A Further Approximation The AB Equation

It is often useful to have an approximate relation for VPIE s, especially when complete vibrational analysis is impossible. The AB approximation serves that purpose, and sometimes gives more physical insight than do detailed, but very complicated calculations using Equation 5.24. It is based on the observation that ordinarily condensed phase vibrations fall in two groups the first containing the high frequencies, m 1 (most often the internal modes, uj = hcvj/kT), where the zero point (low temperature) approximation is appropriate, and 

The second group contains the low frequencies, m 1, which are to be treated in the high temperature approximation. This, we have seen, accounts for excitation into upper levels by expanding I n(s/s ) 1 in even powers of u (see Chapter 4) 

5 Condensed Phase Isotope Effects Isotope Effects in Non-ideal Gases 

The A defined in Equation 5.30 is not to be confused with the Helmholtz free energy. Should the A frequencies be limited to the external hindered translations and rotations, vi g = vi g = 0, and this is an additional simplification. In some molecules, however, there are isotope sensitive low lying internal modes (often internal rotations or skeletal bends). In that cases both terms in Equation 5.30 contribute. 

A is generally positive and predicts (P/ P). B results from the shifts in internal force constants on condensation. An increased force constant on condensation leads in the direction of a normal VPIE, a decrease towards an inverse effect (P P). Keep aware of the different temperature dependences. At low enough temperature A/T2 must predominate. The IE is normal and proportional to 1 /T2. At intermediate temperatures the B term, which can be positive, but more often is negative (see Section 5.4.1), may dominate. This accounts for the commonly observed crossover to inverse IE s. At higher temperatures yet, both terms decay to zero. The temperature dependence of VPIE can thus be complicated. 

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