Ewald summation
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g. electrostatic interactions) in periodic systems. It was first developed as the method for calculating the electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge neutrality of the molecular system to accurately calculate the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.[1]
Derivation[edit]
Ewald summation rewrites the interaction potential as the sum of two terms,
The long-range interaction energy is the sum of interaction energies between the charges of a central unit cell and all the charges of the lattice. Hence, it can be represented as a double integral over two charge density fields representing the fields of the unit cell and the crystal lattice
Here, is the Dirac delta function, , and are the lattice vectors and , and range over all integers. The total field can be represented as a convolution of with a lattice function
Since this is a convolution, the Fourier transformation of is a product
For brevity, define an effective single-particle potential
Since this is also a convolution, the Fourier transformation of the same equation is a product
The energy can now be written as a single field integral
Using Plancherel theorem, the energy can also be summed in Fourier space
where in the final summation.
This is the essential result. Once is calculated, the summation/integration over is straightforward and should converge quickly. The most common reason for lack of convergence is a poorly defined unit cell, which must be charge neutral to avoid infinite sums.
Particle mesh Ewald (PME) method[edit]
Ewald summation was developed as a method in theoretical physics, long before the advent of computers. However, the Ewald method has enjoyed widespread use since the 1970s in computer simulations of particle systems, especially those whose particles interact via an inverse square force law such as gravity or electrostatics. Recently, PME has also been used to calculate the part of the Lennard-Jones potential in order to eliminate artifacts due to truncation.[2] Applications include simulations of plasmas, galaxies and molecules.
In the particle mesh method, just as in standard Ewald summation, the generic interaction potential is separated into two terms . The basic idea of particle mesh Ewald summation is to replace the direct summation of interaction energies between point particles
where and represent the Fourier transforms of the potential and the charge density (this is the Ewald part). Since both summations converge quickly in their respective spaces (real and Fourier), they may be truncated with little loss of accuracy and great improvement in required computational time. To evaluate the Fourier transform of the charge density field efficiently, one uses the fast Fourier transform, which requires that the density field be evaluated on a discrete lattice in space (this is the mesh part).
Due to the periodicity assumption implicit in Ewald summation, applications of the PME method to physical systems require the imposition of periodic symmetry. Thus, the method is best suited to systems that can be simulated as infinite in spatial extent. In molecular dynamics simulations this is normally accomplished by deliberately constructing a charge-neutral unit cell that can be infinitely "tiled" to form images; however, to properly account for the effects of this approximation, these images are reincorporated back into the original simulation cell. The overall effect is called a periodic boundary condition. To visualize this most clearly, think of a unit cube; the upper face is effectively in contact with the lower face, the right with the left face, and the front with the back face. As a result, the unit cell size must be carefully chosen to be large enough to avoid improper motion correlations between two faces "in contact", but still small enough to be computationally feasible. The definition of the cutoff between short- and long-range interactions can also introduce artifacts.
The restriction of the density field to a mesh makes the PME method more efficient for systems with "smooth" variations in density, or continuous potential functions. Localized systems or those with large fluctuations in density may be treated more efficiently with the fast multipole method of Greengard and Rokhlin.
Dipole term[edit]
The electrostatic energy of a polar crystal (i.e. a crystal with a net dipole in the unit cell) is conditionally convergent, i.e. depends on the order of the summation. For example, if the dipole-dipole interactions of a central unit cell with unit cells located on an ever-increasing cube, the energy converges to a different value than if the interaction energies had been summed spherically. Roughly speaking, this conditional convergence arises because (1) the number of interacting dipoles on a shell of radius grows like ; (2) the strength of a single dipole-dipole interaction falls like ; and (3) the mathematical summation diverges.
This somewhat surprising result can be reconciled with the finite energy of real crystals because such crystals are not infinite, i.e. have a particular boundary. More specifically, the boundary of a polar crystal has an effective surface charge density on its surface where is the surface normal vector and represents the net dipole moment per volume. The interaction energy of the dipole in a central unit cell with that surface charge density can be written[3]
History[edit]
The Ewald summation was developed by Paul Peter Ewald in 1921 (see References below) to determine the electrostatic energy (and, hence, the Madelung constant) of ionic crystals.
Scaling[edit]
Generally, different Ewald summation methods give different time complexities. Direct calculation gives , where is the number of atoms in the system. The PME method gives .[4]
See also[edit]
References[edit]
- ^ Kolafa, Jiri; Perram, John W. (September 1992). "Cutoff Errors in the Ewald Summation Formulae for Point Charge Systems". Molecular Simulation. 9 (5): 351–368. doi:10.1080/08927029208049126.
- ^ Di Pierro, M.; Elber, R.; Leimkuhler, B. (2015), "A Stochastic Algorithm for the Isobaric-Isothermal Ensemble with Ewald Summations for all Long Range Forces.", Journal of Chemical Theory and Computation, 11 (12): 5624–5637, doi:10.1021/acs.jctc.5b00648, PMC 4890727, PMID 26616351
- ^ Herce, HD; Garcia, AE; Darden, T (28 March 2007). "The electrostatic surface term: (I) periodic systems". The Journal of Chemical Physics. 126 (12): 124106. Bibcode:2007JChPh.126l4106H. doi:10.1063/1.2714527. PMID 17411107.
- ^ Darden, Tom; York, Darrin; Pedersen, Lee (1993-06-15). "Particle mesh Ewald: An N ⋅log( N ) method for Ewald sums in large systems". The Journal of Chemical Physics. 98 (12): 10089–10092. Bibcode:1993JChPh..9810089D. doi:10.1063/1.464397. ISSN 0021-9606.
- Ewald, P (1921). "Die Berechnung optischer und elektrostatischer Gitterpotentiale". Ann. Phys. 369 (3): 253–287. Bibcode:1921AnP...369..253E. doi:10.1002/andp.19213690304.
- Darden, T; Perera, L; Li, L; Pedersen, L (1999). "New tricks for modelers from the crystallography toolkit: the particle mesh Ewald algorithm and its use in nucleic acid simulations". Structure. 7 (3): R55–R60. doi:10.1016/S0969-2126(99)80033-1. PMID 10368306. S2CID 40964921.
- Frenkel, D., & Smit, B. (2001). Understanding molecular simulation: from algorithms to applications, Academic press.