Beyond the first order of the Hyperspherical Harmonic Expansion Method

In this paper we demonstrate the inadequacy of the first order of the Hyperspherical Harmonic Expansion Method, the L_m approximation, for the calculation of the binding energies, charge form factors and charge densities of doubly magic nuclei like ¹⁶O and ⁴⁰Ca. We then extend the Hyperspherical Expansion Method to many-fermion systems, consisting of an arbitrary number of fermions, and develop an exact formalism capable of generating the complete optimal subset of the hyperspherical harmonic basis functions. This optimal subset consists of those hyperspherical harmonic basis functions directly connected to the dominant first term in the expansion, the hyperspherical harmonic of minimal order L_m, through the total interaction between the particles. The required many-body coefficients are given using either the Gogny or Talmi-Moshinsky coefficients for the two-body operators. Using the two-body coefficients the weight function generating the orthogonal polynomials associated with the optimal subset is constructed.