Anharmonic Force Fields

For these vibrations, the quantization scheme of Section 4.2 can be carried over without any modification (Iachello and Oss, 1991a). The potentials in each stretching coordinate 5 are in an anharmonic force field approximation represented by Morse potentials. The boson operators (Ot,xt) correspond to the quantization of anharmonic Morse oscillators, with classical Hamiltonian... 

The earliest anharmonic force field calculations on polyatomic molecules were made by Pliva and co-workers.1-2 More recently, Kuchitsu and Morino and co-workers,4-7 Overend and co-workers,8-10 Cihla and Chedin,11 and Hoy, Mills, and Strey12 have developed the techniques of calculation and applied them to a wide variety of molecules many other workers have reported calculations on particular molecules, as discussed in Section S. A recent review by Pliva12 contains reference to most published calculations on particular molecules up to 1973. 

The primary motive for attempting calculations of this kind is simply our desire to determine the potential function V(r) more accurately and over a wider range of co-ordinate space. Even if our immediate ambition is only to determine the equilibrium configuration and the harmonic force field, our ability to withdraw this information from spectroscopic data is limited by the need to make corrections arising from the cubic and quartic anharmonic force field. 

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. 

The relation of the anharmonic force field to the spectroscopic observables for a polyatomic molecule is similar to the calculation described above for a... 

The discussion so far may be summarized as follows. There are two reasons for using curvilinear co-ordinates to represent the anharmonic force field of a polyatomic molecule, despite their apparent complexity. The first is that it is only in this way that we obtain cubic and quartic force constants which are independent of isotopic substitution. The second is that in terms of curvilinear bond-stretching and angle-bending co-ordinates we obtain the simplest expression for the force field, in the sense that cubic and quartic interaction terms are minimized. The first reason is compulsive the second reason is not compulsive, but it does make the curvilinear co-ordinates very desirable. 

Table 3 Spectroscopic constants used in anharmonic force field calculations, and their relation to the force field... 

There are many other spectroscopic constants, as listed in Table 3, which provide information on the anharmonic force field in particular cases. Some examples are discussed in Section 5. 

Although it is not our purpose to review this subject here, one aspect of this step in the calculation must be mentioned because of its importance in determining the anharmonic force field and its relationship to the contact... 

The observed a values are of course generally an important source of information on the cubic anharmonic force field. However, in the presence of a Coriolis resonance the particular a values involved are dominated by the harmonic Coriolis contribution arising from equation (66), and analysis of a Coriolis resonance essentially gives information on the constant or... 

The problem is similar to that involved in harmonic force field calculations, but more difficult in almost all respects. In simple cases one may attempt to solve directly, or graphically, for some of the anharmonic 0 values using the observed values of the spectroscopic constants in equations like (61) and (62). These may then be related to / values through the L tensor as described on pp. 124—132. However, such methods are of only limited value. The more general method of calculation is to attempt an anharmonic force field refinement, in which a trial force field is refined, usually in a large non-linear least-squares calculation, to give the best agreement between the observed and... 

Table 4 Molecules for which anharmonic force field calculations have been reported...

Table 6 Anharmonic force fields in curvilinear internal co-ordinates for COa and for CS2 ...

Table 5 suggests that one might hope to determine all the constants in the most general anharmonic force field without too much difficulty. The comparison of Suzuki s with Chedin and Cihla s results in Table 6 gives some feel for the reliability of the results obtained. These two calculations were made in different ways (see the original references) although both refined the force field to fit all observed vibrational levels and rotational constants, Suzuki used an up-to-quartic force field, where Chedin and Cihla used an up-to-sextic force... 

Linear Unsymmetric Triatomic Molecules.—Reducing the symmetry from Daoh to Coot, as in NaO, OCS, and HCN, increases the number of parameters in the general quartic force field to 2re + 4/2 + 6f3 + 9/i Table 7 shows their relationship to the primary spectroscopic observables. It is clear that problems of insufficient data to determine the general force field are already on the horizon for example, data from at least two different isotopic species must be combined in order to determine frrr, frrit, fruit, and fium from the observed values of a and a . In practice, of course, substitutions like 14N for 15N tend to change the spectroscopic constants by only a small fraction, and conversely the observed data on the constants of such isotopic species tend to give nearly parallel information on the force field to that obtained from the parent species. For these reasons the anharmonic force field of molecules like N20 is much less well determined than that of C02. These effects are apparent in the uncertainties obtained on the force constants in the refinement calculations referred to in Table 4. 

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