The subject of this chapter is the quantum mechanical analysis of the interaction of electromagnetic radiation with atomic transitions. The analysis is based on the Schrödinger wave equation, and in the first section, the gauge-invariant form of the external electromagnetic field is introduced. The electric dipole interaction and the long-wavelength approximation for the analysis of this interaction are discussed. The perturbative analysis of both single-photon and two-photon electric dipole interactions is presented, and density matrix analysis is introduced. The interaction of radiation with the resonances of atomic hydrogen is then discussed. The analysis is performed for both coupled and uncoupled representations. In the last section of the chapter, the radiative interactions for multielectron atoms are discussed. The Wigner–Eckart theorem and selection rules for transitions between levels characterized by coupling are developed. The effect of hyperfine splitting on radiative transitions is also briefly discussed.

The chapter begins with the introduction of the two-particle Schrödinger wave equation (SWE) and the solution of this equation for the hydrogen atom. The orbital angular momentum of the electron results from the SWE solution. The Pauli spinors are introduced, and the SWE wavefunctions are modified to account for the spin of the electron. The structure of multielectron atoms is then discussed. The discussion is focused on low-Z atoms for which Russell–Saunders or LS coupling is appropriate. Alternate coupling schemes are briefly discussed. Angular momentum coupling algebra, the Clebsch–Gordan coefficients, and 3j symbols are then introduced. The Wigner–Eckart theorem is discussed, and the use of irreducible spherical tensors for evaluation of quantum mechanical matrix elements is discussed in detail.

The classical theory of the interaction of light with the electron clouds of atoms and molecules will be discussed in this chapter. The discussion will begin with the interaction of a steady electric field with a collection of point charges, leading to the development of terms describing the electric dipole and quadrupole moments. The classical Lorentz model is then introduced to describe interaction of an oscillating electric field with the electron cloud of an atom, and the concepts of absorption and emission are introduced. The propagation of a light wave through a medium with electric dipoles is then discussed. Finally, the classical theory of radiation from an oscillating dipole is discussed.

Laser absorption spectroscopy is widely used for sensitive and quantitative detection of trace species. In this chapter, the density-matrix approach is used to introduce laser absorption spectroscopy. Spectroscopic quantities that characterize the absorption process are defined, and the relationships among these quantities are discussed. Broadening processes for spectral line shapes are also discussed, and the Doppler, Voigt, and Galatry profiles are introduced. The chapter concludes with a detailed example calculation featuring NO absorption.

The interaction of electromagnetic radiation with single-photon resonances in diatomic molecules is discussed in this chapter. The properties of the electric dipole moment of the molecule are determined primarily by the electron cloud that binds the two nuclei together, and these properties can be understood by considering a reference frame fixed to the molecule. However, the response of the molecule must be averaged over all possible orientations of the molecule in the laboratory frame. Using irreducible spherical tensors greatly simplifies the orientation averaging of the molecular response. The Born–Oppenheimer approximation is invoked to initially account for the effect of the electronic, vibrational, and rotational modes of the molecule. Corrections are applied to account for the coupling and interactions of the different modes, including Herman–Wallis effects. Tables of rotational line strengths are presented for singlet, doublet, and triplet electronic transitions. These tables incorporate the use of Hund’s case (a) basis state wavefunctions for increased insight into radiative interactions for levels intermediate between Hund’s cases (a) and (b).

Coherent anti-Stokes Raman scattering (CARS) spectroscopy is a technique that has been widely applied for temperature measurements in combustion and for microscopic imaging of cell structures. CARS spectroscopy is discussed in detail in this chapter as an example of a nonlinear optical technique. The concept of the nonlinear susceptibility is introduced, and the derivation of the susceptibility tensor appropriate for CARS spectroscopy is described in detail. A key aspect of this derivation is the incorporation of the electric dipole transition matrix elements for the Raman scattering process into the susceptibility tensor. CARS spectral modeling and collisional narrowing of CARS spectral features are discussed in detail. The emerging field of femtosecond CARS is discussed. The chapter concludes with detailed examples of CARS intensity calculations.

The structure of diatomic molecules is discussed in this chapter. The electronic structure of diatomic molecules is then discussed in detail. The coupling of the orbital and spin angular momenta of electrons and the angular momentum associated with nuclear rotation are discussed, with an emphasis on Hund’s cases (a) and (b). The rotational wavefunctions for diatomic molecules in the limits of Hund’s cases (a) and (b) and in the case intermediate between Hund’s cases (a) and (b) are then discussed in detail. For molecules that are of importance in combustion diagnostics, such as OH, CH, CN, and NO, the electronic levels are intermediate between Hund’s cases (a) and (b). We use Hund’s case (a) as the basis wavefunctions, and linear combinations of these wavefunctions are used to represent wavefunctions for electronic levels intermediate between cases (a) and (b). The choice of case (a) wavefunctions as the basis set is typical in the literature although case (b) wavefunctions can also be used as a basis set.

Raman scattering spectroscopy is widely used in analytical chemistry, for structural analysis of materials and molecules and, most importantly for our purposes, as a gas-phase diagnostic technique. Raman scattering is a two-photon scattering process, and the mathematical treatment of Raman scattering is very similar to the mathematical treatment of two-photon absorption. Many of the molecules of interest for quantitative gas-phase spectroscopy are diatomic molecules with non-degenerate 1Σ ground electronic levels, including N2, CO, and H2. In this chapter, the theory of Raman scattering is developed based on Placzek polarizability theory and using irreducible spherical tensor analysis. Herman–Wallis effects are discussed in detail. The chapter concludes with detailed examples of Raman scattering signal calculations.