We describe an implementation of Hedins 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 Greens 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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