# Collisional excitation of diatomic molecules

Hi,
assuming I have the potential for a diatomic molecule in the groundstate i=o and excited state i=1

$V_i( r ) = D( 1 − exp[ −β ( r − r_i)] )^2+ C_i$

Using collisional excitation with fast electrons.
What is the vibrational quantum number of the state the molecule will most likely be excited to?
All under low pressure gas discharge.
I first assumed I could use Franck Condon principle, but this is not the case as I am not radiating on the molecules and thus dipole approximation is not allowed.

I cant use Lippmann–Schwinger equation either because I have no futher information on the interaction.
Can anyone help me?
Thank you,
kuecken