A log-derivative propagation scheme for the exact solution of two-state curve crossing problems

Abstract

A new method is presented for the exact quantum solution of certain two-state curve crossing problems, where electronic state ‖1〉 is energetically open at one end of the range of integration, while state ‖2〉, and, possibly, state ‖1〉, is energetically open at the other end of the range of integration. The method involves the use of log-derivative propagators, but differs from the usual log-derivative integration scheme in that one must propagate through the range of integration not only the log-derivative matrix but also a transformation matrix which permits one to reconstruct the initial wave function after the integration is completed. The method is numerically stable and, in a ‘‘solution following’’ approximation to the log-derivative propagators, converges as the fourth power of the step size. Application is made to several model problems. In one case the exact results are compared with the predictions of earlier semiclassical analyses [P. V. Coveney, M. S. Child, and A. Bárány, J. Phys. B 18, 457 (1985)]. The method is completely general, and can be applied to arbitrary potentials.