The key feature introduced by the tensor interaction of  type (see below for a classification of tensor forces) is that it couples channels of different orbital angular momentum $l$. Specifically, for particular values of the conserved quantities $J$, the total angular momentum, and $\pi$, the parity, whenever $\pi = (-1)^{J+1}$, then two values of orbital angular momentum, $l=J-1$ and $J+1$, are coupled by .

For total angular momentum $J$ and orbital angular momentum $l'$ the radial wavefunction $\psi^J_{l'l}$ satisfies the coupled equations, $$\left[ \frac{{\rm d}^2}{{\rm d} r^2} + k^2 -\frac{2\mu}{\hbar^2}(l'J|V|l'J) 
-\frac{l'(l'+1)}{r^2}
\right] \psi^J_{l'l}(k,r) 
= \sum_{l'' \ne l'} \frac{2\mu}{\hbar^2}(l'J|V|l''J)  \psi^J_{l''l}(k,r)
\label{eq:cde}$$ where $\mu$ is the reduced mass of the system and $(l'J|V|l''J)$, a function of $r$, is the matrix element of the inter-nuclear interaction $V$ integrated over all angular and internal degrees of freedom. The second subscript, $l$, on $\psi$ identifies the incoming orbital angular momentum. This is determined by imposing on the solution of the coupled equations, the following asymptotic boundary conditions : $$\psi^J_{l'l}(k,r) \rightarrow \delta_{l'l} I_{l'}(kr) - S^J_{l'l} O_{l'}(kr).
\label{eq:boundary}$$ Here, $I_l(r)$ and $O_l(r)$ are the incoming and outgoing asymptotic Coulomb radial wavefunctions, often written $H_l(r)^{*}$ and $H_l(r)$ respectively as in Satchler [@satchlerbook], namely: $$I_l(kr)=G_l(kr)-{\rm i} F_l(kr);\qquad I_l(kr)=G_l(kr)-{\rm i} 