Methods for Hartree-Fock and Density Functional Theory Electronic Structure Calculations with Linearly Scaling Processor Time and Memory Usage

  • Emanuel H. Rubensson
  • Elias Rudberg
  • Pawel Salek
Chapter

Abstract

We discuss algorithms that can be used to calculate electron densities using computer resources – memory and processor time – that increase only linearly with system size. We focus on the Hartree-Fock and density functional theories and calculations using Gaussian basis sets. However, many of the approaches discussed here are applicable also for other methods and for any local basis. Particular attention is directed towards error control and how to avoid the use of the ad-hoc selected parameters and threshold values often associated with computational approximations employed to reach linear scaling. The discussed aspects include multipole methods, linear scaling computation of the Hartree-Fock exchange and density functional theory exchange-correlation matrices, hierarchic representation of sparse matrices, and density matrix purification. The article also describes how these different parts are put together to achieve linear scaling for the entire Hartree-Fock or density functional theory calculation, controlling errors in the self-consistent field procedure by considering rotations of the occupied subspace.

Keywords

Density functional theory Density matrix purification Multipole approximations Parametrized minimization Sparse matrix Two-electron integrals 

References

  1. 1.
    Goedecker S (1999) Rev Mod Phys 71:1085CrossRefGoogle Scholar
  2. 2.
    Bowler DR, Miyazaki T, Gillan MJ (2002) J Phys Condens Matter 14:2781Google Scholar
  3. 3.
    Wu SY, Jayanthi CS (2002) Phys Rep 358:1CrossRefGoogle Scholar
  4. 4.
    Rudberg E, Rubensson EH, Sałek P Ergo (2009) Version 2.0: a quantum chemistry program for large scale self-consistent field calculations. http://www.ergoscf.org
  5. 5.
    Rudberg E, Rubensson EH, Sałek P (2008) J Chem Phys 128:184106CrossRefGoogle Scholar
  6. 6.
    Rubensson EH (2008) Matrix algebra for quantum chemistry. PhD thesis, Department of Theoretical Chemistry, Royal Institute of Technology, StockholmGoogle Scholar
  7. 7.
    Rudberg E (2007) Quantum chemistry for large systems. PhD thesis, Department of Theoretical Chemistry, Royal Institute of Technology, StockholmGoogle Scholar
  8. 8.
    Rubensson EH, Rudberg E, Sałek P (2008) J Math Phys 49:032103CrossRefGoogle Scholar
  9. 9.
    Helgaker T, Jørgensen P, Olsen J (2000) Molecular electronic-structure theory Wiley, ChichesterGoogle Scholar
  10. 10.
    Pulay P (1980) Chem Phys Lett 73:393CrossRefGoogle Scholar
  11. 11.
    Pulay P (1982) J Comput Chem 3:556CrossRefGoogle Scholar
  12. 12.
    Benzi M, Kouhia R, Tuma M (2001) Comput Meth Appl Mech Eng 190:6533CrossRefGoogle Scholar
  13. 13.
    Rubensson EH, Rudberg E, Sałek P (2007) J Comput Chem 28:2531CrossRefGoogle Scholar
  14. 14.
    Rubensson EH, Bock N, Holmström E, Niklasson AMN (2008) J Chem Phys 128:104105CrossRefGoogle Scholar
  15. 15.
    Lea Thøgersen (2005) Optimization of densities in Hartree-Fock and density-functional theory, Atomic orbital based response Theory, and Benchmarking for radicals. PhD thesis, Department of Chemistry, University of Aarhus, AarhusGoogle Scholar
  16. 16.
    Kudin KN, Scuseria GE (2007) Math Model Num Anal 41:281CrossRefGoogle Scholar
  17. 17.
    Zerner MC, Hehenberger M (1979) Chem Phys Lett 62:550CrossRefGoogle Scholar
  18. 18.
    Cancès E, Le Bris C (2000) Int J Quantum Chem 79:82CrossRefGoogle Scholar
