Rigid-rotor and harmonic-oscillator

The derivation above may be generalized to wave functions other than electronic ones. By evaluation of transition dipole matrix elements for rigid-rotor and harmonic-oscillator rotational and vibrational wave functions, respectively, one arrives at the well-known selection rules in those systems that absorptions and emissions can only occur to adjacent levels, as previously noted in Chapter 9. Of course, simplifications in the derivations lead to many forbidden transitions being observable in the laboratory as weakly allowed, both in the electronic case and in the rotational and vibrational cases. 

To arrive at K and k, our task is to express the following terms appearing in eqns. (5.15) and 5.16) the partition functions (Q) of reactants, products and of the activated complex, the heat of reaction at absolute zero, AHq, the enthalpy of activation at absolute zero, Hq, and the tunnelling correction factor, P. For an ideal gas the total partition function can be expressed within the rigid-rotor and harmonic oscillator (RRHO) approximation as a product... 

For a symmetric rotor, in the present approximation, only the z component of in, the vibrational angular momentum, needs to be considered. The problem may be treated as a perturbation employing zero-order wave functions which are products of rigid rotor and harmonic oscillator functions. When the molecule is in a state such that vka + Vkb — 1, where Qka and Qw> are degenerate, it is necessary to solve the secular determinant... 

Selected values of the enthalpy Hj-Hg, entropy Sj, and free-energy function (GT-Hg)TT, which have been calculated within the rigid-rotor and harmonic-oscillator approximations, are compiled in the following Table 29.1.1-2. The assumed molecular constants, except for the multiplicities of the electronic states (not given here), are presented on p. 2. 

Recently, the above approach has been applied to systems with a collection of harmonic oscillators coupled with rigid rotors and systems with a collection of anharmonic oscillators by Tou and Lin. For a system with a collection of w harmonic oscillators. . . , g, . . . g ) coupled... 

With the above approach we can combine the use of curvilinear normal coordinates with the Eckart frame. When we do so, the harmonic oscillator, rigid rotor, and, to lowest order, the Coriolis and centrifugal coupling contributions to H have exactly the same form as those found for the more commonly used Watson Hamiltonian (58). 

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. 

It should be understood that we start from an exact solution to the rigid rotor and the exact solution to the harmonic oscillator for a perfect parabola and then apply common sense corrections to the idealized solutions by fitting a polynomial to experimental data so the corrections are empirical rather than quantum mechanical. We use a formula based on quantum mechanics plus some empirical terms (with Shoemaker-Garland-Nibler notation). 

Molecules in the gas phase have rotational freedom, and the vibrational transitions are accompanied by rotational transitions. For a rigid rotor that vibrates as a harmonic oscillator the expression for the available energy levels is ... 

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. 

In Chapter 3, a formula was presented which connects the normal vibrational frequencies of two rigid-rotor-harmonic-oscillator isotopomers with their respective atomic masses m , molecular masses Mi and moments of inertia (the Teller-Redlich product rule). If this identity is substituted into Equation 4.77, one obtains... 

In Section 4.8, Equations 4.78,4.79 and Table 4.1 develop the connections between the harmonic oscillator rigid rotor partition function and isotope chemistry as expressed by the reduced partition function ratio, RPFR = (s/s ) f. RPFR is defined in Equation 4.79 as the product over oscillators of ratios of the function [u exp(—u/2)/ (1 - exp(u))]... 

At all but very high temperatures it is necessary to employ the complete equation because the vibrational frequencies for all these molecules are quite high. (Notice at room temperature u(H2) 21, and u(HI) 11). Harmonic oscillator rigid rotor calculated equilibrium constants are shown in Fig. 4.4. As expected the low temperature limiting value, while bounded, is significantly different from unity. At extremely high temperature Equation 4.95 applies and the isotope exchange constant is... 

That result is included in Fig. 4.4. For precise comparison with experiment harmonic oscillator rigid rotor results should be corrected for the effects of nonclassical rotation and anharmonicity. In the region of the maximum (Fig. 4.4) these corrections (see Appendix 4.2), which are temperature dependent, lower the calculated results by several percent. The spectroscopic data employed for the calculation reported in Fig.4.4 are shown in Table4.2. 

The partition function ratios needed for the calculation of the isotope effect on the equilibrium constant K will be calculated, as before, in the harmonic-oscillator-rigid-rotor approximation for both reactants and transition states. One obtains in terms of molecular partition functions q... 

Calculations of isotope effects and isotopic exchange equilibrium constants based on the Born-Oppenheimer (BO) and rigid-rotor-harmonic-oscillator (RRHO) approximations are generally considered adequate for most purposes. Even so, it may be necessary to consider corrections to these approximations when comparing the detailed theory with high precision high accuracy experimental data. 

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