Non-periodic finite-element formulation of Kohn-Sham density functional theory

Journal of the Mechanics and Physics of Solids

Volumn 58, Issue 2, 2010, Pages 256-280

  Suryanarayana, Phanish ^a   Gavini, Vikram ^b   Blesgen, Thomas ^c   Bhattacharya, Kaushik ^a   Ortiz, Michael ^a  

^a California Institute of Technology   (United States)

^b UNIVERSITY OF MICHIGAN   (United States)

^c MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

convergence; Density functional theory; Finite elements; Kohn Sham

Indexed keywords

COARSE GRAINING; COMPUTATIONAL COSTS; CONVERGENCE RATES; DISLOCATIONS AND CRACKS; DOMAIN DECOMPOSITIONS; FINITE ELEMENT; FINITE ELEMENT APPROXIMATIONS; FINITE ELEMENT FORMULATIONS; NUMERICAL QUADRATURE; PSEUDOPOTENTIAL CALCULATION; REAL-SPACE; SADDLE POINT PROBLEMS; TEST CASE; VARIATIONAL PROBLEMS; WELLPOSEDNESS;

CONVERGENCE OF NUMERICAL METHODS; CRACK TIPS; DOMAIN DECOMPOSITION METHODS; PROBABILITY DENSITY FUNCTION;

DENSITY FUNCTIONAL THEORY;

EID: 73449088685     PISSN: 00225096     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmps.2009.10.002     Document Type: Article

References (67)

