Harmonic oscillator model, with rigid rotor

Principal axes can easily be identified in a molecule which possesses symmetry elements e.g., symmetry axes that coincide with principal ones, and a symmetry plane that is oriented perpendicularly to one of the principal axes. The simplest models discussed here are rigid rotor - harmonic oscillator models, which can be extended on demand to better fit the spectral data. For a more complete coverage, the reader is referred to other text books. As a first introduction to infrared rotation-vibration spectra the author prefers Barrow (1962). The topic is discussed in greater details by publications such as by Allen and Cross (1963), Herzberg (1945, 1950), and Hollas (1982). 

At this introductory stage we can carry the comparison with the treatment of ordinary molecules further. In the first approximation these are described by the rigid rotor — harmonic oscillator model. In the next approximation an improvement is achieved by using effective operators with properties as described above. Similarly we may expect that the semirigid rotor — harmonic oscillator model for nonrigid molecules may be improved by introducing effective operators of the form,... 

To predict the properties of materials from the forces on the atoms that comprise them, you need to know the energy ladders. Energy ladders can be derived from spectroscopy or quantum mechanics. Here we describe some of the quantum mechanics that can predict the properties of ideal gases and simple solids. This will be the foundation for chemical reaction equilibria and kinetics in Chapters 13 and 19. Our discussion of quantmn mechanics is limited. We just sketch the basic ideas with the particle-in-a-box model of translational freedom, the harmonic oscillator model for vibrations, and the rigid rotor model for rotations. 

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

Equations (3.28)—(3.31) are based on the harmonic-oscillator/rigid-rotor model. The nonseparable vibrational-rotational i>A, JA levels, with anhar-monicity and vibration-rotation coupling included, may be calculated from Eq. (3.19). The Boltzmann terms for the energy levels, with the (2JA + 1)... 

Thus, the harmonic oscillator—rigid rotor model leads to equations which predict that an absorption band will be made up of lines spaced 2B apart, with a single line of wavenumber cOp at the band center. This cOg line is not observed for diatomic molecules because it represents a forbidden transition. Instead, the band center is observed simply as an intensity minimum in the contour of the band. 

---