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

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.