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Continuous and Discrete Algorithms in Quantum Chemistry: Polynomial Sets, Spin Networks and Sturmian Orbitals

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Computational Science and Its Applications – ICCSA 2013 (ICCSA 2013)

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Abstract

An effort is accounted for in the present paper to exhibit the recently actively investigated connection between the search and use of ”orbitals” as basis sets in applied quantum mechanics and current advances in the mathematics of special functions and orthogonal polynomials, which are in turn motivated by the developments of the quantum theory of angular momentum. The latter theory in modern applications forms the basis for the class of ”spin-network” algorithms. These ”orbitals” enjoy important properties regarding orthogonality and completeness. In configuration space, they are often designated as Kepler-Coulomb Sturmian orbitals, in momentum space they are intimately connected with hyperspherical harmonics. The paper contains a brief presentation including also computational results and a discussion oriented towards the numerical use of these orbitals.

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References

  1. Nikiforov, A., Suslov, S., Uvarov, V.: Classical Orthogonal Polynomials of a Discrete Variable. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  2. Aquilanti, V., Cavalli, S., Coletti, C.: Angular and hyperangular momentum recoupling, harmonic superposition and Racah polynomials: a recursive algorithm. Chem. Phys. Lett. 344, 587 (2001)

    Article  Google Scholar 

  3. Ragni, M., Bitencourt, A.C.P., da S. Ferreira, C., Aquilanti, V., Anderson, R.W., Littlejohn, R.G.: Exact computation and asymptotic approximations of 6j symbols: Illustration of their semiclassical limits. Int. J. Quantum Chem. 110, 731 (2010)

    Article  Google Scholar 

  4. Aquilanti, V., Marinelli, D., Marzuoli, A.: Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials. J. Phys. A: Math. Theor. 46, 175303 (2013)

    Article  Google Scholar 

  5. Bitencourt, A.C.P., Marzuoli, A., Ragni, M., Anderson, R.W., Aquilanti, V.: Exact and Asymptotic Computations of Elementary Spin Networks: Classification of the Quantum–Classical Boundaries. In: Murgante, B., Gervasi, O., Misra, S., Nedjah, N., Rocha, A.M.A.C., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2012, Part I. LNCS, vol. 7333, pp. 723–737. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  6. Anderson, R.W., Aquilanti, V., Bitencourt, A.C.P., Marinelli, D., Ragni, M.: The screen representation of spin networks: 2D recurrence, eigenvalue equation for 6j symbols, geometric interpretation and hamiltonian dynamics. In: Murgante, B., Misra, S., Carlini, M., Torre, C.M., Quang, N.H., Taniar, D., Apduhan, B.O., Gervasi, O. (eds.) ICCSA 2013, Part II. LNCS, vol. 7972, pp. 46–59. Springer, Heidelberg (2013)

    Google Scholar 

  7. Ragni, M., Littlejohn, R.G., Bitencourt, A.C.P., Aquilanti, V., Anderson, R.W.: The Screen representation of spin networks. Images of 6j symbols and semiclassical features. In: Murgante, B., Misra, S., Carlini, M., Torre, C.M., Quang, N.H., Taniar, D., Apduhan, B.O., Gervasi, O. (eds.) ICCSA 2013, Part II. LNCS, vol. 7972, pp. 60–72. Springer, Heidelberg (2013)

    Google Scholar 

  8. Coletti, C., Calderini, D., Aquilanti, V.: d-dimensional Kepler-Coulomb Sturmians and Hyperspherical Harmonics as Complete Orthonormal Atomic and Molecular Orbitals. Adv. Quantum Chem. (2013) (in press)

    Google Scholar 

  9. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Function. Dover, New York (1964)

    Google Scholar 

  10. Aquilanti, V., Bitencourt, A.C.P., da S. Ferreira, C., Marzuoli, A., Ragni, M.: Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity. Physica Scripta 78, 058103 (2008)

    Google Scholar 

  11. Aquilanti, V., Bitencourt, A.C.P., da S. Ferreira, C., Marzuoli, A., Ragni, M.: Combinatorics of angular momentum recoupling theory: spin networks, their asymptotics and applications. Theor. Chem, Acc. 123, 237–247 (2009)

