a “well ordered” classification as discussed in Ref. [@Takayanagi65; @Green75; @Alexander77]. We choose $v_1 > v_2$ or when $v_1=v_2$, $j_1 \ge j_2$. The time-independent Schr[ö]{}dinger equation yields a set of close-coupling equations, $$\begin{gathered}
\left\{ - \frac{\hbar^2}{2 \mu} \frac{d^2}{d R^2} +
\frac{\hbar^2 l(l+1)}{2 \mu R^2}
+ \varepsilon_{vj} - E \right\}
\ F_{vjl}^{JM\varepsilon_I \varepsilon_P}(R)  \\
+ \sum_{v'j'l'}^{*} 
  {\cal U}^{JM\varepsilon_I \varepsilon_P}_{vjl,v'j'l'}(R)
\ F_{v'j'l'}^{JM\varepsilon_I \varepsilon_P}(R) = 0,
\label{CC2}\end{gathered}$$ with the interaction potential matrix, $$\begin{gathered}
{\cal U}^{JM\varepsilon_I \varepsilon_P}_{vjl,v'j'l'}(R) = 
2 \Delta_{vj_1j_2} \Delta_{v'j'_1j'_2} [ {\cal U}^{JM\varepsilon_I}_{vjl,v'j'l'}(R) \\
+ \varepsilon_P (-1)^{j'_1+j'_2+j'_{12}+l'} 
{\cal U}^{JM\varepsilon_I}_{vjl,\bar{v}'\bar{j}'l'}(R)].\end{gathered}$$ Because of the indistinguishability of the molecules and the summation over the “well ordered” states, the size of the set of close-coupling equations  is smaller than the set .

The close-coupling equations  or  are solved for each value of $R$ using the log-derivative matrix propagation method of Johnson [@Johnson73] and Manolopoulos [@Manolopoulos86]. The log-derivative matrix is propagated to a matching distance where asymptotic boundary conditions are applied to obtain the scattering matrix, $S$, for given values of $J$, $\varepsilon_I$ and $\varepsilon_P$. For distinguishable molecule collisions, the state-to-state cross section is given by $$\begin{gathered}
\sigma_{v_1j_1v_2j_2 \to v'_1j'_1v'_2j'_2}(E_c) = \frac{\pi}{(2j_1+1)(2j_2+1)\text{k}^2} \\
\sum_{j_{12}j'_{12}ll'J\varepsilon_I}^{} (2J+1) 
|\delta_{vjl,v'j'l'} 
 - S^{J\varepsilon_I}_{vjl,v'j'l'}(E_c)|^{2},\end{gathered}$$ where the wave vector $\text{k}^2=2 \mu E_c/\hbar^2$. For indistinguishable molecules, 