Vibrational partition functions, calculation

In the infinite sum each successive term is smaller than the previous by a constant factor ( -hujVT which is <1), and can therefore be expressed in a closed form. Only the vibrational frequency is needed for calculating the vibrational partition function for a harmonic oscillator, i.e. only the force constant and the atomic masses are required. 

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... 

Values of and Qb can be calculated for molecules in the gas phase, given structural and spectroscopic data. The transition state differs from ordinary molecules, however, in one regard. Its motion along the reaction coordinate transforms it into product. This event is irreversible, and as such occurs without restoring force. Therefore, one of the components of Q can be thought of as a vibrational partition function with an extremely low-frequency vibration. The expression for a vibrational partition function in the limit of very low frequency is... 

Calculate the vibrational partition function with respect to the vibrational ground state (i.e. the lowest occupied state) and the fraction of molecules in the ground state at 300, 600 and 1500 K for the following molecules, using kTjh= 208.5 cm at 300 K ... 

Numerical evaluation of Equation 14.35 first requires the calculation of the isotopic vibrational partition function ratio in the numerator for the reactant. This can be obtained by applying the methods of Chapter 4 to the relevant H and D vibrational frequencies. The vibrational D/H partition function ratio is larger than unity. The vibrational partition function ratio in the denominator of the right hand side... 

Rotational and vibrational partition functions can be computed from the geometry and vibrational frequencies that are calculated for a molecule or TS. The entropy can then be obtained from these partition functions. Thus, electronic structure calculations can be used to compute not only the enthalpy difference between two stationary points but also the entropy and free energy differences. 

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]. 

Calculation of partition functions requires spectroscopic quantities for the rotational and vibrational partition functions. The quantities required are moments of inertia, rotational symmetry numbers and fundamental vibration frequencies for all normal modes of vibration. The translational terms require the mass of the molecule. All terms depend on temperature. Calculation of partition functions is routine for species for which a detailed spectroscopic analysis has been made. 

The partition function calculation for the activated complex is more tricky. If the surface is known the quantities quoted above can be found from the dimensions and the curvature of the surface around the critical configuration. If this is not possible, then estimates of Qf can be made by analogy with a molecule of similar structure. The vital feature is that the free translation along the reaction coordinate has already been accounted for and must not be included in the calculation, and so the activated complex has one degree of vibrational freedom less than that for a molecule with the same number of atoms. 

Calculate the rate constant, at T = 300 K, according to transition-state theory. The vibrational partition functions for vibrations with wave numbers larger than 1000 cm-1 can be set to 1. 

The vibrational contribution to varies with temperature and can be calculated from the vibrational partition function using... 

We assume that there is no free internal rotation in the molecule, and the contribution from the torsional oscillation (v. = 182.5 cm" ) is included in the vibrational partition function. Extended Huckel calculations (7) show that the potential... 

This expression may be used for the vibrational partition function of a diatomic molecule at all temperatures the only approximation involved is that the oscillations are supposed to be harmonic in nature. The anhar-monicity correction must be made for precision calculations, but its effect is not large. The only property of the molecule required for the evaluation of Q, by equation (16.30) is the vibration frequency, which can be obtained from a study of its spectrum. The values of this frequency for a number of diatomic molecules are given in Table IX. ... 

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

The vibrational frequencies of the three modes of H20 are 1654 cm-1, 3825 cm-1 and 3935cmJ. The vibrational partition function at 300 K, calculated ... 

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

The rates and mechanisms of chemical reactions can be predicted, in principle, by the standard methods of statistical thermodynamics, in terms of the partition functions of reactants and the transition-state complex. However, the range of applicability of this transition-state (absolute rate) theory is severely limited by the fact, that an evaluation of the vibrational partition function for the transition state requires a detailed consideration of the whole PES for the reaction. Thus, a calculation of the absolute rate constants is possible only for relatively simple systems. This indicates a need for more approximate, empirical methods of treating chemical reactions and formulating the reactivity theory, which would allow... 

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