Burger Steven K, Yang Weitao
Department of Chemistry, Duke University, Durham, North Carolina 27708-0346, USA.
J Chem Phys. 2006 Jun 14;124(22):224102. doi: 10.1063/1.2202830.
We present the use of an optimal combined explicit-implicit method for following the reaction path to high accuracy. This is in contrast to most purely implicit reaction path integration algorithms, which are only efficient on stiff ordinary differential equations. The defining equation for the reaction path is considered to be stiff, however, we show here that the reaction path is not uniformly stiff and instead is only stiff near stationary points. The optimal algorithm developed in this work is a combination of explicit and implicit methods with a simple criterion to switch between the two. Using three different chemical reactions, we combine and compare three different integration methods: the implicit trapezoidal method, an explicit stabilized third order algorithm implemented in the code DUMKA3 and the traditional explicit fourth order Runge-Kutta method written in the code RKSUITE. The results for high accuracy show that when the implicit trapezoidal method is combined with either explicit method the number of energy and gradient calculations can potentially be reduced by almost a half compared with integrating either method alone. Finally, to explain the improvements of the combined method we expand on the concepts of stability and stiffness and relate them to the efficiency of integration methods.
我们展示了一种用于高精度追踪反应路径的最优显隐式组合方法的应用。这与大多数纯隐式反应路径积分算法形成对比，后者仅在刚性常微分方程上有效。反应路径的定义方程被认为是刚性的，然而，我们在此表明反应路径并非处处刚性，而是仅在驻点附近才是刚性的。本文所开发的最优算法是显式和隐式方法的组合，并带有一个在两者之间切换的简单准则。使用三种不同的化学反应，我们组合并比较了三种不同的积分方法：隐式梯形法、在代码DUMKA3中实现的显式稳定三阶算法以及在代码RKSUITE中编写的传统显式四阶龙格 - 库塔法。高精度结果表明，当隐式梯形法与任何一种显式方法相结合时，与单独使用任何一种方法进行积分相比，能量和梯度计算的次数有可能减少近一半。最后，为了解释组合方法的改进，我们详细阐述了稳定性和刚性的概念，并将它们与积分方法的效率联系起来。