    Article  Google Scholar 

  12. Aquilanti, V., Coletti, C.: 3nj-symbols and harmonic superposition coefficients: an icosahedral abacus. Chem. Phys. Lett. 344, 601 (2001)

    Article  Google Scholar 

  13. De Fazio, D., Cavalli, S., Aquilanti, V.: Orthogonal Polynomials of a discrete variable as expansion basis sets in quantum mechanics. The hyperquantization algorithm. Int. J. Quantum Chem. 93, 91–111 (2003)

    Article  Google Scholar 

  14. Aquilanti, V., Haggard, H.M., Littlejohn, R.G., Yu, L.: Semiclassical analysis of Wigner 3j-symbol. J. Phys. A 40, 5637–5674 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Aquilanti, V., Haggard, H.M., Hedeman, A., Jeevanjee, N., Littlejohn, R.G., Yu, L.: Semiclassical mechanics of the Wigner 6J-symbol. Journal of Physics A 45, 065209 (2012)

    Google Scholar 

  16. De Fazio, D., Aquilanti, V., Cavalli, S., Aguilar, A., Lucas, J.M.: Exact state-to-state quantum dynamics of the F + HD → HF + D reaction on model potential energy surfaces. J. Chem. Phys. 129, 064303 (2008)

    Google Scholar 

  17. Aquilanti, V., Capecchi, G.: Harmonic analysis and discrete polynomials. From semiclassical angular momentum theory to the hyperquantization algorithm. Theor. Chem. Acc. 104, 183 (2000)

    Article  Google Scholar 

  18. Aquilanti, V., Cavalli, S., Grossi, G., Anderson, R.W.: Stereodirected States in Molecular Dynamics: A Discrete Basis Representation for the Quantum Mechanical Scattering Matrix. J. Phys. Chem. 95, 8184–8193 (1991)

    Article  Google Scholar 

  19. Anderson, R.W., Aquilanti, V.: The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors. J. Chem. Phys. 124, 214104 (2006)

    Article  Google Scholar 

  20. Koekoek, R., Swarttouw, R.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, TU Delft, The Netherlands (1998), Anonymous ftpsite:ftp.twi.tudelft.nl, directory:/pub/publications/tech-reports

    Google Scholar 

  21. Askey, R.: Duality for classical orthogonal polynomials. J. Comp. App. Math. 178, 37–43 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Aquilanti, V., Cavalli, S., Coletti, C.: Hyperspherical Symmetry of Hydrogenic Orbitals and Recoupling Coefficients among Alternative Bases. Phys. Rev. Lett. 80, 3209 (1998)

    Article  Google Scholar 

  23. Garcia, A.G., Hernandez-Medina, M.A.: Discrete Sturm–Liouville problems, Jacobi matrices and Lagrange interpolation series. J. Math. Anal. Appl. 280, 221–231 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Safouhi, H., Hoggan, P.E.: New methods for accelerating the convergence of molecular electronic integrals over exponential type orbitals. Mol. Phys. 101, 19 (2003)

    Article  Google Scholar 

  25. Hoggan, P.E.: General two-electron exponential type orbital integrals in polyatomics without orbital translations. Int. J. Quantum Chem. 109, 2926 (2009)

    Article  Google Scholar 

  26. Fernández Rico, J., López, R., Aguado, A., Ema, I., Ramírez, G.: Reference program for molecular calculations with Slater-type orbitals. J. Comp. Chem. 19, 1284 (1998)

    Article  Google Scholar 

  27. Fernández Rico, J., López, R., Ema, I., Ramírez, G.: Electrostatic potentials and fields from density expansions of deformed atoms in molecules. J. Comp. Chem. 25, 1347 (2004)

    Article  Google Scholar 

  28. Ruiz, M.B., Peuker, K.: Analytical expressions of exchange and three-center nuclear attraction integrals over 1s and 2s Slater orbitals with different exponents. Chem. Phys. Lett. 412, 244 (2005)

    Article  Google Scholar 

  29. Shull, H., Löwdin, P.O.: Superposition of Configurations and Natural Spin Orbitals. Applications to the He Problem. J. Chem. Phys. 30, 617 (1959)

