Molecular Rotational Partition Function

Evaluation of die rotational components of the internal energy and entropy using the partition function of Eq. (10.19) gives 

As was also previously noted in Section 9.3.1, the completely general rigid-rotor Schrodinger equation for a molecule characterized by three unique axes and associated moments of inertia does not lend itself to easy solution. However, by pursuing a generalization of the classical mechanical rigid-rotor problem, one can derive a quantum mechanical approximation that is typically quite good. Within that approximation, the rotational partition function becomes 

it must be noted that evaluating the rotational components of U and. S requires relatively little in the way of molecular information. All that is required is the principal moments of inertia, which derive only from the molecular structure. Thus, any methodology capable of predicting accurate geometries should be useful in the construction of rotational partition functions and the thermodynamic variables computed therefrom. Also again, the units chosen for quantities appearing in the partition function must be consistent so as to render q dimensionless. 

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To calculate the molecular rotational partition function for an asymmetric, linear molecule, we use Eq. 8.16 for the energy level of rotational state /, and Eq. 8.18 for its degeneracy. As discussed in Section 8.2, rotational energy levels are very closely spaced compared to k/jT unless the molecule s moment of inertia is very small. Therefore, for most molecules, replacing the summation in Eq. 8.50 with an integral introduces little error. Thus the... 

Consider the molecular rotational partition function for the CO molecule, a linear diatomic molecule. The moment of inertia of CO is / = 1.4498 x 10-46 kg-m2, and its rotational symmetry number is a = 1. Thus, evaluating Eq. 8.65 at T = 300 K, we find the rotational partition function to be... 

Although we will not provide a derivation, the molecular rotational partition function for a general polyatomic molecule is also simple in form,... 

Here /, are the three moments of inertia. The symmetry index a is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations), for H2O it is 2, for NH3 it is 3, for benzene it is 12 etc. The rotational partition function requires only information about the atomic masses and positions (eq. (12.14)), i.e. the molecular geometry. 

But molecular gases also have rotation and vibration. We only make the correction for indistinguishability once. Thus, we do not divide by IV l to write the relationship between Zro[, the rotational partition function of N molecules, and rrol, the rotational partition function for an individual molecule, if we have already assigned the /N term to the translation. The same is true for the relationship between Zv,h and In general, we write for the total partition function Z for N units... 

In a different approach, Bottinga (1969a) evaluated the nonclassical rotational partition function for water vapor, molecular hydrogen, and methane through the asymptotic expansion of Strip and Kirkwood (1951) ... 

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... 

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). 

It is interesting to consider the relationship between the reaction degeneracy and the molecular symmetries in canonical transition state theories. In the latter, the rate constants are expressed in terms of the partition functions, including the rotational partition functions, so that the molecular symmetries are automatically included. On the other hand, in the microcanonical TST, the rotational density of states is often not part of the rate constant expression (see discussion of rotational effects in the following chapter). Thus, the reaction degeneracy must be included separately. 

Forst (1991) introduced a particularly simple approach in which the partition function, rather than the vibrational frequencies are switched. Furthermore, the switching for the transitional modes is between the vibrational and the rotational partition functions. If is the partition function of a transitional mode in the reactant molecule and 2rot is the partition function of the product rotation which is correlated to the molecular vibration, and S(R) is the switching function, then the logarithm of the partition function as a function of R is given by ... 

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. 

Thus, reaction rate coefficients can be estimated from the thermochemistry of the transition states, whose molecular properties can be calculated with quantum chemical programs. In calculating reaction rate coefficients, the only negative second derivative of energy with respect to atomic coordinates (called imaginary vibrational frequency ) from the transition state is ignored, so that there are only 37/-7 molecular vibrations in the transition structure (37/ — 6 if linear) and all internal and external symmetry numbers have to be included in the rotational partition functions (then any reaction path degeneracy is usually included automatically). 

The requirement for symmetric or antisymmetric wave functions also applies to a system containing two or more identical composite particles. Consider, for example, an molecule. The nucleus has 8 protons and 8 neutrons. Each proton and each neutron has i = j and is a fermion. Therefore, interchange of the two nuclei interchanges 16 fermions and must multiply the molecular wave function by (—1) = 1. Thus the molecular wave function must be symmetric with respect to interchange of the nuclear coordinates. The requirement for symmetry or antisymmetry with respect to interchange of identical nuclei affects the degeneracy of molecular wave functions and leads to the symmetry number in the rotational partition function [see McQuarrie (2000), pp. 104-105]. 

For present purposes only the form of Eq. (3.27) is required. Detailed formulation of Z can be found in Ref. (47) For chains with independent rotational potentials Z is equal to the bond rotational partition function. For high molecular weight, with int dependent rotational potential, Z is the largest eigenvalue of the statistical weight matrix describing this interdependence. 

Equation 3.21 is used to calculate the rotational partition function. Each rotational energy level is 2Z+1 times degenerate because of the vectorial quantization of angular momentum, so it must be counted as many times in the summation. On the other hand, whenever the molecular symmetry is such that identical atoms exchange their positions because of a molecular rotation, the summation must be divided by an appropriate symmetry number, a [2]. Then ... 

We just mentioned molecular rotation. Is there a rotational partition function If yes— how does it look like From classical mechanics we know that the Hamilton function of a rigid body freely rotating in space is given by... 

D Anna et al. (2003) showed that the NO3 reaction with CH2O proceeds by H-atom abstraction so that the sole products of this reaction are HNO3 HCO. Mora-Diez and Boyd (2002) examined the mechanism of the reactions between NO3 and formaldehyde and acetaldehyde theoretically comparisons with experiment are consistent with a direct abstraction mechanism. Alv ez-Idaboy et al. (2001a) reached a similar conclusion for NO3 reaction with formaldehyde, acetaldehyde, propanal, n-butanal, and 2-methylpropanal. Their calculations showed that all reactions proceed via abstraction of the a-carbonyl H-atom the dependence of the rate constant on molecular size was shown to be attributable to the increase in the internal rotational partition function with the size of the aldehyde. 

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