The assumption that the basis-set incompleteness error can be corrected in terms of a costefficient density functional correction term was shown to be well-founded and benchmarks including liquids, solids as well as proteins in solution with up to 3000 atoms revealed that memory requirements and computational efficiency are improved by at least one order of magnitude. Kumar et al. showed that ADMM achieves good accuracy for ground-and excited-state energies, even though rather large errors were found for polarizabilities and hyperpolarizabilities. The least complex variant of ADMM is based on a simple least-squares fitting projection of molecular orbitals onto the smaller auxiliary basis. More sophisticated ADMM variants have been proposed including density matrix purification and projections enforcing orthogonality or charge constraints. Comparing amongst the developed ADMM variants, additional constraints as well as an adequate choice of the exchange functional for the correction term were shown to further improve accuracy for total energies while retaining a comparable efficiency. With respect to other sophisticated exact exchange algorithms including the pair-atomic resolution-of-the-identity method (PARI-K) and the chain-ofspheres algorithm (COSX), total as well as orbital, reaction and atomization energies were found to be one to three orders of magnitude less accurate. However, ADMM achieves a computational efficiency comparable to the cost required for density fitting Coulomb algorithms. ADMM was thus classified as promising model with an "impressive speedup", encouraging further development to improve accuracy and calling for careful optimization of the required auxiliary basis sets. For even larger system sizes of several thousands of atoms it is necessary to go from ab initio excited-state calculations towards semi-empirical tight binding approximations. In recent years, a class of tight binding approaches was suggested by Grimme et al., based on the idea of reducing the computational effort of electron repulsion integrals while retaining an adequate description of the physics of electronic interactions. More specifically, Coulomb and exchange contributions are approximated using an electron repulsion operator that captures the correct long-range 1/R asymptotics by construction but allows for a short range empirical parametrization. It was shown that this choice provides a balanced description of valence, charge-transfer as well as Rydberg states. The developed methods were dubbed simplified TDDFT (sTDDFT) or simplified Tamm-Dancoff approximation (sTDA) corresponding to the related ab initio electronic structure method and were extended to treat e.g. spin-flip excitations or to be combined with the idea of range separation. In contrast to other common tight binding approaches like DFTB, sTDA and sTDDFT involve only a limited number of global parameters which enables their straightforward application over the whole periodic table. Benchmark results demonstrated that the simplified approaches achieve computational savings of at least two orders of magnitude while the loss in accuracy is minor with an average deviation of 0.2 -0.3 eV for excitation energies when being compared to conventional TDDFT or experiment. If based on a semiempirical ground state reference, sTDA errors were found to be slightly larger with mean absolute deviations in the range of 0.3 -0.5 eV. Most importantly, accuracy was found to be consistent for both the low-energy valence as well as the high-energy Rydberg transitions. Simplified approaches were also suggested for the Bethe-Salpeter equation (BSE) and the GW approximation. These methods provided GW ionization potentials within the GW100 test set differing only by 0.2 eV while enabling cubic scaling with system size. Deviations in sBSE excitation energies amount up to 0.5 eV and are thus in an error range comparable to sTDA and sTDDFT. A closely related TDDFT+TB approach has been suggested by Visscher et al. which relies on the same monopole approximation for the electron repulsion integrals, but is -in contrast to sTDA and sTDDFT -not designed for hybrid, but pure density functionals. The mentioned advantages of ADMM and sTDA regarding their physically correct description of exact exchange qualify both approaches as promising for an efficient calculation of excited-state properties. Excited-state gradients within the framework of linear response TDDFT were pioneered for molecular systems by Van Caillie and Amos as well as Furche and Ahlrichs. This work was extended to periodic systems and plane wave basis sets. Furthermore, TDDFT excited-state properties for exact exchange have been extended to global and local hybrid functionals as well as range separation. Implementations are also available for tight binding approaches including long-range corrections. However, algorithms are most often restricted to molecular systems. We implemented sTDA and ADMMapproximated hybrid density functional excited-state gradients based on the Gaussian and plane wave (GPW) framework within the CP2K program package. The GPW methods allow for a natural extension of algorithms to periodic boundary conditions and thus enable the treatment of extended systems. The implementation is based on earlier works of Iannuzzi et al. and Strand et al. already featuring the calculation of ADMM excitation energies for model systems of up to 1000 atoms. As outlined in section 2.1, the CP2K implementation is based on the Sternheimer equations, thus depending solely on the occupied molecular and atomic orbital space. ? ? Such a formulation requires e.g. adjustments in the formulation of the ADMM equations and the sTDA eigenvalue problem, see sections 2.2 and 2.3. Going from molecular to periodic systems also requires Coulomb interactions to be treated using Ewald summation techniques and the minimum image approximation to capture exact exchange, see section 2.4. We tested our excited-state gradient implementations using a molecular benchmark set of Jacquemin et al. which is one of the state-of-the-art test sets for excited-state properties. ADMM-approximated hybrid functional and semi-empirical sTDA kernels are compared to conventional hybrid functional TDDFT exploiting the therein provided EOM-CCSD geometries and ES data. Finally, in section 3.2, the computational efficiency of the proposed algorithms is demonstrated by applications on porous covalent organic framework (COF) materials taken from the CURATED COFs database. 2 Theoretical background