Polyatomic Systems in Approximation The Cell Model

Using harmonic oscillator partition functions to describe both internal and external modes, the logarithmic Q ratios introduced above, ln(Qg/Qc/QgQc/) = ln(Qc/QcO + ln(Qg7Qg), become 

In the condensed phase the sum is over all 3n frequencies, but in the ideal vapor phase the six external (zero) frequencies do not contribute to the IE s, the sum is over the remaining 3n — 6 internals. For condensed rare gases the harmonic assumption is highly approximate, and this is also true for the lattice modes of polyatomics. However as molecular size increases the relative contribution of the external modes becomes less and less important relative to internals. 

To illustrate Equations 5.24a and 5.24b we consider the contribution of a single vibrational mode to VPIE. Comparing CH and CD stretching modes for a typical hydrocarbon at room temperature (300 K) (vCh 3,000 cm-1 in the gas, red shifting 10cm-1 on condensation), we approximate RPFR as 

In Equation 5.25 the ratio of G matrix elements has been obtained using a diatomic approximation (Gi/G/) = [(1/12) + (l/2)]/[(l/12) + (1/1)]. Although in the gas phase the frequency of each isotopomer can be measured to high precision, say 0.05 cm-1 or better, such precision is impossible in the liquid because of inherent broadening caused by intermolecular forces. Except in special cases band centers cannot be located to better than 0.5 cm-1, that limit is imposed by the nature of the liquid state. There is an identical uncertainty for each isotopomer, so spectroscopic precision is about 

---