Aguilar-Mogas Antoni, Giménez Xavier, Bofill Josep Maria
Departament de Química Física i Centre especial de Recerca en Química Teòrica, Universitat de Barcelona i Parc Científic de Barcelona, Martí i Franquès 1, Barcelona, Spain.
J Chem Phys. 2008 Mar 14;128(10):104102. doi: 10.1063/1.2834930.
The intrinsic reaction coordinate curve (IRC), normally proposed as a representation of a reaction path, is parametrized as a function of the potential energy rather than the arc-length. This change in the parametrization of the curve implies that the values of the energy of the potential energy surface points, where the IRC curve is located, play the role of reaction coordinate. We use Caratheodory's relation to derive in a rigorous manner the proposed parametrization of the IRC path. Since this Caratheodory's relation is the basis of the theory of calculus of variations, then this fact permits to reformulate the IRC model from this mathematical theory. In this mathematical theory, the character of the variational solution (either maximum or minimum) is given through the Weierstrass E-function. As proposed by Crehuet and Bofill [J. Chem. Phys. 122, 234105 (2005)], we use the minimization of the Weierstrass E-function, as a function of the potential energy, to locate an IRC path between two minima from an arbitrary curve on the potential energy surface, and then join these two minima. We also prove, from the analysis of the Weierstrass E-function, the mathematical bases for the algorithms proposed to locate the IRC path. The proposed algorithm is applied to a set of examples. Finally, the algorithm is used to locate a discontinuous, or broken, IRC path, namely, when the path connects two first order saddle points through a valley-ridged inflection point.
内禀反应坐标曲线（IRC）通常被视为反应路径的一种表示，它是以势能而非弧长的函数来参数化的。曲线参数化方式的这种变化意味着，IRC曲线所在的势能面各点的能量值起到了反应坐标的作用。我们利用卡拉西奥多里关系严格推导了所提出的IRC路径的参数化。由于这种卡拉西奥多里关系是变分法理论的基础，那么这一事实使得我们能够从该数学理论出发重新构建IRC模型。在这个数学理论中，变分解的性质（最大值或最小值）是通过魏尔斯特拉斯E函数给出的。正如克雷韦特和博菲尔[《化学物理杂志》122, 234105 (2005)]所提出的，我们将魏尔斯特拉斯E函数作为势能的函数进行最小化，以便从势能面上的任意曲线找到两个极小值之间的IRC路径，然后连接这两个极小值。我们还通过对魏尔斯特拉斯E函数的分析，证明了用于确定IRC路径的算法的数学基础。所提出的算法应用于一组示例。最后，该算法用于确定一条不连续或间断的IRC路径，即当路径通过一个谷脊型拐点连接两个一阶鞍点时。
