Frequencies of Isotopic Molecules

With the same set of force constants, the frequency of the B2 vibration is calculated as 

If the secular equation is third order, it gives rise to a cubic equation  

As stated in Sec. 1.14, the vibrational frequencies of isotopic molecules are very useful in refining a set of force constants in vibrational analysis. For large molecules, isotopic substitution is indispensable in making band assignments, since only vibrations involving the motion of the isotopic atom will be shifted by isotopic substitution. 

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

This rule has been confirmed by using pairs of molecules such as H2O and D2O and CH4 and CD4. The rule is also applicable to the product of vibrational frequencies belonging to a single symmetry species. 

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This type of potential field is called a generalized valence force (GVF) field. It consists of stretching and bending force constants, as well as the interaction force constants between them. When using such a potential field, four force constants are needed to describe the potential energy of a bent XY2 molecule. Since only three vibrations are observed in practice, it is impossible to determine all four force constants simultaneously. One method used to circumvent this difficulty is to calculate the vibrational frequencies of isotopic molecules (e.g., D2O and HDO for H2O), assuming the same set of force constants.+ This method is satisfactory, however, only for simple molecules. As molecules become more complex, the number of interaction force constants in the GVF field becomes too large to allow any reliable evaluation. 

Such relations between the frequencies of isotopic molecules are highly useful in making band assignments. 

The Sum Rule. In addition to the product rule, which relates the products of frequencies of any pair of isotopically related molecules, there exist certain sum rules which relate the sums of the squares of the frequencies of isotopic molecules. The basis of the rales is the fact that the sum of the squares of the freiiuencies is a linear function of the reciprocal masses of the atoms. If, therefore, several isotopic molecules can be found and geometrically superimposed with appropriate signs in such a way that the atoms vanish at all positions, the corresponding linear combination, i.e., superposition, of the frequency sums should vanish. Thus if... 

The Product Ride. The vibration frequencies of isotopic molecules are related by a rule which is a generalization of the methane example given above. Suppose that the kinetic and potential energies are expressed in terms of external symmetry coordinates (Chap. 6)... 

Obviously, there is an isotope effect on the vibrational frequency v . For het-eroatomic molecules (e.g. HC1 and DC1), infrared spectroscopy permits the experimental observation of the molecular frequencies for two isotopomers. What does one learn from the experimental observation of the diatomic molecule frequencies of HC1 and DC1 To the extent that the theoretical consequences of the Born-Oppenheimer Approximation have been correctly developed here, one can deduce the diatomic molecule force constant f from either observation and the force constant will be independent of whether HC1 or DC1 was employed and, for that matter, which isotope of chlorine corresponded to the measurement as long as the masses of the relevant isotopes are known. Thus, from the point of view of isotope effects, the study of vibrational frequencies of isotopic isomers of diatomic molecules is a study involving the confirmation of the Born-Oppenheimer Approximation. 

The temperature ranges in which these simple behaviours are approximated depend on the vibrational frequencies of the molecules involved in the reaction. For the calculation of a partition function ratio for a pair of isotopic molecules, the vibrational frequencies of each molecule must be known. When solid materials are considered, the evaluation of partition function ratios becomes even more complicated, because it is necessary to consider not only the independent internal vibrations of each molecule, but also the lattice vibrations. 

Since there are two force constants and four frequencies, one can test the quality of this force field by determining k and kg/f separately from Eqs. (12) and (13) and from Eqs. (14) and (15). However, for some tetrahedral molecules, it is found that the latter two equations yield imaginary values for kmA kg, and this is an indication that stretch-stretch, stretch-bend, and bend-bend interaction terms may be important in U. Unfortunately addition of these terms leads to more complex expressions than Eqs. (12) to (15) and to more force constants than can be determined from the four observed frequencies. Use of frequencies of isotopic forms of the molecule is often helpful in such cases and can permit more accurate determination of potential functions. ... 

The study of the rotation-vibration spectra of polyatomic molecules in the gas phase can provide extensive information about the molecular structure, the force field and vibration-rotation interaction parameters. Such IR-spectra are sources of rotational information, in particular for molecules with no permanent dipole moment, since for these cases a pure rotational spectrum does not exist. Vibrational frequencies from gas phase spectra are desirable, because the molecular force field is not affected by intermolecular interactions. Besides, valuable support for the assignment of vibrational transitions can be obtained from the rotational fine structure of the vibrational bands. Even spectra recorded with medium resolution can contain a wealth of information hot bands , for instance, provide insight into the anharmonicity of vibrational potentials. Spectral contributions of isotopic molecules, certainly dependent on their abundance, may also be resolved. 

Unknown values on the right side of Eq. (7) are only the force constants. These constants are connected with the oscillation frequencies of the molecules. Therefore, the equilibrium isotopic fractionation is principally determinable if the corresponding spectroscopic data is available. Whereas this data for low-molecular isotopic molecules is very often known, it is unknown for high-molecular species. From the reduced partition function it is possible to deduce the following principles about the magnitude and trend of an isotopic fractionation ... 

The observed vibration frequencies of a molecule depend on two features of the molecular structure the masses and equilibrium geometry of the molecule and the potential eneigy surface, or force field, governing displacements from equilibrium. These are described as kinetic and potential effects, respectively for a polyatomic molecule the form and the frequency of each of the 3N—6 normal vibrations depend on the two effects in a complicated way. The object of a force field calculation is to separate these effects. More specifically, if the kinetic parameters are known and the vibration frequencies are observed spectroscopically, the object is to deduce the potential eneigy surface. A major difficulty in this calculation is that the observed frequencies are often insufficient to determine uniquely the form of the potential energy surface, and it is necessary to use data on the frequency shifts observed in isotopically substituted molecules or data on vibration/rotation interaction constants observed in high resolution spectra in order to obtain a unique solution. 

The theory of calculation of ratios of isotopic partition functions has been developed in detail for exchange equilibria in gases. The formulation of Bigeleisen and Goeppert-Mayer,10 which writes everything in terms of the vibrational frequencies of the molecules, will be reviewed. Classically QAJQAx is where the tn/s are... 

The effect of isotopical replacement was also investigated for the Ci Csg structure corresponding to the natural contents of the isotope (about 1.1 %). Because of infiingement of symmetry all vibrational frequencies of such molecules became IR active [90]. In line with the case of clusters characterized by large n considered above, the calculations have shown that the intensity of such peaks is very small in comparison with the intensity of the IR modes of a homogeneous fullerene molecule. Therefore, the most typical effect from replacement of one or several C atoms in Cgo is a smoothing of IR peak (about 1/6 from the initial value) corresponding to the vibration mode of Tiu symmetry [90]. 

A number of unstable and transient species have been synthesized via matrix cocondensation reactions, and their structure and bonding have been studied by vibrational spectroscopy. The principle of the method is to cocondense two solute vapors (atom, salt, or molecule) diluted by an inert gas on an IR window (IR spectroscopy) or a metal plate (Raman spectroscopy) that is cooled to low temperature by a cryocooler. Solid compounds can be vaporized by conventional heating (Knudsen cell), laser ablation, or other techniques, and mixed with inert gases at proper ratios [128]. In general, the spectra of the cocondensation products thus obtained exhibit many peaks as a result of the mixed species produced. In order to make band assignments, the effects of changing the temperature, concentration (dilution ratios), and isotope substitution on the spectra must be studied. In some cases, theoretical calculations (Sec. 1.24) must be carried out to determine the structure and to make band assignments. Vibrational frequencies of many molecules and ions obtained by matrix cocondensation reactions are listed in Chapter 2. 

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