Rules on the changes of approximate symmetry measures along reaction paths
Rules on the changes of approximate symmetry measures along reaction paths
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近似对称性测度沿反应路径变化的规则
DOI：
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发表时间：
2006
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通讯作者：
P. Mezey
中科院分区：
文献类型：
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作者：
P. Mezey
一个定理被证明，描述了当核几何形状沿着反应路径变化时，非对称分子结构的不同对称元素逐渐出现的差异。各个对称元素的逐渐建立是根据对称缺陷度量沿着通向临界点的最陡下降路径的收敛速率来描述的。该临界点可以是势能超曲面的任何指数的最小值或鞍点，只要最陡下降路径通向它即可。如果在临界点的二次邻域内，局部内坐标以 Hessian 矩阵的特征向量给出，则该定理断言，对于沿着临界点的二次邻域内的最速下降路径移位的任何扭曲核排列，沿着 Hessian 的最低正特征值的特征向量存在的对称元素以相对于其他扭曲的恢复速率的指数速率恢复最快。对称元素。尽管这只是明显的对比，但请注意，该结果可能与已知的内部坐标值的收敛行为形成对比，因为沿着相同的特征向量，实际的内部坐标值在收敛到临界点时恢复最慢。分子可能的核构型的完整集合永远不可能是向量空间，但是，使用流形理论方法可以引入一致的局部坐标系集合。因此，该定理首先证明了根据简化的核构型空间（核构型的流形 M）的度量定义的对称性缺陷测度，然后证明了具有有限次多项式依赖于核构型的界限的非常一般的对称性缺陷测度族。指出了该定理在几何优化中的作用的一些含义。 † 本文谨献给 Michael A. Robb 教授。
A theorem is proven, describing the differences in the gradual emergence of different symmetry elements of non-symmetric molecular structures as the nuclear geometries change along reaction paths. The gradual establishment of individual symmetry elements is described in terms of the rates of convergence of symmetry deficiency measures along steepest descent paths leading to a critical point. This critical point may be a minimum or a saddle point of any index of the potential energy hypersurface, as long as the steepest descent path leads to it. If within a quadratic neighbourhood of the critical point the local internal coordinates are given in terms of the eigenvectors of the Hessian matrix, then the theorem asserts that, for any distorted nuclear arrangement displaced along a steepest descent path within the quadratic neighbourhood of a critical point, the symmetry element that is present along the eigenvector of the lowest positive eigenvalue of the Hessian is restored the fastest, at an exponential rate relative to the restoration rate of other distorted symmetry elements. Although it is only an apparent contrast, note that, this result may appear in contrast to the known convergence behaviour of internal coordinate values, since along the same eigenvector the actual internal coordinate value is restored the slowest, while converging to the critical point. The complete set of possible nuclear configurations of a molecule can never be a vector space, however, using the manifold theoretical approach consistent sets of local coordinate systems can be introduced. Accordingly, the theorem is first proven for a symmetry deficiency measure defined in terms of the metric of the reduced nuclear configuration space (manifold M of nuclear configurations), and then for a very general family of symmetry deficiency measures with bounds of finite degree polynomial dependence on nuclear configurations. Some implications of the role of the theorem in geometry optimizations are pointed out. † This paper is dedicated to Professor Michael A. Robb.