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Hypergeometric Polynomials, Hyperharmonic Discrete and Continuous Expansions: Evaluations, Interconnections, Extensions

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Computational Science and Its Applications – ICCSA 2019 (ICCSA 2019)

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Abstract

The important mathematical subject of special functions and orthogonal polynomials found in the last decades a systematization regarding those of hypergeometric type. The growth of these developments are due to interconnections with quantum angular momentum theory which is basic to that of spin-networks, of recent relevance in various branches of physics. Here we consider their power as providing expansion basis sets such as specifically are needed in chemistry to represents potential energy surfaces, the achievements being discussed and illustrated. A novel visualization of key members of the polynomial sets attributes a central role to the Kravchuk polynomials: its relationship with Wigner’s rotation matrix elements are here emphasized and taken as exemplary for computational and analytical features. The sets are considered regarding progress on the formulation of a discretization technique, the hyperquantization, which allows to efficiently deal with physical problems where quantum mechanical operators act on continuous manifolds, to yield discrete grids suitable for computation of matrix elements without need of multidimensional integration.

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Change history

  • 22 January 2021

    The original version of this chapter was inadvertently published without two authors who contributed to the chapter also. The missing authors were Noelia Faginas-Lago and Andrea Lombardi. Their names and affiliations have now been added and the correct sequence of the authors is: Cecilia Coletti, Federico Palazzetti, Roger W. Anderson, Vincenzo Aquilanti, Noelia Faginas-Lago and Andrea Lombardi.

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Authors and Affiliations

  1. Dipartimento di Farmacia, Università “G. D’Annunzio” Chieti-Pescara, Chieti, Italy

    Cecilia Coletti

  2. Dipartimento di Chimica, Biologia e Biotecnologie, Università di Perugia, via Elce di Sotto 8, 06123, Perugia, Italy

    Federico Palazzetti, Vincenzo Aquilanti, Noelia Faginas-Lago & Andrea Lombardi

  3. Department of Chemistry, University of California, Santa Cruz, CA, 95064, USA

    Roger W. Anderson

  4. Istituto di Struttura della Materia-Consiglio Nazionale delle Ricerche, 00016, Rome, Italy

    Vincenzo Aquilanti

Authors
  1. Cecilia Coletti
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  2. Federico Palazzetti
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  3. Roger W. Anderson
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  4. Vincenzo Aquilanti
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  5. Noelia Faginas-Lago
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  6. Andrea Lombardi
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Correspondence to Federico Palazzetti .

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Editors and Affiliations

  1. Covenant University, Ota, Nigeria

    Sanjay Misra

  2. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  3. University of Basilicata, Potenza, Italy

    Beniamino Murgante

  4. Saint Petersburg State University, Saint Petersburg, Russia

    Elena Stankova

  5. Saint Petersburg State University, Saint Petersburg, Russia

    Vladimir Korkhov

  6. Polytechnic University of Bari, Bari, Italy

    Carmelo Torre

  7. University of Minho, Braga, Portugal

    Ana Maria A.C. Rocha

  8. Monash University, Clayton, VIC, Australia

    David Taniar

  9. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  10. Polytechnic University of Bari, Bari, Italy

    Eufemia Tarantino

Appendix: Discrete and Continuous Harmonic Expansions of Intermolecular Interaction

Appendix: Discrete and Continuous Harmonic Expansions of Intermolecular Interaction

Experiments carried out by the Stern-Gerlach magnet in the early 1990s, shown that under supersonic conditions seeded O2 would undergo variation in speed and redistribution of the internal states (for the case of oxygen rotational and vibrational cooling), producing as additional effect the molecular alignment, i.e. the possibility of forcing the molecule to rotate in a preferential plane. Alignment was initially induced to various diatomic molecules and linear hydrocarbons, then extended to disk-shaped molecules like benzene. Current developments are attempting to exploit the helicoidal motion of chiral molecules. The technique determined significant advances in the area of the molecular beam scattering, especially for what concerns the phenomenology of anisotropies related to van der Waals interactions.

A conceptually important advance of the asymptotic (semiclassical) discretization of continuous function exemplified in this work lead to hyperquantization algorithms. They are based on the observation that discrete analogues of hyperspherical harmonics can be defined by means of the 3nj symbols. The method was initially employed to solve the Schrödinger equation for the prototypical reaction F + H2 and generalized to include other triatomic systems and various quantum effects. Exact representations have been also employed to describe steric effects in quantum mechanics, such as the interpretation of reactive scattering resonances. Tools refined for the quantum mechanical treatment of few body systems have been employed to give a reformulation of the classical mechanics protocol, to be applied in many-body problems characterizing atomic and molecular clusters.

More conventionally, spherical and hyperspherical harmonics are implemented in quantum chemical calculations to compute energy as a function of a properly defined distance and of one or more angles, according to the complexity of the system. This approach is inspired by the open-shell – closed-shell atom–atom interactions, where the angle θ defines a minimal model for collinear (θ = 0) and perpendicular (θ = π/2) configurations and had been initially to the atom – diatom case [60, 69, 88]. The method consists in an exact transformation of quantum chemical (or experimental) input data related to a minimum set of configurations, the “leading configurations”, whose choice relies upon geometrical and physical characteristics of the system, by a multipolar expansion [89].

Extension of the simplest case, the abovementioned triatomic system, have been done for four- and five- body problems. For example, systems formed by two diatomic molecules can be described by two Jacobi vectors lying along the chemical bonds and a third vector which joins the centers-of-mass of the two molecules. Hyperspherical harmonics have also been applied to describe interactions of floppy molecules, characterized by having an active torsional motion, as a prototype of enantiomeric change in chiral molecules [71, 73].

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Coletti, C., Palazzetti, F., Anderson, R.W., Aquilanti, V., Faginas-Lago, N., Lombardi, A. (2019). Hypergeometric Polynomials, Hyperharmonic Discrete and Continuous Expansions: Evaluations, Interconnections, Extensions. In: Misra, S., et al. Computational Science and Its Applications – ICCSA 2019. ICCSA 2019. Lecture Notes in Computer Science(), vol 11624. Springer, Cham. https://doi.org/10.1007/978-3-030-24311-1_34

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