In the field of ab-initio calculation of the properties of atoms, molecules and solids, the solution of the electronic Schrödinger equation, an operator eigenvalue equation for the Hamiltonian of the system, plays a major role. Of utmost significance is the lowest eigenvalue of this Hamiltonian, representing the ground state energy of the system. To meet the requirements of the multitude of possible applications of the electronic Schrödinger equation, the last decades have seen the development of a variety of different methods designed to approximate the solution of this extremely high-dimensional minimization problem. The present work delivers a mathematical analysis for aspects of some of these methods used in the context of quantum chemistry calculation. Three approaches used in the algorithmic treatment of the electronic Schrödinger equation are analysed in detail: A "direct minimization" scheme used in Hartree-Fock, Kohn-Sham and in CI calculations, the Coupled Cluster method, being of high practical significance in calculations where high accuracy is demanded, and the common acceleration technique DIIS.