Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2011002201
pages 83-99

LINEAR SCALING SOLUTION OF THE ALL-ELECTRON COULOMB PROBLEM INSOLIDS

J. E. Pask
Condensed Matter and Materials Division, Lawrence Livermore National Laboratory, Livermore,CA 94550
N. Sukumar
University of California Davis
S. E. Mousavi
Department of Civil and Environmental Engineering, University of California, Davis, CA 95616

ABSTRACT

We present a linear scaling formulation for the solution of the all-electron Coulomb problem in crystalline solids. The resulting method is systematically improvable and well suited to large-scale quantum mechanical calculations in which the Coulomb potential and energy of a continuous electronic density and singular nuclear density are required. Linear scaling is achieved by introducing smooth, strictly local neutralizing densities to render nuclear interactions strictly local, and solving the remaining neutral Poisson problem for the electrons in real space. Although the formulation includes singular nuclear potentials without smearing approximations, the required Poisson solution is in Sobolev space H1, as required for convergence in the energy norm. We employ enriched finite elements, with enrichments from isolated atom solutions, for an efficient solution of the resulting Poisson problem in the interacting solid. We demonstrate the accuracy and convergence of the approach by direct comparison to standard Ewald sums for a lattice of point charges and demonstrate the accuracy in all-electron quantum mechanical calculations with an application to crystalline diamond.

REFERENCES

  1. Arias, T. A., Multiresolution analysis of electronic structure: Semicardinal and wavelet bases. DOI: 10.1103/RevModPhys.71.267

  2. Babu&#353;ka, I. and Melenk, J. M., The partition of unity method. DOI: 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N

  3. Batcho, P. F., Computational method for general multicenter electronic structure calculations. DOI: 10.1103/PhysRevE.61.7169

  4. Beck, T. L., Real-space mesh techniques in density-functional theory. DOI: 10.1103/RevModPhys.72.1041

  5. Blöchl, P. E., Projector augmented-wave method. DOI: 10.1103/PhysRevB.50.17953

  6. Brandt, A., Multilevel adaptive solutions to boundary-value problems. DOI: 10.1090/S0025-5718-1977-0431719-X

  7. Bylaska, E. J., Holst, M., and Weare, J. H., Adaptive finite element method for solving the exact Kohn-Sham equation of density functional theory. DOI: 10.1021/ct800350j

  8. Challacombe, M., White, C., and Head-Gordon, M., Periodic boundary conditions and the fast multipole method. DOI: 10.1063/1.474150

  9. Duffy, M. G., Quadrature over a pyramid or cube of integrands with a singularity at a vertex. DOI: 10.1137/0719090

  10. Ewald, P. P., Die Berechnung optischer und elektrostatischer Gitterpotentiale. DOI: 10.1002/andp.19213690304

  11. Fuchs, K., A quantum mechanical investigation of the cohesive forces of metallic copper. DOI: 10.1098/rspa.1935.0167

  12. Genovese, L., Deutsch, T., Neelov, A., Goedecker, S., and Beylkin, G., Efficient solution of Poisson's equation with free boundary conditions. DOI: 10.1063/1.2335442

  13. Goedecker, S. and Ivanov, O. V., Linear scaling solution of the Coulomb problem using wavelets. DOI: 10.1016/S0038-1098(97)10241-1

  14. Greengard, L. and Rokhlin, V., A fast algorithm for particle simulations. DOI: 10.1016/0021-9991(87)90140-9

  15. Havu, P., Havu, V., Puska, M. J., and Nieminen, R. M., Nonequilibrium electron transport in two-dimensional nanostructures modeled using Green's functions and the finite-element method. DOI: 10.1103/PhysRevB.69.115325

  16. Hohenberg, P. and Kohn, W., Inhomogeneous electron gas. DOI: 10.1103/PhysRev.136.B864

  17. Ihm, J., Zunger, A., and Cohen, M. L., Momentum-space formalism for the total energy of solids. DOI: 10.1088/0022-3719/12/21/009

