabla}^{\rm S}
\mbox{\boldmath$\cal B$}^{\rm S} \overrightarrow{\nabla}^{\rm S}
+
\frac{1}{2} E_{\alpha} \overleftarrow{\nabla}^{\rm S}
\mbox{\boldmath$\cal B$}^{\rm S} \overrightarrow{\nabla}^{\rm S}
\nonumber\\
&=&
