content_image/361542.jpg content_image/361545.jpg content_image/361548.jpg content_image/361551.jpg
In region II, we represent the long-range (in R , at fixed r and γ ) potential V$_{LR}$ for the interaction between H$_{2}$ and H − as [51]: 

<img src='content_image/361542.jpg'>

where the first term D$_{as}$ ( r ) is the sum of H − and H$_{2}$( r ) energies at a given internuclear distance r of the H$_{2}$ molecule. The second term is the interaction between the electric charge of H − with the quadrupole moment Q ( r ) of H$_{2}$ (taken from Ref. [52]), and the third term is the interaction of the dipole moment of H$_{2}$ induced by H − , involving the second order Legendre polynomial P$_{2}$ (cos γ ). The functions α$_{0}$ ( r ) and α$_{2}$ ( r ) are the isotropic and anisotropic polarizabilities of H$_{2}$, for which we used the analytical functions given in Ref. [11] that were obtained by fitting the numerical values from Ref. [53]. The dispersion energy varying as 1 /R 6 and other smaller terms are neglected, which is a good approximation because the long-range expansion is only used for R > 56 . 4 a.u. 

In region I we used the following extrapolation formula in r for fixed R and γ : 

<img src='content_image/361545.jpg'>

where a ( R, γ ) and b ( R, γ ) are functions of R and γ that are obtained considering the two ab initio energies V ( r = 0 . 8; R, γ ) and V ( r = 1; R, γ ) calculated at first two values of the coordinate r . In this way, we obtain the quantities a and b given on a two-dimensional grid of points in the ( R , γ ) space. Then we used the 2D B-spline interpolation to obtain smooth two-dimensional functions a ( R, γ ) and b ( R, γ ). 

In region III, we extrapolate the PES in r and at fixed R and γ using a dispersion-like expression: 

<img src='content_image/361548.jpg'>

where the D$_{0}$ ( R, γ ) and C$_{6}$ ( R, γ ) (always positive) coefficients are obtained in a way similar to the coefficients a and b , considering the two last points V ( r = 2 . 2 a.u.; R, γ ) and V ( r = 2 . 4 a.u.; R, γ ). They are also interpolated using the 2D B-spline method for arbitrary values of R and γ . 

The behavior of the PES in region IV is described by the short-range (in R ) repulsive expression at given values of r and γ : 

<img src='content_image/361551.jpg'>