  18. Jones, R. O. and Gunnarsson, O., The density functional formalism, its applications and prospects. DOI: 10.1103/RevModPhys.61.689

  19. Juselius, J. and Sundholm, D., Parallel implementation of a direct method for calculating electrostatic potentials. DOI: 10.1063/1.2436880

  20. Kohn, W. and Sham, L. J., Self-consistent equations including exchange and correlation effects. DOI: 10.1103/PhysRev.140.A1133

  21. Kudin, K. N. and Scuseria, G. E., A fast multipole algorithm for the efficient treatment of the Coulomb problem in electronic structure calculations of periodic systems with Gaussian orbitals. DOI: 10.1016/S0009-2614(98)00468-0

  22. Kudin, K. N. and Scuseria, G. E., Linear-scaling density-functional theory with Gaussian orbitals and periodic boundary conditions: Efficient evaluation of energy and forces via the fast multipole method. DOI: 10.1103/PhysRevB.61.16440

  23. Kudin, K. N. and Scuseria, G. E., Revisiting infinite lattice sums with the periodic fast multipole method. DOI: 10.1063/1.1771634

  24. Kurashige, Y., Nakajima, T., and Hirao, K., Gaussian and finite-element coulomb method for the fast evaluation of coulomb integrals. DOI: 10.1063/1.2716638

  25. Lehtovaara, L., Havu, V., and Puska, M., All-electron density functional theory and time-dependent density functional theory with high-order finite elements. DOI: 10.1063/1.3176508

  26. Losilla, S. A., Sundholm, D., and Juselius, J., The direct approach to gravitation and electrostatics method for periodic systems. DOI: 10.1063/1.3291027

  27. Madelung, E., Das elektrische Feld in Systemen von regelm&#228;&szlig;ig angeordneten Punktladungen.

  28. Melenk, J. M. and Babu&#353;ka, I., The partition of unity finite element method: Basic theory and applications. DOI: 10.1016/S0045-7825(96)01087-0

  29. Merrick, M. P., Iyer, K. A., and Beck, T. L., Multigrid method for electrostatic computations in numerical density-functional theory. DOI: 10.1021/j100033a017

  30. Modine, N. A., Zumbach, G., and Kaxiras, E., Adaptive-coordinate real-space electronic-structure calculations for atoms, molecules, and solids. DOI: 10.1103/PhysRevB.55.10289

  31. Mousavi, S. E. and Sukumar, N., Generalized Duffy transformation for integrating vertex singularities. DOI: 10.1007/s00466-009-0424-1

  32. Murakami, H., Sonnad, V., and Clementi, E., A 3-dimensional finite-element approach towards molecular SCF computations. DOI: 10.1002/qua.560420418

  33. Nikolaev, A. V. and Dyachkov, P. N., Solution of periodic Poisson's equation and the Hartree-Fock approach for solids with extended electron states: Application to linear augmented plane wave method. DOI: 10.1002/qua.10189

  34. Pask, J. E., Klein, B. M., Fong, C. Y., and Sterne, P. A., Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach. DOI: 10.1103/PhysRevB.59.12352

  35. Pask, J. E. and Sterne, P. A., Finite element methods in ab initio electronic structure calculations. DOI: 10.1088/0965-0393/13/3/R01

  36. Pask, J. E. and Sterne, P. A., Real-space formulation of the electrostatic potential and total energy of solids. DOI: 10.1103/PhysRevB.71.113101

  37. Pickett, W. E., Pseudopotential methods in condensed matter applications. DOI: 10.1016/0167-7977(89)90002-6

  38. Rudge, W. E., Generalized Ewald potential problem. DOI: 10.1103/PhysRev.181.1020

  39. Singh, D. J. and Nordstrom, L., Planewaves, Pseudopotentials, and the LAPW Method.

  40. Soler, J. M., Artacho, E., Gale, J. D., Garc&#237;a, A., Junquera, J., Ordej&#243;n, P., and S&#225;nchez-Portal, D., The SIESTA method for ab initio order-N materials simulation. DOI: 10.1088/0953-8984/14/11/302

