Cut-off radius

The real-space sum is particularly easy to parallelize as it simply involves a spatial decomposition with appropriate attention to the overlap between adjacent regions due to the cut-off radius. Performance for the real-space contribution alone is given in Fig. 4. 

In the case of computer simulations of fluids with directional associative forces a less intuitive but computationally more convenient potential model has been used [14,16,106]. According to that model the attraction sites a and j3 on two different particles form a bond if the centers of reacting particles are within a given cut-off radius a and if the orientations of two spheres are constrained as follows i < 6 i and [tt - 2 < The interaction potential is... 

The use of a cut-off distance reduces the fonnal scaling in the large system limit from atom - atoni since the non-bonded contributions now only are evaluated within the locSl sphere determined by the cut-off radius. However, a cut-off distance of 10 A is so large that the large system limit is not achieved in practical calculations. The actual scaling is thus more like where n is perhaps 1.5-1.8. In static applications,... 

Note that can be chosen to be distance dependent. A common approach is to assume that yi is a constant for a distance smaller than a cut-off radius fcut.DPD and to set y,y = 0 otherwise. As calculating random numbers may be a task of relatively significant computational effort in force-field-based MD simulations, it may be sensible to make rcutjDPD smaller than the cut-off radius... 

Most experimental data on flashback are plotted as a function of the average flashback velocity, navF, as shown in Fig. 4.34. It is possible to estimate penetration distance (quenching thickness) from the burner wall in graphs such as Fig. 4.34 by observing the cut-off radius for each mixture. 

Each pseudopotential is defined within a cut-off radius from the atom center. At the cut-off, the potential and wavefunctions of the core region must join smoothly to the all-electron-like valence states. Early functional forms for pseudopotentials also enforced the norm-conserving condition so that the integral of the charge density below the cut-off equals that of the aU-electron calculation [42, 43]. However, smoother, and so computationally cheaper, functions can be defined if this condition is relaxed. This idea leads to the so called soft and ultra-soft pseudopotentials defined by Vanderbilt [44] and others. The Unk between the pseudo and real potentials was formaUzed more clearly by Blochl [45] and the resulting... 

Fig. 4. Spacial distributions of the strength of the correlations as calculated by the correlated motion coefficient CMi(2.0,5.0) for all atoms. The circle at the right-hand side of the figure denotes the size of the cut-off radius r, which is chosen here to be 2.0

All in all, the WPPM approach can provide a simultaneous structure and microstructure refinement, based on physical models of the phases under study, without using any arbitrary profile function. Considering the terms of Equation (26), refinement parameters to be optimized in a least-squares analysis are relatively few, namely, mean (p) and variance (cr) of a suitable distribution of coherent domain sizes, dislocation density (p), effective outer cut-off radius (R ) and character (/e, effective fraction of edge dislocations), twin fault (P), deformation fault (a) and APB (y) probabilities. 

According to the Wilkens-Krivoglaz approach,the FT of the diffraction profile produced by a system of dislocations with average density p, Burgers vector b and effective outer cut-off radius R, can be written as ... 

Figure 5. Charge density distribution for the nuclide as given by the Fourier-Bessel expansion with parameters taken from [35, p. 264] [ rms radius a, cut-off radius R, see also Eq. (91) ].

Fig. 14. A side view along the >> axis of bonding of cyclic Sis-siloxane molecules cage type and ladder type siloxane molecules at maximum possible occupancy on an a-quartz (001) surface (3x4 unit cell) after manual matching on the X-ray structure, followed by energy minimization with DISCOVER/COMPASS at periodic-boundary conditions, with cut-off radius 9.5 A (white vertical lines mark the computational box), a, c, e methyl derivatives, b, d, f iso-octyl derivatives, a, b cyclic Sis-siloxanes, c, d cage molecules (type VII according to Ref. [75]), e, f ladder molecules. Specifications of occupancy (per 3x4 unit cell), where A= number of bound siloxane molecules, R = number of aliphatic side-chains, F == number of remaining free OH groups on the quartz surface, E = bond enthalpy (kcal moP unit cell), a iV = 4, / = 12, F = 0, A = -319.5, b V = 4, F = 12, F= 0, AF = -1 50.9, c 4,F = 36, F= 8, A = -1950, d V= 1, A = 9, F= 11,...

The disclination is supposed to have a core whose energy is not known. To allow for this, we postulate a cut-off radius around the disclination and integrate for distances greater than / <, to obtain... 

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