\left( \mbox{\boldmath$\cal B$}^{\rm S} \overrightarrow{\nabla}^{\rm S}
H_{\alpha\alpha'}^{\rm S} \right) \cdot
\overrightarrow{\nabla}^{\rm S}
\;,
\label{eq:Lspin}\end{aligned}$$ with the average adiabatic Hamiltonian $$H_{\alpha\alpha'}^{\rm S}
=
H_{\rm S}+\frac{1}{2}\left(E_{\alpha}+E_{\alpha'}\right)
\;.$$

The transition operator for the spin bath is given by $$\begin{aligned}
\begin{array}{lll}
{\cal T}^{\bf S}_{\alpha\alpha',\beta\beta'}
&=&
d^{\bf S}_{\alpha\beta}
\cdot\left( \mbox{\boldmath$\cal B$}^{\rm S} \overrightarrow{\nabla}^{\rm S}
H_{\rm S}\right) \delta_{\beta'\alpha'}
+
\frac{1}{2}\Delta E_{\alpha\beta}
d^{\bf S}_{\alpha\beta}
\cdot\left(
\mbox{\boldmath$\cal B$}^{\rm S}
\overrightarrow{\nabla}^{\rm S}
\right)
\delta_{\alpha'\beta'}
\\
&+& 
d^{{\bf S}*}_{\alpha'\beta'} 
\cdot
\left( \mbox{\boldmath$\cal B$}^{\rm S}
\overrightarrow{\nabla}^{\rm S} H_{\rm S} \right)
\delta_{\alpha\beta}
+\frac{1}{2}\Delta E_{\alpha'\beta'}
d^{{\bf S}*}_{\alpha'\beta'}
\cdot
\left( \mbox{\boldmath$\cal B$}^{\rm S} \overrightarrow{\nabla}^{\rm S} \right)
\delta_{\alpha\beta}
\;.
\end{array}
\label{eq:Jspin}\end{aligned}$$

The limit $d_{\alpha\alpha}^{\bf S}\to 0$ of the spin transition operator in Equation (\[eq:Jspin\]) provides the form of the standard transition operator for canonical conjugate coordinates, given in Equation (\[eq:T\_aa’bb’\]). Finally, because of the spin nature of the bath, one finds a higher order transition operator (which does not appear in the case of canonical conjugate bath coordinates): $$\begin{aligned}
{\cal S}_{\alpha\alpha',\beta\beta'}
&=&
\frac{1}{2} \Delta E_{\alpha\sigma} d^{\bf S}_{\alpha\sigma}
\mbox{\boldmath$\cal B$}^{\rm S} d^{\bf S}_{\sigma\beta}\delta_{\alpha'\beta'}
+\frac{1}{2} \Delta E_{\alpha\beta} d^{\bf S}_{\alpha\beta}
\mbox{\boldmath$\cal B$}^{\rm S} d^{{\bf S}*}_{\alpha'\beta'}
\nonumber\\
&+& \frac{1}{2}\Delta E_{\alpha'\sigma'} d^{{\bf S}*}_{\alpha'\sigma'}
\mbox{\boldmath$\cal B$}^{\rm S}
d^{{\bf S}*}_{\sigma'\beta'}\delta_{\alpha\beta}
+\frac{1}{2}\Delta E_{\alpha'\beta'} d^{{\bf S}*}_{\alpha'\beta'}
\mbox{\boldmath$\cal B$}^{\rm S} d^{\bf S}_{\alpha\beta}
\nonumber\\
&-&\frac{1}{2}\left(E_{\alpha}+E_{\alpha'}\right)\overleftarrow{\nabla}^{\bf S}
