Rigid Rotor Harmonic Oscillator Approximation RRHO

2 RIGID ROTOR HARMONIC OSCILLATOR APPROXIMATION (RRHO) 

Consider the vibration and rotation of a diatomic molecule. Since the molecule is rotating in space, the Hamiltonian is best written in terms of spherical coordinates. The potential V(r) depends only on the separation of the atoms, and it develops from the electrons and the chemical bonding that occurs between the atoms. The Schroedinger equation for a rotating and vibrating diatomic molecule is 

The term t. in Equation 6-5 represents the reduced mass of the molecule. The operator in spherical coordinates is given in Equation 3-11 and is substituted into Equation 6-5. 

Recall that the operator (the legendrian) is only in terms of the angular coordinates (see Equation 3-12). As a result, the differential equation is separable in terms of the radial and angular parts. The wavefunction must then be a product of an angularfunction, Y m, and a radial function, R. 

Because the radial and angular parts are separable and the molecule rotates freely in space, the angular part of Equation 6-6 is identical to the Particle-on-a-Sphere model problem developed in Section 3.2. Hence, the angular functions Yim are the spherical harmonics (Equation 3-19). The solution of the A operator applied to Y is known and given in Equation 3-20. 

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Figure 6-8. The contribution of the first anharmonicity, centrifugal distortion, and rotation-vibration coupling for H Cl vibration/rotation energy levels relative to the energy values computed from the rigid rotor harmonic oscillator approximation (RRHO). The numbers in parenthesis correspond to the contribution of each correction term. The constants were obtained from Table 6-2.

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. 

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. 

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]. 

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. 

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. 

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... 

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