content_image/343806.jpg content_image/343807.jpg content_image/343808.jpg content_image/343809.jpg content_image/343805.jpg content_image/343803.jpg
The detached electron wave function Ψ + M is given by: 

<img src='content_image/343806.jpg'>

Spherical polar coordinates ( r$_{1}$, θ$_{1}$, φ$_{1}$ ), ( r$_{2}$, θ$_{2}$, φ$_{2}$ ) and ( r$_{3}$, θ$_{3}$, φ$_{3}$ ) for the detached-electron relative to each center are used. The emitted waves, Ψ + 1 , Ψ + 2 and Ψ + 3 , can be expressed using Ref. [20] by: 

<img src='content_image/343807.jpg'>

<img src='content_image/343808.jpg'>

<img src='content_image/343809.jpg'>

where C = 4 kBi ( k 2 $_{b}$+ k $^{2}$)$_{2}$, k$_{b}$ is related to the binding energy E$_{b}$ of H − by E$_{b}$ = k 2 b 2 , and B is a normalization constant, having value of 0 . 31552. 

The detached-electron wave function Ψ + $_{M}$( r, θ, φ ) from the three-center system can be obtained after the substitution of each center wave function in Eq. (1). The problem can be further simplified by using the idea that L is much larger than d . Let ( r, θ, φ ) be the spherical coordinates of the detached electron. For the phase terms, r$_{1}$ ≈ r − d cos θ , r$_{2}$ ≈ r + d cos θ and r$_{3}$ ≈ r are used, while in all other places r$_{1}$ ≈ r$_{2}$ ≈ r$_{3}$ ≈ r and θ$_{1}$ ≈ θ$_{2}$ ≈ θ$_{3}$ ≈ θ . With these approximations Ψ + $_{M}$( r, θ, φ ) reduces to: 

<img src='content_image/343805.jpg'>

## III. DERIVATION OF THE TOTAL CROSS SECTION FOR THE TRIATOMIC NEGATIVE ION 

The detached-electron flux in the radial direction for the triatomic negative ion is calculated as in [1]: 

<img src='content_image/343803.jpg'>

A large imaginary spherical surface Γ enclosing the triatomic negative ion is used to investigate the behavior of the total cross section. From the electron flux crossing the surface a generalized differential cross section dσ ( q ) ds can be obtained [20]: 

