Hindered Pseudorotation

If there is an angular dependence of the potential function, it is still possible to separate the Schrodinger equation approximately in polar coordinates if the angular barriers are much lower than the barrier to planarity. As mentioned earlier, Gwinn et al.34 have given an excellent treatment of this case. This has been applied to the interpretation of the microwave and far infrared spectra of tetrahydrofuran and 1,3-dioxolane36, 38 Equation (3.33) may be transformed to mass weighted polar coordinates 

If the conditions mentioned above are met, the Hamiltonian may be averaged over the radial coordinate, yielding the following Schrodinger equation 

The energy origin has been translated so that E 0. Equation (3.44) is of the same form as the Schrodinger equation for a two-fold internal rotor. Higher order terms than four-fold may arise from 

The latter will also introduce pseudocentrifugal distortion terms. 

In some molecules even an approximate separation of variables is not possible. Cyclopentanone is a good example of such a molecule it is discussed as a special case in Section IV. B. 

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The term pseudorotation was first appUed to cyclopentane like inversion, it has an atomic analogue in 5-coordinate compounds (e.g. PF5). ) The name means false rotation , and it is therefore appropriate for any conformational process which results in a conformation superposable on the original, and which differs from the original in being apparenUy rotated about one or more axes. Pseudorotation, in analogy with real molecular and internal rotations, can be free, as in cyclopentane, or more or less hindered, eis in cycloheptane and higher cycloalkanes. In moderately to severely hindered pseudorotation, it is appropriate to consider distinct stable conformations which are pseudorotation partners, and these cases are often amenable to study by dynamic nmr methods. When the barrier to pseudorotation is very low, or in the limit when pseudorotation is free, it is not really justified to talk about separate stable conformations (e.g. the C2 and Cg forms of cyclopentane), because strictly there is only one conformation, and the pseudorotation is simply a molecular vibration. 

Gwinn and co-workers34 have given an excellent exposition of the theory appropriate to treating hindered pseudorotation, with particular attention to the use of an angular Hamiltonian for small-barrier cases. For the special case of pure pseudo-... 

Fig. 4.28. Angular potential function hindering pseudorotation in tetrahydrofuran. (Reproduced from Engerholm, G. G., Luntz, A. C., Gwinn, W. D., Harris, D. O. J. Chem. Phys. 50. 2446(1969). ...

The value of the pseudorotational constant, 3.25 cm-1, was taken from the earlier far-infrared study6). Figure 4.28 depicts the potential function hindering pseudorotation. It is seen that the largest barrier encountered in one cycle is 55 cm-1, indicating that the approximation that the pseudorotation barrier be much less than the barrier to planarity has been met in this case. 

Nine- and Ten-membered Rings.—Conformational energies and interconversion barriers have been calculated for cyclononane. The [3,3,3]-conformation was found to be the most stable. N.m.r. indicated that cyclonona-l,2,6-triene adopts the unsymmetrical twist-boat-chair conformation (19) and undergoes a hindered pseudorotation with an activation energy of 12.9 kcal mol to achieve C2 time-averaged symmetry. ... 

Figure 12. Dynamic SSB (pseudorotation) produced by combined JTE plus PJTE in C. and >2 structures correspond to the APES minima and saddle points, respectively. Equivalent structures are congruent upon the symmetry operation. The pseudorotation coordinate Q(b ) is shown in the center. Pseudorotation or hindered pseudorotation in such systems takes place because the barriers between their minima are small (Reprinted with permission from Ref. 52. Copyright 1999, Elsevier Science Publishers).

Hindered pseudorotation, coupled states for the definition of coupling parameters see [04Maml, 03Mel]. Parameters of the potential along the pseudopotential path (in cm ) [03Mel]. 

Spherical polar coordinates are used for conformational representation of pyranose rings in the C-P system. Unlike the free pseudorotation of cyclopentane, the stable conformations of cyclohexane conformers are in deeper energy wells. Even simong the (less stable) equatorial (6 = 90 ) forms, pseudorotation is somewhat hindered. Substitutions of heteroatoms in the ring and additions of hydroxylic or other exocyclic substituents further stabilize or destabilize other conformers compared to cyclohexane. A conformational analysis of an iduronate ring has been reported based on variation of < ) and 0 (28), and a study of the glucopyranose ring... 

Figure 3. The conformational sphere for pyranoid rings. The perfect chairs are at the north and south poles (0=0 and 180 , respectively). The boat and skew (B and S designations) at the equator permit pseudorotation that is slightly hindered, at least for cyclohexane. The envelopes, E (also called sofas and half-boats), and half-chairs, H, are not observed for rings coiqposed of saturated carbon and oxygen atoms, but are iiqportant forms for rings with unsaturated carbon atoms. The aiqplitude of puckering corresponds to the radius of the sphere.

There are perhaps only two research areas in which the quantitative estimation of steric factors is routinely done, namely for hindered biphenyls (Cooke and Harris, 1967) and rotational or pseudorotation barriers in ethanes and cyclic compounds (Allinger et al., 1967). There are indications, however, that this interest is widening. Simonetta and Favini (1966) performed conformational calculations on the Cope rearrangement via chair or boat transition states and indicated that the orbital preference emphasized in a previous section should be supplemented with a steric factor. 

As already mentioned, in some molecular crystals, a hindered or nearly free rotation of entire molecules is observed, e.g. of benzene molecules in crystals of benzene, or of molecular groups, e.g. of CH3 groups in crystals of methyl naphthalene. These motions are stochastic and are termed pseudorotations or reorientations. These two terms denote the two limiting approximations, that of free rotation and that of a fixed orientation of the molecules or molecular groups. Experimental methods which have proved useful for the investigation of these stochastic motions are nuclear-spin magnetic resonance (NMR) [24] and quasielastic neutron scattering [35, 36]. 

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