The Teller-Redlich Product Rule

5 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 

The subscripts 1 and 2 refer to the two isotopomers being compared. On the right hand side of the equation the product is over the N atoms of the molecule rather than the 3N Cartesian coordinates. From Equation 3.49, one obtains the Teller-Redlich product rule after some rearrangement 

For a non-linear molecule, there are only 3N — 5 frequencies and two moments of inertia which are equal. 

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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. 

A2 An Equality for Use in the Derivation of the Teller-Redlich Product Rule... 

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 MMI (mass moment of inertia), EXC (excitation factor), and ZPE (zero point energy) terms are defined on successive lines of Equation 4.145. For reactions involving heavier isotopes the effects are no longer concentrated in the ZPE term and it is convenient to apply the Teller-Redlich product rule (Section 3.5.1) and eliminate the moments of inertia by using Equations 4.79,4.79a, and 4.141, thus obtaining an equivalent relation... 

For the determination of energy factored parameters, full isotopic substitution is bound to be useless. This is because, in the energy factored approximation, such substitution is predicted to multiply each frequency by a factor of [fij/j, ), where fx is the reduced mass mcfnol[nic+ o] of carbon monoxide, independent of the values of the parameters concerned. However, a great deal of information can be obtained from the spectra of partially substituted species, for which the Teller-Redlich product rule takes the particularly simple form... 

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. 

Changes in moments of inertia have generally been neglected in studies of ionic decompositions, i.e. 7a(I) 7b(I) 7C(I) has been put equal to 7a(n) 7b(ii> 7c(Ii). They will probably receive more attention in the future as interest in rotational energy effects grows. If, however, the moments are cancelled, the Teller — Redlich product rule for competing decompositions of the same ion reduces to... 

Application of the Teller-Redlich product rule to eq. (11) shows that the limiting value is the reduced mass factor for motion along the reaction coordinate.3 The limiting value for the C -[Pg.31]

The magnitude and nature of the primary effect will be examined in terms of eq. (11) however, the present remarks are not limited to systems in which active rotations are absent. The discussion is based on calculations using the C2 model for the light molecule and the C2 model, modified to correspond to the measured frequencies for ethane- (Table VI) for the C2-di model. The complex for H rupture, is the semirigid complex 4, specified in II,B2 and Table I. The D-rupture complex (Table VI) was constructed in the same way and fits the Teller-Redlich product rule. The difference in critical energies for the two prototype reactions is Aeo = 1.38 keal. mole-1. The density of states for the molecules and the sum of states for the complexes are shown in Figures 3 and 4, respectively. 

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. 

Equation IV.8 can be rearranged through the use of Eq. II.5 and the Teller-Redlich product rule to give... 

If one or more isotopic substitutions are performed on the moleciile, new frequencies will be obtained, as well as new G matrix elements. Within the Born-Oppenheimer approximation, however, the F matrix elements will transfer intact to the new molecule. The amotint of new information which can be obtained about the elements in the F matrix in this way is, however, limited by several isotope rules >rtiich the sets of harmonic frequencies of each symmetry type must obey. One of these, the form of the Teller-Redlich product rule which applies to two isotopic variants having the same molecular symmetry, may be deduced immediately from the secular equation Itself. When the nxn secular determinant is expanded in polynomial form, the constant term, which must be equal to the product of the roots, n. ... 

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... 

A general quantitative relation involving the isotope effect, similar in form to Eq. (3.51), is the Teller-Redlich product rule. It is based on the assumption that the product of the zero order wavenumber ratios (v /vj... 

From the invariance of the force constant matrix , under isotopic substitution, one gets the Teller - Redlich product rule... 

Alternatively, the necessity for knowing the moments of inertia can be avoided by making use of the Teller-Redlich product rule (22a) which allows the isotope effects to be expressed as a function of vibrational frequencies alone. This is the basis of the treatment of Bigeleisen and Mayer (3), according to which the exchange equilibrium constant is ... 

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