Problems with underdetermined structures

We have seen that the random orientations of molecules in the gas phase give rise to diffraction data sets that are one-dimensional, and consequently we might not have access to enough data to determine the molecular structure completely. Similar interatomic distances can appear under the same peak in the radial distribution curve, in which case it is the weighted average of the distances that is obtained from the experiment rather than the individual distances. Similarly, the amplitudes of vibration cannot all be determined. We have also seen that the area of a peak in the radial distribution curve depends on the atomic numbers of the atoms involved. Thus, contributions to the scattering from atoms with low atomic numbers, particularly hydrogen, are small, and it is often impossible to determine their positions accurately. 

In addition, assumptions about symmetry are often made in analysis of diffraction data (both gas and condensed phases) and of rotation constants. Consider Ceo, discussed in the previous section. Of course, it makes sense to assume that this molecule has icosahedral symmetry, but this is still an assumption. Electron diffraction cannot prove the symmetry, although large differences in distances between otherwise identical atom pairs would be obvious. If we were to define the structure without symmetry restriction, we would have to refine (3 x 60) — 6 = 174 structural parameters and this would clearly be an underdetermined problem. The assumption of symmetry, which is not just a guess but is supported in this case by theory and spectroscopic data, reduces the problem to the very tractable problem of refining only two parameters. 

In cases such as these, it is important that care is taken not to over-interpret the available results. However, the problem can be alleviated by introducing further information into the stmctural refinement process. This can either be experimental observations from other structural methods (notably rotation constants), or data from calculations (mostly quantum mechanical calculations) or - in the simplest form - an educated guess. 

Problems can escalate rapidly as the number of atoms in the molecule increases, but the ease of determining a particular stmcture depends on how the various distances happen to fall in relation to each other. In silyl 

Molecular structure and radial distribution curve for perchloric acid, HCIO4, showing the positions of the two Cl—O bonded distances and the two non-bonded O- O distances. Redrawn from [7] with permission of The Royal Society of Chemistry. 

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Whatever model is used, the problem with equation (1) is that it is intrinsically underdetermined since it contains the product of interaction and spectral density terms. The usual solution is to examine relaxation at different temperatures and assume a function (usually exponential) for the temperature dependence of the correlation times. This is problematic in foods, as the structure of food is typically very temperature dependent. The alternative is directly to determine the spectral density function of the material by determining 7, over a wide range of frequencies. This may be done by using fast field cycling NMR. 

Multiple Conformations Ensembles. When multiple conformations are present, the problem rapidly becomes underdetermined. For example, if a nucleotide can exist in two major conformational states, with x(N)P(N) and x(S)P(S) where the glycosidic torsion angles in the N and S sugar states are not equal, each conformation will have its own characteristic NOEs and coupling constants. For n bases, there are 2" conformations, and the contribution of each one to the ensemble will depend on its population. Even for a dinucleotide, there are at least 4 conformations (viz. SS, SN, NS, NN), which multiplies the number of structural parameters to be determined fourfold, plus four equilibrium constants. There are not enough independent NMR data to determine all of these parameters and the problem is underdetermined. In this situation, the best that can be hoped for is to derive a set of structures that in some way represent the ensemble of structures that is present in solution. 

With a model that has on the order of 180 factors, we need to solve for over 16,000 covariances. Factor returns series include, in many cases, less than 30 to 40 periods. With such a small sample size compared to the number of factors, we have a severely underdetermined problem and are virtually assured that the covariance forecasts will show a large degree of spurious linear dependence among the factors. One consequence is that it becomes possible to create portfolios with artificially low risk forecasts. The structure of these portfolios would be pecu-... 

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