1. Anantharaman, A., Cances, E., 2008. On Kohn-Sham models with LDA and GGA exchange-correlation functionals. URL: 〈http://arxiv.org/abs/0809.5139v1〉.
2. Arroyo M., and Ortiz M. Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. International Journal for Numerical Methods in Engineering 65 (2006) 2167-2202
3. Bachelet G.B., Hamann D.R., and Schlüter M. Pseudopotentials that work: from H to Pu. Physical Review B 26 8 (1982) 4199-4228
4. Bernstein N., Kermode J.R., and Csanyi G. Hybrid atomistic simulation methods for materials systems. Reports on Progress in Physics 72 2 (2009) 026501 (25pp.)
5. Bey J., and Aachen R. Simplicial grid refinement: on Freudenthal's algorithm and the optimal number of congruence classes. Numerische Mathematik 85 (2000) 1-29
6. Bowler D., Choudhury R., Gillan M., and Miyazaki T. Recent progress with large-scale ab initio calculations: the CONQUEST code. Physica Status Solidi B-Basic Solid State Physics 243 5 (2006) 989-1000
7. Car R., and Parrinello M. Unified approach for molecular dynamics and density-functional theory. Physical Review Letters 55 22 (1985) 2471-2474
8. Castro A., Appel H., Oliveira M., Rozzi C.A., Andrade X., Lorenzen F., Marques M.A.L., Gross E.K.U., and Rubio A. Octopus: a tool for the application of time-dependent density functional theory. Physica Status Solidi B-Basic Solid State Physics 243 11 (2006) 2465-2488
9. Ceperley D.M., and Alder B.J. Ground state of the electron gas by a stochastic method. Physical Review Letters 45 7 (1980) 566-569
10. Chelikowsky J.R., Troullier N., and Saad Y. Finite-difference-pseudopotential method: electronic structure calculations without a basis. Physical Review Letters 72 8 (1994) 1240-1243
11. Choly N., Lu G., Weinan R., and Kaxiras E. Multiscale simulations in simple metals: a density-functional-based methodology. Physical Review B 71 9 (2005) 094101
12. Ciarlet P. The Finite-element Method for Elliptic Problems (2002), SIAM, Philadelphia, PA
13. Dal Maso G. An Introduction to Γ -convergence (1993), Birkhäuser, Boston
14. Dennis J., and Schnabel R. Numerical Methods for Unconstrained Optimization and Nonlinear Equations (1996), SIAM, Philadelphia, PA
15. Engel E., Höck A., Schmid R.N., Dreizler R.M., and Chetty N. Role of the core-valence interaction for pseudopotential calculations with exact exchange. Physical Review B 64 12 (2001) 125111
16. Finnis M. Interatomic Forces in Condensed Matter (2003), Oxford University Press, Oxford
17. Fiolhais C., Perdew J.P., Armster S.Q., MacLaren J.M., and Brajczewska M. Dominant density parameters and local pseudopotentials for simple metals. Physical Review B 51 20 (1995) 14001-14011
18. Freund R.W. A transpose-free quasi-minimal residual algorithm for non-hermitian linear systems. SIAM Journal on Scientific Computing 14 2 (1993) 470-482
19. Galli G., and Parrinello M. Large scale electronic structure calculations. Physical Review Letters 69 24 (1992) 3547-3550
20. García-Cervera C.J., Lu J., and Weinan R. A sub-linear scaling algorithm for computing the electronic structure of materials. Communications in Mathematical Sciences 5 4 (2007) 999-1026
21. García-Cervera C.J., Lu J., Xuan Y., and Weinan R. Linear-scaling subspace-iteration algorithm with optimally localized nonorthogonal wave functions for Kohn-Sham density functional theory. Physical Review B (Condensed Matter and Materials Physics) 79 11 (2009) 115110
22. Gavini V., Bhattacharya K., and Ortiz M. Quasi-continuum orbital-free density-functional theory: a route to multi-million atom non-periodic DFT calculation. Journal of the Mechanics and Physics of Solids 55 4 (2007) 697-718
23. Gavini V., Knap J., Bhattacharya K., and Ortiz M. Non-periodic finite-element formulation of orbital-free density functional theory. Journal of the Mechanics and Physics of Solids 55 4 (2007) 669-696
24. Gilbarg D., and Trudinger N. Elliptic Differential Equations of Second Order (1983), Springer, New York
25. Goedecker S. Linear scaling electronic structure methods. Reviews of Modern Physics 71 4 (1999) 1085-1123
26. Gonze X., Beuken J.M., Caracas R., Detraux F., Fuchs M., Rignanese G.M., Sindic L., Verstraete M., Zerah G., Jollet F., Torrent M., Roy A., Mikami M., Ghosez P., Raty J.Y., and Allan D.C. First-principles computation of material properties: the abinit software project. Computational Materials Science 25 (2002) 478-492 (15pp.)
27. Govind N., Wang Y.A., and Carter E.A. Electronic-structure calculations by first-principles density-based embedding of explicitly correlated systems. The Journal of Chemical Physics 110 16 (1999) 7677-7688
28. Gygi F.m.c., and Galli G. Real-space adaptive-coordinate electronic-structure calculations. Physical Review B 52 4 (1995) R2229-R2232
29. Hehre W.J., Stewart R.F., and Pople J.A. Self-consistent molecular-orbital methods. I. Use of Gaussian expansions of Slater-type atomic orbitals. The Journal of Chemical Physics 51 6 (1969) 2657-2664
30. Hohenberg P., and Kohn W. Inhomogeneous electron gas. Physical Review 136 3B (1964) B864-B871
31. Huber K. Constants of Diatomic Molecules (1972), McGraw-Hill, New York
32. Ismail-Beigi S., and Arias T.A. New algebraic formulation of density functional calculation. Computer Physics Communications 128 1-2 (2000) 1-45
