Force calculation

Josh Barnes and Piet Hut. A hierarchical o(n log n) force-calculation algorithm. Nature (London), 324 446-449, December 1986. 

Moving responsibility for the force computation away from the patches required a move away from pure message-driven execution to dependency-driven execution in which patches control the data (atomic coordinates) needed for compute objects to execute. A compute object, upon creation, registers this dependency with those patches from which it needs data. The patch then triggers force calculation by notifying its dependent compute objects when the next timestep s data is available. Once a compute object has received notification from all of the patches it depends on, it is placed in a prioritized queue for eventual execution. 

Molecular dynamics conceptually involves two phases, namely, the force calculations and the numerical integration of the equations of motion. In the first phase, force interactions among particles based on the negative gradient of the potential energy function U,... 

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

The force decomposition algorithm maps all possible interactions to processors and does not require inter-processor communication during the force calculation phase of MD simulation. However, to obtain the net force on each particle for the update phase would need global communication. In this section, we will present parallel algorithms based on force decomposition. 

For this algorithm, each processor is assigned atoms, so the force calculation time is O( ). Using the communication scheme mentioned above, each processor communicates with ( /P — 1) processors in each row and column. Thus the total number of terms being communicated per step is (-/P — 1)( ). Therefore, 0(N) CPU time is required in communicating the net force per step. Therefore,... 

Since this approach maps all possible interactions to processors, no communication is required during force calculation. Moreover, the row assignments are completed before the first step of the simulation. The computation of the bounds for each processor require O(P ) time, which is very negligible compared to N (for N S> P). The communication required at the end of each step to update the position and velocity vectors is done by reducing force vectors of length N, and therefore scales as 0 N) per node per time step. Thus the overall complexity of this algorithm is. 

FIG. 2 Interaction forces between glass surfaces upon compression in ethanol-cyclohexane mixtures. The dashed and solid lines represent the van der Waals force calculated using the nonretarded Hamarker constants of 3 X 10 1 for glass/cyclohexane/glass and 6 X 10 J for glass/ethanol glass, respectively. 

The main difficulty in the theoretical study of clusters of heavy atoms is that the number of electrons is large and grows rapidly with cluster size. Consequently, ab initio "brute force" calculations soon meet insuperable computational problems. To simplify the approach, conserving atomic concept as far as possible, it is useful to exploit the classical separation of the electrons into "core" and "valence" electrons and to treat explicitly only the wavefunction of the latter. A convenient way of doing so, without introducing empirical parameters, is provided by the use of generalyzed product function, in which the total electronic wave function is built up as antisymmetrized product of many group functions [2-6]. 

Examination of the NMR spectrum of thiane 3,3,5,5-d4 oxide enabled the estimation of the axial/equatorial equilibrium constant . The value was found to be 1.62 (at — 90 °C), corresponding to a free-energy difference of 0.175 kcal mol , which is in good agreement with field force calculations. ... 

Limitations It is desirable to have an estimate for the smallest particle size that can be effectively influenced by DEP. To do this, we consider the force on a particle due to DEP and also due to the osmotic pressure. This latter diffusional force will randomize the particles and tend to destroy the control by DEP. Eigure 20-31 shows a plot of these two forces, calculated for practical and representative conditions, as a function of particle radius. As we can see, the smallest particles that can be effectively handled by DEP appear to be in range of 0.01 to 0.1 pm (100 to 1000 A). 

Chipot, C. Kollman, P. A. Pearlman, D. A., Alternative approaches to potential of mean force calculations free energy perturbation versus thermodynamic integration. Case study of some representative nonpolar interactions, J. Comput. Chem. 1996,17, 1112-1131... 

Maragliano, L. Ferrario, M. Ciccotti, G., Effective binding force calculation in dimeric proteins, Mol. Simul. 2004, 30, 807-816... 

Hamaker constants can sometimes be calculated from refractive igdex data by the Lifshitz equations (8), but it now appears that Y values are closely related to refractive indices and are a direct measure of the Lifshitz attractions. In Equation 1 a correction factor f for "retardation" of dispersion forces is shown which can be determined from Figure 2, a graph of 1/f at various values of H and a as a function of Xj, the characteristic wavelength of the most energetic dispersion forces, calculable and tabulated in the literature (9). 

Hartsough, D. S. and Merz Jr., K. M. Potential of mean force calculations on the SN1 fragmentation of tert-butyl chloride, J.Phys. Chem., 99 (1995), 384-390... 

The combination of the transport current and magnetic field surroimd-ing superconductors creates stresses on the assembly through the generation of Lorentz forces. Calculation of coil stresses for final coil... 

FIGURE 14.22 Model-calculated percentage (a) increase in Os (zonal average) in July and (b) the corresponding contributions to instantaneous radiative forcing calculated as a function of latitude and altitude from preindustrial times to the present (adapted from Chalita et al., 1996). 

Table 14.4 summarizes the estimated total direct radiative forcing calculated for the period from preindustrial times to 1992 for C02, CH4, N20, and O, (IPCC, 1996). The estimate for CH4 includes the effects due to its impacts on tropospheric ozone levels or on stratospheric water vapor, both of which are generated during the oxidation of methane. That shown for 03 is based on the assumption that its concentration increased from 25 to 50 ppb over the Northern Flemi-sphere. The total radiative forcing due to the increase in these four gases from preindustrial times to the present is estimated to be 2.57 W m 2. 

Table 14.5 shows the direct radiative forcing calculated for some CFCs, HCFC-22, and some other chlorine-containing gases due to the increase in their atmospheric concentrations from preindustrial times (when the concentrations of most of them were zero) to 1992. The direct radiative forcing due to the two most commonly used CFCs in the past, CFC-11 and CFC-12 (see Chapter 12.C.1), is approximately 8% of the total radiative forcing due to C02, CH4, N20, and 03 (Table 14.4). 

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