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Implementation and Evaluation

We will consider an Ewald implementation which is a modified version of the ewald code written for Berend Smit's Molecular Simulation course webpage. This code simply computes the Ewald energy for a cubic lattice, given an appropriate number of particles, and a value for (which is called in the code), and a value for , the maximum integer index for enumerating -vectors. 5

The units used in a system with electrostatics differ depending on community. So far, we have assumed that the units of electrostatic potential are charge , divided by length , because we write potential as , where is measured in units of and distance in units of . Energy is therefore written in units of over , and force in units of over . If we want the final energy in more familiar units, we can choose and , and use the standard prefactor to convert from charge squared per length'' to energy''. For example, in SI units, (C /m)/J. In this implementation, we use a length of and and measure energy such that .

We will examine two configurations, both with = 8 = 512 particles, with alternating + and - charges. One configuration has the particle on a cubic lattice with lattice spacing , which is the standard NaCl crystal structure. We will call this the crystal'' configuration. The other is like the crystal, only each particle is displaced by a random amount from its lattice position with a maximum displacement of 0.3. We will call this the liquid'' configuration. We compute the total electrostatic energy via the Ewald sum technique for various values of and maximum -vector index. As we increase the number of -vectors taken in the sum, we would like to show that the total energy converges to a certain value. We will measure this in terms of the Madelung constant, : (341)

The following results were obtained for the crystal when :

alpha kmax u_real    u_fourier     u_self    u_total     M
1.20  2    -.206695  0.297020E-33  -0.67703  -0.88372    1.7674
1.20  3    -.206695  0.509667E-33  -0.67703  -0.88372    1.7674
1.20  4    -.206695  0.994016E-02  -0.67703  -0.87378    1.7476
1.20  5    -.206695  0.994016E-02  -0.67703  -0.87378    1.7476
1.20  6    -.206695  0.994016E-02  -0.67703  -0.87378    1.7476
This shows that, for the perfectly periodic crystal, very few -vectors are needed to reach a converged energy.

For the liquid, the results are somewhat different, again for :

alpha kmax u_real    u_fourier     u_self    u_total     M
1.20  2   -.244210   0.210535E-01  -0.67703  -0.90018    1.8004
1.20  3   -.244210   0.349887E-01  -0.67703  -0.88625    1.7725
1.20  4   -.244210   0.482931E-01  -0.67703  -0.87294    1.7459
1.20  5   -.244210   0.510164E-01  -0.67703  -0.87022    1.7404
1.20  6   -.244210   0.519170E-01  -0.67703  -0.86932    1.7386
1.20  7   -.244210   0.521388E-01  -0.67703  -0.86910    1.7382
1.20  8   -.244210   0.521741E-01  -0.67703  -0.86906    1.7381
1.20  9   -.244210   0.521802E-01  -0.67703  -0.86906    1.7381
1.20 10   -.244210   0.521808E-01  -0.67703  -0.86906    1.7381

Considering the converged results for various values of , we see that is not too sensitive to , once : Madelung constant vs. Ewald parameter for a system of 512 ions.
Note that, in principle, doesn't depend on . However, errors due to too low a choice for indicate that the Fourier sum is poorly converged, because of the factor .   Next: Ewald Summation: Suggested Exercises Up: Long-Range Interactions: The Ewald Previous: Ewald Forces
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