    Article  Google Scholar 

  30. Avery, J.: Hyperspherical Harmonics and Generalized Sturmians. Kluwer Academic Publishers, Dordrecht (2000)

    MATH  Google Scholar 

  31. Aquilanti, V., Cavalli, S., Coletti, C., Di Domenico, D., Grossi, G.: Hyperspherical harmonics as Sturmian orbitals in momentum space: a systematic approach to the few-body Coulomb problem. Int. Rev. Phys. Chem. 20, 673 (2001)

    Article  Google Scholar 

  32. Aquilanti, V., Caligiana, A., Cavalli, S., Coletti, C.: Hydrogenic Orbitals in Momentum Space and Hyperspherical Harmonics. Elliptic Sturmian Basis Sets. Int. J. Quantum Chem. 92, 212 (2003)

    Article  Google Scholar 

  33. Randazzo, J.M., Ancarani, L.U., Gasaneo, G., Frappicini, A.L., Colavecchia, F.D.: Optimal Sturmian basis functions for atomic three-body systems. Phys. Rev. A 81, 042520 (2010)

    Google Scholar 

  34. Randazzo, J.M., Frappicini, A.L., Colavecchia, F.D., Gasaneo: Discrete sets of Sturmian functions applied to two-electron atoms. Phys. Rev. A 79, 022507 (2009)

    Google Scholar 

  35. Mitnik, D.M., Colavecchia, F.D., Gasaneo, G., Randazzo, J.M.: Computational methods for Generalized Sturmans basis. Comp. Phys. Comm. 182, 1145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Avery, J., Avery, J.: Generalized Sturmian Solutions for Many-Particle Schrödinger Equations. J. Phys. Chem. A 108, 8848 (2004)

    Article  Google Scholar 

  37. Avery, J., Avery, J.: Coulomb Sturmians as a basis for molecular calculations. Mol. Phys. 110, 1593 (2012)

    Article  Google Scholar 

  38. Aquilanti, V., Cavalli, S., Coletti, C., Grossi, G.: Alternative Sturmian Bases and Momentum Space Orbitals: an Application to the Hydrogen Molecular Ion. Chem. Phys. 209, 405 (1996)

    Article  Google Scholar 

  39. Calderini, D., Cavalli, S., Coletti, C., Grossi, G., Aquilanti, V.: Hydrogenoid orbitals revisited: From Slater orbitals to Coulomb Sturmians. J. Chem. Sci. 124, 187 (2012)

    Article  Google Scholar 

  40. Fock, V.: Zur Theorie des Wasserstoffatoms. Z. Phys. 98, 145 (1935)

    Article  Google Scholar 

  41. Macek, J.: Properties of autoionizing states of He. J. Phys. B: Atom. Molec. Phys. 1, 831 (1968)

    Article  Google Scholar 

  42. Smith, F.T.: Generalized Angular Momentum in Many-Body Collisions. Phys. Rev. 120, 1058 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  43. Smith, F.T.: A Symmetric Representation for Three Body Problems. I. Motion in a Plane. J. Math. Phys. 3, 735 (1962)

    Article  MATH  Google Scholar 

  44. Ballot, L., Fabre de la Ripelle, M.: Application of the hyperspherical formalism to the trinucleon bound state problems. Ann. Phys. 127, 62 (1980)

    Article  Google Scholar 

  45. Aquilanti, V., Cavalli, S., Coletti, C., De Fazio, D., Grossi, G.: In: Tsipis, C.A., Popov, V.S., Herschbach, D.R. (eds.) New Methods in Quantum Theory, pp. 233–250. Kluwer (1996)

    Google Scholar 

  46. Aquilanti, V., Cavalli, S., De Fazio, D.: Hyperquantization algorithm. II. Implementation for the F+H2 reaction dynamics including open-shell and spin-orbit interactions. J. Chem. Phys. 109, 3805 (1998)

    Article  Google Scholar 

  47. Aquilanti, V., Cavalli, S., Coletti, C.: The d-dimensional hydrogen atom: hyperspherical harmonics as momentum space orbitals and alternative Sturmian basis sets. Chem. Phys. 214, 1 (1997)

    Article  Google Scholar 

  48. Aquilanti, V., Cavalli, S., Grossi, G.: Hyperspherical coordinates for molecular dynamics by the method of trees and the mapping of potential energy surfaces for triatomic systems. J. Chem. Phys. 85, 1362 (1986)

