  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. B42, 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
($N_\uparrow=N_\downarrow$), and for clusters with odd number of electrons,
only one electron remains uncompensated ($N_\uparrow-N_\downarrow=1$). 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.
