RRHO approximation

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. 

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... 

In summary, to calculate rate and equilibrium constants we need to calculate AG and AGo. This can be done within the RRHO approximation if the geometry, energy and force constants are known for the reactant, TS and product. The translational and rotational contributions are trivial to calculate, while the vibrational frequencies require the full force constant matrix (i.e. all energy second derivatives), which may be a significant computational effort. 

J. Chao, R. C. Wilhoit and B. J. Zwolinski, Ideal gas thermodynamic properties of ethane and propane , J. Phys. Chem. Ref. Data, 2, 427 (1973). Review and evaluation of structural parameters (including vibrational frequencies and internal rotation properties) tabulation of thermodynamic properties [C°, S°, H° — H°), (H° — H )/T, - G°-Hl)/T, AfG°,AfH°, logK ] for 0< T (K)< l500 calculated by statistical thermodynamic methods [rigid-rotor harmonic oscillator (RRHO) approximation]. 

T(K) < 1500 calculated by statistical thermodynamic methods (RRHO approximation). 

Comparison with experiment obviously entails thermal corrections. The RRHO approximation will cause some errors, the largest of which will be neglect of the population of the various low-energy conformers. If we neglect the difference between the rovibrational partition functions of the different conformers, then the conformer contribution to the enthalpy function hcf29s = Ht=298 — Eq is easily found as [96]... 

As before, simple approximations to P and P° can be obtained within the RRHO approximation. 

Using the designation J rather than / describes molecular rotation, and the degenerate m, states are designated as Mj. The result in Equation 6-22 is called the rigid rotor harmonic oscillator (RRHO) approximation for a diatomic molecule. 

Substitution of Equations 6-23 and 6-24 into Equation 6-22 results in the following expression for the rotational/vibrational energy of a diatomic molecule at the RRHO approximation ... 

Equations 6-26 and 6-27 represent the predictions for the absorption lines for an infrared spectmm of a diatomic molecule based on the RRHO approximation. 

Table 6-1. The predicted position, AE, and separation of peaks, A(AE), of a diatomic molecule in an infrared spectrum using the RRHO approximation.

The RRHO approximation predicts that the infrared spectrum of a diatomic molecule will have a number of peaks all of equal separation (2Bo) except one larger gap of 4Bo between the set of peaks where J is increasing by one (R-branch) and the set where J is decreasing by one (P-branch). As can be seen by Figure 6-2, the RRHO approximation predicts the infrared spectrum of a diatomic molecule remarkably well in spite of the harmonic oscillator approximation for vibrational motion and the sevare truncation of the power series in Equation 6-18. The distinct gap between the P and R-branches can be clearly seen. However, note that the distance of separation between the peaks in the infrared spectrum has some variation whereas the... 

Figure 6 2. An FTIR spectrum of hydrogen chloride ( H C1) at room temperature (298 K) is shown. The general pattern of absorption peaks is predicted by the RRHO approximation however, increasing deviation from the RRHO approximation occurs as the initial J state increases.

RRHO approximation predicts it to be constant. The RRHO approximation does not take into account that the rotational motion cannot be entirely separated from the vibrational motion of the molecule, and the vibrational motion is not strictly harmonic oscillations. This will be taken into account in the following sections by including additional terms in the series expansion of (Equation 6-18) and by correcting for some... 

The RRHO approximation and analysis of the infrared spectrum of formulates a picture of the vibration-rotation energy levels of a diatomic molecule. The energy difference between vibrational energy levels is large with respect to the rotational energy levels. A vibrational state v will have an infinite manifold of J rotational states. This is depicted in Figure 6-4. 

One method for solving Equation 646 is to use perturbation theory. The term in the brackets in Equation 646 can be recognized as the Schroedinger equation for the RRHO approximation (Equation 6-19), and the term bs can be taken as a first-order perturbation. The first-order and higher order corrections to the energy eigenvalues to the RRHO approximation can then be computed using Perturbation Theory. 

The energy eigenvalue expression in Equation 6-50 is the same as for the RRHO approximation (Equation 6-25) except for the last term. The term Do is called the centrifugal distortion constant. Note that it does have the same symbol as dissociation energy however, its context will indicate whether it represents dissociation energy or the centrifugal distortion constant. 

The vibrational/rotational energy states of a diatomic molecule can now be written to include not only the RRHO approximation but also in terms of the correction factors including the first anharmonicity correction, centrifugal distortion, and vibration-rotation coupling (Section 6.3 - 6.5). 

The effect and the order of magnitude of each of the correction terms relative to the RRHO approximation for H Cl are shown Figure 6-8. 

Finally, a word about statistical thermodynamic calculation methods. Most of the published tables used the rigid rotor harmonic oscillator (RRHO) approximation method. These calculations are accurate for most molecules up to 1500 K. The JANAF (1971) calculations were based mainly on the RRHO approximation. When values at temperatures above 3000 K are desired, however, the RRHO values are too low. Unfortunately, anharmonicity constants are still known only for very few molecules. Some publications do include values obtained using anharmonicity corrections (Burcat, 1980 McBride et aL, 1963 McDowell and Kruse, 1963). There are still uncertainties regarding the best way to calculate anharmonic corrections. McBride and Gordon (1967) discuss the alternatives, of which NRRA02 appears to be the best. 

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