Flexibility rotational degrees

Peralkylated (I and II) derivatives have been synthesized that act as bridging ligands in analogous dicationic dinuclear complexes (Fig. 4) The effective chain lengths (population of rotamers) and flexibility (rotational barriers, nature of thiourea bridging unit) in these species depend on the degree of alkylation and the nature of the substituents on nitrogen. Reactions... 

The separation of the repeat units of polymers into "backbone" (BB) and "side group" (SG) portions was discussed in Section 2.D. Geometrical parameters, namely the "rotational degrees of freedom" NBBrot and NSoj.0t, will now be defined as heuristic descriptors of flexibility for the backbones and side groups of polymers, respectively, by using the following simple set of rules ... 

The number of rotational degrees of freedom in the side groups, and the relative ease of rotational motions of the side groups, also affect the chain stiffness. If two side groups have equal volumes, but one of them is much less flexible than the other one, the use of the less flexible side group will normally result in a greater enhancement of chain stiffness. The structural parameter xg accounts for these factors, and is determined by the following rules ... 

At the simplest level, the association between two binding partners is both entropically disfavored due to the loss of translational and rotational degrees of freedom and enthalpically favored due to the interactions formed upon association. However, there is much more involved in protein-ligand complex formation, mostly because the interaction partners are flexible systems and the association occurs in aqueous solution. 

The main variation of MC methods is how the perturbing step is done. For a system composed of spherical particles (atoms), the only variables are the centre of mass of each particle, and the trial moves are simple translations of particles. For rigid non-spherical particles, the three rotational degrees of freedom must also be sampled, while for flexible molecules, it is usually also of interest to sample the internal degrees of freedom (conformations, vibrations). The latter can be done in Cartesian coordinates, or selectively in for example only the torsional variables. 

The final type of manipulator has three rotational degrees of freedom. This is the most complex type to control, but it has increased flexibility. Fig. 2d shows this type of manipulator-the anthropomorphic arm. The work volume of a practical manipulator of this form is shown in Fig. 3. You will notice that it is basically spherical but has missing portions due to the presence of the arm itself and because the rotations cannot achieve a full 360 degrees. The scallops on the inner surface are caused by constraints imposed by the joints. 

Including the torsional stiffaess of the bents adds another degree of freedom to the system, namely, the rotational degree of freedom. Thus, the beam is now indeterminate to two degrees. We will use the classical flexibility method in solving this problem. 

This kind of perfect flexibility means that C3 may lie anywhere on the surface of the sphere. According to the model, it is not even excluded from Cj. This model of a perfectly flexible chain is not a realistic representation of an actual polymer molecule. The latter is subject to fixed bond angles and experiences some degree of hindrance to rotation around bonds. We shall consider the effect of these constraints, as well as the effect of solvent-polymer interactions, after we explore the properties of the perfectly flexible chain. Even in this revised model, we shall not correct for the volume excluded by the polymer chain itself. 

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