Electronic structure calculations have become an indispensable tool in many areas of materials science and quantum chemistry. Even though the Kohn-Sham formulation of the density-functional theory (DFT) simplifies the many-body problem significantly, one is still confronted with several numerical challenges. In this article we present the projector augmented-wave (PAW) method as implemented in the GPAW program package (https://wiki.fysik.dtu.dk/gpaw) using a uniform real-space grid representation of the electronic wavefunctions. Compared to more traditional plane wave or localized basis set approaches, real-space grids offer several advantages, most notably good computational scalability and systematic convergence properties. However, as a unique feature GPAW also facilitates a localized atomic-orbital basis set in addition to the grid. The efficient atomic basis set is complementary to the more accurate grid, and the possibility to seamlessly switch between the two representations provides great flexibility. While DFT allows one to study ground state properties, time-dependent density-functional theory (TDDFT) provides access to the excited states. We have implemented the two common formulations of TDDFT, namely the linear-response and the time propagation schemes. Electron transport calculations under finite-bias conditions can be performed with GPAW using non-equilibrium Green functions and the localized basis set. In addition to the basic features of the real-space PAW method, we also describe the implementation of selected exchange-correlation functionals, parallelization schemes, ΔSCF-method, x-ray absorption spectra, and maximally localized Wannier orbitals.

Synthesis, characterization, and functionalization of self-assembled, ligand-stabilized gold nanoparticles are long-standing issues in the chemistry of nanomaterials. Factors driving the thermodynamic stability of well documented discrete sizes are largely unknown. Herein, we provide a unified view of principles that underlie the stability of particles protected by thiolate (SR) or phosphine and halide (PR ₃ , X) ligands. The picture has emerged from analysis of large-scale density functional theory calculations of structurally characterized compounds, namely Au ₁₀₂ (SR) ₄₄ , Au ₃₉ (PR ₃ ) ₁₄ X ₆ ⁻ , Au ₁₁ (PR ₃ ) ₇ X ₃ , and Au ₁₃ (PR ₃ ) ₁₀ X ₂ ³⁺ , where X is either a halogen or a thiolate. Attributable to a compact, symmetric core and complete steric protection, each compound has a filled spherical electronic shell and a major energy gap to unoccupied states. Consequently, the exceptional stability is best described by a “noble-gas superatom” analogy. The explanatory power of this concept is shown by its application to many monomeric and oligomeric compounds of precisely known composition and structure, and its predictive power is indicated through suggestions offered for a series of anomalously stable cluster compositions which are still awaiting a precise structure determination.

Gold octamers (Au ₈ ) bound to oxygen-vacancy F-center defects on Mg(001) are the smallest clusters to catalyze the low-temperature oxidation of CO to CO ₂ , whereas clusters deposited on close-to-perfect magnesia surfaces remain chemically inert. Charging of the supported clusters plays a key role in promoting their chemical activity. Infrared measurements of the stretch vibration of CO adsorbed on mass-selected gold octamers soft-landed on MgO(001) with coadsorbed O ₂ show a red shift on an F-center–rich surface with respect to the perfect surface. The experiments agree with quantum ab initio calculations that predict that a red shift of the C–O vibration should arise via electron back-donation to the CO antibonding orbital.