SU3HOB-ladder: a Fortran package for harmonic-oscillator brackets in the SU(3) basis from pseudo-spin ladder operators

Abstract

We present SU3HOB-ladder, a Fortran 2008 package computing the general Talmi-Moshinsky harmonic-oscillator transformation brackets for an arbitrary mass-ratio parameter d in the SU(3) basis. The bracket reduces to a single sum over Wigner d-functions of the reflection-containing Talmi-Moshinsky matrix, weighted by products of SU(3) > SO(3) isofactors of the chain U(6) > U(3)xU(2); the isofactors are d-independent, so they are built once per block (E,L) and reused for every d. We obtain them from the U(2) pseudo-spin generators of that same chain: because the U(3) and U(2) labels inside [E] of U(6) are complementary, the highest-weight states are the null space of the raising operator J_+ = a^dag(1).a(2) and the remaining members of each multiplet follow by lowering, so that no SU(3) recoupling machinery, K-matrix or outer-multiplicity resolution is required at any point. We compare this construction with two alternatives that produce the same brackets - the same scheme with isofactors taken from the vector-coherent-state library Su3cgvcs, and the classical Talmi-Moshinsky sum of Kamuntavicius et al. - over every block up to E=50. Evaluating the classical sum with its factorials in logarithmic form rather than as tabulated binomials removes the range limit that otherwise caps it near E ~ 12, and makes it an absolute check on the bracket values over the whole range where its cost is bearable. Against that check the two routes agree to within a factor of a few at every shell, while the ladder construction is faster by three orders of magnitude; it is also the only one of the three usable over the whole range, retaining max|HOB.HOB^T - I| <~ 2x10^-6 at E=50, where the library route has become unusable and the classical route is prohibitively slow.