Sphere-exclusion

The hypothetical enantiophore queries are constructed from the CSP receptor interaction sites as listed above. They are defined in terms of geometric objects (points, lines, planes, centroids, normal vectors) and constraints (distances, angles, dihedral angles, exclusion sphere) which are directly inferred from projected CSP receptor-site points. For instance, the enantiophore in Fig. 4-7 contains three point attachments obtained by ... 

Figure 6 (Left) Schematic of a configuration of particles (black) with exclusion spheres (gray). (Right) The cross-hatched volume of available space that is formed upon removal of the particle is equal to that particle s free volume V. The surface area of the cavity is that particle s free surface area Sf. (Adapted from Ref. 71).

Identify the cavities Each connected cluster of Voronoi polyhedra vertices and edges that avoid overlap with the exclusion spheres in the system identifies a distinct cavity of available space. 

Determine the actual cavity volume and surface area within the Delaunay tetrahedra The overlap of the exclusion spheres with the relevant Delaunay tetrahedra is subtracted analytically, leaving only the actual cavity volume and surface area. This nontrivial calculation is from the multiple overlap of exclusion spheres, but a systematic method for carrying it out is available.70 71... 

The calculation of free volumes and free surface areas requires an efficient way of extending the cavity algorithm. To calculate the free volume of a particle i, the particle and its associated exclusion sphere are effectively removed from a snapshot of the particle configuration. The volume and surface area of the cavity that is produced in the absence of particle i are equal to the free volume and free surface area of particle i, respectively. 

In any sphere packing, it is possible to partition the given volume into occupied and available space. The former is the union of all the exclusion spheres, and the latter is its complement, namely, the volume available for the placement of the center of an additional sphere. The exclusion region of a sphere of diameter concentric sphere of radius a. Exclusion spheres can overlap. At a high enough density, the available space is in general composed of disconnected cavities. 

Fig. 7. A random configuration of atoms (black) surrounded by exclusion spheres (gray). The disconnected pockets of space that lie outside of the generally overlapping exclusion spheres are termed cavities (cross-hatched). A natural choice for the effective exclusion radius for the Lennard-Jones fluid is r.v = ct, the Lennard-Jones diameter.

Figure 2.3 Left panel collision of hard spheres. Upper hard spheres of equal radius d/2 centered at A and B. The excluded volume (dashed), drawn centered at B, is of radius d. Lower hard spheres of unequal diameters, di and dj. The "equivalent" exclusion sphere is of radius d = (di +d2)/2. The hard-sphere potential as a function of the center-to-center separation is shown in the right panel.

Armed with the hard-sphere model we can make the definition of a cross-section more transparent. Imagine that an exclusion sphere is centered around each beam molecule. Corresponding to this sphere is a circle of radius in the plane perpendicular to the beam velocity. Thus the beam molecule sweeps out a cylinder of volume nd Ax as it moves a distance Ax through the target gas. If the center of a target molecule lies within that volume a collision will occur and the beam molecule will be deflected off the x axis and therefore lost to the detector. From the volume of the cylinder swept we see that for the hard-sphere model... 

Derived features permit the construction, from specific atoms or substrucmres, of centroids, lines, planes, normals, and both inclusion and exclusion spheres rms best-fit constraint values may be associated with several of these features. Geometric constraints allow the geometric relationships of atoms, substmctures or derived features to be specified to an arbitrary degree of precision. Constraints may include distances, three-point angles (e.g., valence bond angles) and... 

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