  19. 19.
    Häser M, Ahlrichs R (1989) J Comput Chem 10:104CrossRefGoogle Scholar
  20. 20.
    Rudberg E, Sałek P (2006) J Chem Phys 125:084106CrossRefGoogle Scholar
  21. 21.
    Greengard L, Rokhlin V (1987) J Comput Phys 73:325CrossRefGoogle Scholar
  22. 22.
    Schmidt KE, Lee MA (1991) J Stat Phys 63:1223CrossRefGoogle Scholar
  23. 23.
    Panas I, Almlöf, J, Feyereisen, MW (1991) Int J Quantum Chem 40:797CrossRefGoogle Scholar
  24. 24.
    Panas I, Almlöf J (1992) Int J Quantum Chem 42:1073CrossRefGoogle Scholar
  25. 25.
    White CA, Johnson BG, Gill PMW, Head-Gordon M (1994) Chem Phys Lett 230:8CrossRefGoogle Scholar
  26. 26.
    White CA, Head-Gordon M (1994) J Chem Phys 101:6593CrossRefGoogle Scholar
  27. 27.
    Challacombe M, Schwegler E, Almlöf J (1995) J Chem Phys 104:4685CrossRefGoogle Scholar
  28. 28.
    Challacombe M, Schwegler E (1997) J Chem Phys 106:5526CrossRefGoogle Scholar
  29. 29.
    White CA, Johnson BG, Gill PMW, Head-Gordon M (1996) Chem Phys Lett 253:268CrossRefGoogle Scholar
  30. 30.
    Strain MC, Scuseria GE, Frisch MJ (1996) Science 271:51CrossRefGoogle Scholar
  31. 31.
    Choi CH, Ruedenberg K, Gordon MS (2001) J Comput Chem 22:1484CrossRefGoogle Scholar
  32. 32.
    Sierka M, Hogekamp A, Ahlrichs R (2003) J Chem Phys 118:9136CrossRefGoogle Scholar
  33. 33.
    Watson MA, Sałek P, Macak P, Helgaker T (2004) J Chem Phys 121:2915CrossRefGoogle Scholar
  34. 34.
    Gan CK, Tymczak C, Challacombe M (2004) J Chem Phys 121:6608CrossRefGoogle Scholar
  35. 35.
    Schwegler E, Challacombe M (1996) J Chem Phys 105:2726CrossRefGoogle Scholar
  36. 36.
    Burant JC, Scuseria GE (1996) J Chem Phys 105:8969CrossRefGoogle Scholar
  37. 37.
    Schwegler E, Challacombe M, Head-Gordon M (1997) J Chem Phys 106:9708CrossRefGoogle Scholar
  38. 38.
    Ochsenfeld C, White CA, Head-Gordon M (1998) J Chem Phys 109:1663CrossRefGoogle Scholar
  39. 39.
    Schwegler E, Challacombe M (1999) J Chem Phys 111:6223CrossRefGoogle Scholar
  40. 40.
    Ochsenfeld C (2000) Chem Phys Lett 327:216CrossRefGoogle Scholar
  41. 41.
    Lambrecht DS, Ochsenfeld C (2005) J Chem Phys 123:184101CrossRefGoogle Scholar
  42. 42.
    Aquilante F, Pedersen TB, Lindh R (2007) J Chem Phys 126:194106CrossRefGoogle Scholar
  43. 43.
    Murray CW, Handy NC, Laming GJ (1993) Mol Phys 78:997CrossRefGoogle Scholar
  44. 44.
    Challacombe M (2000) J Chem Phys 113:10037CrossRefGoogle Scholar
  45. 45.
    Treutler O, Ahlrichs R (1995) J Chem Phys 102:346CrossRefGoogle Scholar
  46. 46.
    Lebedev VI, vychisl, Zh (1975) Mat mat Fiz 45:48Google Scholar
  47. 47.
    Lindh R, Malmqvist PA, Gagliardi L (2001) Theor Chem Acc 106:178CrossRefGoogle Scholar
  48. 48.
    Becke AD (1988) J Chem Phys 88:2547CrossRefGoogle Scholar
  49. 49.
    Stratmann RE, Scuseria GE, Frisch MJ (1996) Chem Phys Lett 257:213CrossRefGoogle Scholar
  50. 50.
    Rudberg E, Rubensson EH, Sałek P (2009) J Chem Theory Comput 5:80CrossRefGoogle Scholar
  51. 51.
    Sankey OF, Drabold DA, Gibson A (1994) Phys Rev B 50:1376CrossRefGoogle Scholar
  52. 52.
    Bekas C, Kokiopoulou E, Saad Y, SIAM J (2008) Matrix Anal Appl 30:397Google Scholar
  53. 53.
    Drabold DA, Sankey OF (1993) Phys Rev Lett 70:3631Google Scholar
  54. 54.
    Wang LW, Zunger A (1994) J Chem Phys 100:2394CrossRefGoogle Scholar
  55. 55.
    Gao B, Jiang J, Liu K, Wu Z, Lu W, Luo Y (2007) J Comput Chem 29:434CrossRefGoogle Scholar
  56. 56.
    McWeeny R (1956) Proc R Soc London Ser A 235:496CrossRefGoogle Scholar
  57. 57.
