Construction of SO(5)⊃SO(3)SO(5)⊃SO(3) spherical harmonics and Clebsch–Gordan coefficients

The SO(5)⊃SO(3)SO(5)⊃SO(3) spherical harmonics form a natural basis for expansion of nuclear collective model angular wave functions. They underlie the recently-proposed algebraic method for diagonalization of the nuclear collective model Hamiltonian in an SU(1,1)×SO(5)SU(1,1)×SO(5) basis. We present a computer code for explicit construction of the SO(5)⊃SO(3)SO(5)⊃SO(3) spherical harmonics and use them to compute the Clebsch–Gordan coefficients needed for collective model calculations in an SO(3)SO(3)-coupled basis. With these Clebsch–Gordan coefficients it becomes possible to compute the matrix elements of collective model observables by purely algebraic methods.Program summaryProgram title: GammaHarmonicCatalogue identifier: AECY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AECY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 346 421No. of bytes in distributed program, including test data, etc.: 16 037 234Distribution format: tar.gzProgramming language: Mathematica 6Computer: Any which supports MathematicaOperating system: Any which supports Mathematica; tested under Microsoft Windows XP and LinuxClassification: 4.2Nature of problem: Explicit construction of SO(5) ⊃ SO(3) spherical harmonics on S4S4. Evaluation of SO(3)-reduced matrix elements and SO(5) ⊃ SO(3) Clebsch–Gordan coefficients (isoscalar factors).Solution method: Construction of SO(5) ⊃ SO(3) spherical harmonics by orthonormalization, obtained from a generating set of functions, according to the method of Rowe, Turner, and Repka [1]. Matrix elements and Clebsch–Gordan coefficients follow by construction and integration of SO(3) scalar products.Running time: Depends strongly on the maximum SO(5) and SO(3) representation labels involved. A few minutes for the calculation in the Mathematica notebook.References: [1] D.J. Rowe, P.S. Turner, J. Repka, J. Math. Phys. 45 (2004) 2761.