Intrinsic geometry

Another entity that we shall need belongs to the realm of intrinsic geometry geodesic curvature. Consider a surface x, a point P on x and a curve on x passing through P. The curvature vector of at P joins P to the centre of curvature of This curvature vector may be decomposed into mutually orthogonal components. These components are given by projection of the... 

We have seen that the local constraint on the surface curvatures, set by the surfactant parameter, can be treated within the context of differential geometry, which deals with the intrinsic geometry of the surface. In contrast, the global constraint, set by the composition of the mixture, is dependent upon the extrinsic properties of the surface, which need not be related to its intrinsic characteristics. (For example, the surface to volume ratio of a set of parallel planes can assume any value by suitably tuning the spacing bebveen the planes. Similarly, the ratio of surface area to external volume i.e. the volume of space outside each sphere closer to that sphere than any other) of a lattice of spheres depends upon the separation between the spheres.)... 

However, in other situations a connection can be made between the surface to volume ratio of a surface and the intrinsic geometry of that surface. For example, the ratio of surface area to internal volume of a sphere or cylinder depends only upon the curvatures, since it is a function only of the radii. It turns out that this connection can be extended to certain h) erbolic surfaces, leading to accurate estimates of the relation between the global and local geometric characteristics of these surfaces. 

Under this assumption, the (global) surface to volume ratio can be estimated from the (local) intrinsic geometry alone using the relations derived already for parallel surfaces. The volume is tiled without overlap or gaps by a dense foliation of parallel surfaces from the original (homogeneous) surface. For example, the (internal) volume, V, of a sphere is related to the area of the sphere. A, by the local relation (4.3), valid eveiywhere on the sphere. 

However, a two-dimensional (2D) interface separating three-dimensional (3D) domains has two independent curvatures, which can be either concave or convex. The product of those curvatures determines the intrinsic geometry both convex (or concave) leads to an elliptic cap , one vanishing curvature gives a planar, cylindrical or conical parabolic sheet, and opposite curvatures to a saddle-shaped hyperbolic surface (Figure 16.2). 

Since V q is assumed to be known, its derivatives are readily obtainable. The Cartesian second derivatives of V are obtained from the ab initio calculation, if necessary scaled to the experimental frequencies and eigenvectors. The F< and Qa,ij are obtained from Taylor expansions of each type of coordinate (internal and Cartesian) in terms of the other about the minimum of V. After the F,y have been determined, the intrinsic geometry parameters, Rio, are obtained by solving the following system of linear equations with respect to Rj — Rjo ... 

TTie second stage of the procedure is equally important the transformation is now applied to other conformers of the set of model molecules. (For example, in the first implementation, to obtain an SDFF for polyethylene, the transformation was applied to the ab initio results for the four stable conformers of rt-pentane and the ten stable conformers of n-hexane.) There are at least three reasons for doing this. First, since the initial K q may not properly describe the nonbonded interactions, inclusion of the different conformers provides the opportunity to optimize the nonbonded parameters consistent with the philosophy that the intrinsic geometry parameters, RiO, and (to first approximation) MM force constants, F,y, should be about the same for all structures. Second, while the conformation dependence of some kinds of F,y is well... 

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