Exact treatment of Poisson’s equation in solids with space-filling cells

Gonis, A.; Sowa, Erik C.; Sterne, P. A.

doi:10.1103/physrevlett.66.2207

Cited by 30 publications

(24 citation statements)

References 6 publications

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“…which is not fulfilled in the so-called moon region between the near VP cells, 2,3,9 shown by light (pink) shading in Fig. 1, or, in other words, the complement of the VP and its bounding sphere with radius r BS .…”

Section: A Computationally Efficient and Accurate Poisson Solvermentioning

confidence: 98%

“…In general, the present method is at least 10(l int + 1) 2 N VP times faster than any of the existing schemes. 2,3,8 The factor (l int + 1) 2 comes from an additional internal L sum (typically l int max = 6l ext max ), and the factor 10 is from use of isoparametric integration versus shape functions, if used. In particular, l ext ∼ 8-10 will provide ∼10 4 N VP speedup for a system with N VP sublattices.…”

Section: General Commentsmentioning

confidence: 99%

“…whereρ (R) is a truncated density centered at site R. Computational time in most methods [1][2][3][4][5][6][7][8][9] arises from the use of L ≡ {l,m} multipole [spherical-harmonics Y L ( r )] expansions. Evaluation of intercell potential [term two in Eq.…”

Section: Introductionmentioning

confidence: 99%

“…III, attempting to separate two of three (r,r ,R) degrees of freedom. In most existing methods, [1][2][3][4][5][6][7][8][9] an additional multipole expansion of Y L ( r + R) is performed yielding conditionally convergent nested L sums (internal vs external: l int max > 3l ext max ) due to the nearest-neighbor sites, and relevant in the light-shaded (pink) region in Fig. 1.…”

Section: Introductionmentioning

confidence: 99%

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Accurate and fast numerical solution of Poisson's equation for arbitrary, space-filling Voronoi polyhedra: Near-field corrections revisited

2011

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We present an accurate and rapid solution of Poisson's equation for space-filling, arbitrarily shaped, convex Voronoi polyhedra (VP); the method is O(NVP), where NVP is the number of distinct VP representing the system. In effect, we resolve the long-standing problem of fast but accurate numerical solution of the near-field corrections, contributions to the potential due to near VP-typically those involving multipole-type conditionally convergent sums, or use of fast Fourier transforms. Our method avoids all ill-convergent sums, is simple, accurate, efficient, and works generally, i.e., for periodic solids, molecules, or systems with disorder or imperfections. We demonstrate the practicality of the method by numerical calculations compared to exactly solvable models. We present an accurate and rapid solution of Poisson's equation for space-filling, arbitrarily shaped, convex Voronoi polyhedra (VP); the method is O(N VP ), where N VP is the number of distinct VP representing the system. In effect, we resolve the long-standing problem of fast but accurate numerical solution of the near-field corrections, contributions to the potential due to near VP-typically those involving multipole-type conditionally convergent sums, or use of fast Fourier transforms. Our method avoids all ill-convergent sums, is simple, accurate, efficient, and works generally, i.e., for periodic solids, molecules, or systems with disorder or imperfections. We demonstrate the practicality of the method by numerical calculations compared to exactly solvable models. Keywords Materials Science and Engineering Disciplines Condensed Matter Physics | Engineering Physics | Materials Science and Engineering Comments

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Section: A Computationally Efficient and Accurate Poisson Solvermentioning

confidence: 98%

Section: General Commentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Accurate and fast numerical solution of Poisson's equation for arbitrary, space-filling Voronoi polyhedra: Near-field corrections revisited

2011

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“…For example, "exact" linear muffin-tin orbital (EMTO) method 7 uses an approach from Gonis et al 8 to overcome VP integration issues for the Poisson potential, but it is extremely slowly convergent; various KKR-based codes, such as the linear-scaling multiple-scattering (LSMS), 9 utilizes shape-functions to perform VP integrations, which, as we show, is slowly convergent and limited in accuracy; the full-potential linear augmented wave 10 (FLAPW) method avoids VP integrals (via non-overlapping muffin-tins and Fourier methods over entire unit cell), but never determining site-VP-specific properties and requiring a larger number of spherical harmonic basis functions and huge number of plane-waves.…”

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confidence: 99%

Efficient isoparametric integration over arbitrary space-filling Voronoi polyhedra for electronic structure calculations

et al. 2011

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A numerically efficient, accurate, and easily implemented integration scheme over convex Voronoi polyhedra (VP) is presented for use in ab-initio electronic-structure calculations. We combine a weighted Voronoi tessellation with isoparametric integration via Gauss-Legendre quadratures to provide rapidly convergent VP integrals for a variety of integrands, including those with a Coulomb singularity. We showcase the capability of our approach by first applying to an analytic chargedensity model achieving machine-precision accuracy with expected convergence properties in milliseconds. For contrast, we compare our results to those using shape-functions and show our approach is greater than 10 5 faster and 10 7 more accurate. A weighted Voronoi tessellation also allows for a physics-based partitioning of space that guarantees convex, space-filling VP while reflecting accurate atomic size and site charges, as we show within KKR methods applied to Fe-Pd alloys.

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