Rigid rotors, molecular

Rigid rotor The simple model of molecular rotation assuming that two atoms are connected by rigid bonds when rotating. 

This simplification was already introduced in Chapter 2. In the rigid rotor approximation there is no rotational-vibrational interaction. The molecular Schrodinger... 

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... 

The statement applies not only to chemical equilibrium but also to phase equilibrium. It is obviously true that it also applies to multiple substitutions. Classically isotopes cannot be separated (enriched or depleted) in one molecular species (or phase) from another species (or phase) by chemical equilibrium processes. Statements of this truth appeared clearly in the early chemical literature. The previously derived Equation 4.80 leads to exactly the same conclusion but that equation is limited to the case of an ideal gas in the rigid rotor harmonic oscillator approximation. The present conclusion about isotope effects in classical mechanics is stronger. It only requires the Born-Oppenheimer approximation. 

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... 

The thermodynamic properties were computed with the molecular geometry and vibrational frequencies given above assuming an ideal gas at 1 atm pressure and using the harmonic-oscillator rigid-rotor approximation. These properties are given for the range 0-2000°K in the Appendix (Table AI). 

Less symmetric molecules require a considerably more complicated treatment, but in the end their spectral transitions arc functions of their three moments of inertia (see Section 10.3.5). From a computational standpoint, then, prediction of rotational spectral lines depends only on the moments of inertia, and hence only on the molecular geometry. Thus, any method which provides good geometries will permit an accurate prediction of rotational spectra within the regime where tlie rigid-rotor approximation is valid. 

In the (classical) high-temperature limit, when hi/s -C kBT, the exponential can be expanded to first order and the average vibrational energy is kBT. The approximation hvs -C kis l is, however, not well satisfied for typical molecular vibrational frequencies, except at temperatures that exceed several thousand degrees. The average rotational energy of a rigid rotor is... 

Surprisingly, the enthalpy of combustion of isoxazole was determined only very recently.270 For isoxazole, AH° (298.15 K) = —(394.70 + 0.12) kcalth mol-1, from which the enthalpy of formation in the gas phase was derived as AHf (g) = 18.78 0.13 kcalth mol-1. The enthalpies of combustion of 3-amino-5-methylisoxazole and 5-amino-3,4-dimethylisoxazole have also been determined.271 Thermodynamic parameters for isoxazole have been derived from vibrational spectra using the harmonic oscillator-rigid rotor approximation.272,273 Analysis of the rotational spectra of isotopic forms of isoxazole, studied by double resonance modulated microwave spectroscopy, has given the molecular dimensions shown in Fig. 1.274,275... 

Conventional RRKM theory estimates for N" (E,J) are then obtained via a convolution with the rigid rotor rotational energy levels, E, oi(J,K). for the molecular structure at the stationary point ... 

It follows from the preceding discussion that the equilibrium constant for complex formation evaluated using the rigid rotor-harmonic oscillator approximation, with molecular constants derived from ab initio SCF calculations with a medium basis set (of DZ quality), is not very accurate. Comparison of the AG° values calculated using extended and medium basis sets indicates that the major uncertainty in AG is derived from AH . TASP is not as dependent on the basis set used. Furthermore, it is evident that the entropy term plays an extremely important rote in complex formation neglecting it may result not only in quantitative, but even in qualitative failure. 

The rigid rotor model assumes that the intemuclear distance Risa constant. This is not a bad approximation, since the amplitude of vibration is generally of the order of 1% of i . The Schrbdinger equation for nuclear motion then involves the three-dimensional angular-momentum operator, written J rather than L when it refers to molecular rotation. The solutions to this equation are already known, and we can write... 

The thermodynamic functions were estimated from those in the present table for HgS(g) (6 ) by adding those for DgS(g) and subtracting those for HgS(g), where both the added and subtracted functions were generated using the rigid-rotor harmonic oscillator approximation. In this calculation the molecular constants for DgS were taken from reference (2). 

IR-frequencies were determined by Christe et al. [72CHR/SCH] and, together with molecular constant data, they computed thermodynamic properties of SeFsC g) using the rigid-rotor, harmonic-oscillator approximation. The following temperature dependence... 

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