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We describe an implementation of Hedin's GW approximation for molecules and clusters, the complexity of which scales as O(N 3 ) with the number of atoms. Our method is guided by two strategies: i) to respect the locality of the underlying electronic interactions and ii) to avoid the singularities of Green's functions by manipulating, instead, their spectral functions using fast Fourier transform methods. To take into account the locality of the electronic interactions, we use a local basis of atomic orbitals and, also, a local basis in the space of their products. We further compress the screened Coulomb interaction into a space of lower dimensions for speed and to reduce memory requirements. The improved scaling of our method with respect to most of the published methodologies should facilitate GW calculations for large systems. Our implementation is intended as a step forward towards the goal of predicting, prior to their synthesis, the ionization energies and electron affinities of the large molecules that serve as constituents of organic semiconductors.

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Two self-consistent schemes involving Hedin's GW approximation are studied for a set of sixteen different atoms and small molecules. We compare results from the fully self-consistent GW approximation (SCGW ) and the quasi-particle self-consistent GW approximation (QSGW ) within the same numerical framework. Core and valence electrons are treated on an equal footing in all the steps of the calculation. We use basis sets of localized functions to handle the space dependence of quantities and spectral functions to deal with their frequency dependence. We compare SCGW and QSGW on a qualitative level by comparing the computed densities of states (DOS). To judge their relative merit on a quantitative level, we compare their vertical ionization potentials (IPs) with those obtained from coupled-cluster calculations CCSD(T). Our results are futher compared with "one-shot" G0W0 calculations starting from Hartree-Fock solutions (G0W0-HF). Both self-consistent GW approaches behave quite similarly. Averaging over all the studied molecules, both methods show only a small improvement (somewhat larger for SCGW ) of the calculated IPs with respect to G0W0-HF results. Interestingly, SCGW and QSGW calculations tend to deviate in opposite directions with respect to CCSD(T) results. SCGW systematically underestimates the IPs, while QSGW tends to overestimate them. G0W0-HF produces results which are surprisingly close to QSGW calculations both for the DOS and for the numerical values of the IPs.

We present a study of the optical response of compact and hollow icosahedral clusters containing up to 868 silver atoms by means of time-dependent density functional theory. We have studied the dependence on size and morphology of both the sharp plasmonic resonance at 3-4 eV (originated mainly from sp-electrons), and the less studied broader feature appearing in the 6-7 eV range (interband transitions). An analysis of the effect of structural relaxations, as well as the choice of exchange correlation functional (local density versus generalised gradient approximations) both in the ground state and optical response calculations is also presented. We have further analysed the role of the different atom layers (surface versus inner layers) and the different orbital symmetries on the absorption cross-section for energies up to 8 eV. We have also studied the dependence on the number of atom layers in hollow structures. Shells formed by a single layer of atoms show a pronounced red shift of the main plasmon resonances that, however, rapidly converge to those of the compact structures as the number of layers is increased. The methods used to obtain these results are also carefully discussed. Our methodology is based on the use of localised basis (atomic orbitals, and atom-centered and dominant-product functions), which bring several computational advantages related to their relatively small size and the sparsity of the resulting matrices. Furthermore, the use of basis sets of atomic orbitals also allows the possibility of extending some of the standard population analysis tools (e.g. Mulliken population analysis) to the realm of optical excitations. Some examples of these analyses are described in the present work.