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Hypergeometric orthogonal polynomials as expansion basis sets for atomic and molecular orbitals:The Jacobi ladder

机译：超越正交多项式作为原子和分子轨道膨胀基集：雅各梯

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In this survey we account for basic mathematical ingredients for dealing with quantum chemical problems.We focus on comprehensive previous work(Coletti et al,2013,pp.74-127,Ref 1)documenting relationships with the Askey scheme,a classification of the orthogonal polynomials sets of hypergeometric type.A reduction of the scheme is proposed individuating nine fundamental functional sets which have their counterparts in quantum mechanics;they occur in the general Kepler-Coulomb problem:as well known basis sets for expansions of orbitals in quantum chemistry and in the treatment of specific atomic and molecular applications.A novelty of the approach,with respect to this extensively covered topic,is the establishment of this representation for Kravchuk polynomials,on the mathematical side and,correspondingly,of the spherical top wavefunctions on the physical side:the latter are explicitly connected with the Wigner's rotation matrix of angular momentum theory.Novel presentations of the Askey-type hierarchy of hypergeometrical orthonormal basis sets relevant in quantum mechanics and the relationships connecting them are established by powerful tools:from the mathematical viewpoint,the Askey duality and asymptotic analysis;from a physical viewpoint,the symmetry by transposition and semiclassical limits.A new three-by-three matrix visualization illustrates the set of correspondences to assist further work on the path connecting classical and quantum physics and discrete and continuous mathematics that is presented elsewhere(Coletti et al.,2019,Ref.46).This is pictured as a bridge where Racah polynomials and harmonic oscillator wavefunctions are the corner stones,while the rotation matrix of Wigner is the keystone.Here,the path is illustrated as the steps of a stairway that we define as the Jacobi ladder,where going up and down is insightful for applications.Extension to the full Askey scheme,object of future work,is briefly noted:some reference is made to our recent progress in spherical to hyperspherical manifold representations involving the q-scheme of Askey and related orthogonal polynomials as possible orthonormal basis sets in quantum mechanics.
机译：在本次调查中，我们考虑了处理量子化学问题的基本数学成分。我们专注于全面的上一项工作（Coletti等，2013，PP.74-127，Ref 1）与ASKey方案的关系，对正交的分类多项式超大距型。提出了该方案的减少，提出了在量子力学中的九个基本函数集中的分组;它们发生在一般的Kepler-Coulomb问题中：众所周知的基础集，用于量子化学中的轨道扩展的基础集。关于特定原子和分子应用的治疗。关于这种广泛覆盖的话题的方法是对kravchuk多项式的这种代表性，在数学侧和相应地，在物理方面的球形顶部波力发生时建立这一表现形式：后者与角色势理论的Wigner旋转矩阵明确连接。askey的呈现 - 在量子力学中相关的HeatherMetrical基础组的类型层次结构和连接它们的关系是由强大的工具建立的：从数学观点，ASKEY二元性和渐近分析;从物理观点来看，通过换位和半透明限制的对称性.A新的三个 - 三个矩阵可视化示出了一组相应的，以帮助进一步研究连接其他地方（Coletti等，2019，Ref.46）的传统和量子物理学和离散和连续数学的路径。这是如此Racah多项式和谐波振荡器波力的桥梁是拐角石头，而Wigner的旋转矩阵是梯形的旋转矩阵。该路径被示出为我们定义为雅各的梯子的步骤，上下朝向下降对于应用程序。简要介绍了未来工作的全部askey方案的细长，并注意到我们最近进展的一些参考在球形到高度球形的歧管表示，涉及askey和相关正交多项式的q-schreation在量子力学中可能的正交基础。

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