Evaluating accuracy of Hessian-based predictor-corrector integrators

Abstract

Direct dynamics simulations are a useful and general approach for studying the atomistic properties of complex chemical systems because they do not require fitting an analytic potential energy function. Hessian-based predictor-corrector integrators are a widely used approach for calculating the trajectories of moving atoms in direct dynamics simulations. We employ a monodromy matrix to propose a tool for evaluating the accuracy of integrators in the trajectory calculation. We choose a general velocity Verlet as a different object. We also simulate molecular with hydrogen(CO₂) and molecular with hydrogen (H₂O) motions. Comparing the eigenvalues of monodromy matrix, many simulations show that Hessian-based predictor-corrector integrators perform well for Hessian updates and non-Hessian updates. Hessian-based predictor-corrector integrator with Hessian update has a strong performance in the H₂O simulations. Hessian-based predictor-corrector integrator with Hessian update has a strong performance when the integrating step of the velocity Verlet approach is tripled for the predicting step. In the CO₂ simulations, a strong performance occurs when the integrating step is a multiple of five.

Keywords

monodromy matrix / eigenvalue / Hessian-based predictor-corrector / velocity Verlet

References

[1] BUNKERD L. Classical trajectory methods [J]. Methods Comput Phys, 1971, 10(1): 287-325

[2] PESLHERBEG H, WANGH, HASEW L. Monte Carlo sampling for classical trajectory simulations[J]. Adv Chem Phys, 1999, 105(1): 171-201

[3] ALLENM D, TILDESLEYD JComputer simulation of liquids [M], 1987, NY, Oxford Press

[4] VOTHG A. Computer simulation of proton solvation and transport in aqueous and biomolecular systems [J]. Acc Chem Res, 2006, 39(1): 143-150

[5] SUNL, HASEW L. Born-Oppenheimer direct dynamics classical trajectory simulations [J]. Rev Comput Chem, 2003, 19(1): 79-146

[6] MARXD, HUTTERJ. Ab initio molecular dynamics: Theory and implementation [C]. Modern Methods and Algorithms of Quantum Chemistry. Julich, Germany: John von Neumann Institute for Computing, 2000329477

[7] HERBERTJ M, HEAD-GORDONM. Accelerated, energyconserving Born–Oppenheimer molecular dynamics via Fock matrix extrapolation [J]. Phys Chem Chem Phys, 2005, 7(1): 3269-3275

[8] LIX C, TULLYJ C, SCHLEGELH B, FRISCHM J. Ab inito Ehrenfest dynamics [J]. J Chem Phys, 2005, 123(1): 084106

[9] ISBORNC M, LIX, TULLYJ C. Time-dependent density functional theory Ehrenfest dynamics: Collisions between atomic oxygen and graphite clusters [J]. J Chem Phys, 2007, 1261134307

[10] WUH, LUS-f, ZUOQ-w, COLMENARESE. A high accuracy computing reduction algorithm based on data reuse for direct dynamics simulations [C]. The 2016 International Conference on Computational Science and Computational Intelligence (CSCI'16). Las Vegas, USA: IEEE, 20161243-1247

[11] WUH, LUS-f, ZHUN-j, LIUJ-l, COLMENARESE. A high order predictor-corrector integration algorithm for first principle chemical dynamics simulations [J]. Journal of Theoretical and Computational Chemistry, 2016, 15(1): 1-12

[12] LIUX-c, ZHANGD-f, HENGG, WUHeng. The comparative of a family of high order predictor-corrector algorithms for first principle chemical dynamics simulation [J]. International Journal of Enhanced Research in Science, Technology & Engineering, 2015, 4(10): 65-75

[13] WUH, RANHMANM, WANGJ, LOURDERAJU, HASEW L, ZHUANGY. Higher-accuracy schemes for approximating the Hessian from electronic structure calculations in chemical dynamics simulations [J]. J Chem Phys, 2010, 133(1): 074101

[14] HRATCHIANH P, SCHLEGELH B. Using Hessian updating to increase the efficiency of a Hessian based predictor-corrector reaction path following method [J]. J Chem Theory Comput, 2005, 1161-69

[15] SCHLEGELH B. Geometry optimization [J]. Wiley Interdisciplinary Reviews: Computational Molecular Science, 2011, 1(5): 790-809

[16] WUH, LUS-fei. Evaluating the accuracy of a third order of a third order Hssian-based predictor-corrector integrator [C]. The 30th European Simulation and Modelling Conference. Las Palmas, Spain: EUROSES, 2016275278

[17] GUTZWILLERM CChaos in classical and quantum mechanics [M], 1991129

[18] ODELLA, DELINA, JOHANSSONB, BOCKN, CHALLACOMBEM, NIKLASSONA M N. Higher-order symplectic integration in Born–Oppenheimer molecular dynamics [J]. J Chem Phys, 2009, 131(1): 244106

[19] HUX, HASEW L, PIRRAGLIAT. Vectorization of the general Monte Carlo classical trajectory program VENUS [J]. J Comput Chem, 1991, 1211014-1024

[20] VALIEVM, BYLASKAE J, WANGD, NIEPLOCHAJ, APRAE, WINDUST L, DEJONGWA. NWChem: A comprehensive and scalable open-source solution for large scale molecular simulations [J]. Comput Phys Commun, 2010, 181(1): 1477-1489