The First Quantum Correction

In the previous section, it was shown that (si /s2)f goes to unity at high temperature when u tends to zero as the temperature increases. Thus, at high temperature we replace e-u by 1 and (1 — e-u) by u. If an additional term is carried in the expansion for e-u when u is very small, one obtains a deviation of (si/s2)f from unity which scales as h2 and this term gives the first order correction to the classical mechanical value of (si/s2)f, which is unity. 

The first quantum correction is deduced by recognizing that for small u 

Thus to order u2, for small u (high temperature) one obtains 

Thus the first correction to the classical statistical mechanics at high temperature goes as h2. There are higher order corrections. The result obtained here is identical to that found by J. Kirkwood for a harmonic oscillator. The approach to the 

The fy s, the force constants in ordinary Cartesian coordinates, are the ones obtained from the Born-Oppenheimer potential, and are independent of isotopic substitution. Remember that the m s in Equation 4.96 are the masses of the atoms in the molecule. 

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We have seen that Equation 4.95 for (s2/si)f involves the difference between the sums of the squares of the frequencies for two isotopomers. Consider now two isotopic atoms X and with masses ma and mp and two isotopomers AX and AXP with mp > ma. Then, from Equations 4.95 and 4.99, according to the first quantum correction... 

Thus, in the first quantum correction approximation, the isotope effect reflects the change in the force constants at the position of isotopic substitution between the two molecules involved in the isotopic fractionation. Moreover, the fractionation is such that the light atom enriches in the molecular species that has the smaller force constant. While this statement has been derived here only at high temperature, it can be generalized to state that isotope effects are probes for force constant changes at the position of isotopic substitution. That is what isotope effects are all about. 

It has been previously noted that the first quantum correction to the classical high temperature limit for an isotope effect on an equilibrium constant is interesting. Each vibrational frequency makes a contribution c[>(u) to RPFR and this contribution can be expanded in powers of u with the first non-vanishing term proportional to u2/24, the so called first quantum correction. Similarly, for rates one introduces the first quantum correction for the reduced partition function ratios, includes the Wigner correction for k /k2 and makes use of relations like Equation 4.103 for small x and small y, to find a value for the rate constant isotope effect (omitting the noninteresting symmetry number term)... 

As written Equation 4.150 applies to the case of a single isotopic substitution in reactant A with light and heavy isotopic masses mi and m2, respectively. Equation 4.150 shows that the first quantum correction (see Section 4.8.2) to the classical rate isotope effect depends on the difference of the diagonal Cartesian force constants at the position of isotopic substitution between the reagent A and the transition state. While Equations 4.149 and 4.150 are valid quantitatively only at high temperature, we believe, as in the case of equilibrium isotope effects, that the claim that isotope effects reflect force constant changes at the position of isotopic substitution is a qualitatively correct statement even at lower temperatures. 

An estimate of the VPIE of monatomics can be obtained from the first quantum correction using the Wigner high temperature approximation (appropriate because the level spacing in the quantized intermolecular well is small compared to the thermal energy, hv/kT 1, see Chapters 3 and 4)... 

The use of the correction term f extends the method of the first quantum correction to values of u where the first quantum correction itself is inadequate. Equation II. 11 becomes consequently... 

We now consider the same model for the reaction but calculate the temperature independent factor from the considerations with regard to potential surfaces given in Section III. In addition, it is convenient to avoid the calculation of the frequency shift, Avv and evaluate the quantum correction through an approximation (see Eq. 11.30) based on the method of the first quantum correction. One. obtains at 4006K... 

The first quantum correction to the third cluster integral is... 

Here m and m are the light and heavy atom masses respectively and d CG - Cl. For a diatomic with both atoms Isotopes of the same element, d 0. Friedman obtained an expression for the first quantum correction to the configurational part of the partition function,... 

Equation (III-55) shows that the first quantum correction diminishes the partition function. 

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