Models of Internal Motions

two basic models of rotation of a methylene group about a C-C bond are discussed. One simple model involves a stochastic rotational diffusion about a bond this occurs if the potential barrier between several equilibrium sites is smaller than ksT, In other words, rotation about a bond occurs by successive random jumps of small amplitude. This rotation is specified by a correlation time Tj — being the diffusion constant 

The other basic model applies to rotational jumps among a few equilibrium sites when the potential barriers are larger than ksT. For aliphatic chains, the j + l)th bond (see Fig. 4.1) is again taking one of three possible orientations (t, g, g ) with respect to the plane defined by the jth and (j — l)th bonds. The potential energy of a trans (t) conformer is about 2.1 kJ/mol [8.10] below that of the gauche states which are assumed 

FIGURE 8.1. Potential energy curve and kinetic coefficients describing jump motion about a bond among three equilibrium sites (1,2,3). 

The jump between and g states is allowed but is sterically less favorable. Now W j/W2j = exp [— Eb—Ea)/RT], but is taken to be proportional to the square of aj [8.3]. It has been decided to write Wij = (jjDj/ and Wsj = cr jDj/3 with W2j = Dj/S. Now, aj is less than one when Etg 0. A zero value of Etg cjj = 1) corresponds to rotational jumps among three equivalent sites and to identical equilibrium probabilities of occupation for and g. Furthermore, the time constants A2j and A3 reduce 

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The joint three-site problem has been treated by P. A. Hart and C. F. Anderson (unpublished) using an approach for each P—O bond of the phosphodiester (or any other bond pairs of the phosphodiester) that is analogous to that of Tsutsumi (1979). To see how the analog of Eq. (2) takes shape, I focus on the phosphodiester model of Fig. 4 and die potential-energy profile of Fig. 5. The model of internal motion allows rotation to occur about each P—O bond randomly, independendy, and discretely. For each bond the potential energy profile of Fig. 5 is required in which /, Fj 3 and 1 21 = IFj, = W 2 Overall motion is required to be isotropic. As usual, direction cosines and P—H distances are computed using bond angjes and distances, determined by X-ray analysis or estimated by other means, for the nine joint conformations. 

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