Ab initio classical trajectories on the Born-Oppenheimer surface: Hessian-based integrators using fifth-order polynomial and rational function fits

Classical trajectories can be computed directly from electronic structure c alculations without constructing a global potential-energy surface. When th e potential energy and its derivatives are needed during the integration of the classical equations of motion, they are calculated by electronic struc ture methods. In the Born-Oppenheimer approach the wave function is converg ed rather than propagated to generate a more accurate potential-energy surf ace. If analytic second derivatives (Hessians) can be computed, steps of mo derate size can be taken by integrating the equations of motion on a local quadratic approximation to the surface (a second-order algorithm). A more a ccurate integration method is described that uses a second-order predictor step on a local quadratic surface, followed by a corrector step on a better local surface fitted to the energies, gradients, and Hessians computed at the beginning and end points of the predictor step. The electronic structur e work per step is the same as the second-order Hessian based integrator, s ince the energy, gradient and Hessian at the end of the step are used for t he local quadratic surface for the next predictor step. A fifth-order polyn omial fit performs somewhat better than a rational function fit. For both m ethods the step size can be a factor of 10 larger than for the second order approach without loss of accuracy. (C) 1999 American Institute of Physics. [S0021-9606(99)30131-8].