Ewald Sums

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Nonbond Energies of Periodic Systems

The Ewald technique (Tosi 1964) is a method for computation of nonbond energies of periodic systems. Crystalline solids are the most appropriate candidates for Ewald summation, but the technique can also be applied (with less confidence) to amorphous solids and solutions.

Figure 2-10 depicts the electrostatic energy for quartz as computed by various techniques supported by the Discover program. If life were simple, all the techniques would converge to the same value at high cutoff distances. Note, however, that the direct atom-based approach yields results that fluctuate wildly as the cutoff increases, even for rather large cutoffs. The problem is that the sum is only conditionally convergent. As the cutoff increases, charges of opposite sign are taken into account and the partial sum is modified significantly. Worse, reordering the terms of a conditionally convergent series can yield arbitrary results. The trick then is to find physically and chemically meaningful orderings of the series. The cell-based and group-based techniques provided by the Discover program are natural candidates. Unfortunately, as Figure 2-10 indicates, they yield somewhat different values. This is due to the different cutoff conventions employed. The group-based technique computes the result for a sphere, while the cell-based technique computes the result for a parallelepiped that preserves the shape of the unit cell. A standard Ewald calculation that does not take the dipole moment of the unit cell into account yields yet another value. Fortunately, an Ewald calculation that includes the effect of the dipole moment agrees with the group-based calculation.

For van der Waals energy, the energy sum is absolutely convergent, and no chaotic behavior arises from the direct approach. Even so, as Figure 2-11 indicates, the convergence of the dispersive energy is slower than might be expected. Even with a cutoff distance of 30 Å, the error is a significant fraction of 0.1 kcal mol-1. (Note that the Ewald calculation is less costly for comparable accuracy.) The repulsive energy, on the other hand, converges at a cutoff distance of only 15 Å and needs no special treatment. (Atom-based calculations for much larger systems, however, show that sometimes even the repulsive error can exhibit a surprisingly high error at a cutoff of 12 Å.)

Theory of Ewald Technique

The Ewald approach to improving convergence is to multiply a general lattice sum:

Eq. 2-10:
by a convergence function (r), which decreases rapidly with r. Of course, to preserve equality, one must then add a term equal to the product of 1 - (r) with the lattice sum:

Eq. 2-11:
Here, the first term converges quickly, because m(r) decreases rapidly. Ewald's insight was that the second term can be Fourier transformed to provide a rapidly converging sum over the reciprocal lattice. The sum over L in Eq. 2-11 runs over all lattice vectors, but the i = j terms must be omitted when L = 0.

The convergence functions used by the Discover program are, for the electrostatic energy:

Eq. 2-12:
and for the dispersive energy:

Eq. 2-13:
The electrostatic convergence function 1 was also used by Catlow and Norgett (1976) and Karasawa and Goddard (1989). The dispersive convergence function 6 was recommended and used by Karasawa and Goddard. The convergence parameter plays a similar role in both cases. As increases, the real-space sum converges more rapidly and the reciprocal space sum converges more slowly. (That is, a large implies a heavy computational load for reciprocal space, and a small implies a heavy computational load for real space.) Cutoffs must be adjusted accordingly, and processing time is affected by the cutoffs. A value of that balances processing in the real and reciprocal spaces proves to be optimal. The same value of can be used for both the dispersive and electrostatic energy, and thus they can be combined for greater efficiency. The Discover program automatically chooses so as to balance the computational loads for real and reciprocal space.

The Ewald energy expression for the electrostatic energy is:

Eq. 2-14:
where the prime means that the sums are for all h except h = 0; a = | ri - rj - RL |; b = h/2; h = | h |; = det(H) = cell volume; ri = Hsi; and h = 2(HT)-1n (reciprocal lattice vectors).

In Eq. 2-14, the first term corresponds to the real-space sum, and the second term corresponds to the reciprocal-space sum. The similarity between the Ewald real-space sums and the full real-space sums facilitates the use of existing Discover nonbond machinery for Ewald calculations. The last term in Eq. 2-14 arises from the exclusion of the i = j terms when L = 0, since the reciprocal space sum includes these terms. The third term in Eq. 2-14 corresponds to h = 0 in the second term, while the fourth term is from exclusion of the i = j terms when L = 0. Note that the reciprocal-space term of Eq. 2-14 involves a summation over i and j. If the geometric combination rule -Bij = sqrt(Bii Bjj ) holds, then the sum over i and j can be reduced to an analogous sum over i. This provides a substantial performance improvement (N 1.5 instead of N 2.5, where N is the number of atoms per unit cell).

Most implementations of the electrostatic Ewald drop the h = 0 term in the sum over h in Eq. 2-14. At first glance it appears to be 0 because of the neutrality of the unit cell. More careful inspection reveals that it is not always 0, and its value depends upon how limits are taken (Deem et al. 1990). Thus, the Discover 95.0 program assumes a spherical cutoff, to include a correction for this term.

Accuracy of Ewald Calculations

The Ewald method allows you to select, before running the calculation, a level of accuracy for the calculation. An Ewald calculation with accuracy = 1 e-4 is comparable in performance to an atom-based calculation with a large cutoff (19 Å) over the range that has been tested. It should also be noted that the Ewald results are significantly more accurate. Ewald processing time grows as N 1.5, where N is the number of atoms in the unit cell. In addition, increases in accuracy do not require unreasonable increases in the Ewald lattice cutoff. For acetic acid, for example, increasing the accuracy by 2 orders of magnitude (from 1 e-2 to 1 e-4 ), with constant repulsive cutoff, increased processing time only about 1.5 fold (from 22.08 to 35.34 seconds) and increased the Ewald lattice cutoff less than 20% (from 11.7 to 13.4 Å).

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