This textbook aims at explaining the basic ideas and concepts underlying the theory of orthogonal operators and its application to atomic spectroscopy. Embedded between two more general parts on angular momentum theory and relativity, the main principles are presented in chapter 14 of part II. With orthogonal operators, standard deviations of energy fits are frequently reduced by an order of magnitude and only elementary linear algebra is needed to project ab initio results of any provenance onto an orthogonal basis of operators. The framework of orthogonal operators is surely fortified with the support of Hartree-Fock calculations combined with perturbation theory; B-splines are a strong tool to calculate the occurring one- and two-electron excited states. These ab initio aspects are covered as well in part II for this reason. Second quantization in the coupled form as an elegant method to handle a wide variety of angular momentum problems is used throughout the book in all three parts. In part III, second quantization is also used to show how a jj → SL transformation yields fully relativistic results in SL−coupling. Hyperfine structure and electromagnetic radiation are treated in some detail to illustrate the potential of this formalism.