Isotopic independence

The vibrational frequencies of isotopic isotopomers obey certain combining rules (such as the Teller-Redlich product rule which states that the ratio of the products of the frequencies of two isotopic isotopomers depends only on molecular geometry and atomic masses). As a consequence not all of the 2(3N — 6) normal mode frequencies in a given isotopomer pair provide independent information. Even for a simple case like water with only three frequencies and four force constants, it is better to know the frequencies for three or more isotopic isotopomers in order to deduce values of the harmonic force constants. One of the difficulties is that the exact normal mode (harmonic) frequencies need to be determined from spectroscopic information and this process involves some uncertainty. Thus, in the end, there is no isotope independent force field that leads to perfect agreement with experimental results. One looks for a best fit of all the data. At the end of this chapter reference will be made to the extensive literature on the use of vibrational isotope effects to deduce isotope independent harmonic force constants from spectroscopic measurements. 

It is important to point out here, in an early chapter, that the Born-Oppenheimer approximation leads to several of the major applications of isotope effect theory. For example the measurement of isotope effects on vapor pressures of isotopomers leads to an understanding of the differences in the isotope independent force fields of liquids (or solids) and the corresponding vapor molecules with which they are in equilibrium through use of statistical mechanical theories which involve vibrational motions on isotope independent potential functions. Similarly, when one goes on to the consideration of isotope effects on rate constants, one can obtain information about the isotope independent force constants which characterize the transition state, and how they compare with those of the reactants. 

As will be shown further on, the most interesting isotope effects are quantum effects. Since the most important quantized motions in molecules are vibrations it makes sense that isotope effects yield information about the isotope independent surface (the vibrational force field) on which these quantized motions take place. 

One remembers that Eeiec is the isotope independent potential energy surface for vibration in the Born-Oppenheimer approximation, Eeiec = V. Note... 

In the use of Equation 3.A2.3, it should be remembered that the fq force constants, which are the elements of F, are isotope independent, as is the determinant. 

In Equation 4.100, (fxx + fyy + fzz) is the sum of the three diagonal Cartesian force constants at the position of isotopic substitution, this sum being isotope independent. 

As already noted, in the Born-Oppenheimer approximation, the nuclear motion of the system is subject to a potential which expresses the isotope independent electronic energy as a function of the distortion of the coordinates from the position of the transition state. An analysis of the motions of the N-atom transition state leads to three translations, three rotations (two for a linear molecule), and 3N - 6 (3N- 5 for a linear transition state) vibrations, one which is an imaginary frequency (e.g. v = 400icm 1 where i = V—T), and the others are real vibrational frequencies. The imaginary frequency corresponds to motion along the so-called reaction... 

K(T) is the probability of finding two particles closer than they would be in a random distribution (i.e. it is the probability of pair formation, the factor 2 occurs because it takes two particles to form one pair). K is the equilibrium constant for pair formation and has units of volume. The first term bo = (2/3) net3 is the volume excluded by the short range repulsive part of the potential and is isotope independent. The VCIE is therefore associated with the isotope effect on (B — bo) and can be written... 

Fig. 5.5 PlotofT2 ln(fc/fg) = T2 I n (fc / fv) for some isotopically substituted cthylcncs. The solid lines are theoretically calculated from the isotope independent force field in Table 5.7 (Reused with permission from Bigeleisen, J., Fuks, S., Ribnikar, S. V., and Yato, Y., J. Chem. Phys. 66, 1689 (1977). Copyright 1977, American Institute of Physics)...

Fig. 5.8 VPIE s of Waters and Ices. The points are experimental from various sources. The lines are calculated using an isotope independent harmonic force field consistent with spectroscopic information (Van Hook, W. A. J. Phys. Chem 72,1234 (1968))... 

It is important to point out once again that explanations (rationalizations) of isotope effects which employ arguments invoking hyperconjugation and/or steric effects are completely equivalent to the standard interpretation of KIE s in terms of isotope independent force constant differences, reactant to transition state. In turn, these force constant differences describe isotope dependent vibrational frequencies and frequency differences which are not the same in reactant and transition states. The vibrational frequencies determine the partition functions and partition function ratios in the two states and thus define KIE. The entire process occurs on an isotope independent potential energy surface. This is not to claim that the... 

For harmonic oscillators recall that the ZPE s, (ZPE = (l/2)hc(//p,)1/2), and ZPE differences scale proportionally to (1/p-h) and (1/ jid), respectively. The q s are oscillator reduced masses and / is the isotope independent force constant. Thus, writing equations analogous to Equation 10.19 for tritium substitution, and taking the ratio, we obtain kH/kT = (kH/kD)x where x, the Swain-Schaad exponent in the harmonic approximation is expressed... 

The isotope independent potential energy surface was evaluated using a mixed quantum mechanics/molecular mechanics (QM/MM) method. The system (N atoms) was partitioned into Nqm quantum mechanical atoms and Nmm classical mechanical atoms. Nqm consisted of the 15 atom substrate (phospho-D-glycerate)... 

The Born-Oppenheimer isotope independent potential energy surface calculated with the bath atoms frozen in place as outlined in the paragraph above was employed by the authors to compare TST and VTST rate constants and kinetic isotope effects. The results are shown in Table 11.9. 

Fig. 12.2 Dipole moment as a function of internuclear distance for a diatomic oscillator. For the case shown re is greater than the r value at the maximum of the curve. Consequently (9p/9r)e < 0. The value of r(max) may also be larger than re, whence (9 i/9r)c > 0 (and this is the case for both CO and HC1). Since the isotope effect [ -

In Equation 12.8 Be is the rotational constant, Be = h/(8jt2I), (I is the moment of inertia), coe is the vibrational frequency, 27T(oe = (k/ix)1, (k the vibrational force constant and x the reduced mass), re the equilibrium bond length (isotope independent to reasonable approximation), and ae is the vibration-rotation interaction constant... 

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