Semiclassical analysis of the electronic shell structure in metal clusters

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Abstract
The efficiency of the independent-electron approach to computing the electronic shell structure of metal clusters is tested, quantum mechanically as well as semiclassically. In this approach, which is much less time consuming than self-consistent methods, the quantum size-to-size variations of the shape of the electronic effective potential are disregarded. We show that, provided the potential surface profile is fitted to the results of self-consistent jellium calculations for some large cluster, the independent-electron model yields results very similar to those of the fully self-consistent jellium model. As an example, in the case of trivalent metals, we show that the calculated values of the usual observables, such as the second difference of the total energy, the ionization potential, or the shell correction energy, are almost identical in both descriptions. Using simple mathematical ingredients, we derive the semiclassical formalism of the density of states which provides an intuitive understanding of the electronic shell and supershell structures in terms of classical closed orbits. A graphical analysis of the semiclassical formula is presented, allowing us to establish a clear correspondence between each closed-orbit contribution with a particular set of quantum levels. From examples involving spherical potential shapes appropriate to clusters, we demonstrate that the semiclassical approach is a fairly good quantitative theory, in addition to its evident usefulness for immediate qualitative analysis. As an application, we investigate the influence of the potential surface profile on the electronic shell structure. This study shows that the electronic structure, in particular, the beat location in the supershell pattern, is very sensitive to the potential surface diffuseness. This could explain the large discrepancy between the theoretical predictions and the experimental results for gallium clusters.