A semiclassical model for the calculation of nonadiabatic transition probabilities for classically forbidden transitions

The calculation of multidimensional semiclassical wave functions in the forbidden region using real valued coordinates

A method that uses only real valued coordinates is presented for integrating the many dimensional semiclassical wave function into the forbidden region. The procedure first determines a surface of caustic points by running the set of trajectories that define the wave function in the allowed region. In the forbidden region, the momentum and the action integral are both complex functions of position, and their imaginary parts vanish on the caustic surface. The direction of the imaginary part of the momentum p(I) can be chosen to the perpendicular to the caustic surface at all points on that surface. Equations are derived for integrating the values of the real and imaginary parts of the momentum along the curves that follow the direction of p(I). The equations for the change in the action integral and the prefactor for the semiclassical wave function along these curves are also obtained, allowing for the determination of the semiclassical wave function in the forbidden region. Calculations are performed for a two dimensional problem, and the semiclassical wave function is found to be is excellent agreement with the results of exact quantum calculations.

A singularity free surface hopping expansion for the multistate wave function

A version of a surface hopping wave function for nonadiabatic multistate problems, which is free of turning point singularities, is derived and tested. The primitive semiclassical form of the particular surface hopping method considered has been shown to be highly accurate, even for classically forbidden processes. However, this semiclassical wave function displays the usual singular behavior at turning points and caustics in the classical motion. Numerical data has shown that this somewhat reduces its accuracy when the energy is near the crossing energy of the diabatic electronic surfaces. The singularity free version of this surface hopping wave function is derived by partitioning the x-axis into a large number of small steps for one dimensional problems. The adiabatic electronic energy surfaces are approximated to be linear functions within each step. The matching conditions required by the continuity of the wave function and its derivative at each step boundary provide the needed conditions to obtain the amplitudes for changes in electronic state and/or reflection of the trajectory for the motion of the nuclei. This leads to a form of the surface hopping wave function that is free of turning point singularities. The method is tested for a one dimensional model problem, and it is found to be highly accurate at all energies considered, even when the energy is near the crossing energy.