Deterministic and Statistical Approaches to Quantum Chemistry

The field of quantum chemistry has recently undergone a series of paradigm shifts, including a boom in machine learning applications that target the electronic structure problem. Along with these technological innovations, the community continues to identify shortcomings in traditional KS-DFT approaches and develop improved approximations. The original work presented in this thesis addresses a selection of open questions along these two lines. Specifically, the thesis is structured to reflect the ongoing advancement of traditional (deterministic) approaches toward more recent examples exploiting (statistical) non-linear regression techniques. The first section of this thesis analyzes the performance of approximate density functionals and dispersion correction schemes on chemical situations that are not well-represented in standard benchmark datasets of van der Waals complexes. First, we discuss how the synergy between delocalization error and London dispersion interactions in asymmetrically charged radical cation dimers remains problematic even for recent density functionals. Solutions are provided to improve the description of these systems that are typical charge-carrier in organic electronic materials. While this first chapter focuses on non-covalent interactions between molecules in the ground-state, very little is known about the consequences of an incomplete treatment of London dispersion interactions upon photo-excitation. Using the prototypical stilbene photoswitch as a working example, the second chapter demonstrates that neglecting these interactions in the excited states leads to qualitative failures in the description of the photodeactivation process. The conclusions presented here apply broadly to any photoswitch functionalized with large and polarizable side chains. In contrast to traditional (deterministic) quantum chemistry, machine learning-based variants are still in their infancy when facing the challenge of targeting fundamental, albeit complex, quantum chemical objects by encoding their symmetries and properties. The subsequent chapters describe the development and application of machine-learning techniques to predict the molecular electron density [p(r)] using an atom-centered decomposition compatible with symmetry-adapted Gaussian process regression (SA-GPR). Concrete applications of the framework are shown for a chemically rich set of dimers, whose predicted electron densities serve to compute covalent and non-covalent interaction fingerprints, electrostatic potentials, as well as quantitative interaction energies. The transferability of the model is demonstrated by the accurate prediction of p(r) for a set of pentapeptides. Combining transferability and accuracy, our regression framework grants access to the density information of complex chemical systems at a fraction of the traditional ab-initio computational cost. The final chapter exploits the complementarity of both strategies and proposes a machine learning framework capable of quantifying the deviation of approximate density functionals from the piecewise linearity condition of exact DFT. The predicted curvature information is applied both for restoring the correspondence between the Kohn-Sham HOMO eigenvalue and the first ionization potential in optimally tuned DFT functionals as well as to provide a large-scale analysis of the relationship between the deviation from the piecewise-linearity condition and the chemical and structural pattern