Published online zzz PACS 71.15.Ap, 71.15.Dx A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.

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Electronic states of one monolayer high Na nanoislands on the Cu͑111͒ surface are studied as a function of the nanoisland size. Properties of nanoislands such as one-electron states, the electron density, and the associated potential are obtained self-consistently within the density-functional formalism using a one-dimensional pseudopotential for the Cu͑111͒ substrate and the jellium model for Na. A wave packet propagation method is used to study the energies and lifetimes of quasistationary states localized at Na islands. For very large islands, island-localized states merge into the two-dimensional continuum of the Na quantum well state. Thus, we assign the quasistationary states studied as arising from the quantization of the two-dimensional quantum well continuum due to the finite island size. The scattering at the island boundaries results in the energy-conserving resonant electron transfer into the continuum of the substrate states broadening the island-localized states into resonances.

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