Comparison of density functional approximations in the jellium model for metal clusters

We calculated the exchange, correlation and total energies of clusters of a lkali metals with N = 1 - 150 atoms in the spherical jellium model. The cal culations were made using the Kohn-Sham method with exchange and correlatio n energies evaluated in the meta-generalized gradient approximation (MGGA), proposed by J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, in the general ized gradient approximation (GGA) of J. P. Perdew, K. Burke and M. Emzerhof , and in the Local Density Approximation (LDA). We evaluated the relative deviations of MGGA and GGA. energies with respect to LDA. Exchange energies of MGGA and GGA are more negative than the LDA e xchange energy and become closer to this as the cluster size increases. On the other hand, the GGA and MGGA correlation energies, which are almost ide ntical, are less negative than LDA. The deviations of GGA and MGGA exchange -correlation energies with respect to LDA are smaller than those of the exc hange and correlation energies separately. For clusters with 18 and 20 atoms we have compared our jellium results with Variational and Diffusion Monte-Carlo results. Errors of LDA for exchange and correlation tend to cancel so that the total exchange-correlation energ y is close to the Monte-Carlo results. Similar cancellations occur with GGA and MGGA. We also examined the validity of the liquid drop model.