Total potential and split potential for r −6. When not using the Ewald sum, the whole area under the blue curve is solved in real space, whereas only the area under the green curve is solved in real space when using the Ewald sum. The error in the calculation is related to the area under the curves beyond the cutoff. Using the Ewald technique is thus more accurate.
Real space error in the forces measured and estimated with Eq. (20) for (top) bulk system and (bottom) interfacial system. Measured errors are depicted as solid lines, estimated errors as dashed lines. From top to bottom, the real space cutoff is (blue) 2.0σ, (green) 3.0σ, and (red) 4.0σ. The estimate works well for the bulk system, but fails for the interfacial system.
Measured and estimated reciprocal space errors for (top) a bulk system and (bottom) an interfacial system. The graphs on left show increasing interpolation order from P = 3 at the top to P = 6 at the bottom, with a fixed grid containing 32 grid points in each direction. The graphs on the right show grid density increasing from 22 points in each direction at the top to 26 points at the bottom, for fixed interpolation order P = 5. Measured errors are depicted as solid lines. Errors estimated with Eqs. (21) and (24) are depicted as dashed lines, those estimated with Eqs. (21) and (22) are depicted as dotted lines.
Scaling of the Ewald sum and the PPPM method for dispersion interactions.
Surface tension and density of hexane as a function of the total error in the calculated forces. The arrows point in the direction of increasing Ewald parameter.
Surface tension, density, and errors in the forces in simulations of hexane. In the lower left graph, the red squares correspond to the real space error, while the triangles and circles correspond to the reciprocal space error when using the fine grid and coarse grid, respectively. The circles in the lower right graph correspond to the reciprocal space error. The Ewald parameter is β = 0.28 Å−1 in all figures on the right side. Dotted lines are error estimates calculated with Eqs. (20)–(22).
Measured density profiles for simulations of LJ particles: (top) simulations with cutoff r c = 2.5σ and (bottom) simulations at a reduced temperature of T* = 1.1.
Surface tensions and densities simulated when using a PPPM or a plain cutoff for dispersion. Results obtained when not using a long-range solver strongly depend on the chosen cutoff and approach the results of the simulation with the PPPM with increasing cutoff size.
Coefficients required for the calculation of the reciprocal space error estimate. Reprinted with permission from M. Deserno and C. Holm, J. Chem. Phys. 109, 7694 (1998). Copyright 1998 American Institute of Physics.
Results of the validation runs for the LJ particles. Uncertainties given in parentheses.
Results of the validation runs for the SPC/E water. Simulation marked with a dagger was run with higher precision.
Results of the validation runs for the OPLS hexane.
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