Symbolic-Numerical Algorithm for Large Scale Calculations the Orthonormal \(\mathrm {SU(3)}\) BM Basis

  • A. Deveikis
  • A. A. GusevEmail author
  • V. P. Gerdt
  • S. I. Vinitsky
  • A. Góźdź
  • A. Pȩdrak
  • Č. Burdik
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)


In this paper we proposed a new symbolic, non-standard recursive and fast orthonormalization procedure of linearly independent vectors but as in other approaches not orthonormal based on the Gram-Schmidt orthonormalization algorithm. Our adaptation of the Gram-Schmidt orthonormalization procedure provide simple analytic formulas for the \(\mathrm {SU(3)}\) Bargmann-Moshinsky basis orthonormalization coefficients and do not involve any square root operation on the expressions coming from the previous iterative computation steps. This distinct features of the proposed orthonormalization algorithm may make the large scale symbolic calculations feasible. We demonstrate efficiency of our procedure by benchmark large-scale calculations of the non-canonical BM basis with the highest weight vectors of \(\mathrm {SO(3)}\) irreducible representations.



The work was partially supported by the Bogoliubov-Infeld program, Votruba-Blokhintsev program, the RUDN University Program 5-100 and grant of Plenipotentiary of the Republic of Kazakhstan in JINR. AD is grateful to Prof. A. Góźdź for hospitality during visits in Institute of Physics, Maria Curie-Skłodowska University (UMCS).

The authors thank the both referees for their useful comments, remarks and suggestions.


  1. 1.
    Saha, A., et al.: Spectroscopy of a tetrahedral doubly magic candidate nucleus \(^{160}_{70}\)Yb\(^{90}\). J. Phys. G Nucl. Part. Phys. 46, 055102 (2019)CrossRefGoogle Scholar
  2. 2.
    Deveikis, A., Gusev, A.A., Gerdt, V.P., Vinitsky, S.I., Góźdź, A., Pȩdrak, A.: Symbolic algorithm for generating the orthonormal Bargmann–Moshinsky basis for \(\rm SU(3)\) group. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2018. LNCS, vol. 11077, pp. 131–145. Springer, Cham (2018). Scholar
  3. 3.
    Vinitsky, S., et al.: On generation of the Bargmann-Moshinsky basis of SU(3) group. J. Phys. Conf. Ser. 1194, 012109 (2019)CrossRefGoogle Scholar
  4. 4.
    Gozdz, A., Pedrak, A., Gusev, A.A., Vinitsky, S.I.: Point symmetries in the nuclear SU(3) partner groups model. Acta Phys. Polonica B Proc. Suppl. 11, 19–28 (2018)CrossRefGoogle Scholar
  5. 5.
    Bargmann, V., Moshinsky, M.: Group theory of harmonic oscillators (II). Nucl. Phys. 23, 177–199 (1961)CrossRefGoogle Scholar
  6. 6.
    Moshinsky, M., Patera, J., Sharp, R.T., Winternitz, P.: Everything you always wanted to know about \(SU(3)\supset O(3)\). Ann. Phys. (N.Y.) 95, 139–169 (1975)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Alisauskas, S., Raychev, P., Roussev, R.: Analytical form of the orthonormal basis of the decomposition \(SU(3)\supset O(3)\supset O(2)\) for some \((\lambda,\mu )\) multiplets. J. Phys. G Nucl. Phys. 7, 1213–1226 (1981)CrossRefGoogle Scholar
  8. 8.
    Varshalovitch, D.A., Moskalev, A.N., Hersonsky, V.K.: Quantum Theory of Angular Momentum. Nauka, Leningrad (1975). (also World Scientific (1988))Google Scholar
  9. 9.
    Raychev, P., Roussev, R.: Matrix elements of the generators of SU(3) and of the basic O(3) scalars in the enveloping algebra of SU(3). J. Phys. G Nucl. Phys. 7, 1227–1238 (1981)CrossRefGoogle Scholar
  10. 10.
    Cseh, J.: Algebraic models for shell-like quarteting of nucleons. Phys. Lett. B 743, 213–217 (2015)CrossRefGoogle Scholar
  11. 11.
    Dytrych, T., et al.: Efficacy of the SU(3) scheme for ab initio large-scale calculations beyond the lightest nuclei. Comput. Phys. Commun. 207, 202–210 (2016)CrossRefGoogle Scholar
  12. 12.
    Pan, F., Yuan, S., Launey, K.D., Draayer, J.P.: A new procedure for constructing basis vectors of SU(3)\(\supset \)SO(3). Nucl. Phys. A 743, 70–99 (2016)CrossRefGoogle Scholar
  13. 13.
    Asherova, R.M., Smirnov, Y.F.: On asymptotic properties of a quantum number \(\Omega \) in a system with SU(3) symmetry. Repts. Math. Phys. 4, 83–95 (1973)CrossRefGoogle Scholar
  14. 14.
    Draayer, J.P., Akiyama, Y.: Wigner and Racah coefficients for SU3. J. Math. Phys. 14, 1904–1912 (1973)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • A. Deveikis
    • 1
  • A. A. Gusev
    • 2
    Email author
  • V. P. Gerdt
    • 2
  • S. I. Vinitsky
    • 2
    • 3
  • A. Góźdź
    • 4
  • A. Pȩdrak
    • 5
  • Č. Burdik
    • 6
  1. 1.Department of Applied InformaticsVytautas Magnus UniversityKaunasLithuania
  2. 2.Joint Institute for Nuclear ResearchDubnaRussia
  3. 3.RUDN UniversityMoscowRussia
  4. 4.Institute of PhysicsMaria Curie-Skłodowska UniversityLublinPoland
  5. 5.National Centre for Nuclear ResearchWarsawPoland
  6. 6.Department of Mathematics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical UniversityPragueCzech Republic

Personalised recommendations