Anharmonic Force Constant Refinements

Anharmonic Force Constant Refinements.—The preceding parts of this Section 4 constitute an outline of how the vibration-rotation spectrum of a molecule may be calculated from a knowledge of the force field in some set of geometrically defined internal co-ordinates, denoted V(r) in general in this Report [but denoted V(X) in the special discussion on pp. 126—132], In practice we wish to solve the reverse problem we observe the vibration-rotation spectra, and we wish to deduce the force field. 

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 

The refinement calculation may be carried out in a variety of ways, and a few general remarks should be made before we consider particular examples. We wish to determine re, /2, /3, and /4, where these denote symbolically the equilibrium structure (which may be thought of as the linear force field), the quadratic, cubic, and quartic force field. (Terms higher than quartic are not considered here.) Each set of data depends on all constants up to a certain order, as shown in Table 3 for example, Ae, Be, and Ce depend only on re, the oo values depend on re and /2, the a values on re, /2, and /3, and the x values on re, /2, /3, and /4. Ideally one should refine all data simultaneously to all force constants (including the equilibrium structure), but in practice the calculation has to be broken down into steps. Thus usually the equilibrium structure re, or some approximation to re, is determined first from the rotational constants then the quadratic force field /2 is determined from the o , , and r values holding re constrained then the cubic force field /, is determined from the values holding re and /2 constrained and finally the quartic force field /, is determined from the x values holding re, /2, and/3 constrained. (This should be compared with the discussion for diatomic molecules at the end of Section 3.) 

Often the original structure determination will have involved some uncorrected vibrational averaging effects it may be an r0 or an r, structure.28 However, once /2, or some approximation to /3, has been determined it is possible to correct r to re and obtain an improved equilibrium structure (in most cases this correction can be made directly from the x values without going through a cubic anharmonic calculation, but in some cases the calculation will allow unobserved a. values to be determined, perhaps for other isotopic species, etc.). Similarly, it is often true that the harmonic field /2 is calculated from the observed fundamentals (the v values) rather than the harmonic vibration wavenumbers (the to values), for want of information on the corrections. However, once /4, or some approximation to/4, has been determined, it may be used to calculate a complete set of x values and hence to calculate all the corrections to obtain the co values. Thus the calculation of re and may be improved from a knowledge of /3 and /4. 

Some calculations have been reported in which fz and /4 and occasionally /a, /3, and/4, have all been refined simultaneously, although almost all calculations have been made with re constrained. It is not clear that simultaneous refinement of /2,/3, and /4 has any advantage. Hoy et al.12 have observed that there is a technical advantage in constraining re and/2 while refining/3 and/4, 

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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. 

The most general force field of a molecule would include anharmonic as well as harmonic terms. However, with the limited experimental information generally available for refining an empirical force field for complex molecules, the harmonic approximation is the only feasible one at present. This means that, for the isolated molecule, we need to know the force constants, Fy, in the quadratic term of the Taylor series expansion of the potential energy, V ... 

In attempts to refine theoretical anharmonic, usually quartic, force fields the most useful set of experimental data includes vibrational fundamentals, overtones, and combination bands, as well as rotational constants of the individual vibrational levels. 

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