Particle number dependence of size and energy quantities in sodium clusters

Abstract

The effective radial electronic potentials for neutral sodium clusters, which were determined by Ekardt on the basis of the local density approximation and the jellium model, are parameterized by means of the (symmetrized) Woods-Saxon and “wine-bottle”-symmetrized Woods-Saxon potentials with the aim of investigating the dependence of size and energy quantities on the cluster particle number. The potential parameters are determined by various least-squares fitting procedures. It is found that for the radius R of the above potentials, complex expressions are more appropriate than the standard one R = r 0N 1 3 for relatively small values of N. Furthermore, N-power expansions are derived for those complex expressions of R, as well as for the rms radius of the potential. It is also found that improved results in these cases are obtained with an expression of the form R=R 0N 1 3 + b , which is still very simple. There is also investigated the variation of energy quantities, such as the single-particle energies of the 1s and 1p states, the level spacing | E 1p − E 1s| and the average energy-level spacing, with respect to the particle number N. Expressions for the first three of these quantities with N-dependent terms of the form aN t-2 3 + βN −1 give good results. Finally, the total energy per particle is considered with the aim of deriving its asymptotic expression for large values of the particle number.