Analytic solutions to two-state collision problems for the case of exponential coupling

Analytic solutions to two-state collision problems are obtained for systems in which the interaction matrix element of the Hamiltonian displays an exponential variation with time. When the difference in the diagonal matrix elements is either constant or governed by the same exponential function, exact analytic solutions can be found. When it is in the form of a constant plus an exponential term, the case of practical importance, an approximate solution is obtained based upon these exact solutions. The solution is used to calculate cross sections for fine-structure transitions in atomic collisions (Na-He, F-Xe, F-${\mathrm{H}}^{+}$).