b Instrucciones de Diseño para la Ordenanza 6.

Since we have to perform simulations of systems with a huge number of particles, we have to think about acceleration methods for our simulation program. The computing time of the original Verlet algorithm scales as N2_{, because the separations of all particle}

pairs must be calculated for every time step in the minimum image convention. Only after doing this, all interactions whose separation exceeds rcut can be neglected. Verlet

3.9 Acceleration methods: cell and neighbor list 51

Figure 3.3:The Verlet neighbor list: The center particle (circle) is only interacting with the particles inside the cut off sphere with radius rcut(triangles). All particles within the

list sphere with radius rlistare stored in the neighbor list (diamonds + triangles) and

the separations are calculated every time step. The distances to all other particles (stars) are only calculated for setting up and updating the neighbor list.

proposed a method to decrease the computer time by storing all particles with positions inside or close to the cut off sphere of the center particle in an array [6], the so called neighbor list. In practice the cut off sphere of every particle is surrounded by an additional sphere with radius rlist > rcut, shown in figure 3.3. For setting up the list at the start

of the simulation, we have to calculate all particle separations like for the pure Verlet algorithm. The simulation can now run for some steps by calculating only the separations of the neighboring particles, since only these particles are able to enter the cut off sphere. If the number of neighbors of one particle is small, compared to the total particle number, this method reduces the computation effort to order N . After a definite time particles from outside the list sphere will be able to enter the cut off sphere and the neighbor list has to be updated. For this reason the two largest displacements of every time step are stored and as soon as the sum of these exceeds the distance rlist− rcut, the neighbor

list is updated. The optimal choice of rlist− rcut is dependend on almost all simulation

parameters. We abstain from determining the optimal list radius for every simulation, as our experience shows a choice of rlist− rcut = 0.5 is always close to the optimum.

The neighbor list method works well for simulation boxes bigger than the cut off sphere, but still of same magnitude. For huge systems of which the half box length L/2 exceeds rcut several times the set-up of the neighbor list becomes predominant and the computing

52 3 Molecular dynamics simulations

Figure 3.4:Combined cell and neighbor list: The simulation box is divided into cells at least of size rlist. For setting up the neighbor list, only particles in the same or neighboring

cells of the considered particle have to be respected (shaded region), the diamonds outside can be neglected.

acceleration method.

Auerbach et al. proposed to combine the Verlet neighbor list with a cell list [29]. This allows to use it even for very huge systems and reduces the computing time to order N again. A schematic picture in two dimensions is given in figure 3.4. If the length of the simulation box exceeds at least four times rlist, the box is divided into cells with size equal

to or slightly larger than the list radius rlist. Before setting up the neighbor list, we store

the cell number for each particle in an additional array, so we have only to calculate the separations of particles within the same or neighboring cell (shaded cells). So we can save computer time neglecting the calculations of separations to particles in the non-shaded region from the outset, since the computational effort for setting up the cell list is of order N , too. This method is not only useful for systems with huge particle numbers, for very low densities this saves computer time even for smaller particle numbers.

Bibliography

[1] M.P. Allen and D.J. Tildesley. Computer Simulation of Liquids. Oxford University Press, 1989.

[2] K. Binder and G. Ciccotti, editors. Monte Carlo and Molecular Dynamics of Con- densed Matter Systems, volume 49 of Conference Proceedings. Italian Physical Society, July 1995.

[3] D.C. Rapaport. The Art of Molecular Dynamics Simulation. Cambridge University Press, 2004.

[4] R. Hentschke, E.M. Aydt, B. Fodi, and E. St¨ockelmann. Molekulares Modellieren mit Kraftfeldern. http://constanze.materials.uni-wuppertal.de.

[5] C.W. Gear. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall Series in Automatic Computation, Engelwood Cliffs, 1971.

[6] L. Verlet. Computer ”experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Physical Review, 159(1):98–103, 1967.

[7] W. C. Swope, H. C. Andersen, P. H. Berens, and R. Wilson. A computer simulation method for the calculation of equilibrium constants for the formation of physical clus- ters of molecules: Application to small water clusters. Journal of Chemical Physics, 76(1):637–649, 1982.

[8] K. Singer. Thermodynamic and structural properties of liquids modelled by 2- Lennard-Jones centres pair potentials. Molecular Physics, 33(6):1757–1795, 1977. [9] G.R. Luckhurst and P. Simpson. Computer simulation studies of anisotropic sys-

tems. VIII. The Lebwohl-Lasher model of nematogens revisited. Molecular Physics, 47(2):251–265, 1982.

