Anharmonicity and Nonrigid Rotator Corrections

Anharmonicity and Nonrigid Rotator Corrections With the rigid rotator and harmonic oscillator approximations, the combined energy for rotation and 

With c given in cms I, both Ce and B have the units of cm which is the unit most commonly used by spectroscopists. 

To correct for these effects, the combined energy due to rotation and vibration is often written as 

The vibrational and rotational constants are now written as a c and Be. They may be thought of as the values that correspond to the equilibrium interatomic distance of minimum potential energy.J The first and third terms are expressions 

Values for the constants for a few diatomic molecules are given in Tables 10.1 and 10.2. 

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For diatomic molecules, B0 is the rotational constant to use with equation (10.125), while Be applies to equation (10.124). They are related by Bq = Be 2 - The moment of inertia 70(kg-m2) is related to 50(cm ) through the relationship /0 = h/ 8 x 10 27r22 oc), with h and c expressed in SI units. For polyatomic molecules, /a, /b, and Iq are the moments of inertia to use with Table 10.4 where the rigid rotator approximation is assumed. For diatomic molecules, /0 is used with Table 10.4 to calculate values to which we add the anharmonicity and nonrigid rotator corrections. 

For diatomic molecules, lj0 is the vibrational constant to use with equation (10.125) for calculating anharmonicity and nonrigid rotator corrections, while J)e and tDe-Ve... 

Table A4.5 summarizes the equations for calculating anharmonicity and nonrigid rotator corrections for diatomic molecules. These corrections are to be added to the thermodynamic properties calculated from the equations given in Table A4.1 (which assume harmonic oscillator and rigid rotator approximations).

The following equations are used to calculate the anharmonicity and nonrigid rotator corrections to the thermodynamic properties of diatomic molecules. 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

Table A4.5 Anharmonic oscillator and nonrigid rotator corrections...

For diatomic molecules, corrections can be made for the assumption used in the derivation of the rotational partition function that the rotational energy levels are so closely spaced that they can be considered to be continuous. The equations to be used in making these corrections are given in Appendix 6. Also given are the equations to use in correcting for vibrational anharmonicity and nonrigid rotator effects. These corrections are usually small.22... 

Analyses of linear nonrigid rotators, anharmonic oscillators, and vibrating rotators, yielding first-order corrections for nonrigidity, anharmon-icity, and vibration-rotation interaction (nonseparability of vibrational and rotational modes), respectively, have also been completed and are conventionally used in obtaining corrections (which are most important at elevated temperatures) to the simple product form of the molecular partition... 

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