Gold cluster beyond hollow cage: Double-shell Au$_{58}$

Gold clusters were found to have planar and hollow cage-like structures due to the strong relativistic effect. By using first principles calculation, we take Au$_{58}$ as an example to demonstrate that gold cluster can show shell structure. Au$_{58} $ reaches its highest stability with an optimal inner core of $10$ atoms. Particularly, a double-shell structure with a hollow inner shell shows remarkable robustness. It is significant to consider this shell structure as a descendant of the hollow cage structures found previously, such as tetrahedral Au$_{16}$, icosahedral Au$_{32}$, tubular Au$_{50}$ and so on, for this implies a possible evolution from planar, to cage, to shells and finally to the compact structure as the number of atoms in the cluster increasing.