Orientation space

As we mentioned in the last paragraph, for most experiments with solution samples there is much less uncertainty in the k2 parameter than is often supposed, or suspected. Even for chromophores with orientations solidly fixed, a large fraction of the relative orientation space of the chromophores transition moments are such that K2 is often not too far from 2/3 [6, 10, 89], It is unlikely that the donor and acceptor molecules will be oriented such that the extreme values of k2 apply because the orientation configurational space for values close to these extreme values is relatively small [6], However, this does not discount, especially for fixed orientations and distances between D and A molecules, that k2 can assume a particular value very different from 2/3, and then this must be known to make a reasonable estimate of rDA. 

It is noteworthy that the critical value given by Eq. (2.5.11) is exactly half as large as for two quadrupole orientations on a square lattice (cf. Eq. (2.5.7)). This is a vindication of the inference that a halved critical temperature results from a corresponding change in the orientation space dimensionality. Thus, one might with good reason anticipate that the critical temperature for a triangular lattice of quadruples with arbitrary planar orientations should also be approximately half as... 

Therefore, we assume that the barrier is high and the probability flux Q couples two compact spots in the orientational space that are localized at the poles of the unit sphere. Accordingly, on the total flux the requirement of nondivergency is imposed that is, it is assumed that in the whole coordinate interval except for the vicinities [0, f> ] and [If2,71 ] of the poles, the quantity B sin if Qn is constant. Applying this condition to defined by relation (4.37), one comes out with the equation that couples the gradients of the energy and of the distribution function ... 

Although combiBUILD has been shown to be useful when the binding orientation of the scaffold is known, that is not always the case. The combinatorial DOCK algorithm has been developed as a more general tool for examining virtual libraries [52], Combinatorial DOCK places more emphasis on searching orientational space, that is how possible combinatorial compounds are oriented in the active site, and less emphasis on searching conformational space. 

Figure 3. Orientation space map for a molecule with essential groups A, B, and C. Each axis represents the distance between two points.

Our trajectories are sampled with the help of a simulated annealing protocol. But how can we test that the sampling is appropriate One measure that can help us to assess the quality of the simulation is the distribution of the orientation of the initial momentum vector. If we sample effectively the space of initial conditions (by sampling complete trajectories), then the momentum vectors should cover all of the orientation space. Alternatively, the vectors of the initial direction of the momentum behave as random vectors (with norm of one). In... 

Here Vp is used to denote that the gradient operates with respect to the orientation space of p,32 and D is the diffusion coefficient for rotational Brownian motion. 

Solid-state photoreactions are featured by their chemo-, regio-, and stereoselectivities, which are often quite different from those in solution (1). These features originate from the crystal structure of the parent molecule that is ordered with respect to packing, distance, mutual orientation, space symmetry, and molecular conformation. Reactions in crystals normally proceed with a minimum of atomic and molecular movement as a result of physical restraints by the crystal lattice (topochemi-cal principle) (2). To predict and control the crystal structure and reactivity by designing a chemical structure (crystal engineering) is one of the most attractive challenges in modern solid-state photochemistry (3). 

If all particle orientations are assumed equally probable, one finds upon integrating over orientation space that the usual form of Pick s law in physical space applies ... 

Shape of the Cavity and Guest Orientation Space Qroup P3 21 ... 

Organization of spaces Allocating and keeping open functional and process-oriented spaces in DMU applications. The major problem is the consideration of concurrent requirements concerning the spaces. 

In a liquid, as a result of inter-molecular interactions, molecules are continuously rotating and translating. Thus, if we could tag a molecule and study its detailed motion, we would find it executing a random Brownian motion not only in the three-dimensional positional spaee, but also a similar motion in the three-dimensional orientational space. For simplicity, let us first consider the motion of a tagged prolate-shaped moleeule in a solvent of spherical molecules, as shown in Figure 3.A.1 below. 

The set of all the possible g rotations is called the orientation space, or the Euler space, with such a representation of each particular rotation being a point in this three-dimensional space. 

Assume first that the orientation space is divided into small boxes Agi. To characterize the orientations of the grains of a given specimen one searches in each box for the relative volume of the grains whose orientations fall into this box. When the size of Agi tends towards zero one obtains the definition of the ODF, /( ), which is,... 

These two first examples of ODF are particularly simple. Usually the industrial processes involved in the fabriction of materials (solidification, heat treatments, phase transformations, deformation, etc.) lead to much more complicated crystallographic textures. This is illustrated in Fig. 11 where the whole ODF of a deformed aluminum sheet is presented. In this case there are also significant deviations from the random distribution with peaks and holes in the whole orientation space. 

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