Integration procedures Verlet

In our search for a suitable algorithm, we tested four different integration procedures. The general analysis leading to each of these procedures is discussed by Ralston and Wilf.47 The first we tried was used by Verlet in his study of liquid argon.44 It replaces Eq. (A.I) and (A.2) by... 

This procedure is then repeated after each time step. Comparison with Eq. (2) shows that the result is the velocity Verlet integrator and we have thus derived it from a split-operator technique which is not the way that it was originally derived. A simple interchange of the Ly and L2 operators yields an entirely equivalent integrator. 

Figure 1 A stepwise view of the Verlet integration algorithm and its variants, (a) The basic Verlet method, (b) Leap-frog integration, (c) Velocity Verlet integration. At each algorithm dark and light gray cells indicate the initial and calculating variables, respectively. The numbers in the cells represent the orders m the calculation procedures. The arrows point from the data that are used in the calculation of the variable that is being calculated at each step.

The use of Verlet and Singer s algorithm makes it necessary to use extra care in integrating the equation of motion. Ciccotti et aL have shown how to do it in the case of Verlet s algorithm. As to the rotational equation of motion, we followed a similar procedure using the quantities... 

The non-adiabatic quantum simulation procedures we employ have been well described previously in the literature, so we describe them only briefly here. The model system consists of 200 classical SPC flexible water molecules," and one quantum mechanical electron interacting with the water molecules via a pseudopotential. 2 The equations of motion were integrated using the Verlet algorithm with a 1 fs time step in the microcanonical ensemble, and the adiabatic eigenstates at each time step were calculated with an iterative and block Lanczos scheme. Periodic boundary conditions were employed using a cubic simulation box of side 18.17A (water density 0.997 g/ml). 

The careful reader should have realized that we choose not to break up this operator with another Trotter factorization, as was done for the extended system case. In practice, one does not multiple-time-step the modified velocity Verlet algorithm because it will, in general, have a unit Jacobian. Thus, one would like the best representation of the operator that can be obtained in closed form. However, even in the case of a modified velocity Verlet operator that has a nonunit Jacobian, multiple-time-stepping this procedure can be costly because of the multiple force evaluations. Generally, if the integrator is stable without multiple-time-step procedures, avoid them. The solution to this first-order inhomogeneous differential equation is standard and can be found in texts on differential equations (see, e.g.. Ref. 53). 

FIGURE 6.11 TCFIs for (a) water/water, (b) water/t-butanol, and (c) t-butanol/t-butanol obtained from simulation of water/t-butanol mixtures using the Verlet method (crosses) versus the water mole fraction Xj, compared with TCFIs obtained from experimental data nsing the Wooley/O Connell procedure, where either the Wilson (black line), NRTL (red Une), or mM (green line) models were employed for obtaining the activity coefficient derivatives. Note that the NRTL and mM model approaches infinity since they predict a phase split. (Calculated values from R. J. Wooley and J. P. O Connell, 1991, A Database of Flnctuation Thermodynamic Properties and Molecular Correlation-Function Integrals for a Variety of Binary Liquids, Fluid Phase Equilibria, 66, 233.) (See color insert.)... 

I think it is fair to say that the merits and demerits of DPD are still debated. In my opinion, the DPD technique does have a problem with the hydrodynamics, which relaxes in the same time and distance scale as the dissolved particles. In reality, because of the near incompressibility of the solvent, the hydrodynamics relaxes essentially instantaneously on that particle s timescale of structural evolution. One other problem of the technique, as pointed out by Marsh and Yeomans, is that the temperature of the system depends on the value of the time step (as the dissipative force is inversely proportional to the square root of the time step). In an interesting article, Lowe looked at DPD from the perspective of another thermostatting procedure, but which conserves momentum and enhances the viscosity. Besold et al. examined the various integration schemes used in DPD and found differences in the response fimctions and transport coefficients. These artefacts can be largely suppressed by using velocity-Verlet-based schemes in which the velocity dependence of the dissipative forces is taken into account. 

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