We introduce an analytical model designed to capture the most important features of the electronic matrix elements arising in non-Born-Oppenheimer couplings between a bound anion state and a neutral-molecule-plus-ejected-electron state. In this particle-in-a-radial-box model, vibrations are assumed to cause modulations in the depth (U-0) and length (L) parameters of the box. The most important elements of this model are that L is chosen to reproduce the proper dependence of the radial size of the anion's orbital on electron binding energy, and U-0 is chosen to produce the correct electron affinity. Within this model, which is shown to be consistent with trends seen in ab initio calculations of associated electron ejection rates, the coupling matrix elements can be evaluated analytically to provide closed-form expressions for how the rates depend upon (1) the kinetic energy of the ejected electron, (2) the energy spacing between the anion and neutral energy surfaces as a function of geometry, (3) the difference in the slopes of the anion and neutral energy surfaces, and (4) overlaps of the neutral's vibration-rotation wave function with the spatial derivative of that of the anion.