Harmonic oscillator/rigid rotor model

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,... 

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. 

It should be noted that, in these approximations [usually referred to as the rigid rotor-harmonic oscillator (RRHO) model], all the kinds of motions— electronic vibrational, and rotational—are strictly separated. 

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. 

A major success of statistical mechanics is the ability to predict the thermodynamic properties of gases and simple solids from quantum mechanical energy levels. Monatomic gases have translational freedom, which we have treated by using the particle-in-a-box model. Diatomic gases also have vibrational freedom, which we have treated by using the harmonic oscillator model, and rotational freedom, for which we used the rigid-rotor model. The atoms in simple solids can be treated by the Einstein model. More complex systems can require more sophisticated treatments of coupled vibrations or internal rotations or electronic excitations. But these simple models provide a microscopic interpretation of temperature and heat capacity in Chapter 12, and they predict chemical reaction equilibria in Chapter 13, and kinetics in Chapter 19. 

Thus far, exactly soluble model problems that represent one or more aspects of an atom or molecule s quantum-state structure have been introduced and solved. For example, electronic motion in polyenes was modeled by a particle-in-a-box. The harmonic oscillator and rigid rotor were introduced to model vibrational and rotational motion of a diatomic molecule. 

Taking into account corrections to the harmonic oscillator-rigid rotor model, vibrational-rotational energies for a diatomic molecule can be represented by... 

Weakly bound complexes display unusual structural and dynamical properties resulting from the shape of their intermolecular potential energy surfaces. They show large amplitude internal motions, and do not conform to the dynamics and selection rules based on the harmonic oscillator/rigid rotor models (4). Consequently, conventional models used in the analysis of the spectroscopic data fail, and the knowledge of the full intermolecular potential and dipole/polarizability surfaces is essential to determine the assignments of the observed transitions. 

The construction of a Hamiltonian is normally an easy problem. The solution of the Schrodinger equation, on the contrary, represents a serious problem. It can be solved exactly for several model cases a particle in a box (one-, two- or three-dimensional), harmonic oscillator, rigid rotor, a particle passing through a potential barrier, hydrogen atom, etc. In most applications only an approximate solution of the Schrodinger equation is attainable. 

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)... 

The vibrational and rotational components can be calculated from the harmonic oscillator and rigid rotor models, for example, whose expressions can be found in many textbooks of statistical thermodynamics [20]. If a more sophisticated correction is needed, vibrational anharmonic corrections and the hindered rotor are also valid models to be considered. The translational component can be calculated from the respective partition function or approximated, for example, by 3I2RT, the value found for an ideal monoatomic gas. 

So far we have used the models of the rigid rotor, and the harmonic or anharmonic oscillator to describe the internal dynamics of the diatomic molecnle. Since the period for rotational motion is of the order 10 " s, and that for vibrational motion is 10 s, the... 

The above treatment of hindered rotors assumes that a given mode can be approximated as a one-dimensional rigid rotor, and studies for small systems have shown that this is generally a reasonable assumption in those cases (82). However, for larger molecules, the various motions become increasingly coupled, and a (considerably more complex) multidimensional treatment may be needed in those cases. When coupling is significant, the use of a one-dimensional hindered rotor model may actually introduce more error than the (fully decoupled) harmonic oscillator treatment. Hence, in these cases, the one-dimensional hindered rotor model should be used cautiously. 

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