First quantum correction

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

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)... 

Pack RT (1983) First quantum corrections to second virial coefficients for anisotropic interactions Simple, corrected formula. J Chem Phys 78 7217-7222... 

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 make use of the modified first quantum correction, Eq. 11.15, and obtain... 

In addition to hydrogen isotope effects on polyatomic molecules, there have been extensive investigations in complex systems using principally the isotopes of carbon, nitrogen, and oxygen. For such systems the theoretical analysis can be simplified through the use of the G(u) formula and its approximation through the modified first quantum correction. Carbon isotope effects in decarboxylation may be taken as typical examples of such studies. The reactions and rate constants for the decarboxylation of mono- and dibasic adds may be defined by the set of equations ... 

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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