Ab initio classical trajectories on the Born–Oppenheimer surface: Updating methods for Hessian-based integrators

Abstract

For the integration of the classical equations of motion in the Born-Oppenheimer approach, each time the energy and gradient of the potential energy surface are needed, a properly converged wave function is calculated. If Hessians ͑second derivatives͒ can be calculated, significantly larger steps can be taken in the numerical integration of the equations of motion without loss of accuracy. Even larger steps can be taken with a Hessian-based predictor-corrector algorithm. Since updated Hessians are used successfully in quasi-Newton methods for geometry optimization, it should be possible to improve the performance of trajectory calculations using updated Hessians. The Murtagh-Sargent ͑MS͒ update, the Powell-symmetric-Broyden ͑PSB͒ update and Bofill's update ͑a weighted combination of MS and PSB͒ were tested, and Bofill's update was found to be the best. Slightly smaller step sizes were needed with Hessian updating to maintain good conservation of the energy, but this was more than compensated by the reduction in total computational cost. An overall factor of 3 in speed-up was obtained for trajectories of systems containing 4 to 6 heavy atoms computed at the HF/3-21G level.

Ab Initio Molecular Dynamics with Born-Oppenheimer and Extended Lagrangian Methods Using Atom Centered Basis Functions

“…This approach was pioneered by Helgaker Figure 1 illustrates a Hessian-based predictor-corrector method that we developed a few years ago. [31][32][33] Given a Hessian from an electronic structure calculation, a predictor step is taken on the local quadratic surface. The Hessian is then recalculated and a fifth order polynomial or a rational function is fitted to the energies, gradients and Hessians at the beginning and end points of this predictor step.…”

Accelerated direct semiclassical molecular dynamics using a compact finite difference Hessian scheme

“…60-65 and references therein), and some have been used in direct dynamics simulations. [66][67][68] Most of the schemes are based on a first-order Taylor expansion which, for optimization, is the equation that quasi-Newton methods are based upon. The accuracy of the first-order Taylor expansion is sufficient for optimization, as evidenced by fast convergence of quasi-Newton methods.…”

A Single Transition State Serves Two Mechanisms. The Branching Ratio for CH2O•- + CH3Cl on Improved Potential Energy Surfaces

“…The Hessian is calculated analytically and followed by five updates before it is recalculated analytically again. 29 As in our previous work, 3,15 the trajectories were started from the transition state and the initial conditions were chosen to correspond to a thermal distribution at 298 K. Motion along the transition vector was in the direction toward the products and was sampled from a thermal distribution. 30 Rotational energies were sampled from a thermal distribution of a symmetric top.…”