axis are conserved during the collision. In the following, we will discuss the close-coupling equations for both distinguishable and indistinguishable molecule - molecule collisions.

For collisions between two distinguishable molecules, the diabatic functions, $$\begin{aligned}
\phi^{JM\varepsilon_I}_{vjl}(\vec{r}_1,\vec{r}_2,\hat{R}) 
= \chi_{vj_1j_2}(r_1,r_2) \big\langle \hat{r}_1 \hat{r}_2 \hat{R} \big| j l J M \big\rangle ,
\label{diabfuncbasis}\end{aligned}$$ serve as basis functions to expand the total wave function, $$\begin{gathered}
\Psi(\vec{r}_1,\vec{r}_2,\vec{R}) = \\
\frac{1}{R} 
\sum_{v,j,l,J,M}^{} F_{vjl}^{JM\varepsilon_I}(R) \phi^{JM\varepsilon_I}_{vjl}(\vec{r}_1,\vec{r}_2,\hat{R}) , 
\label{PSI}\end{gathered}$$ where $v \equiv v_1 v_2$ denotes the vibrational quantum numbers and $j \equiv j_1 j_2 j_{12}$ denotes the rotational quantum numbers of the molecules. The function $\chi_{vj_1j_2}(r_1,r_2) = \chi_{v_1j_1}(r_1) \chi_{v_2j_2}(r_2)$ is the product of the vibrational wave functions of the two diatomic fragments and the angular function, $$\begin{gathered}
\langle \hat{r}_1 \hat{r}_2 \hat{R}  \big| j l J M \rangle =  \\
\sum^{}_{m_{j_1}, m_{j_2}, m_{j_{12}}, m_l} \langle  j_1 m_{j_1} j_2 m_{j_2} \big| j_{12} m_{j_{12}} \rangle  
\big\langle j_{12} m_{j_{12}} l m_l \big| J M \big\rangle \\
Y_{j_1 m_{j_1}}(\hat{r}_1) Y_{j_2 m_{j_2}}(\hat{r}_2) Y_{l m_l}(\hat{R}),\end{gathered}$$ denotes the rotational wave function of the two molecules in the total angular momentum representation. In this representation, the rotational angular momenta $\vec{j}_1$ and $\vec{j}_2$ are coupled to give the total rotational momentum $\vec{j}_{12}$ which is subsequently coupled to the orbital angular 