Extension and Simple Proof of Lekner's Summation Formula for Coulomb Forces

Sperb, René P.

doi:10.1080/08927029408021982

Cited by 37 publications

(32 citation statements)

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“…The calculation of the sum over the Bernoulli numbers can be accelerated by making the cutoff depend on u y ρ using the error formula (20).…”

Section: Program 43 (Calculation Of the Near Formula)mentioning

confidence: 99%

“…It employs modified Bessel functions, but is only useful for particles separated in the z-direction, and is only calculable pairwise. Sperb [20] derived a different formula, which is also applicable for particles lying in the same z-plane. A combination of theses two formula has been used recently in MD simulations, because it is easy to implement [21] for small number of particles, but suffers the drawback of being an O(N 2 ) method, and therefore the system sizes which can be investigated are limited.…”

Section: Introductionmentioning

confidence: 99%

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MMM2D: A fast and accurate summation method for electrostatic interactions in 2D slab geometries

Arnold

Holm

2002

Computer Physics Communications

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We present a new method, in the following called MMM2D, to accurately calculate the electrostatic energy and forces on charges being distributed in a two dimensional periodic array of finite thickness. It is not based on an Ewald summation method and as such does not require any fine-tuning of an Ewald parameter for convergence. We transform the Coulomb sum via a convergence factor into a series of fast decaying functions which can be easily evaluated. Rigorous error bounds for the energies and the forces are derived and numerically verified. Already for small systems our method is much faster than the traditional 2D-Ewald methods, but for large systems it is clearly superior because its time demand scales like O(N 5/3 ) with the number N of charges considered. Moreover it shows a rapid convergence, is very precise and easy to handle.

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“…The calculation of the sum over the Bernoulli numbers can be accelerated by making the cutoff depend on u y ρ using the error formula (20).…”

Section: Program 43 (Calculation Of the Near Formula)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

MMM2D: A fast and accurate summation method for electrostatic interactions in 2D slab geometries

Arnold

Holm

2002

Computer Physics Communications

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“…(8), then an alternative expression which converges faster than the sums over the Bessel function, for the configurations leading to a slow convergence behaviour is derived and used as an alternative to the slow convergence [8,9] ; with this method, some cautions are needed to take properly into account the electroneutrality of the system [10].…”

Section: Introductionmentioning

confidence: 99%

“…This last property of Lekner sums introduces some complications in numerical implementations of Lekner sums. There are mainly two methods to improve the convergence rate of these summations : the Lekner-cyclic [4,7] and Lekner-Sperb [8,9,10] methods.…”

Section: Introductionmentioning

confidence: 99%

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Yukawa potentials in systems with partial periodic boundary conditions. II. Lekner sums for quasi-two-dimensional systems

Mazars

2007

Molecular Physics

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Yukawa potentials may be long ranged when the Debye screening length is large. In computer simulations, such long ranged potentials have to be taken into account with convenient algorithms to avoid systematic bias in the sampling of the phase space. Recently, we have provided Ewald sums for quasi-two dimensional systems with Yukawa interaction potentials [M. Mazars, J. Chem. Phys., 126, 056101 (2007) and M. Mazars, Mol. Phys., Paper I]. Sometimes, Lekner sums are used as an alternative to Ewald sums for Coulomb systems. In the present work, we derive the Lekner sums for quasi-two dimensional systems with Yukawa interaction potentials and we give some numerical tests for pratical implementations. The main result of this paper is to outline that Lekner sums cannot be considered as an alternative to Ewald sums for Yukawa potentials. As a conclusion to this work : Lekner sums should not be used for quasi-two dimensional systems with Yukawa interaction potentials.

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Crystal Structure

Enkelmann¹

1998

Electronic Materials: The Oligomer Approach

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Extension and Simple Proof of Lekner's Summation Formula for Coulomb Forces

Abstract: A summation formula is proven for a lattice sum occurring frequently in molecular dynamics calculations. It has a much faster convergence rate than the original sum. An important application is in the case of Coulomb forces.

Cited by 37 publications

References 1 publication

MMM2D: A fast and accurate summation method for electrostatic interactions in 2D slab geometries

MMM2D: A fast and accurate summation method for electrostatic interactions in 2D slab geometries

Yukawa potentials in systems with partial periodic boundary conditions. II. Lekner sums for quasi-two-dimensional systems

Crystal Structure

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