Analytic evaluation of the Born-Oppenheimer amplitude for rearrangement collisions

We present a closed-form expression for the Born-Oppenheimer amplitude for arbitrary initial state to arbitrary final state transition. So far these amplitudes have been evaluated analytically for transitions among some low-lying states (1s, 2s, 2p) with the help of parametric differentiations which is rather tedious. The present method offers an elegant way to avoid these parametric differentiations to calculate the Jackson-Schiff transition amplitudes between states of arbitrary quantum numbers. Analytic expressions of our basic integral are checked against a well known result for a special case. We demonstrate numerically the need for the analytic evaluation of these amplitudes especially for the transition between states of large quantum numbers. (author)