Series expansions for Franck-Condon factors. I. Linear potential and the reflection approximation

A series expansion for Franck-Condon factors is obtained for a bound-to-continuum transition in the case of a linear potential for the continuum state. The first term in the series is the result obtained from the ``reflection'' approximation. A similar series is obtained for a bound-to-bound transition. The effect of nonlinearity in the potential is considered, and an approximate correction factor is derived. These results are compared with exact calculations on real systems.