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In this paper, we have considered the mechanical stability of a jellium system in the presence of spin degrees of freedom and have generalized the stabilized jellium model, introduced by J. P. Perdew, H. Q. Tran, and E. D. Smith [ Phys. Rev. B 42, 11627 (1990)], to a spin-polarized case. By applying this generalization to metal clusters (Al, Ga, Li, Na, K, Cs), we gain additional insights about the odd-even alternations, seen in their ionization potentials. In this generalization, in addition to the electronic degrees of freedom, we allow the positive jellium background to expand as the clusters' polarization increases. In fact, our self-consistent calculations of the energetics of alkali metal clusters with spherical geometries, in the context of density functional theory and local spin density approximation, show that the energy of a cluster is minimized for a configuration with maximum spin compensation (MSC). That is, for clusters with even number of electrons, the energy minimization gives rise to complete compensation (N ↑ = N ↓ ), and for clusters with odd number of electrons, only one electron remains uncompensated (N ↑ −N ↓ = 1). It is this MSC-rule which gives rise to alternations in the ionization potentials. Aside from very few exceptions, the MSC-rule is also at work for other metal culsters (Al, Ga) of various sizes. 36.40, 71.10, 31.15.E Typeset using REVT E X

In this work, we have applied the self-compressed stabilized jellium model to predict the equilibrium properties of isolated thin Al, Na and Cs slabs. To make a direct correspondence to atomic slabs, we have considered only those L values that correspond to n-layered atomic slabs with 2≤n≤20, for surface indices (100), (110), and (111). The calculations are based on the density functional theory and self-consistent solution of the Kohn-Sham equations in the local density approximation. Our results show that firstly, the quantum size effects are significant for slabs with sizes smaller than or near to the Fermi wavelength of the valence electrons λ(F), and secondly, some slabs expand while others contract with respect to the bulk spacings. Based on the results, we propose a criterion for realization of significant quantum size effects that lead to expansion of some thin slabs. For more justification of the criterion, we have tested it on Li slabs for 2≤n≤6. We have compared our Al results with those obtained from using all-electron or pseudo-potential first-principles calculations. This comparison shows excellent agreements for Al(100) work functions, and qualitatively good agreements for the other work functions and surface energies. These agreements justify the way we have used the self-compressed stabilized jellium model for the correct description of the properties of simple metal slab systems. On the other hand, our results for the work functions and surface energies of large- n slabs are in good agreement with those obtained from applying the stabilized jellium model for semi-infinite systems. In addition, we have performed the slab calculations in the presence of surface corrugation for selected Al slabs and have shown that the results are worsened.

In the framework of spherical geometry for jellium and local spin density approximation, we have obtained the equilibrium r s values,r s (N, ζ), of neutral and singly ionized "generic" N -electron clusters for their various spin polarizations, ζ. Our results reveal thatr s (N, ζ) as a function of ζ behaves differently depending on whether N corresponds to a closed-shell or an openshell cluster. That is, for a closed-shell one,r s (N, ζ) is an increasing function of ζ over the whole range 0 ≤ ζ ≤ 1, and for an open-shell one, it has a decreasing part corresponding to the range 0 < ζ ≤ ζ 0 , where ζ 0 is a polarization that the cluster assumes in a configuration consistent with Hund's first rule. In the context of the stabilized spin-polarized jellium model, our calculations based on these equilibrium r s values,r s (N, ζ), show that instead of the maximum spin compensation (MSC) rule, Hund's first rule governs the minimum-energy configuration. We therefore conclude that the increasing behavior of the equilibrium r s values over the whole range of ζ is a necessary condition for obtaining the MSC rule for the minimum-energy configuration; and the only way to end up with an increasing behavior over the whole range of ζ is to break the spherical geometry of the jellium background. This is the reason why the results based on simple jellium with spheroidal or ellipsoidal geometries show up MSC rule.

The stabilized jellium model (SJM) provides us with a method for calculating the volume changes of different simple metals as functions of the spin polarization, ζ, of the delocalized valence electrons. Our calculations show that for bulk metals, the equilibrium Wigner-Seitz (WS) radius, r̄sB(ζ), is always an increasing function of the polarization, i.e., the volume of a bulk metal always increases as ζ increases, and the rate of increase is higher for higher-electron-density metals. Using the SJM along with the local spin-density approximation, we have also calculated the equilibrium WS radius, r̄s(N,ζ), of spherical jellium clusters, at which the pressure on the cluster with given number of total electrons N with spin configuration ζ vanishes. Our calculations for Cs, Na, and Al clusters show that r̄s(N,ζ) as a function of ζ behaves differently depending on whether N corresponds to a closed-shell or an open-shell cluster. For a closed-shell cluster, it is an increasing function of ζ over the whole range 0⩽ζ⩽1, whereas for open-shell clusters it has a decreasing behaviour over the range 0⩽ζ⩽ζ0, where ζ0 is a polarization such that the cluster has a configuration consistent with Hund's first rule. The results show that for all neutral clusters with the ground-state spin configuration, ζ0, the inequality r̄s(N,ζ0)⩽r̄sB(0) always holds (self-compression) but, at some polarization ζ1>ζ0, the inequality changes direction (self-expansion). However, the inequality r̄s(N,ζ)⩽r̄sB(ζ) always holds and equality is achieved in the limit N→∞.