Rigid-rotator calculations

These surfaces are all based on some combination of ab initio electronic structure calculations plus fitting. The AD and BM surfaces are based respectively in whole or in part on extended-basis-set single-configuration self-consistent-field calculations, whereas the RB and RBST calculations are based on calculations including electron correlation by Moller-Plesset fourth-order perturbation theory. For the rigid-rotator calculations R., the intramolecular internuclear distances R- and R ... 

Under some circumstances the rotationally anisotropy may be even further simplified for T-R energy transfer of polar molecules like HF (41). To explore this quantitatively we performed additional rigid-rotator calculations in which we retained only the spherically symmetric and dipole-dipole terms of the AD potential, which yields M = 3 (see Figures 1, 3, and 4). These calculations converge more rapidly with increasing N and usually yield even less rotationally inelastic scattering. For example Table 2 compares the converged inelastic transition probabilities... 

Fig. 2. Thermally averaged effective cross sections (in a ) for H + H2(0,j) H + H2(0,j ) as functions of j . The cross sections are averaged over a thermal distribution of relative translational energy at 444 K and 875 K. The Green-Truhlar results are rigid-rotator calculations, but the present calculations include vibrational motion. The arrow at the lower limit of one of the error bars indicates our uncertainty as to that lower limit. If only one trajectory contributed to the cross section we defined the uncertainty to be 100%.

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. 

Figure 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

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... 

By starting with this partition function and going through considerable mathematical manipulation, one arrives at the following equations for calculating the corrections to the rigid rotator and harmonic oscillator values calculated from Table 10,4, U... 

E10.6 For the diatomic molecule Na2, 5 = 230.476 J-K-1-mol" at T= 300 K, and 256.876 J-K-,-mol-1 at T= 600 K. Assume the rigid rotator and harmonic oscillator approximations and calculate u, the fundamental vibrational frequency and r, the interatomic separation between the atoms in the molecule. For a diatomic molecule, the moment of inertia is given by l pr2, where p is the reduced mass given by... 

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).

We have carried out converged rigid-rotator dynamics calculations for four potentials ... 

Finally, non-rigid reorientation of the macromolecule causes a decrease in the relaxation rates with respect to what is calculated on the assumption of rigid rotation (see Section II.B.2). When this effect is not taken into account, this decrease is usually reflected in a longer proton metal ion distance. 

Expressions for the partition function can be obtained for each type of energy level in an atom or molecule. These relationships can then be used to derive equations for calculating the thermodynamic functions of an ideal gas. Table 11.4 or Table A6.1 in Appendix 6 summarize the equations for calculating the translational, rotational, and vibrational contributions to the thermodynamic functions, assuming the molecule is a rigid rotator and harmonic oscillator.yy Moments of inertia and fundamental vibrational frequencies for a number of molecules are given in Tables A6.2 to A6.4 of Appendix 6. From these values, the thermodynamic functions can be calculated with the aid of Table 11.4. 

Table A6.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the translational, rotational, and vibrational degrees of freedom. The equations assume that the rigid rotator and harmonic oscillator approximations are valid.

For diatomic molecules, <2>o is the vibrational constant to use with the equations in Table A6.1 to calculate the thermodynamic values for a diatomic molecule, assuming the rigid rotator and harmonic oscillator approximations are valid. The vibrational constants Qe and u>exe are the values to use with the equations in Table A6.5 to calculate the anharmonicity and non-rigid rotator corrections. They are related to u>o by... 

This expression is roughly consistent with the band structure of rotation-vibration spectra. Since the rotational quantum number J assumes integral values, the lines comprising a rotation-vibration band of a rigid rotator are equally spaced. The separation of such lines allows calculation of the moment of inertia of the molecule without the necessity for exploring the far infrared. 

Based on the above calculations, both the free energy function (EEE) and the standard enthalpy of formation AH° (298) of NiCp were obtained, the first from its structure and normal mode of vibration by applying the harmonic-oscillator, rigid-rotator approximation, and the second from a Born-Haber thermodynamic cycle.The thermodynamic description of NiCp, together with that of the other participating compounds, permitted a second series of partial equilibrium calculations of the... 

To conclude this section it is possible to state that the origin and magnitude of errors in the calculated barriers are, at least for simple molecules, well understood. The knowledge accumulated permits to decide whether a rigid rotation model and a basis set without d--functions are appropriate to a particular molecule. Valuable information on this topic was contributed by systematic 4-31G studies on... 

Spectroscopic constants used in calculating corrections to rigid rotator-harmonic oscillator approximation (cm ) ... 

Spectroscopic constants used In calculating corrections to the rigid-rotator harmonic oscillator model (cm" ) ... 

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