Vibrational partition functions

Electric charge, generalized thermodynamic quantity, canonical partition function, vibrational coordinate Gas constant, resistance, radius Molar refraction... 

Molecule Rotational partition function Vibrational partition function rot (K) vibr (K)... 

Since translational and internal energy (of rotation and vibration) are independent, the partition function for the gas can be written... 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

For translational, rotational and vibrational motion the partition function Ccin be calculated using standard results obtained by solving the Schrodinger equation ... 

The eigenvalues (coa of the mass weighted Hessian matrix (see below) are used to compute, for each of the 3N-7 vibrations with real and positive cOa values, a vibrational partition function that is combined to produce a transition-state vibrational partition function ... 

The total partition function may be approximated to the product of the partition function for each contribution to the heat capacity, that from the translational energy for atomic species, and translation plus rotation plus vibration for the diatomic and more complex species. Defining the partition function, PF, tlrrough the equation... 

This factor can be obtained from the vibration partition function which was omitted from the expression for the equilibrium constant stated above and is, for one degree of vibrational freedom where vq is the vibrational frequency in the lowest energy state. 

If hvQ is small compared with kT, the partition function becomes kT/hvQ. The function kT jh which pre-multiplies the collision number in the uansition state theoty of the bimolecular collision reaction can therefore be described as resulting from vibration of frequency vq along the transition bond between the A and B atoms, and measures the time between each potential n ansition from reactants to product which will only occur provided that die activation energy, AEq is available. 

The classical value is attained by most molecules at temperatures above 300 K for die translation and rotation components, but for some molecules, those which have high heats of formation from die constituent atoms such as H2, die classical value for die vibrational component is only reached above room temperature. Consideration of the vibrational partition function for a diatomic gas leads to the relation... 

Here Zint is the intramolecular partition function accounting for rotations and vibrations. However, in equilibrium, the chemical potential in the gas phase is equal to that in the adsorbate, fi, so that we can write the desorption rate in (I) as... 

The partition functions vib. and for use in Eq. (5-23) can be evaluated by quantum mechanical arguments. We will subsequently require vib. which is given by Eq. (5-25), where v is the vibrational frequency. 

Q is the partition function of the transition state with this special vibrational mode factored out. 

Next, Ah and Ad are written in terms of partition functions (see Section 5.2), which are in principle calculable from quantum mechanical results together with experimental vibrational frequencies. The application of this approach to mechanistic problems involves postulating alternative models of the transition state, estimating the appropriate molecular properties of the hypothetical transition state species, and calculating the corresponding k lko values for comparison with experiment.""- " "P... 

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. 

For a polyatomic molecule the total vibrational energy may be written as a sum of energies for each vibration, and the partition function as a product of partition functions. 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

C) The error in AE" /AEq is 0.1 kcal/mol. Corrections from vibrations, rotations and translation are clearly necessary. Explicit calculation of the partition functions for anharmonic vibrations and internal rotations may be considered. However, at this point other factors also become important for the activation energy. These include for example ... 

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... 

We will use the harmonic oscillator approximation to derive an equation for the vibrational partition function. The quantum mechanical expression gives the vibrational energies as... 

We have seen earlier that for a linear polyatomic molecule, the vibrational motions can be divided into (3rj — 5) fundamentals, where rj is the number of atoms. For a nonlinear molecule (3rj - 6) fundamentals are present. In either case, each fundamental vibration can be treated as a harmonic oscillator with a partition function given by equations (10.100) and (10.101). Thus. 

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... 

We have seen that for the electronic partition function there is no closed form expression (as there is for translation, rotation, and vibration) and one must know the energy and degeneracy of each state. That is. 

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... 

One can define a special equilibrium constant K (special in that it lacks the partition function for the unique vibration), giving for the rate constant... 

---