Burger Steven K, Yang Weitao
Department of Chemistry, Duke University, Box 90346, Durham, North Carolina 27708-0346, USA.
J Chem Phys. 2006 Dec 28;125(24):244108. doi: 10.1063/1.2402166.
The diagonally implicit Runge-Kutta framework is shown to be a general form for constructing stable, efficient steepest descent reaction path integrators, of any order. With this framework tolerance driven, adaptive step-size methods can be constructed by embedding methods to obtain error estimates of each step without additional computational cost. There are many embedded and nonembedded, diagonally implicit Runge-Kutta methods available from the numerical analysis literature and these are reviewed for orders two, three, and four. New embedded methods are also developed which are tailored to the application of reaction path following. All integrators are summarized and compared for three systems: the Muller-Brown [Theor. Chem. Acta 53, 75 (1979)] potential and two gas phase chemical reactions. The results show that many of the methods are capable of integrating efficiently while reliably keeping the error bound within the desired tolerance. This allows the reaction path to be determined through automatic integration by only specifying the desired accuracy and transition state.
对角隐式龙格 - 库塔框架被证明是构建任意阶稳定、高效最速下降反应路径积分器的通用形式。借助该框架，可通过嵌入方法构建由容差驱动的自适应步长方法，从而无需额外计算成本就能获得每一步的误差估计。数值分析文献中有许多嵌入式和非嵌入式对角隐式龙格 - 库塔方法，本文对二阶、三阶和四阶的此类方法进行了综述。还开发了专门适用于反应路径跟踪应用的新嵌入式方法。针对三个系统总结并比较了所有积分器：穆勒 - 布朗势[《理论化学学报》53, 75 (1979)]以及两个气相化学反应。结果表明，许多方法能够高效积分，同时可靠地将误差限制在所需容差范围内。这使得仅通过指定所需精度和过渡态就能通过自动积分确定反应路径。
