Random rotation

Examples of mesomorphic forms characterized by disorder in the conformation of the chains have already been described in Section 2.6. For instance, a mesomorphic form is present in the high-temperature form I of polytetrafluoro-ethylene.106,107 In this phase the chains are in disordered conformation due to the presence of helix reversals along the chains.108-110 Moreover, intermolec-ular disorder is also present due to the random rotations of the chains around the chain axes.109 A long-range three-dimensional order is present only in the pseudohexagonal placement of the chain axes.107,109... 

Figure 2. Fonnation of ring and oblique texture patterns, a - several randomly rotated artificial crystallites and its Fourier transform (inset) b - reciprocal space with rings and zero tilt Ewald sphere c - 60° tilt of the Ewald sphere (reflection centers lie on the ellipse) d - the diffraction pattern as it is seen on the image plane.

Use of Proton and 13C NMR at temperatures from 27 to 400 °C provide very detailed information as to the nature of these motions [30], Thus, it has been shown that even at 300 °C the phenylene ring displays a rapid 180° flipping motion. Above the transition temperature of 350 °C the ester unit also begins to rotate in the form of 180° flips as a result of lattice expansion (see Fig. 7). Furthermore, the entire repeat unit participates in a synchronous motion. This should be interpreted as a jumping motion rather than free or random rotation. 

A new treatment for S = 7/2 systems has been undertaken by Rast and coworkers [78, 79]. They assume that in complexes with ligands like DTPA, the crystal field symmetry for Gd3+ produces a static ZFS, and construct a spin Hamiltonian that explicitly considers the random rotational motion of the molecular complex. They identify a magnitude for this static ZFS, called a2, and a correlation time for the rotational motion, called rr. They also construct a dynamic or transient ZFS with a simple correlation function of the form (BT)2 e t/TV. Analyzing the two Hamiltonians (Rast s and HL), it can be shown that at the level of second order, Rast s parameter a2 is exactly equivalent to the parameter A. The method has been applied to the analysis of the frequency dependence of the line width (ABpp) of GdDTPA. These results are compared to a HL treatment by Clarkson et al. in Table 2. 

In addition to translational Brownian motion, suspended molecules or particles undergo random rotational motion about their axes, so that, in the absence of aligning forces, they are in a state of random orientation. Rotary diffusion coefficients can be defined (ellipsoids of revolution have two such coefficients representing rotation about each principal axis) which depend on the size and shape of the molecules or particles in question28. 

Unlike other pharmacophore identification routines, the conformational search is performed on-the-fly in GASP and represents an integral part of the program. Each compound is input a single, low-energy conformation and random rotations and translations are applied in order to explore the conformational variation prior to superposition. 

One MC pass is defined as N attempted MC steps, once for each of the N molecules. A single MC step consists of an attempted random translational displacement of the com, a random rotational displacement around a randomly chosen axis and a random intramolecular rotation around the central C-C bond joining the two rings. In the simulation of the liquid beiphenyl, prior to equilibration at 400 K, the liquid was kept at 900 K for a few hundred MC passes. For all the condensed phases, equilibration was carried out for about 10000 MC passes and the averages were calculated over an equivalent number of MC passes. 

The activation energy barriers AG of our reactions are made up of two parts an enthalpy of acti vation AH, which tells us about the energy required to bring atoms together against the strain am repulsive forces they usually have, and an entropy of activation AS, which tells us about how easy i is to form an ordered transition state from a wriggling and randomly rotating molecule. 

The Monte Carlo simulations were performed in the NVT ensemble and, in order to minimize the possibility of sampling regions of the zeolitic stmcture not accessible to the molecules, the initial distribution of the hydrocarbon molecules were chosen to be idaitical to the corresponding one of the thermalized configuration, at 300 K. The Metropolis [33] algorithm was then used to generate up to 8000 configurations. Three different steps, with equal probability, were considered random translation of the center of mass, random rotation of the whole molecule [34] and a perturbation on any of the internal coordinates of the molecules. 

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