Accurate models of electron correlation are key to understanding and predicting important physical characteristics that underpin the development of many modern quantum technologies. One of the most widely used approaches to modeling cor- relation is the GW approximation within many-body perturbation theory (MBPT). There are a large number of ‘flavors’ of the GW approximation, with varying levels of computational cost and accuracy. Our aim is to elucidate the various deficiencies and develop novel corrections in order to ameliorate such failings.
In this thesis we study simple model systems, whose properties can be determined exactly by numerical solution of the Schrödinger equation, that exhibit the key physical properties present in real quantum systems such as atoms and molecules. We then are able to compare the exact results to that of existing methods to identify the fundamental source of shortcomings, using this insight to develop new approximations. We find that a common systematic error across all flavors of the GW approximation is the ‘self-screening’ error. We develop a conceptually and computationally simple correction that removes the unwanted effect of this error on the charge density and ionization potential.
We demonstrate that MBPT methods exhibit Kohn’s concept of advantageous ‘nearsightedness’, unlike the computationally cheaper Kohn-Sham density functional theory (KS-DFT). We attribute this to the non-locality of the potentials used in MBPT. We highlight that this allows approximations to more easily encapsulate advanced aspects of exchange and correlation to be made within MBPT. Hybrid functionals contain these non-local potentials and so benefit from this nearsightedness, and when enforced to obey physically justified constraints, yield extremely accurate densities and ionization potentials.
We also extend our investigation to the time-dependent properties of correlated systems. A form of time-dependent many-body perturbation theory, that brings together the simplicity of time-dependent DFT (TDDFT) and sophisticated correlation effects of MBPT is investigated. We show that this approach significantly outperforms common approximations to TDDFT without requiring the more onerous computational cost of non-equilibrium Green’s function methods.
