Development of low-scaling methods to calculate ground state energies and analytical gradients based on the adiabatic-connection fluctuation-dissipation theorem

New methods to efficiently calculate energetics and first order-properties for mean-field and correlated electronic structure theories are presented. In linear-scaling short-range hybrid calculations new integral screening criteria exploiting the locality of the attenuated Coulomb operator are introduced that allow to significantly increase the performance of these density functionals. This enables short-range hybrid calculations at a similar cost as pure semi-local density functional theory (DFT) calculations with increased accuracy due to the admixture of exchange. At the level of correlated electronic structure theory, the realm of systems which can be calculated with a linear-scaling random-phase-approximation (RPA) method is extended with a novel multi-node parallel algorithm. Furthermore, more efficient quadrature schemes are used based on the equivalence of the employed integral transforms with the Fourier transform of the non-interacting polarizability. In combination with a new way to introduce Cholesky orbitals this results in a more effient and numerically more accurate linear-scaling RPA correlation energy method with improved memory requirements. Combining these techniques further with an approach to calculate analytical first order properties using quantities from many body perturbation theory, allows to present a low-scaling method that gives access to RPA gradients for molecular systems with several hundred atoms. Moreover linear- and low-scaling methods to calculate different beyond RPA correlation energies are devised, exploiting the locality of exchange type contractions. This enables to apply these more accurate RPA methods to significantly larger systems, which allows to demonstrate the gain in accuracy for large, dispersion dominated systems. The newly developed methods extend the scope of RPA and RPA with exchange correlation energies, and first order RPA properties significantly, while the accuracy is under full numerical control. Furthermore, there is no overhead to the respective canonical algorithm, making the presented methods competitive also for small system sizes. Finally, a new beyond RPA scheme is presented that combines the benefits of plain RPA and RPA with second-order screened exchange, leading to a more balanced, highly accurate post Kohn-Sham method