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.

We present a modification of the ∆SCF method of calculating energies of excited states, in order to make it applicable to resonance calculations of molecules adsorbed on metal surfaces, where the molecular orbitals are highly hybridized. The ∆SCF approximation is a density functional method closely resembling standard density functional theory (DFT), the only difference being that in ∆SCF one or more electrons are placed in higher lying Kohn-Sham orbitals, instead of placing all electrons in the lowest possible orbitals as one does when calculating the ground state energy within standard DFT. We extend the ∆SCF method by allowing excited electrons to occupy orbitals which are linear combinations of Kohn-Sham orbitals. With this extra freedom it is possible to place charge locally on adsorbed molecules in the calculations, such that resonance energies can be estimated, which is not possible in traditional ∆SCF because of very delocalized Kohn-Sham orbitals. The method is applied to N2, CO and NO adsorbed on different metallic surfaces and compared to ordinary ∆SCF without our modification, spatially constrained DFT and inverse-photoemission spectroscopy (IPES) measurements. This comparison shows that the modified ∆SCF method gives results in close agreement with experiment, significantly closer than the comparable methods. For N2 adsorbed on ruthenium (0001) we map out a 2-dimensional part of the potential energy surfaces in the ground state and the 2π-resonance. From this we conclude that an electron hitting the resonance can induce molecular motion, optimally with 1.5 eV transferred to atomic movement. Finally we present some performance test of the ∆SCF approach on gas-phase N2 and CO, in order to compare the results to higher accuracy methods. Here we find that excitation energies are approximated with accuracy close to that of time-dependent density functional theory. Especially we see very good agreement in the minimum shift of the potential energy surfaces in the excited state compared to the ground state.

We present a model for desorption induced by ͑multiple͒ electronic transitions ͓DIET ͑DIMET͔͒ based on potential energy surfaces calculated with the delta self-consistent field extension of density-functional theory. We calculate potential energy surfaces of CO and NO molecules adsorbed on various transition-metal surfaces and show that classical nuclear dynamics does not suffice for propagation in the excited state. We present a simple Hamiltonian describing the system with parameters obtained from the excited-state potential energy surface and show that this model can describe desorption dynamics in both the DIET and DIMET regimes and reproduce the power-law behavior observed experimentally. We observe that the internal stretch degree of freedom in the molecules is crucial for the energy transfer between the hot electrons and the molecule when the coupling to the surface is strong.