    Heath MT (1997) Scientific computing: an introductory survey. McGraw-Hill, SingaporeGoogle Scholar
  58. 58.
    Li XP, Nunes RW, Vanderbilt D (1993) Phys Rev B 47:10891CrossRefGoogle Scholar
  59. 59.
    Millam JM, Scuseria GE (1997) J Chem Phys 106:5569CrossRefGoogle Scholar
  60. 60.
    Daw MS (1993) Phys Rev B 47:10895CrossRefGoogle Scholar
  61. 61.
    Challacombe M (1999) J Chem Phys 110:2332CrossRefGoogle Scholar
  62. 62.
    Helgaker T, Larsen H, Olsen J, Jørgensen P (2000) Chem Phys Lett 327:397CrossRefGoogle Scholar
  63. 63.
    Dyan A, Dubot P, Cenedese P (2005) Phys Rev B 72:125104CrossRefGoogle Scholar
  64. 64.
    Larsen H, Olsen J, Jørgensen P, Helgaker T (2001) J Chem Phys 115:9685CrossRefGoogle Scholar
  65. 65.
    Shao Y, Saravanan C, Head-Gordon M, White CA (2003) J Chem Phys 118:6144CrossRefGoogle Scholar
  66. 66.
    Goedecker S, Colombo L (1994) Phys Rev Lett 73:122CrossRefGoogle Scholar
  67. 67.
    Goedecker S, Teter M (1995) Phys Rev B 51:9455CrossRefGoogle Scholar
  68. 68.
    Baer R, Head-Gordon M (1997) J Chem Phys 107:10003CrossRefGoogle Scholar
  69. 69.
    Bates KR, Daniels AD, Scuseria GE (1998) J Chem Phys 109:3308CrossRefGoogle Scholar
  70. 70.
    Liang W, Saravanan C, Shao Y, Baer R, Bell AT, Head-Gordon M (2003) J Chem Phys 119:4117CrossRefGoogle Scholar
  71. 71.
    Silver RN, Roeder H, Voter AF, Kress JD (1996) J Comput Phys 124:115CrossRefGoogle Scholar
  72. 72.
    Palser AHR, Manolopoulos DE (1998) Phys Rev B 58:12704CrossRefGoogle Scholar
  73. 73.
    Niklasson AMN (2002) Phys Rev B 66:155115CrossRefGoogle Scholar
  74. 74.
    Niklasson AMN, Tymczak CJ, Challacombe M (2003) J Chem Phys 118:8611CrossRefGoogle Scholar
  75. 75.
    Holas A (2001) Chem Phys Lett 340:552CrossRefGoogle Scholar
  76. 76.
    Mazziotti DA (2003) Phys Rev E 68:066701CrossRefGoogle Scholar
  77. 77.
    Xiang HJ, Liang WZ, Yang J, Hou JG, Zhu Q (2005) J Chem Phys 123:124105CrossRefGoogle Scholar
  78. 78.
    Pino R, Scuseria GE (2002) Chem Phys Lett 360:117CrossRefGoogle Scholar
  79. 79.
    Rubensson EH, Rudberg E, Sałek P (2008) J Chem Phys 128:074106CrossRefGoogle Scholar
  80. 80.
    Paterson MS, Stockmeyer L, SIAM (1973) J Comput 2:60Google Scholar
  81. 81.
    Saravanan C, Shao Y, Baer R, Ross PN, Head-Gordon M (2003) J Comput Chem 24:618CrossRefGoogle Scholar
  82. 82.
    Maslen PE, Ochsenfeld C, White CA, Lee MS, Head-Gordon M (1998) J Phys Chem A 102:2215CrossRefGoogle Scholar
  83. 83.
    Rubensson EH, Rudberg E, Sałek P (2009) J Comput Chem 30:974CrossRefGoogle Scholar
  84. 84.
    Rubensson EH, Rudberg E, Sałek P (2007) Proceedings of PARA’06, Springer LNCS 4699:90Google Scholar
  85. 85.
  86. 86.
    Automatically tuned linear algebra software (ATLAS). http://mathatlas.sourceforge.net/
  87. 87.
  88. 88.
  89. 89.
    Elmroth E, Gustavson F, Jonsson I, Kågström B (2004) SIAM Rev 46:3CrossRefGoogle Scholar
  90. 90.
    Becke AD (1993) J Chem Phys 98:1372CrossRefGoogle Scholar
  91. 91.
    Spartan’02 (2002) Molecular modeling package by Wavefunction, Inc.Google Scholar
  92. 92.
    Challacombe M (2000) Comp Phys Commun 128:93CrossRefGoogle Scholar
  93. 93.
    Bowler DR, Miyazaki T, Gillan MJ (2001) Comp Phys Commun 137:255CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Emanuel H. Rubensson
    • 1
  • Elias Rudberg
    • 1
  • Pawel Salek
    • 2
  1. 1.Division of Scientific Computing, Department of Information TechnologyUppsala UniversityUppsalaSweden
  2. 2.ul. Zaporoska 8/4KrakówPoland

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