Vibrational Product Rule

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. 

Two Important Rules for Harmonic Vibrational Frequencies 3.5.1 The Teller-Redlich Product Rule... 

There are two important rules involving harmonic vibrational frequencies that are well known to spectroscopists. They are important in the present context because they permit the simplification of some of the statistical mechanics results for iso-topomers in Chapter 4. The first rule, the Teller-Redlich (TR) product rule, follows straightforwardly from Equation 3.A1.13 (Appendix 3.A1) if one remembers that A = 4n2vf and that there are six frequencies for the non-linear molecule which... 

In Chapter 3, a formula was presented which connects the normal vibrational frequencies of two rigid-rotor-harmonic-oscillator isotopomers with their respective atomic masses m , molecular masses Mi and moments of inertia (the Teller-Redlich product rule). If this identity is substituted into Equation 4.77, one obtains... 

This result indicates that (s2/si)f (compare Equations4.78 and 4.93) is just the quantum effect on the molecular partition functions of the normal mode vibrations. This result has now been derived without the explicit use of the Teller-Redlich product rule. 

Considerations like the above led the German statistical mechanician L. Waldman independently to an equation similar to the (si/s2)f equation of Bigeleisen and Mayer. The foregoing can be regarded as an independent proof of the Teller-Redlich product rule but this statement depends on the assumption of no rotational-vibrational interaction. 

Equation 4.140 is the exact analogue of Equation 4.76 for a stable molecule except there is one less vibrational degree of freedom. It must now be noted that the derivation of the Teller-Redlich Product Rule, applies equally well to a transition state as to a stable molecule. Thus, when the Teller-Redlich Product rule is introduced into the expression for q2/qi of a transition state, the ratio of vibrational frequencies includes the isotopic ratio of the imaginary frequencies in the transition state. One can then write for transition state isotopic ratios, analogously to Equation 4.78... 

The last step in Urey s derivation is the application of the Redlich-Teller product rule (e.g., Angus et al. 1936 Wilson et al. 1955), which relates the vibrational frequencies, moments of inertia, and molecular masses of isotopically substituted molecules. For CIO,... 

Calculations of 180 EIEs upon reactions of natural abundance O2 require the normal mode stretching frequencies for the 160—160 and 180—160 isotopologues (16 16j/ and 18 16, ). These values can often be obtained directly from the literature or estimated from known force constants. DFT calculations can be used to obtain full sets of vibrational frequencies for complex molecules. Such calculations are actually needed to satisfy the requirements of the Redlich-Teller product rule. In the event that the full set of frequencies is not employed, the oxygen isotope effects upon the partition functions change and are redistributed in a manner that does not produce a physically reasonable result. 

A secondary motive is our general desire to verify and extend our understanding of vibration-rotation interactions in molecular spectra, and particularly to interpret data on different isotopic species in a consistent manner. Consider, for example, a constants (which measure the dependence of the rotational constant B on the vibrational quantum numbers vr) determined experimentally for several isotopic species of the same molecule. It is clear that these constants are not all independent, since they are related to the potential function which is common to all isotopic species. However, the consistency of the data and of our theoretical formulae can only be tested through a complete anharmonic force field calculation (there are at this time no known relationships between the a values analogous to the Teller-Redlich product rule). Similar comments apply to many other vibration-rotation interaction constants. 

In the intramolecular situation of M+ decomposing to mj" and mJi, the vibrational frequencies and moments of inertia of the reactant are, of course, the same for each decomposition and the frequencies and moments of the transition states differ. The significant fact is that, again within the Bom — Oppenheimer approximation, the Teller — Redlich product rule applies to transition states. Choosing vibrational frequencies and moments of inertia of the transition state of one decomposition, therefore immediately fixes, within certain limits, frequencies and moments of the transition state of the other [see eqn. (29)]. The Teller — Redlich product rule has the following general form for transition states. 

The consequence of the restrictions imposed by the product rule on vibrational frequencies of transition states and critical energies of decompositions is that there is generally very little leeway in the mechanistic interpretation of intramolecular kinetic isotope effects. There are tight constraints upon the type of transition state structure consistent with a given intramolecular kinetic isotope effect. 

