Computation box

The overall asymmetry index for molecule k may be taken as the average of aU the pairwise contacts to it, and the overall asymmetry index in a computational box as the average of all molecule-molecule indices ... 

An adaptation of the Box method, however, seems to offer the advantage of improved efficiency while still being susceptible to automatic computation. Box s approach may be divided into two stages. The first, to which he has applied the name method of steepest ascents, is primarily for the purpose of approximately locating the optimum response. The second is a more intensive investigation in the local region of the optimum. This will permit a precise determination of the optimum and also indicate the behavior of the response in its neighborhood. 

In the NEGF-DFT algorithm, the total number of valence electrons is obtained by Ne = f dr p(r) = Tr(pS). Here the trace is over all the orbitals included into the computation box. It is often useful to perform a Mulliken population analysis by representing Ne as a summation ... 

Fig. 8. Voltage drop Vi, /, z) in the scattering region is presented for bias of 0.5 V applied to the left Au electrode. The electrode separation is of 11.7 A, and the C60 sits in the middle of the tunnel junction. Here, we define the voltage drop in the scattering region as Vj,(r) = Vn(r Vj,) — Vjj(r Vj, = 0). The presented Vj,(y, z) is an average over the horizontal planes, i.e., f dxV, (x,y,z)/(xt — xt>), where xt, and xt are the coordinates of the bottom and top of the computation box in the x direction. The dashed vertical lines highlight the C60 location in the molecular junction. We project the positions of the left and right electrodes and the molecule onto the surface Vf,(y, z). The corresponding edges are shown by the bold solid lines.

Boundary conditions in simulations with the objective to study equilibrium properties of a bulk fluid should be chosen so as to minimize the finite-size effects and boundary effects. One possible approach to this is to replicate the computational box and use periodic boundary conditions [141], thereby making the simulated system pseudo-infinite. 

The chosen computational box should be space-filling and it is replicated throughout space in all directions. While there are several different space-filling shapes [112] the cubic box is the simplest and most commonly used. 

The particles in the central computational box is surrounded by image particles residing in each of the periodic replicas of the central box. The image particles move in exactly the same way as the particles in the central computational box. The periodic boundary conditions are implemented so that when a particle moves out of the central computational box during the course of the simulation, then its periodic image reappears at the opposite side of the central computational box. 

Using the minimum image distance criteria ensures that the distance between two particles varies continuously as particles move out of the central computational box and reappears at the opposite side. Furthermore, the periodic boundary conditions has the effect of restraining unphysical density fluctuations. However, it also means that particles in the central computational box will never be more than half the box length L apart and phenomena with a characteristic length-scale longer than this will be suppressed [142,143]. 

Fig. 15. Bonding of a cyclic Si3-siloxane molecule on an a-quartz (001) surface (a, c, e) and bonding of a cyclic Sis-siloxane molecule on a rhombohedral calcite (101) surface (b, d, f). a detail of Fig. ISc, c manual matching on the X-ray structure, e after energy minimization with DISCOVER/COMPASS (white lines mark the computational boxes), b detail of Fig. 15d, d manual matching on the X-ray structure, f energy minimization with DISCOVER/COMPASS, g calcite with characteristic Ca-Ca distances (green spheres) and calcium carbonate anions (C atoms gray, O atoms redX h as Fig. 15f, side view along the y axis (two rings removed for fi ee view).

We let two electron-proton plasma populations (with a density difference of a factor of three) collide in the reference frame of the denser population. In this frame we continuously inject the less dense population with a bulk Lorentz factor, T = 15 in the -direction. The computational box consists of 125 x 125 x 2000 gridzones or 37 x 37 x 600Ag where Ae is the electron skin depth c/oje. Using 16 particles pr. cell this adds up to almost 109 particles. 

Fig. 9.31. Computational box used to compute the energy of an intrinsic stacking fault in A1 (adapted from Wright et al. (1992)).

A crucial step in the optimization of an MD code is the location of all pairs of particles within the cut-off distance Rc. An all-pairs neighbour search would be of 0(N2) so some kind of cell-based search is generally employed. The computational box of side L that contains the N particles is divided by a fine 3-D grid into subcells of side rsc and number... 

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,...

FIGURE 26.13 (a) The computational box divided into cells with size of the interaction range rcut- (h) The Verlet list of neighbors is searched in the sphere with radius shghtly larger than Tcut-... 

FIGURE 26.15 (a) The schematics of the geometrical decomposition of the computational box. The decomposition of the computational boxes for red blood cells flow in a capillary without (b) and with (c) load-balancing [102]. 

FIGURE 26.17 (a) Classical periodic boundary conditions, (b) checker-board periodic boundary conditions (CPBC), (c) and (d) various shapes of computational box simulated by CPBC. 

In order to avoid siuface effects for condensed phase simulations, periodic boundary conditions are applied. The central computational box is replicated infinitely in all dimensions. A detailed description can be found in the textbooks of Allen and Tildesley [10] as well as of Frenkel and Smit [11],... 

Unfortunately, when using ACID, your computer can be a primary source of background noise. There really isn t much you can do about the whir of fans and the click of the hard disks without isolating your computer box from the monitor, mouse, and microphones. You may be able to place the box in another room and run the wires to your semi-isolated environment. Another idea is to isolate your computer in a cabinet beneath your desk, making sure that your computer continues to have adequate air circulation for cooling purposes. If you are really a techno-geek, you could install a whisper-quiet water-cooling system on your PC. 

Numerical solutions depend strongly on the choice of grid. Using any reservoir simulator, compute the flowfields about stand-alone fractures and shales, say, centered in a large computational box. Do your results satisfy the symmetries and antisymmetries derived here for pressure and normal velocity A single straight shale in an infinite medium has a perfectly antisymmetric disturbance pressure field. Show that this antisymmetry is... 

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