Vector parametrization of the N-body problem in quantum mechanics: Polyspherical coordinates.
Vector parametrization of the N-body problem in quantum mechanics: Polyspherical coordinates.
复制标题
量子力学中 N 体问题的矢量参数化：多球坐标。
DOI：
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
Iung
中科院分区：
文献类型：
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作者：
Chapuisat;Iung
分离质心运动后，n体系统的构型可以完全用N-1个相对位置向量表示。许多通常用于描述分子构型的坐标集合可以被看作是球坐标，对于各种矢量，集合在一起。球面角是局部的；也就是说，它们是为从一个矢量到另一个矢量变化的帧定义的。这种性质的每一组特定的坐标（多球坐标）由三个欧拉角组成，用于物体固定框架的整体旋转，以及3N-6个内部坐标：N-1个矢量长度，N-2个矢量对之间的平面角度，以及N-3个矢量之间围绕第三个矢量的二面角。本文旨在开发这种类型的参数化的一个例子，其中身体固定框架z轴平行于一个矢量。导出了所描述系统的量子力学动能算符。研究了算子在函数基集角部分的作用（欧拉角为Wigner旋转矩阵元，内角为球次谐波），并详细描述了表示动能算子的矩阵结构。讨论了现有矢量参数化和多球坐标的优缺点。主要的优点是在数值上计算动能算子的矩阵元素：所有角度的积分都是解析实现的，因此数值上的努力只集中在N-1径向坐标上。径向基函数将根据物理环境（碰撞或振动，或任何其他）来选择。因此，所提出的角基集构成了动能算子的充分有限基表示，并与势能的离散变量表示相结合，可能为三以上粒子系统的动力学研究提供有效的配置框架。
The configuration of an N-body system can be entirely represented by N-1 relative position vectors after separation of the center-of-mass motion. Many of the sets of coordinates that are commonly used for describing molecular configurations can be viewed as spherical coordinates, for the various vectors, collected together. The spherical angles are local; i.e., they are defined for frames that change from one vector to another. Each particular set of coordinates of that nature (polyspherical coordinates) consists of three Euler angles for the overall rotation of the body-fixed frame and 3N-6 internal coordinates: the N-1 vector lengths, N-2 planar angles between pairs of vectors, and N-3 dihedral angles between two vectors around a third one. This article aims at developing an example of this type of parametrization, where the body-fixed-frame z axis is parallel to one vector. The quantum-mechanical kinetic-energy operator for the system so described is derived. The operator action on the angular part of the functional basis set is studied (Wigner rotation matrix elements for the Euler angles and spherical harmonics for the internal angles), and the structure of the matrix representing the kinetic-energy operator is described in detail. The advantages and drawbacks of the present vector parametrization and the polyspherical coordinates are discussed. The principal advantage is in numerically calculating the matrix elements of the kinetic-energy operator: The integration over all angles turns out to be analytically achieved, so that the numerical effort is to be concentrated only on the N-1 radial coordinates. Radial basis functions are to be selected according to the physical context (collisonal or vibrational, or any other). Thus the angular basis set proposed constitutes an adequate finite-basis representation for the kinetic-energy operator and, combined with a discrete-variable representation for the potential energy, is likely to provide an efficient collocation framework for the dynamical study of more-than-three particle systems.