33. Kleinman L., and Bylander D.M. Efficacious form for model pseudopotentials. Physical Review Letters 48 20 (1982) 1425-1428
34. Kohn W. Nobel lecture: electronic structure of matter-wave functions and density functionals. Reviews of Modern Physics 71 5 (1999) 1253-1266
35. Kohn W., and Sham L.J. Self-consistent equations including exchange and correlation effects. Physical Review 140 4A (1965) A1133-A1138
36. Kotochigova S., Levine Z.H., Shirley E.L., Stiles M.D., and Clark C.W. Local-density-functional calculations of the energy of atoms. Physical Review A 55 1 (1997) 191-199
37. Kresse G., and Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical Review B 54 16 (1996) 11169-11186
38. Langreth D.C., and Mehl M.J. Beyond the local-density approximation in calculations of ground-state electronic properties. Physical Review B 28 4 (1983) 1809-1834
39. Lu G., Tadmor E.B., and Kaxiras E. From electrons to finite elements: a concurrent multiscale approach for metals. Physical Review B (Condensed Matter and Materials Physics) 73 2 (2006) 024108
40. Mauri F., Galli G., and Car R. Orbital formulation for electronic-structure calculations with linear system-size scaling. Physical Review B 47 15 (1993) 9973-9976
41. Modine N.A., Zumbach G., and Kaxiras E. Adaptive-coordinate real-space electronic-structure calculations for atoms, molecules, and solids. Physical Review B 55 16 (1997) 10289-10301
42. Nogueira F., Fiolhais C., He J., Perdew J.P., and Rubio A. Transferability of a local pseudopotential based on solid-state electron density. Journal of Physics: Condensed Matter 8 3 (1996) 287-302
43. Parr R., and Yang W. Density-functional Theory of Atoms and Molecules (1989), Oxford University Press, Oxford
44. 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. Physical Review B 59 19 (1999) 12352-12358
45. Pask J.E., and Sterne P.A. Finite element methods in ab initio electronic structure calculations. Modelling and Simulation in Materials Science and Engineering 13 3 (2005) R71-R96
46. Payne M.C., Teter M.P., Allan D.C., Arias T.A., and Joannopoulos J.D. Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients. Reviews of Modern Physics 64 4 (1992) 1045-1097
47. Perdew J.P., Chevary J.A., Vosko S.H., Jackson K.A., Pederson M.R., Singh D.J., and Fiolhais C. Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation. Physical Review B 46 11 (1992) 6671-6687
48. Perdew J.P., and Wang Y. Accurate and simple analytic representation of the electron-gas correlation energy. Physical Review B 45 23 (1992) 13244-13249
49. Perdew J.P., and Zunger A. Self-interaction correction to density-functional approximations for many-electron systems. Physical Review B 23 10 (1981) 5048-5079
50. Pickett W.E. Pseudopotential methods in condensed matter applications. Computer Physics Reports 9 3 (1989) 115-197
51. Rappe A.M., Rabe K.M., Kaxiras E., and Joannopoulos J.D. Optimized pseudopotentials. Physical Review B 41 2 (1990) 1227-1230
52. Roothaan C.C.J. New developments in molecular orbital theory. Reviews of Modern Physics 23 2 (1951) 69-89
53. Rudin W. Functional Analysis (1991), Mc Graw-Hill, New York
54. Saad Y. Numerical Methods for Large Eigenvalue Problems (1992), Manchester University Press
55. Saad Y., and Schultz M.H. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 7 3 (1986) 856-869
56. Segall M.D., Lindan P.J.D., Probert M.J., Pickard C.J., Hasnip P.J., Clark S.J., and Payne M.C. First-principles simulation: ideas, illustrations and the CASTEP code. Journal of Physics: Condensed Matter 14 11 (2002) 2717-2744
57. Skylaris C.-K., Haynes P.D., Mostofi A.A., and Payne M.C. Introducing ONETEP: linear-scaling density functional simulations on parallel computers. The Journal of Chemical Physics 122 8 (2005) 084119
58. Soler J.M., Artacho E., Gale J.D., García A., Junquera J., Ordejón P., and Sánchez-Portal D. The siesta method for ab initio order-n materials simulation. Journal of Physics: Condensed Matter 14 11 (2002) 2745-2779
59. Struwe M. Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (1990), Springer, Berlin
60. Thoutireddy, P., 2002. Variational arbitrary Lagrangian-Eulerian method. Ph.D. Thesis, California Institute of Technology.
61. Troullier N., and Martins J.L. Efficient pseudopotentials for plane-wave calculations. Physical Review B 43 3 (1991) 1993-2006
62. Tsuchida E. Ab initio molecular-dynamics study of liquid formamide. The Journal of Chemical Physics 121 10 (2004) 4740-4746
63. van der Vorst H.A. BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 13 2 (1992) 631-644
64. Vanderbilt D. Soft self-consistent pseudopotentials in a generalized eigenvalue formalism. Physical Review B 41 11 (1990) 7892-7895
65. Veillard A., and Clementi E. Correlation energy in atomic systems. v. Degeneracy effects for the second-row atoms. The Journal of Chemical Physics 49 5 (1968) 2415-2421
66. Wills J.M., and Cooper B.R. Synthesis of band and model Hamiltonian theory for hybridizing cerium systems. Physical Review B 36 7 (1987) 3809-3823
67. Zumbach G., Modine N.A., and Kaxiras E. Adaptive coordinate, real-space electronic structure calculations on parallel computers. Solid State Communications 99 2 (1996) 57-61