tion method (diatomics-in-molecule –DIM) explicitly uses the ab initio data (wave functions and energies) obtained for fragments (H$_{2}$, H − , H − 2 , H$_{3}$) in separate ab initio calculations, it gives automatically the correct dissociation energy. Since we have calculated PES at relatively large distances R , we can extract ab initio values (labelled with ”ai“) of the corresponding long-range coefficients C ai 3 and C ai 4 and compare them with the theoretical values C th 3 and C th 4 of Eq. (1). We first make a fit of C ai 4 to the ab initio data by fixing C ai 3 to the theoretical value C th 3 . In this way the value of C ai 4 agrees with the value of C th 4 within 0.3% for γ = 0, and 5 % for γ = 90 ° at r = 1 . 4 a.u. Then, we did the opposite: we fitted C ai 3 to the ab initio data by fixing C ai 4 to C th 4 . In this case we found that fitted and theoretical values of C$_{3}$ agree within 0.1 % for γ = 0 and 2 % for γ = 90 ° at r = 1 . 4 a.u. The PES-PS differs significantly from the analytical long-range behavior of Eq. (1). Surprisingly, the long-range behavior of the PES-PS is quite different from ours and the one of PES-SM, and probably inaccurate due to the matching procedure between the long and short distances used by these authors. As mentioned in the introduction, one of the motivations of this study is to describe low-energy collisions between H$_{2}$ and H − , and to investigate the formation of H − 3 in such collisions. Therefore we need an approach which treat both bound and continuum states of the triatomic anion (including rearrangement of nuclei). We use the Smith-Whitten hyperspherical coordinates [54]: a hyper-radius ρ and two hyperangles θ, φ , which can be defined for the three identical particles by the three internuclear distances r$_{i}$, ( i = 1 , 2 , 3) as where ϵ$_{1}$ = 2 π/ 3, ϵ$_{2}$ = − 2 π/ 3, and ϵ$_{3}$ = 0. We also employ the adiabatic separation between hyperradius and hyperangles, which are known to be well adapted to atom-molecule inelastic and reactive scattering involving identical particles. The dynamics is treated within the framework of the slow variable discretization (SVD) method [55–57] that allows us to account easily for non-adiabatic couplings between hyperspherical adiabatic channels. We briefly recall below the main steps of our approach, which is discussed in greater details in Refs. [56, 57]. r$_{i}$ = 3 − 1 / $^{4}$ρ √ 1 + sin θ sin( φ + ϵ$_{i}$ ) , (5) 12 IV. VIBRATIONAL STATES OF H − 3 AND D − 3