the state-to-state cross section is given by a statistically weighted sum of the exchange-permutation symmetrized cross sections, $$\begin{aligned}
\sigma_{v_1j_1v_2j_2 \to v'_1j'_1v'_2j'_2}(E_c)
= W^+ \sigma^{\varepsilon_P=+1} + W^- \sigma^{\varepsilon_P=-1} ,\end{aligned}$$ where $$\begin{gathered}
\sigma^{\varepsilon_P}
= \frac{\pi(1+\delta_{v_1v_2}\delta_{j_1j_2})(1+\delta_{v'_1v'_2}\delta_{j'_1j'_2})}
{(2j_1+1)(2j_2+1)\text{k}^2} \\
\sum_{j_{12}j'_{12}ll'J\varepsilon_I}^{*} (2J+1)
|\delta_{vjl,v'j'l'} 
- S^{J\varepsilon_I\varepsilon_P}_{vjl,v'j'l'}(E_c)|^{2}.\end{gathered}$$ For collisions between para-H$_2$ molecules (nuclear spin $I=0$), $W^+=1, W^-=0$ so that only the $\varepsilon_P=+1$ exchange-permutation symmetry is needed in the close-coupling calculations, while for collisions between ortho-H$_2$ molecules (nuclear spin $I=1$), $W^+=2/3, W^-=1/3$ and both exchange-permutation symmetries $\varepsilon_P=\pm 1$ are needed. The total inelastic cross section is the sum of all inelastic state-to-state cross sections. The state-to-state rate coefficients are obtained by integrating the corresponding cross sections over the Maxwell–Boltzmann distribution of velocities, $$\begin{gathered}
k_{v_1j_1v_2j_2 \to v'_1j'_1v'_2j'_2}(T)
= \frac{1}{k_B T} \left( \frac{8}{\pi \mu k_B T} \right)^{1/2}  \\
\int_0^{\infty} \sigma_{v_1j_1v_2j_2 \to v'_1j'_1v'_2j'_2}(E_c) \, e^{-E_c / (k_B T)} \, E_c \, dE_c.
\label{rate-constant}\end{gathered}$$

Results and discussion
======================

Computational details
---------------------

![ The effective potentials, $V_{\text{eff}}^{J}$, as functions of $R$ for the H$_4$ system for $J=0, \varepsilon_I=+1, \varepsilon_P=+1$. \[SPAG-FIG\]](poteffH4.eps){width="8cm"}

In this study we investigate the scattering of two identical para-H$_2$ molecules using the BMKP PES [@Boothroyd02]. For identical molecule collisions, the rovibrational quantum numbers obey the “well ordered states” classification [@Takayanagi65; @Green75; 