The description of excited states in large extended systems using quantum-mechanical approaches is still a challenge for theoretical spectroscopy. In contrast to the electronic ground state, the quest of finding an appropriate model with well-balanced accuracy-cost ratio for the excited state is complicated by various electronic states of different nature having to be described simultaneously. For instance, this renders the parametrization of transferable classical force fields for excited states an almost impractical task. The cost of highly accurate wave-function methods scales prohibitively with system size making routine large scale applications impossible. When aiming for a robust and efficient tool set for the treatment of extended periodic systems, density functional theory and semi-empirical tight binding approaches therefore represent a suitable compromise. While time-dependent density functional theory (TDDFT) has been established as efficient and broadly applicable for the treatment of excited states, there has also been a recent revival of semi-empirical approaches to extend treatable system sizes from hundreds to thousands of atoms. Both the re-emergence of semi-empirical methods as well as the continuing improvement of density functional approximations are among the most important future developments in computational chemistry. The correct description of exact exchange has proven to be crucial for an adequate treatment of excited states in TDDFT. It has a much greater influence on the geometrical displacement upon excitation, and thus on transition energies and spectra, than the exchange-correlation functional. Benchmarks on molecular systems, including radicals and ions, suggest including 30-40% exact exchange to obtain reliable spectroscopic data. Straight-forward evaluation of the two-electron exact exchange integrals leads to a formal scaling of N 4 with system size N for localized basis sets, emphasizing the need for more cost-efficient approximation schemes. An overview of recent developments and the various existing algorithms can e.g. be found in Ref. One approach is the auxiliary density matrix method (ADMM), where the gain in efficiency is achieved by evaluating a model exact exchange energy within a relatively small auxiliary basis augmented by a correction term based on a local exchange functional.