Density functional theory (DFT) and its time-dependent counterpart (TDDFT) are crucial tools in material discovery, drug design, biochemistry, catalysis, and nanoscience. However, despite its exact theoretical basis, approximations are necessary throughout, from the description of electron exchange and correlation (xc) interactions to the representation of wavefunctions for
ever larger systems and the use of calculated quantities to explain and predict real-world phenomena. To address long-standing problems related to the speed and accuracy of approximations to the xc functional, we develop neural networks to emulate two such approximations, the local density (LDA) and generalized gradient (PBE) approximations, within the DFT code gpaw. We
present a strategy for retraining the network and assess which training data is necessary to optimize performance for total energies over a wide class of molecules and crystals. While certain classes of materials proved difficult to describe, neural network implementations were able to reproduce the LDA and PBE xc functionals with high accuracy and a reasonable computation time. In an effort to develop a more efficient, robust, and accurate method for predicting the optical properties of low-dimensional systems, we introduce the LCAO-TDDFT-k-ω code within gpaw, where a linear combination of atomic orbitals (LCAO) representation of the Kohn-Sham wavefunctions and TDDFT implementation in wavenumber k and frequency ω space provides substantial memory and time savings, and a first order derivative discontinuity correction to the electronic gap brings the optical spectra in line with experimental measurements. Convergence of the basis set, the use of low-dimensional response functions, and different ways to incorporate the energy correction are explored for a series of materials across all dimensions: 0D fullerene and chlorophyll monomers, 1D single-walled carbon nanotubes, 2D graphene and phosphorene monolayers, and 3D anatase and rutile titanium dioxide. We develop a set of visualization tools for resolving the energetic, spatial, and reciprocal space distributions of excitations, and find LCAO-TDDFT-k-ω yields qualitative and semi-quantitative agreement with other TDDFT methods and implementations at a fraction of the time and memory cost. Finally, we introduce a phenomenological hydrodynamic model for the optical conductivity of graphene, with contributions due to universal conductivity, Pauli blocking, and intraband transitions included in a systematic way, is fit empirically with results from TDDFT, and manages to reproduce experimental spectra across a wide range of energies within energy loss equations derived for 2D materials. We find experimental parameters such as the amount of doping in graphene, the size of the collection aperture, and the energy of incoming electrons influence the shape of the spectra in important ways, especially in the energy region accessible to higher resolution probing techniques