    Article  Google Scholar 

  49. Drake, G.W.F.: Atomic, Molecular and Optical Physics Handbook. AIP, New York (1996)

    Google Scholar 

  50. Morton, D.C., Wu, Q., Drake, G.W.F.: Energy Levels for the Stable Isotopes of Atomic Helium (4He I and 3He I). Can. J. Phys. 84, 83–105 (2006)

    Article  Google Scholar 

  51. Avery, J., Avery, J.: The Generalized Sturmian Method for Calculating Spectra of Atoms and Ions. J. Math. Chem. 33, 145 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  52. Avery, J., Coletti, C.: Generalized Sturmians applied to atoms in strong external fields. J. Math. Chem. 27, 43 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  53. Avery, J., Coletti, C.: New Trends in Quantum Systems in Chemistry and Physics, vol. I, p. 77. Kluwer, Dordrecht (Maruani, J., et al., eds.)

    Google Scholar 

  54. Szmytkowski, R.: The Dirac-Coulomb Sturmian functions in the Z = 0 limit: properties and applications to series expansions of the Dirac Green function and the Dirac plane wave. J. Phys. A 33, 427 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  55. Patera, J., Winternitz, P.: A new basis for the representations of the rotation group. Lamé and Heun polynomials. J. Math. Phys. 14, 1130 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  56. Kalnins, E.G., Miller, W., Winternitz, P.: The Group O(4), Separation of Variables and the Hydrogen Atom. S.I.A.M. J. Appl. Math. 30, 630 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  57. Miller, W.: Symmetry and Separation of Variables. Addison-Wesley Publishing Company, Reading Mass (1977)

    MATH  Google Scholar 

  58. Aquilanti, V., Caligiana, A., Cavalli, S.: Hydrogenic elliptic orbitals, Coulomb Sturmian sets, and recoupling coefficients among alternative bases. Int. J. Quantum Chem. 92, 99 (2003)

    Article  Google Scholar 

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Authors and Affiliations

  1. Universitá di Perugia, via Elce di Sotto, 8, 06183, Perugia, Italy

    Gaia Grossi & Vincenzo Aquilanti

  2. Dipartimento di Farmacia, Università G. d’Annunzio, Via dei Vestini, 66100, Chieti, Italy

    Cecilia Coletti

  3. Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126, Pisa, Italy

    Danilo Calderini

Authors
  1. Danilo Calderini
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  2. Cecilia Coletti
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  3. Gaia Grossi
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  4. Vincenzo Aquilanti
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Editors and Affiliations

  1. L-I.S.U.T. - D.A.P.I.t. Facoltà Ingegneria, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano, 10, 85100, Potenza, Italy

    Beniamino Murgante

  2. Covenant University, Canaanland, OTA, Nigeria

    Sanjay Misra

  3. Partimento di Scienze e Tecnologie per LAgricoltura, le Foreste, la Natura e lEnergia, Università degli Studi della Tuscia, Via S. Camillo de Lellis, snc, 01100, Viterbo, Italy

    Maurizio Carlini

  4. Dipartimento di Scienze dell’Ingegneria Civile e dell’Architecttura, Politecnico di Bari, Via Orabona, 4, 70125, Bari, Italy

    Carmelo M. Torre

  5. International University VNU-HCM, Quarter 6, Linh Trung, Thu Duc, Ho Chi Minh City, Vietnam

    Hong-Quang Nguyen

  6. School of Business Systems, Monash University, 3800, Clayton, VIC, Australia

    David Taniar

  7. Department of Intelligent Informatics, Kyushu Sangyo University, 2-3-1 Matsukadai, Higashi-ku, 813-8503, Fukuoka, Japan

    Bernady O. Apduhan

  8. Department of Mathematics and Computer Science, University of Perugia, Via Vanvitelli, 1, 06123, Perugia, Italy

    Osvaldo Gervasi

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Calderini, D., Coletti, C., Grossi, G., Aquilanti, V. (2013). Continuous and Discrete Algorithms in Quantum Chemistry: Polynomial Sets, Spin Networks and Sturmian Orbitals. In: Murgante, B., et al. Computational Science and Its Applications – ICCSA 2013. ICCSA 2013. Lecture Notes in Computer Science, vol 7972. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39643-4_3

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  • DOI: https://doi.org/10.1007/978-3-642-39643-4_3

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