The cluster size dependence of the calculated binding energies per atom for a cluster of radius without a bubble (cluster marked as 엯) and for a cluster with a bubble at the equilibrium electron bubble radius ( marked as ∎). The experimental binding energy per atom in the bulk , , is also presented (bulk marked as ◻). The calculated data for the large clusters , and the bulk values of without a bubble, follow a linear dependence vs and are represented by the liquid drop model with the cluster size equation, Eq. (13) (solid line). The dashed curve connecting the data with a bubble was drawn to guide the eye.
The dependence on the bubble radius of the cluster reorganization energy , of the mean density , and of the cluster radius for . These data reflect on the structural manifestations (decrease of ), on the energetic implications (increase of ), and on the cluster expansion (increase of ) with increasing .
The density profiles , where is the bulk density, at different bubble radii for clusters with (a) and with (b), reflecting on the formation of a “helium balloon” with a finite thickness with the bubble center being located at the center of the cluster.
The dependence of the cluster reorganization energy, Eq. (12), on the radius of the bubble. Data are presented for different cluster sizes , which are marked on the curves, and for the bulk.
The potential energy surfaces for the excess electron bubble states in clusters portraying the total energy vs the bubble radius for fixed values of marked on the curves. The full points (●) and open points (엯) represent the results of the computations for the clusters using the density functional method for and the quantum mechanical treatment for , while for the bulk we took . The open point (엯) on each configurational diagram represents the equilibrium radius of the electron bubble. The two straight lines marked represent the energy of the quasi-free-electron. The horizontal line represents the bulk value of , while the second line represents in the cluster of the smallest size of . The values for each for fall between these two nearly straight lines.
The cluster size dependence of the equilibrium electron bubble radii and the total ground state energies , corresponding to the minima of the potential curves of Fig. 5.
The dependence of the energy gap between the quasi-free-electron energy and the total ground state energy at the equilibrium configuration of the electron bubble on the reciprocal value of the cluster radius at this equilibrium configuration for clusters in the range of and for the bulk. A crude extrapolation of this linear dependence of to zero leads to a localization threshold at , which corresponds to .
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