Molecular Vibrational Partition Function

In a polyatomic molecule witli many vibrations, we simplify the vibrational partition function much as the original molecular partition function was simplified we assume that the total vibrational energy can be expressed as a sum of individual energies associated with each mode, in which case, for a non-linear molecule, we have 

we will adopt the convention of including the ZPVE in the zero of energy [Eq. (10.1)], so that each zeroth vibrational level has an energy of zero. In that case, any individual mode s partition function can be written as 

The sum in Eq. (10.26) is well known as a convergent geometric series, so that we may write 

Using Eq. (10.27) for each mode, the full vibrational partition function of Eq. (10.25) can be expressed as 

Note that Eqs. (10.29) and (10.30) take the vibrational frequencies as independent variables, and as such cannot be calculated ab initio without first optimizing a structure at some level of theory and then computing the second derivatives in order to obtain the frequencies within the harmonic oscillator approximation. (Of course, one could avoid the harmonic oscillator approximation (see, for example, Barone 2004), but tlie necessary calculations and 

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From a theoretical perspective, isotope effects are fairly trivially computed. The stationary points on the PES and their electronic energies are independent of atomic mass, as are the molecular force constants. Thus, one simply needs to compute the isotopically dependent zero-point energies and translational, rotational, and vibrational partition functions, and evaluate Eq. (15.33). 

The translational contribution to the molecular partition function, which is calculated using Eq. 8.59, clearly makes the largest contribution. (In obtaining this value, we also made use of the ideal gas law to calculate the volume V = 0.02479 m3 of a mole of gas at this temperature and pressure.) The rotational partition function is evaluated via Eq. 8.67, and the vibrational partition function for each mode is found via Eq. 8.71. Only the very... 

BO-scheme, if no symmetry restrictions are used (a-space optimization) some states may display saddle point character with indices equal to or larger than 1. When such a situation is met the very fact that there are solutions with imaginary frequencies indicate they are not acceptable as physical stationary state solutions. For this reason, it is common practice, when calculating any property related to the molecular spectra, to discard these solutions as it is done in evaluating vibrational partition functions to get chemical rates [19]. 

The detailed theory differs from the transition-state theory in replacing kT /h by v, a specific molecular constant, and in using (zJB(vib), the true vibrational partition function of AB rather than AB(vib), which has one less vibrational degree of freedom. However, since v is expected to be about 10 sec and kT/h = 6 X 10 scc at 300°K and the extra vibrational term in gJn contributes a factor of less than 10, it can be seen... 

To relate these thermodynamic quantities to molecular properties and interactions, we need to consider the statistical thermodynamics of ideal gases and ideal solutions. A detailed discussion is beyond the scope of this review. We note for completeness, however, that a full treatment of the free energy of solvation should include the changes in the rotational and vibrational partition functions for the solute as it passes from the gas phase into solution, AGjnt. ... 

At a given temperature, the isotopic equilibrium constant for Reaction 5 is, of course, fixed. The equilibrium constant for Reaction 6 will depend on the value of the vibrational partition function ratio for the isotopic species of the molecular addition compound. The numerical value of the numerator in Equation 8 will vary from donor to donor and will be smallest for weakly bound molecular addition compounds. As the equilibrium constant for Reaction 6 approaches unity as a limit, the equilibrium constant for Reaction 2 approaches the equilibrium constant for Reaction 5. The equilibrium constant for Reaction 5 thus represents the... 

Problem Calculate the vibrational partition function of (i) molecular hydrogen, (ii) molecular chlorine, at 300° E, assuming them to be harmonic oscillators. 

The value of the equilibrium constant may thus be derived from AHS for the reaction, and the molecular weights, moments of inertia, and the symmetry numbers of the substances taking part. Equations of the type of (33.56) have been employed particularly in the study of isotopic exchange reactions, where the error due to the cancellation of the vibrational partition functions is very small, especially if the temperatures are not too high. ... 

The EIE may be calculated from molecular translational, rotational, and vibrational partition function ratios as described by Bigeleisen and Mayer (Equa-... 

The inclusion of molecular vibrations will effect the partition function which, in addition to the electronic contribution, will contain the vibration part [28]. Each molecule is assumed to behave as a set of (3 - 6) independent harmonic oscillators with the fundamental frequencies vf so that the vibration partition functions are... 

The derived density of states for the translations, rotations, and vibrations can be used in Eq. (6.41) to obtain the corresponding classical partition functions. This will yield an accurate translational partition function at all temperatures of chemical interest because the translational energy level spacings are so dense. It will also yield accurate rotational partition functions at room temperature because molecular rotational constants are typically between 0.01 and 1 cm k However, at the low temperatures achieved in molecular beams, the accuracy of the classical rotational partition function (especially for molecules with high rotational constants, such as formaldehyde or H2 (Bg = 60.8 cm )) is insufficient. The energy level spacing of vibrations (ca. 2000 cm ) are considerably larger than the room temperature of 207 cm " so that even at room temperature, the vibrational partition function must be evaluated by summation in Eq. (6.40). 

For most organic molecules, the substitution of hydrogen for its heavier isotopes materially has little influence on the molecular mass or moment of inertia. The isotopic rate ratio thus reduces, essentially, to a dependence on the vibrational partition functions, viz. 

In the molecular-motion contribution, the molecular partition function is the product of translational, rotational, vibrational, and electronic partition functions. If the molecule in solution is assumed to have the entire volume of the solution available to it, the ratio of gas-phase and solution-phase translational partition functions equals one. Likewise, the electronic partition function ratio will be one. It is unclear what one should use for the rotational partition function in solution, but if this is assumed to have the same form as that in the gas phase, the rotational partition function ratio (which involves the moments of inertia) will be very close to one, since structural changes from gas to solution are slight. Significant contributions to the vibrational partition function are made only by the low-frequency vibrational normal modes, and these modes sometimes show substantial changes in frequency on going from the gas phase to solution. If a vibrational calculation is done in the gas phase and in solutitm, one can calculate AG°oiv m, but the most common procedure is to omit it, assuming that its contribution is negligible. 

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