Restricted rotational isomeric model

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

Figure 2 (a) Sketch of the variables determining the rotation isomeric state model of a statistical coil 9 = bond angle, = bond torsion angle [3], (b) Representation of one of the possible conformations of a two-dimensional statistical coil (33 monomers) with bond-angle restriction ( — 90° <0 <90°) [3],... 

Acyl-2-nitroenamines (554) with = R can exist in four planar isomeric forms (Scheme 16) due to restricted rotation around the C(1)=C(2) and Q2)—C(3) bonds. The combined use of IR and NMR spectroscopy provides a complete picture of the isomeric equilibria of these compounds. Quantum-mechanical calculations of the structure, relative energies, solvation and vibrational spectra of a series of model 2-acyl-2-nitroenamines have confirmed the structural assignment of the isomers either in solution or in the pure state. 

The valence angle model, though more realistic than the freely jointed model, still underestimates the true dimensions of polymer molecules, because it ignores restrictions upon bond rotation arising from short-range steric interactions. Such restrictions are, however, more difficult to quantify theoretically. A simpler procedure is to assume that the conformations of each sequence of three backbone bonds are restricted to the rotational isomeric states that correspond to the potential energy minima such as those shown for n-butane in Fig. 2.3. For the simplest case of polyethylene and for vinylidene-type polymers, the application of the rotational isomeric state theory yields the following equation... 

The present description of the rotational isomeric state model has several parts. First, we will describe the formulation and uses of Z, which initially restricts the focus to the thermodynamic (energetic) part of the rotational isomeric state model. Then we will describe how the structural information (h, 6i, 4>i) is incorporated in the model. Finally, we will combine the thermodynamic and the structural information, which will take us back to equation (2). Along the way we will mention a few other properties, in addition to r )o, that can be successfiilly rationalized with the rotational isomeric state model. Finally, several illustrative applications will be presented. 

Whilst some of these cases have been treated previously, Eichinger s method allows the calculation not only of statistical averages, but also of the distribution function itself. For polymer molecules with real bond angles and restricted bond rotation, the rotational isomeric state (RIS) model has proved powerful. Flory has summarized some of the most important results of the treatment, and Mark has considered the applications, particularly to bulk polymers and networks, pioneered by his group. A recent paper uses the RIS model to calculate the distribution function of the end to-end vector distribution for short polymer chains. 

The characteristic ratio Coo characterizes chain flexibility. It depends on the 6 and torsional potential and is determined by the chemical structure of the monomers [20]. The rotational isomeric state (RIS) model, introduced by P.J. Flory [20] is essentially an adaptation of the one-dimensional Ising model of statistical physics to chain conformations. This model restricts each torsion angle to a discrete set of states (e.g., trans, gauche , gauche"), usually defined around the minima of the torsional potential of a single bond, V((f>) (see Fig. 2d). This discretization, coupled with the locality of interactions, permits calculations of the conformational partition... 

In the next level of approximation, the restriction on bond angle (0, Fig. 2.3) rotation is accounted for within the rotational isomeric state model (i.e. only trans and gauche or gauche conformers are allowed). This leads to a mean-square end-to-end distance... 

The development of a theory for mesogens composed of flexible molecules is complicated by the large number of conformational states which the molecules can occupy. Even within the Flory rotational isomeric state model [60] where the conformations are restricted to just trans and gauche forms the number of discrete conformers, 3 ", adopted by a spacer containing TV groups can become excessive and the extensive calcu-... 

Before embarking on a discussion of the results of these studies let us add one historical note. The difficulty with swinging the polymer tails in a conformational transition has been recognized for many years. A means of circumventing was proposed by Schatzki. Verdier and Stockmayer had earlier invoked a similar principle but used it only to produce Rouse modes. We know now that slow Rouse modes are insensitive to the details of the faster time-scale dynamics. The proposed motions are completely local, and involve going from one equilibrium rotational isomeric state to another by moving only a finite, small number of atoms. Mechanisms of this class have come to be known as crankshaft motions (a term applicable in the strictest sense only to the Schatzki proposal). Because of the limited amount of motion and the simplicity of the dynamics these models are easy to understand, analyze, and simulate. This probably contributes to the continued attention devoted to them. The crankshaft idea has helped to focus attention on the necessity to localize the motion associated with conformational transitions, but complete localization is too restrictive. There are theoretical objections that can be raised to the crankshaft mechanism, but the bottom line is that no signs of it are found in our simulations. 

Prepare models of the four isomeric butenes, C4H8. Note that the restricted rotation about the double bond is responsible for the cis-trans stereoisomerism. Verify this by observing that breaking the n bond of cw-2-butene allows rotation and thus conversion to rran5 -2-butene. Is any of the four isomeric butenes chiral (nonsuperposable with its mirror image) Indicate pairs of butene isomers that are structural (constitutional) isomers. Indicate pairs that are diastereomers. How does the distance between the Cl and C4 atoms in mw5 -2-butene compare with that of the anti conformation of butane Compare the Cl to C4 distance in cw-2-butene with that in the conformation of butane in which the methyls are eclipsed. 

Make a model of 1,2-dimethylcyclopentane. How many stereoisomers are possible for this compound Identify each of the possible structines as either cis or trans. Is it apparent that cis-trans isomerism is possible in this compound because of restricted rotation Are any of the stereoisomers chiral What are the relationships of the 1,2-dimethylcyclopentane stereoisomers ... 

For all the afore-mentioned computations of Kx values Semiyen et al. used Eq. (5.24). The unperturbed mean square end-to-end distance of the linear precursors were calculated via the matrix algebraic methods of Flory and Jemigan [84, 85] using rotational isomeric state models based on detailed structural information [86]. However, Semiyen et al. [62, 63] also improved and applied another mathematical approach to the calculation of Kx, the so-called Direct Computational Method . The JS theory is limited to polymers obeying Gaussian statistics and cycles, free of enthalpic interactions. The Direct Computational Method does not need such restrictions [87-90]. The distances between terminal atoms of chains are calculated for all discrete conformations defined by the rotational isomeric state model. Any correlation between the directions of terminal bonds involved in the cyclization process can be investigated and their effect on Kx assessed. It can take into account favorable and unfavorable correlations between the directions of terminal bonds, as well as any excluded volume effect. Semiyen demonstrated [62, 63, 72] that the Direct Computational Method yields more realistic Kx values for small cyclic oligomers. 

Figure 21-9 Two isomers of 1,2-dichloroethene are possible because rotation about the double bond is restricted. This is an example of geometric isomerism. A ball-and-stick model and a space-filling model are shown for each isomer, (a) The cis isomer, (b) The tram isomer.

More than 30 years ago Warshel proposed, on the basis of semiempirical simulations, an isomerization mechanism that could explain how this process can occur in the restricted space of the Rh binding pocket (Warshel 1976). Since two adjacent double bonds were found to isomerize simultaneously the mechanism reveal a so-called bicycle pedal motion. Due to the concerted rotation of two double bonds in opposite directions the overall conformational change is minimized and hence this mechanism was found to be space-saving. The empirical valence bond (EVB) method (Warshel and Levitt 1976) was used to compute the excited state potential energy surface of the chromophore during a trajectory calculation where the steric effects of the protein matrix were modeled by specific restraints on the retinal atoms. Since then, Warshel and his coworkers have improved the model employing better structural data and new computational developments (Warshel and Barboy 1982 Warshel and Chu 2001 Warshel et al. 1991). The main refinement of the bicycle pedal mechanism was that the simultaneous rotation of the adjacent double bonds is aborted at a twist of 40° and leads to the isomerization of only one bond (Warshel and Barboy 1982). 

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