This book reviews the methods used to calculate the electronic structures of diatomic molecules. Diatomic gaseous molecules are found in many astronomical objects ranging from cool stars, to brown dwarfs, and to extra solar planets. On Earth gaseous diatomic molecules are found in the atmosphere, in exhaust flames, and in coloring materials. Studying the spectra of these molecules are of immense scientific importance with wide applications. This book starts in chapters one and two with a review of the methods we used to calculate the spectra of diatomic molecules. In chapters three to six we show applications for the theory by calculating the spectroscopic properties of four molecules. These are YN, YS, ZrN, and ZrS. The identification of these molecules in space motivated the present study. In chapter six we simulate the spectrum of ZrS at high temperature, we also produce line lists for its absorption spectrum in an attempt to help identify this molecule in interstellar environments. The results of the present work have been published in peer reviewed scientific journals.