Quantum mechanics is one of the most predictive theories that we have at our disposal to explain natural phenomena with atomistic detail. Yet, applying it to complex system, such as chemical reactions, is an extremely demanding task. Approximations justified by physical principles are necessary to reduce the computational cost of these simulations. This dissertation analyzes two classes of problems in molecular quantum mechanics, highlighting the main ingredients necessary to qualitatively and quantitatively describe them. First, we look at how the choice of orbitals influences the description of strong-field ionization processes in simple systems, drawing an analogy to the well-known symmetry dilemma within quantum chemistry. We show that, through a simple model and a mean-field treatment, allowing the wavefunction to simply separate pairs of electrons is not enough to understand the ionization process, requiring additional flexibility to allow for spin rotations.

The second class of problems discussed in this dissertation concerns the description of excited states that arise when promoting one of the electrons in an inner-shell of a system. X-ray spectroscopy is a vibrant field that has been explored in great detail over recent years due to advances in light sources. From a theory perspective, these core-excited states also pose some challenges to existing quantum chemical methods, especially due to lack of relaxation effects after creating a hole in one of the core orbitals. We discuss different approaches to model X-ray emission and absorption, as well as the role of scalar relativistic effects in accurately modeling the core-excitation energies and the spectra of heavy elements. In doing so, we have devised both state-specific and linear-response methods to model core-excited states, expanding the toolbox available to quantum chemists to interpret new experiments in the X-ray range of the electromagnetic spectrum.