\mbox{\boldmath$\cal B$}^{\rm S} \cdot
d^{\bf S}_{\alpha\beta}\delta_{\alpha'\beta'}
-\frac{1}{2} \left(E_{\alpha}+E_{\alpha'}\right)
\overleftarrow{\nabla}^{\bf S} \mbox{\boldmath$\cal B$}^{\rm S}
\cdot d^{{\rm S}*}_{\alpha'\beta'}\delta_{\alpha\beta}
\;.
\label{eq:Sspin}\end{aligned}$$

The adiabatic limit of the spin-bath QCLE in (\[eq:qcsd-dyna\]) can be taken by setting to zero the off-diagonal elements of $d_{\alpha\alpha'}$, which appear in the operators in Equations (\[eq:Jspin\]) and (\[eq:Sspin\]). This is physically reasonable whenever the coupling between the different adiabatic energy surfaces is negligible. One obtains $$\begin{aligned}
\begin{array}{lll}
{\cal T}^{{\bf S},{\rm ad}}_{\alpha\alpha',\beta\beta'}
&=&
\left( d^{\bf S}_{\alpha\alpha} + d^{{\bf S}*}_{\alpha'\alpha'} \right)
\mbox{\boldmath$\cal B$}^{\bf S} \overrightarrow{\nabla}^{\bf S} H_{\rm S}
\delta_{\alpha\beta}\delta_{\beta'\alpha'}
\\
&=&
- i
\left( \phi^{\bf S}_{\alpha\alpha} - \phi^{\bf S}_{\alpha'\alpha'} \right)
\mbox{\boldmath$\cal B$}^{\bf S} \overrightarrow{\nabla}^{\bf S}
\delta_{\alpha\beta}\delta_{\beta'\alpha'}
\;. \end{array}
\label{eq:Jspin-ad}\end{aligned}$$

The geometric phase $$\phi^{\rm S}_{\alpha\alpha}=-id^{\bf S}_{\alpha\alpha}
\label{eq:geo-phase}$$ has been introduced exploiting the purely imaginary character of $d^{\bf S}_{\alpha\alpha}$. Similarly, the higher order transition operator becomes $$\begin{aligned}
{\cal S}_{\alpha\alpha',\beta\beta'}^{\rm ad}
&=&
-\frac{i}{2}
\sum_{I,J}
\left(\phi^{\bf S}_{\alpha\alpha} - \phi^{\bf S}_{\alpha'\alpha'}\right)
\mbox{\boldmath$\cal B$}^{\rm S} \overrightarrow{\nabla}^{\bf S}
\left(E_{\alpha}+E_{\alpha'}\right)\delta_{\alpha\alpha}\delta_{\alpha'\alpha'}
\nonumber\\
\label{eq:Sspin-ad}\end{aligned}$$

Putting everything together, the adiabatic approximation of the spin-bath QCLE may be written as $$\begin{aligned}
\frac{\partial}{\partial t}W^{\bf S}_{\alpha\alpha'} ({\bf S},t)
&=&\left[-i\omega_{\alpha\alpha'}
-i 
\left(\phi^{\bf S}_{\alpha\alpha} - \phi^{\bf S}_{\alpha'\alpha'}\right)
\mbox{\boldmath$\cal B$}
\overrightarrow{\nabla}^{\bf S}
H_{\bf S}^{\alpha\alpha'}
- H_{\rm S}^{\alpha\alpha'}
\overleftarrow{\nabla}^{\bf S}
\mbox{\boldmath$\cal B$}
\overrightarrow{\nabla}^{\bf S}
\right]
W^{\bf S}_{\alpha\alpha'}({\bf S},t).
\label{eq:qcsd-dyna-ad}\end{aligned}$$

