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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. 

## IV. VIBRATIONAL STATES OF H − 3 AND D − 3 

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]. 

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