Teller-Redlich rule

This expression can fortunately be simplified by use of a theorem known as the Teller-Redlich rule, which expresses the molecular mass and moment of inertia ratios in terms of a ratio of a product of all the atomic masses m and the vibrational frequencies ... 

According to the Teller-Redlich rule, only three of the four experimentally obtained quantities ( /], 1 2 4) independent. In this case, therefore, a maximum of three... 

A modification of the Teller-Redlich rule (309) has proved valuable in locating the bands of the partially CO substituted derivatives. The rule predicts that the full isotope shift of 45 cm , expected on replacing a by a group is distributed between the shifts of bands... 

Teller-Redlich rule phys chem For two isotopic molecules, the product of the frequency ratio values of all vibrations of a given symmetry type depends only on the geometrical structure of the molecule and the masses of the atoms, and not on the potential constants. tel-or red lik, rUI ... 

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... 

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... 

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... 

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,... 

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... 

The validity of the Product Rule for the P 2 class is dependent on the isolation of the anion in its lattice. For a XO4 ion for which only X is being substituted, the Redlich-Teller Product Rule can be written as follows ... 

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. 

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 two modes belonging to this species were not detected. On the other hand all six expected C—H modes were identified. The consistency of the assignment was checked by the use of the Redlich-Teller product rule which was satisfied to 0.8%. The reported Raman spectrum is reproduced in Table XIII. There follows the reported assignment of the fundamental infrared frequencies of 1,2,5-thiadiazole in the vapor (Table XIV). The assignments above... 

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. 

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