Q3416- Chapter 16 Electronic Spectroscopy of Diatomic Molecules P.F. Bernath University of Waterloo, Waterloo, Ontario 1 Introduction 3 2 Born–Oppenheimer Approximation 3 3 Separation of Vibration and Rotation 4 4 Electron Spin and Nuclear Spin 5 5 Notation 5 6 Franck–Condon Principle 7 7 Rotational Line Strengths 8 8 Spectroscopic Constants and Potential Energy Curves From Ab Initio Calculations 8 9 Calculation of Relativistic Effects 11 10 Calculation of Lifetimes and Transition Dipole Moments 12 11 Calculation of Bond Energies, Ionization Potentials, and Electron Affinities 13 12 Conclusion 14 Note 14 References 14 1 INTRODUCTION The study of the electronic spectra of diatomics has a long history. Prominent diatomic spectra include the green color of the Swan system of C 2 that can be seen in a Bunsen burner flame, (1) the emission of N 2 in the aurora (2) and the atmospheric A-band absorption of oxygen first seen by Fraunhofer. (3) The spectra of TiO were seen in Handbook of Molecular Physics and Quantum Chemistry, Edited by Stephen Wilson. Volume 3: Molecules in the Physico- chemical Environment: Spectroscopy, Dynamics and Bulk Proper- ties.  2002 John Wiley & Sons, Ltd stars (4) well before any interpretation of the bands could be made. The development of quantum mechanics and the Born – Oppenheimer approximation (5) led to the quantitative understanding of electronic spectra. In this chapter, we begin with a discussion of the basic principles of the electronic spectroscopy of diatomic molecules. (6,7) The rest of the chapter is devoted to a discussion of the ab initio calculation of molecular prop- erties associated with the electronic spectra of diatomic molecules. The calculated properties will be compared with experimental observations for selected examples. 2 BORN–OPPENHEIMER APPROXIMATION The interpretation of the electronic spectra of diatomic molecules begins with the nonrelativistic Hamiltonian in the laboratory coordinate system, namely,  H =− ¯ h 2 2m A ∇ 2 A − ¯ h 2 2m B ∇ 2 B − ¯ h 2 2m e N  i =1 ∇ 2 i +  V (1) for the usual electrostatic potential  V , nuclei A, B with masses m A , m B and N electrons of mass m e . Transforming to the nuclear center-of-mass relative coordinate system and removing the center-of-mass kinetic energy (8) results in the new Hamiltonian,  H =− ¯ h 2 2µ ∇ 2 r − ¯ h 2 2m e N  i =1 ∇ 2 i − ¯ h 2 2(m A + m B ) × N  i,j ∇ i ·∇ j +  V (2)