Raman spectra of 2-deuteriothiophene, 2-deuterioselenophene, and 3-deuterioselenophene were useful for assigning the vibrations of type A because fourteen of the fifteen vibrations of the type are polarized and intense. A deuterium atom distorts the symmetry of the vibrations which are of type B in selenophene or thiophene. The assignment of the in-plane vibrations fits the isotopic product rules and the sum rules (see Table III). These data confirm the similarity of the vibrational spectra of thiophene and selenophene. 

The geometry and the force constants of a molecule determine the isotopic shifts of the vibrational frequencies. However, the products of the vibrational frequencies of two different isotopomers within the same irreducible representation are related by the Teller-Redlich product rule. For two symmetry-equivalent vibrations the following expression is obtained ... 

Calculated Frequencies. Table II contains the normal-mode vibrational frequencies vu of the light isotopic species, and the frequency shifts A Vi = vii — V2i upon isotopic substitution, calculated with the force fields listed in Table I. The force field for NOa" reproduces the observed frequencies and frequency shifts very well, whereas the calculated frequencies and shifts for N02 differ somewhat from those observed. However, we consider the general quadratic potential used in the calculation the best fit to the observed frequencies. The discrepancy is caused by a disagreement of the observed (2) frequencies with the Teller-Redlich product rule, which is, of course, assumed in the calculations. 

The right side of Eq. (5) contains the product of the u-functions for the 3n — 6 vibrations of an n-atomic molecule and this molecule s principle moments of inertia. In most cases the principle moments of inertia are unknown. According to Bigeleisen and Mayer, Eq. (5) can be replaced by a reduced partition function ratio according to Teller and Redlich s product rule ... 

Two important rules hold for the vibrational frequencies of isotopic molecules. The first, called the product rule, can be derived as follows. 

For an interpretation of our results we performed statistical RRKM calculations of the individual decay rate constants of all four competing decay channels at low threshold energy. The latter have been experimentally extracted from the directly measured total decay rate constant (see Fig. 4) and the simultaneously measured branching ratios of the relevant fragment ions /16/. For different isotopically labelled species a good simulation of experimental results is obtained with a single set of parameters for the determination of the frequencies of the activated complex ( solid line in Fig. ). Isotope shifts of the vibrational frequencies were obtained by use of the Teller-Redlich product rule. This points to a high reliability of the set of parameters used and yields detailed information on the structure of the activated complex for the four decay channels under consideration /16/. In... 

For isotopic substitution in which the molecular point group is unchanged, the Teller-Redlich product rule links the two sets of vibrational frequencies. There is one product rule for each symmetry species of the molecule as follows... 

Equation (132) must be modified for linear molecules, which have only two rotational degrees of freedom, and hence an extra vibration. Similar modifications occur in the product rule (133), and the final expression for K/K differs from (135) only in the number of vibrational terms. Analogous modifications for linear molecules (or transition states) must be made in the expressions given later for the kinetic isotope effect. These are not given separately, though the case in which A and B are atoms does of course involve linear molecules throughout. 

Vibrational Selection Rules for the Harmonic OsdUator. If the higher terms in (7) are neglected, and if the vibrational wave function f/y is assumed to be strictly of the form described earlier in this chapter, that is, a product of harmonic oscillator functions, then the selection rules for vibrational transitions are very restrictive indeed. The integral for fix becomes... 

Under certain conditions, these two expressions will give rise to simple relations between the intensities of isotopic molecules, the latter analogous to the product rule, the former to the sum rule for the frequencies. The conditions are (a) that the molecule should have no dipole moment and/or (b) that the symmetry species of the vibrations over which the summation of intensities is carried out should not be the same as that of any rotation of the molecule which moves the permanent dipole moment. If one or both of these conditions are satisfied, dyu/dS. is the same for all isotopic molecules. It then immediately follows that the intensity sum... 

Because of the large number of possible isotopic derivatives of benzene, the application of the product rule has been of great importance in making frequency assignments for this molecule. Moreover, the study of the vibrational spectra of such related molecules allows a large number of the force constants, in particular the interaction constants, to be evaluated. 

---