Lepetit Bruno, Wang Desheng, Kuppermann Aron
Laboratoire Collisions, Agrégats, Réactivité, UMR 5589, CNRS, IRSAMC, Université Paul Sabatier Toulouse 3, 31062 Toulouse Cedex 9, France.
J Chem Phys. 2006 Oct 7;125(13):133505. doi: 10.1063/1.2218515.
A numerical generation method of hyperspherical harmonics for tetra-atomic systems, in terms of row-orthonormal hyperspherical coordinates-a hyper-radius and eight angles-is presented. The nine-dimensional coordinate space is split into three three-dimensional spaces, the physical rotation, kinematic rotation, and kinematic invariant spaces. The eight-angle principal-axes-of-inertia hyperspherical harmonics are expanded in Wigner rotation matrices for the physical and kinematic rotation angles. The remaining two-angle harmonics defined in kinematic invariant space are expanded in a basis of trigonometric functions, and the diagonalization of the kinetic energy operator in this basis provides highly accurate harmonics. This trigonometric basis is chosen to provide a mathematically exact and finite expansion for the harmonics. Individually, each basis function does not satisfy appropriate boundary conditions at the poles of the kinetic energy operator; however, the numerically generated linear combination of these functions which constitutes the harmonic does. The size of this basis is minimized using the symmetries of the system, in particular, internal symmetries, involving different sets of coordinates in nine-dimensional space corresponding to the same physical configuration.
提出了一种基于行正交超球坐标（一个超半径和八个角度）的四原子体系超球谐函数的数值生成方法。九维坐标空间被划分为三个三维空间，即物理旋转空间、运动学旋转空间和运动学不变空间。八角度惯性主轴超球谐函数在物理和运动学旋转角度的维格纳旋转矩阵中展开。在运动学不变空间中定义的其余两角度谐函数在三角函数基中展开，并且在此基中动能算符的对角化提供了高精度的谐函数。选择此三角函数基是为了为谐函数提供数学上精确且有限的展开。单独来看，每个基函数在动能算符的极点处不满足适当的边界条件；然而，构成谐函数的这些函数的数值生成线性组合满足边界条件。利用体系的对称性，特别是内部对称性，最小化此基的大小，内部对称性涉及九维空间中对应于相同物理构型的不同坐标集。
