Classical threshold law for the formation of van der Waals molecules

We study the role of pairwise long-range interactions in the formation of van der Waals molecules through direct three-body recombination processes A + B + B $\rightarrow$ AB + B, based on a classical trajectory method in hyperspherical coordinates developed in our earlier works [J. Chem. Phys. 140, 044307 (2014); J. Chem. Phys. 154, 034305 (2021)]. In particular, we find the effective long-range potential in hyperspherical coordinates with an exact expression in terms of dispersion coefficients of pairwise potentials. Exploiting this relation, we derive a classical threshold law for the total cross section and the three-body recombination rate yielding an analytical expression for the three-body recombination rate as a function of the pairwise long-range coefficients of the involved partners.