Electronic structure theory represents a powerful tool for predicting and interpreting various physical and chemical phenomena, such as energy levels, bonding, and the reactivity of molecular complexes. These are commonly based on the ground-state characteristic, which needs to be approximated due to its complex origin as a partial differential operator eigenvalue problem. Computational chemistry applies models, therefore, that include trade-offs between accuracy and physical feasibility. This thesis comprises essentially three parts, whereby the first one concerns the introduction of a fundamental language in the form of a mathematical framework to establish a common ground for an unambiguous discussion of the main principles in electronic structure theory. The second part investigates a well-known chemical postulate, the "Aufbau principle", to provide a mathematical interpretation that can be used to establish its satisfaction or violation of approximation models. The last part is about a particular multiscale approach within subsystem density functional theory, denoted as frozen-density embedding theory. In this work, a non-decomposable approximation model within a semilocal framework is presented, yielding superior subsystem descriptors in comparison to common decomposable semilocal approximants. Its strength is especially displayed when the distribution of electrons between the subsystem species is prone to the charge leak problem.