Source-linked AI summary
Efficient and accurate three dimensional Poisson solver for surface problems
Luigi Genovese, Thierry Deutsch, Stefan Goedecker
TL;DR
Surface problems require electrostatic potentials for systems periodic in two directions and free in the third. The paper presents a mixed-space Poisson solver that preserves these boundary conditions, with O(N log N) cost and reported accuracy advantages over reciprocal-space alternatives.
Problem
Surface systems are periodic in two dimensions but finite in the third, and existing surface solvers can require screening and large boxes for accurate results beyond the bulk region.
Source: §I · p. 1Method
The solver uses a mixed reciprocal-real-space representation, interpolating scaling functions in the non-periodic direction, and FFT-based operations with O(N log N) cost.
Source: §I · p. 2, §III · p. 3Results
The method is reported much more precise than the compared Poisson solvers; Mortensen accuracy is calculated only for bulk planes, 30% of the total volume, with errors that blow up outside that region.
Source: §IV · p. 4Takeaways & Limitations
The solver explicitly preserves the required surface boundary conditions and is intended for real-space methods using charge densities and potentials on a uniform grid.
Source: §III · p. 3, §IV · p. 5
Abstract
from arXiv · showhide
We present a method that gives highly accurate electrostatic potentials for systems where we have periodic boundary conditions in two spatial directions but free boundary conditions in the third direction. These boundary conditions are needed for all kind of surface problems. Our method has an O(N log N) computational cost, where N is the number of grid points, with a very small prefactor. This Poisson solver is primarily intended for real space methods where the charge density and the potential are given on a uniform grid.
I. INTRODUCTION
Surface systems are periodic in two directions and finite in the third, so their Poisson solvers must handle different boundary conditions by direction. This section motivates a mixed reciprocal-real-space solver that avoids screening approximations and is most efficient for localized densities in the non-periodic direction.
I. INTRODUCTION
- For periodic boundary conditions, reciprocal-space Poisson solving is rapid and simple, with FFT-based O(N log N) scaling in the number of grid points.The density is transformed to reciprocal space, the equation is solved there, and the potential is transformed back.Source: §I · p. 1
- Free-boundary solvers based on reciprocal-space methods retain O(N log N) scaling but have restrictions, including geometry constraints or numerically expensive large boxes.Large periodic volumes also leave long-range interactions between super-cells because of the inverse Laplacian's long-range behavior.Source: §I · p. 1
- Surface systems are periodic in two dimensions but finite in the third, a setting for which efficient Poisson solvers are important.Surface-specific experimental techniques have produced results that can benefit from theoretical prediction and analysis.Source: §I · p. 1
- Surface-specific reciprocal-space solvers use screening functions to subtract artificial super-cell interactions and may require large boxes for accuracy beyond the bulk region.The reported accuracy limitation and overhead are described for previously developed surface solvers.Source: §I · p. 1
- Integral-equation approaches avoid the tuning required by iterative differential-equation methods, such as multigrid.Multigrid requires tuning steps including smoothing, restriction, and prolongation.Source: §I · p. 2
- A mixed reciprocal-real-space solver handles surface boundary conditions by direction without screening approximations; it is most efficient for localized densities along the non-periodic direction.The charge density in that direction is represented with interpolating scaling functions.Source: §I · p. 2
II. INTERPOLATING SCALING FUNCTIONS
Interpolating scaling functions form a grid-based basis with compact support and coefficients given by grid values. Their refinement relation connects representations on grids of spacing h and h/2.
II. INTERPOLATING SCALING FUNCTIONS
- The full basis set consists of translations of the mother function φ by grid spacing h.Source: §II · p. 2
- The refinement relation connects scaling functions on grids with spacing h and h/2.The filter elements h_j characterize the wavelet family, and m is the scaling-function order.Source: §II · p. 2
- The mother function is symmetric with compact support from −m to m, equals one at the origin and zero at other integer grid points, and yields coefficients equal to function values on the grid.Source: §II · p. 2
III. POISSON EQUATION FOR SURFACES BC
The surface problem uses periodic boundaries in x and y and free boundaries in z, with a mixed representation designed to preserve the charge density’s relevant descriptors. A kernel-based FFT procedure solves the discretized Poisson problem, while recursive kernel evaluation supports accuracy and efficiency.
III. POISSON EQUATION FOR SURFACES BC
- Periodic x,y and free z boundaries are handled with plane waves in the periodic directions and interpolating scaling functions along z.The density is represented on a grid in the non-periodic direction.Source: §III · p. 2
- The mixed grid representation preserves the discrete density’s multipoles and Fourier components as those of its continuous representation, allowing grid-valued input.The stated equivalence holds for ℓ1 < Nx, ℓ2 < Ny, and ℓ3 < m.Source: §III · pp. 2–3
- O(N log N) is the total computational cost of the two-dimensional FFT, zero-padded convolution solve, and inverse transform.Real-to-complex FFTs can reduce memory and operations because all quantities are real.Source: §III · p. 3
- Recursive kernel evaluation replaces Nz+1 integrals of range 2m with 2 integrals of range m and 4m integrals of range 1, using smooth integrands.The text contrasts this with inefficient trapezoidal integration around the non-smooth point z = 0.Source: §III · pp. 3–4
- Refinement improves high-λ kernel accuracy, and Fourier-space kernel evaluation takes roughly half the time needed for applying the kernel.For high λ, the kernel is evaluated at lower λ on an enlarged domain before applying the refinement relation.Source: §III · p. 4
IV. NUMERICAL RESULTS AND COMPARISON WITH OTHER METHODS
The method’s error decreases algebraically as h^m, with m set by the interpolating scaling-function order, and it explicitly preserves surface boundary conditions. Comparisons report greater precision than reciprocal-space solvers and accurate, physically sound plane-capacitor potentials; the method has O(N log N) scaling.
IV. NUMERICAL RESULTS AND COMPARISON WITH OTHER METHODS
- The error decreases algebraically as h^m, with m set by the interpolating scaling-function order; the periodic-plane representation converges exponentially faster.Tests use an analytical charge distribution whose potential is localized in the non-periodic direction and goes explicitly to zero at the borders; other analytical functions gave comparable accuracies.Source: §IV · p. 4, Figure 1 · p. 4
- Reciprocal-space solvers are much less precise; Mortensen accuracy is reported only in the central 30% of the non-periodic volume, where errors outside that region blow up.Accurate results across a sufficiently large non-periodic domain would require a very large computational volume for Mortensen.Source: §IV · p. 4, Figure 1 · p. 4
- For a plane capacitor, only this method gives both accurate and physically sound potentials with the surface boundary conditions preserved.Mortensen produces a fully periodic solution in this case; adding a suitable linear function along the non-periodic direction can give it zero electric field at the borders, while Hockney is qualitatively better but inaccurate.Source: §IV · p. 4, Figure 2 · p. 5
- The method obtains accurate surface potentials with O(N log N) scaling and explicitly preserves the required boundary conditions.The conclusion describes it as suitable for applications inside electronic-structure codes.Source: §IV · p. 5
- The reported 128^3-grid timings exclude kernel setup, which takes around 50% of total time; communication to gather the complete potential dominates at large processor counts.The timings are for a Cray XT3 and vary with processor count; the supplied passage gives no individual elapsed-time values.Source: §IV · p. 4, Table I · p. 5