Rotational partition function, polyatomic molecules

The harmonic approximation reduces to assuming the PES to be a hyperparaboloid in the vicinity of each of the local minima of the molecular potential energy. Under this assumption the thermodynamical quantities (and some other properties) can be obtained in the close form. Indeed, for the ideal gas of polyatomic molecules the partition function Q is a product of the partition functions corresponding to the translational, rotational, and vibrational motions of the nuclei and to that describing electronic degrees of freedom of an individual molecule ... 

Nonlinear polyatomic molecules require further consideration, depending on their classification, as given in Section 9.2.2. In the classical, high-temperature limit, the rotational partition function for a nonlinear molecule is given by... 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

We noted in Section 8.2 that only half the values of j are allowed for homonuclear diatomics or symmetric linear polyatomic molecules (only the even-y states or only the odd- y states, depending on the nuclear symmetries of the atoms). The evaluation of qmt would be the same as above, except that only half of the j s contribute. The result of the integration is exactly half the value in Eq. 8.64. Thus a general formula for the rotational partition function for a linear molecule is... 

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

For a nonlinear polyatomic molecule the rotation partition function becomes... 

Similarly, most terms cancel in the calculation of the ratio of rotational partition functions. For diatomic molecules and linear polyatomic molecules, this ratio is given by fjl ... 

O2). The rotational partition function ratio for non-linear polyatomic molecules is ... 

The rotational partition function depends on the shape of the molecule. For a linear polyatomic molecule, it is given by... 

In most cases, for polyatomic molecules the nuclear partition function is again neglected, because it usually has a very small effect on the overall thermodynamic properties of polyatomic molecules. (Indeed, the only reason why we had to consider it for diatomic molecules is because it imposes an obvious, measurable effect on various observations, like spectra and thermodynamic properties to be considered in section 18.8.) In the high-temperature limit, a linear polyatomic molecule has the same rotational partition function as a homonuclear diatomic molecule ... 

A nonlinear polyatomic molecule can have up to three different moments of inertia, labeled 7a. 7b. and 7q. By convention, 7 is less than 7g, which is less than 7. Polyatomic molecules that have some symmetry may have some of their moments of inertia equal. If all three are equal, then the molecule is called a spherical top (see Chapter 14) and the rotational partition function can be written as... 

For a linear polyatomic molecule like acetylene or cyanogen, the rotational energy levels are the same as those of diatomic molecules in Eq. (22.2-18). Equation (25.4-13) can be used for the rotational partition function with the appropriate symmetry number and moment of inertia. The rotational energy levels of nonlinear polyatomic molecules are more complicated than those of diatomic molecules. The derivation of the rotational partition function for nonlinear molecules is complicated, and we merely cite the result ... 

The quantum rotational partition function for a diatomic or linear polyatomic molecule is equal to the classical version divided by and also divided by the symmetry number a, equal to 1 for a heteronuclear diatomic molecule and equal to 2 for a homonuclear diatomic molecule. 

For a polyatomic molecule with moments of inertia 4,4 and 4 along the principal axis, the rotational partition function is... 

Rotational motion of a general (nonlinear) polyatomic molecule accounts for three degrees of freedom. The partition function in this case was given by Eq. 8.67. It is easy to verify that... 

Equations (6)-(7) apply to the rigid rotation of a molecule as a whole. Most polyatomic molecules possess one or more modes of internal rotation. If the internal rotation is unhindered, the partition function is... 

An atomistic approach, which has relevance to the current work, is the previously discussed normal-mode method. In the normal-mode method the constituent monomer units in the cluster are assumed to interact with a reasonable model potential in a fixed structure. From the assumed structure and model potential a normal-mode analysis is jjerformed to determine a vibrational partition function. Rotational and translational partition functions are then included classically. The normal-mode method treats the cluster as a polyatomic molecule and is most appropriate at very low temperatures where anharmonic contributions to the intermolecular forces can be ignored. As we shall show by numerical example, as the temperature is increased, the... 

The complete partition function of a polyatomic molecule may now be represented by the product Qt X Q., where Qt is the tran.slational, including the electronic, factor, as derived above, and Q is the combined rotational and vibrational, i.e., internal, factor. Since In Q is then equal to In Qt + In Qiy equation (24.12) may be written in the form... 

The example given above provides a simple illustration of the use of moments of inertia and vibration frequencies to calculate equilibrium constants. The method can, of course, be extended to reactions involving more complex substances. For polyatomic, nonlinear molecules the rotational contributions to the partition functions would be given by equation (16.34), and there would be an appropriate term of the form of (16.30) for each vibrational mode. ... 

Such representation is useful when it is impossible or impractical to make detailed calculations of partition functions, but when order-of-magnitude estimates would be helpful. Table 2.3 is a tabulation of representative values for these factors computed from equations (2-62), (2-64), or (2-65) and (2-66) using typical values for molecular constants in the temperature range from 300 500°K. The translational contribution per degree of freedom is much larger than rotation or vibration, however, in complicated polyatomic molecules it is possible for the total vibrational contribution to become large as (3iV — 6) becomes a large number. 

For a general polyatomic molecule, the rotational energy levels cannot be written in a simple form. A good approximation, however, can be obtained from classical mechanics, resulting in the following partition function. 

If the rotator is symmetrical this partition function must be reduced by the appropriate symmetry factor (a) to eliminate permutations which are indistinguishable. For a symmetric diatomic molecule such as Nj or Oj, a = 2 while for unsymmetric molecules such as CO or NO, a = 1.0. The corresponding expression for a polyatomic molecule with moments of inertia about the three principal axes is... 

Internal Partition Functions for Polyatomic Molecules.— The internal partition function for a polyatomic molecule comprises contributions from nuclear spin and electronic levels, and from rotational and vibrational degrees of freedom. On the assumption that the corresponding energies are additive and independent, these contributions can be factored, and the corresponding contributions to the thermodynamic functions are additive. 

---