Determination of the harmonic oscillator energy level spacing for atomic clusters

The harmonic oscillator energy level spacing HBAR omega for atomic clusters as a function of the particle number N is expressed analytically in terms of the parameters of a Woods-Saxon (or Symmetrized Woods-Saxon) potential which approximates the effective spherical self-consistent jellium model potential. The expressions derived depend on the particular scheme adopted to approximate the potential by the harmonic oscillator one and on the assumed dependence of the potential radius R on N. It is also observed, considering the case of sodium clusters, that for large N the expressions of HBAR omega are in good agreement with the well known expression of HBAR omega in terms of the Wigner-Seitz radius.