Connection between semi-empirical and ab initio methods in the quantum theory of molecular electronic spectra

In the beginning of the modern quantum theory of atomic and molecular spectra, the semi-empirical methods were exceedingly successful in explaining the structure of the spectra and their general properties, and it was hence somewhat of a surprise that, when the fundamental semi-empirical parameters were later calculated by ab initio methods, one would always find rather different numerical values. A brief review is given of this problem, and some simplified mathematical derivations are introduced. The various concepts of wave operators used in modern quantum theory are discussed, and, after studying the energy-dependent wave operators obtained by the standard partitioning technique, particular attention is devoted to the energy-independent wave operators and the nonlinear form of the Schroedinger equation. After studying the solution of the nonlinear Bloch equation and the Bloch-Lindgren equation, the concept of the model Hamiltonian is reviewed, and on this basis the connection between the semi-empirical approach and the ab initio methods is discussed. Reference is made to the fundamental comparison work carried out by Karl Freed and his associates in Chicago. Some difficulties remaining are briefly mentioned