We model electronic properties of the second monolayer Na adatom islands (quantum dots) on the Cu(111) surface covered homogeneously by the first Na monolayer. An axially-symmetric three-dimensional jellium model, taking into account the effects due to the first Na monolayer and the Cu substrate, has been developed. The electronic structure is solved within the local-density approximation of the density-functional theory using a real-space multigrid method. The model enables the study of systems consisting of thousands of Na-atoms. The results for the local density of states are compared with differential conductance ($dI/dV$) spectra and constant current topographs from Scanning Tunneling Microscopy.
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