The study of a family of Hessian-based predictor- corrector integrators for direct dynamics simulation

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2017-08
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Abstract

Direct dynamics simulations are a very useful and general approach for studying the atomistic properties of complex chemical systems because they make it possible to construct a system’s potential energy surface under an electronic structure theory without the need for fitting an analytic potential energy function. Classical trajectories, an application category of direct dynamics simulations,which obey Newton’s laws, have been proven useful for researching atomic system of motion. Classical trajectories can be computed directly from electronic structure calculations without computing a global potential-energy surface. The potential energy and its derivatives are directly calculated by electronic structure methods during the integration of the classical equations of motion. Because the wave function is converged rather than propagated to generate a more accurate potential-energy surface in direct dynamics simulations, steps of moderate size can be taken by integrating the equations of motion into a local quadratic approximation of the surface (a second-order algorithm) if the analytic second derivatives (Hessians) can be computed. The following discusses our contributions: (I) We describe an accurate integration method that uses a second-order predictor step on a local quadratic surface, followed by a corrector step on a local surface fitted to the energies, gradients, and Hessians computed at the beginning and end points of the predictor step. In the predictor stage, this method utilizes the information (Location, Velocity, Gradient, and Hessian ) of the current time step to predict the location of the next time step. In the corrector stage, this method hires a fifth-order polynomial or a rational polynomial to fit the energy surface. Next, different from previous methods, we propose a new algorithm, which uses those information (Location, Velocity, Gradient, and Hessian) of the current time step and the previous time steps to predict the location of the next time step in the predictor step. Similarly, it uses a second-order predictor step on a local quadratic surface and a corrector step on a local surface fitted to the energies, gradients, and Hessians computed at the beginning and end points of the predictor step. Finally, we show a family of Hessian-based Predictor-Corrector integrates. These integrates make use of some information (Location, Velocity, Gradient and Hessian) about the previous time step and all information about the current time step. The corrector step exploits the fifth-order polynomial fit or the rational polynomial fit. The approach involves four Fourth-order predictors, six Fifth-order predictors, and four Sixth-order predictors. As in the previous methods, the second-order predictor steps on a local quadratic surface and the corrector steps on a better local surface are fitted to the energies, gradients, and Hessians computed at the beginning and end points of the predictor step. (II) We propose a tool for evaluating the accuracy of integrators for the trajectory calculation by employing monodromy matrix. We choose a general approach, the Velocity Verlet integrator, as a referent object. In here, we simulate a molecular without hydrogen ( CO2) and a molecular with hydrogen (H2O) motions. Many simulations show the family of Hessian-based Predictor-Corrector integrators perform well for Hessian update and non Hessian update by comparison of the eigenvalues of monodromy matrix. Compared to traditional methods, the new tool are more intuitive and clear.

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Keywords
PredictorCorrector, Monodromy, Hessian-Based, Polynomial, Integrators
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