[10] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21(6):1087–1092, 1953.

[11] S. C. Harvey, R. K.-Z. Tan, and T. E. Cheatham. The flying ice cube: Velocity rescaling in molecular dynamics leads to violation of energy equipartition. Journal of Computational Chemistry, 19(7):726–740, 1998.

[12] F.J. Vesely. N-particle dynamics of polarizable Stockmayer-type molecules. Journal of Computational Physics, 24:361–371, 1976.

54 Bibliography

[13] H.J.C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, A. DiNola, and J.R. Haak. Molecular dynamics with coupling to an external bath. Journal of Chemical Physics, 81(8):3684–3690, 1984.

[14] H. Fr¨ohlich. Theory of Dielectrics. Clarendon Press, 1949.

[15] J.A. Barker and R.O. Watts. Monte Carlo studies of the dielectric properties of water-like models. Molecular Physics, 26(3):789–792, 1973.

[16] L. Onsager. Electric moments of molecules in liquids. Journal of the American Chemical Society, 58:1486–1493, 1936.

[17] C. Millot, J.-C. Soetens, and M.T.C. Martins Costa. Static dielectric constant of the polarizable Stockmayer fluid. Comparison of the lattice summation and reaction field methods. Molecular Simulation, 18:367–383, 1997.

[18] M. Valisk´o, D. Boda, J. Liszi, and I. Szalai. A systematic Monte Carlo simulation and renormalized perturbation theoretical study of the dielectric constant of the polarizable Stockmayer fluid. Molecular Physics, 101(14):2309–2313, 2003.

[19] M. Valisk´o, D. Boda, J. Liszi, and I. Szalai. The dielectric constant of polarizable fluids from the renormalized perturbation theory. Molecular Physics, 100(20):3239– 3243, 2002.

[20] M. Valisk´o, D. Boda, J. Liszi, and I. Szalai. Relative permittivity of dipolar liq- uids and their mixtures. Comparison of theory and experiment. Physical Chemistry Chemical Physics, 3:2995 – 3000, 2001.

[21] P. Ewald. Die Berechnung optischer und elektrostatischer Gitterpotentiale. Annalen der Physik, 369:253–287, 1921.

[22] J.P. Valleau. The problem of coulombic forces in computer simulation. volume 9 of NRCC Workshop Proceedings, pages 3–8, 1980.

[23] C.G. Gray, Y.S. Sainger, C.G. Joslin, P.T. Cummings, and S.Goldman. Computer simulation of dipolar fluids. Dependence of the dielectric constant on system size: A comparative study of Ewald sum and reaction field approaches. Journal of Chemical Physics, 85(3):1502–1504, 1986.

[24] M. Neumann. Dipole moment fluctuation in computer simulations of polar systems. Molecular Physics, 50(4):841–858, 1983.

[25] M. Neumann and O. Steinhauser. The influence of boundary conditions used in machine simulations on the structure of polar systems. Molecular Physics, 39(2):437– 454, 1980.

[26] M. Neumann. Dielectric properties and the convergence of multipolar lattice sums. Molecular Physics, 60:225–235, 1987.

[27] M. Neumann, O. Steinhauser, and G.S. Pawley. Consistent calculation of the static and frequency-dependent dielectric constant in computer simulations. Molecular Physics, 52(1):97–113, 1984.

Bibliography 55

[28] B. Garz´on, S. Lago, and C. Vega. Reaction field simulations of the vapor-liquid equi- libria of dipolar fluids. Does the reaction field dielectric constant affect the coexistence properties? Chemical Physics Letters, 231(4-6):366–372, 1994.

[29] D. J. Auerbach, W. Paul, A. F. Bakker, C. Lutz, W. E. Rudge, and F. F. Abraham. A special purpose parallel computer for molecular dynamics: Motivation, design, implementation, and application. Journal of Physical Chemistry, 91:4881–4890, 1987.

4 The Stockmayer fluid in literature:

Theory and computer simulation

In this chapter we give an overview over the literature published on the ST fluid. We report both theoretical works and results of computer simulation. Notice that the ST fluid has been investigated for more than 65 years. Therefore our main intention is not completeness with respect of the published works, but rather with respect to the different scientific aspects.