Convergence sphere

The convergence sphere of the multipole expansion for a molecule such as butane may be penetrated by r molecule. 

Fig. 8 Since topological atoms are finite it is possible to monitor their formal convergence. The atomic electron density is totally contained within the blue volume. Since r < r outside the convergence sphere the potential (y(r)) converges exactly in this region (with permission from the American Chemical Society)...

Convergence problems are very common due to the number of orbitals available and low-energy excited states. The most difficult calculations are generally those with open-shell systems and an unfllled coordination sphere. All the techniques listed in Chapter 22 may be necessary to get such calculations to converge. 

Fig. 7.2. Numerical estimates of the overall average elastic constants. The error-bars were calculated as cr-N°5, where N is the number of individual estimates for each of the two overall elastic constants obtained with a given number of spheres in the unit cell and a is the standard deviation. The two horizontal dashed lines drawn through the averages obtained with 64 spheres are meant to facilitate the convergence analysis...

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

As the fluid flows over the forward part of the sphere, the velocity increases because the available flow area decreases, and the pressure decreases as a result of the conservation of energy. Conversely, as the fluid flows around the back side of the body, the velocity decreases and the pressure increases. This is not unlike the flow in a diffuser or a converging-diverging duct. The flow behind the sphere into an adverse pressure gradient is inherently unstable, so as the velocity (and lVRe) increase it becomes more difficult for the streamlines to follow the contour of the body, and they eventually break away from the surface. This condition is called separation, although it is the smooth streamline that is separating from the surface, not the fluid itself. When separation occurs eddies or vortices form behind the body as illustrated in Fig. 11-1 and form a wake behind the sphere. 

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

The convergent method also allows the synthesis of dendrimers of regular shapes other than spheres. Ellipsoid- and rod-shaped dendrimers can be synthesized hy using appropriate core molecules [Tomalia, 2001]. 

From Equation 5-120 all other closure temperature equations may be obtained. The evaluation of G values takes some effort because the series in Equations 5-107 to 5-113 converge slowly. For the limiting case of 7t 0, Dodson (1973) obtained the values of G (shape factor) to be 8.65 for plane sheets with infinite area, 27 for infinitely long cylinders, and 55 for spheres. 

The dendrimer is a new type of cross-linked nanoparticle prepared by repetitive reactions to compose dendritic structure. There are two kinds of preparative methods, divergent (30) and convergent (31) methods, as shown in Figure 12.2.7. The dendrimer by the former method is constructed from the center of sphere to the outer layers, and therefore the surface chemistry is decided by the final reaction. When a dendrimer is formed from a central ammonia molecule, the forth-generation dendrimer of about 3 nm diameter has 48 functional groups at the surface. If a crowd of functional groups is not desirable on the outermost layer, they can be thinned down properly by replacing a two-functionality compound with a one-functionality one. 

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