  41. Strain, M. C., Scuseria, G. E., and Frisch, M. J., Achieving linear scaling for the electronic quantum Coulomb problem. DOI: 10.1126/science.271.5245.51

  42. Strang, G. and Fix, G. J., An Analysis of the Finite Element Method.

  43. Stroud, A. H., Approximate Calculation of Multiple Integrals.

  44. Sukumar, N. and Pask, J. E., Classical and enriched finite element formulations for Bloch-periodic boundary conditions. DOI: 10.1002/nme.2457

  45. Suryanarayana, P., Gavini, V., Blesgen, T., Bhattacharya, K., and Ortiz, M., Non-periodic finite-element formulation of kohn-sham density functional theory. DOI: 10.1016/j.jmps.2009.10.002

  46. Torsti, T., Eirola, T., Enkovaara, J., Hakala, T., Havu, P., Havu, V., H&#246;yn&#228;l&#228;nmaa, T., Ignatius, J., Lyly, M., Makkonen, I., Rantala, T. T., Ruokolainen, J., Ruotsalainen, K., R&#228;s&#228;nen, E., Saarikoski, H., and Puska, M. J., Three real-space discretization techniques in electronic structure calculations. DOI: 10.1002/pssb.200541348

  47. Tsuchida, E. and Tsukada, M., Electronic-structure calculations based on the finite-element method. DOI: 10.1103/PhysRevB.52.5573

  48. Wang, J. and Beck, T. L., Efficient real-space solution of the Kohn-Sham equations with multiscale techniques. DOI: 10.1063/1.481543

  49. Watson, M. A. and Hirao, K., A linear-scaling spectral-element method for computing electrostatic potentials. DOI: 10.1063/1.3009264

  50. Weinert, M., Solution of Poisson&prime;s equation: Beyond Ewald-type methods. DOI: 10.1063/1.524800

  51. White, S. R., Wilkins, J. W., and Teter, M. P., Finite-element method for electronic-structure. DOI: 10.1103/PhysRevB.39.5819

  52. Wigner, E. P. and Seitz, F., On the constitution of metallic sodium. DOI: 10.1103/PhysRev.43.804

  53. Yamakawa, S. and Hyodo, S., Gaussian finite-element mixed-basis method for electronic structure calculations. DOI: 10.1103/PhysRevB.71.035113

  54. Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z., The Finite Element Method: Its Basis and Fundamentals.


Articles with similar content:

EFFECTIVE THERMOELASTIC PROPERTIES OF HETEROGENEOUS THERMOPERISTATIC BAR OF RANDOM STRUCTURE
International Journal for Multiscale Computational Engineering, Vol.13, 2015, issue 1
Valeriy A. Buryachenko, Chen Wanji, Yang Shengqi
AN ADAPTIVE DOMAIN DECOMPOSITION PRECONDITIONER FOR CRACK PROPAGATION PROBLEMS MODELED BY XFEM
International Journal for Multiscale Computational Engineering, Vol.11, 2013, issue 6
Haim Waisman, Luc Berger-Vergiat
MODELING DYNAMIC FRACTURE AND DAMAGE IN A FIBER-REINFORCED COMPOSITE LAMINA WITH PERIDYNAMICS
International Journal for Multiscale Computational Engineering, Vol.9, 2011, issue 6
Wenke Hu, Youn Doh Ha, Florin Bobaru
Calculation of Quasistatic Eigen-Field of a Charge, which Moves Arbitrarily in Cylindrical Drift Chamber
Telecommunications and Radio Engineering, Vol.67, 2008, issue 13
T. Yu. Yatsenko, K. V. Ilyenko, G. M. Gorbik
TWO-SCALE AND THREE-SCALE COMPUTATIONAL CONTINUA MODELS OF COMPOSITE CURVED BEAMS
International Journal for Multiscale Computational Engineering, Vol.16, 2018, issue 6
Z. F. Yuan, Dinghe Li, Jacob Fish