Rotational isomeric state probability

The average dimensions, or other statistical properties of the polymeric chains, are then calculated, allowing for the relative probabilities of the rotational states and making full allowance for the mutual interdependence of adjacent pairs of bond rotational states. Parameters required for the models are normally obtained from molecular structure data and from experimental information relating to the statistical conformations of the polymeric chains. Further references to rotational isomeric state models will be made later. 

In the rotational isomeric state model, these states are not equally probable. Correlations between neighbouring torsional states are included in the model. For example, a consecutive sequence of g+ and g has high energy due to overlap between atoms and therefore is taken to have very low probability in the rotational isomeric state model. The relative probabilities of the states of neighboring torsional angles are used to calculate the mean-square end-to-end distance and C o [Eq. (2.12)]. 

The same approach can be used to deduce average conformations of local portions of the chain. The probability that bond i is in a particular rotational isomeric state, rj, is obtained by dividing Z into the sum of the statistical weights of all conformations where this bond is in the desired state. 

To predict the chemical shifts observed for the methyl carbons in a-PP (see Fig. 20.10), which show sensitivity to heptad stereosequences, we simply have to calculate the trans and gauche probabdities for the backbone bonds in each of the 36 heptad stereosequences. When this is carried out with the Suter—Floiy rotational isomeric state (RIS) conformational model for PP [29] and the resultant probabilities of finding CH3 in a gauche arrangement with its y-substituents (CHs) are multiplied by ycHs—ch = 5.2ppm, we obtain the 5CH3S shown as the stick spec-... 

The next most rapid type of motion is identified with deliberate imprecision as defect diffusion . It is closely similar to the fairly rapid vibrational excursions of limited angular extent that are considered in the theory presented above. Motions of this type can arise either from cumulative low energy distortions of bond angle within one set of potential energy minima, or from thermally accessible local changes of the rotationally isomeric states of a few bonds. The latter changes will probably be correlated, in order to satisfy the... 

Once the rotational isomeric states have been chosen, it is necessary to assign probabilities to each set of s. Consider tiie molecule n-butane. The RIS probabilities for the g states must be equal. If the relative probability of a g state is denoted as o, the absolute probabilities are ... 

---