Classical trajectories can be computed directly from electronic structure calculations without constructing a global potential-energy surface. When the potential energy and its derivatives are needed during the integration of the classical equations of motion, they are calculated by electronic structure methods. In the Born-Oppenheimer approach the wave function is converged rather than propagated to generate a more accurate potential-energy surface. If analytic second derivatives (Hessians) can be computed, steps of moderate size can be taken by integrating the equations of motion on a local quadratic approximation to the surface (a second-order algorithm). A more accurate 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 structure work per step is the same as the second-order Hessian based integrator, since the energy, gradient and Hessian at the end of the step are used for the local quadratic surface for the next predictor step. A fifth-order polynomial fit performs somewhat better than a rational function fit. For both methods the step size can be a factor of 10 larger than for the second order approach without loss of accuracy.