Jellium from small molecules to the bulk

The jellium model of the free-electron gas can account for the increased abundance of alkali metal clusters of a certain size which are observed in mass spectroscopy experiments. This occurrence of so-called magic numbers is related directly to the electronic shell structure of the atomic clusters. Rather than solving the Schrodinger equation self-consistently for jellium clusters, we first consider the two simpler problems of a free-electron gas that is confined either within a sphere of radius, R, or within a cubic box of edge length, L (cf. problem 28 of Sutton (1993)). This corresponds to imposing hard-wall boundary conditions on the electrons, namely 

The sphere radius, R, and edge-length, L, may be written in terms of the radius of the sphere containing one electron, r and the number of monovalent atoms in the cluster AT, as 

The eigenfunctions of the free-electron Schrodinger equation with spherical boundary conditions can be written in separable form like that for the hydrogen atom, eqn (2.48), namely 

The boundary condition eqn (5.1) then determines the eigenvalues E since, for example, for / = 0 we must have sin kR = 0, so that = nn/R, where n is a positive integer. The eleven lowest roots of jt(KR) — 0 are given in Table 5.1. We see that the three roots corresponding to / = 0 are kR = n, 2n, and 3n as expected. Note that l = 4 and l = 5 states are referred to as g and h respectively following as they do after the f states corresponding to 1 = 3. 

The eigenspectrum of the free-electron gas confined within a sphere of 

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Jellium from small molecules to the bulk 109 Table 5.1 The eleven lowest roots / of /)(kR) = 0... 

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