Skip to main content
SpringerLink
Log in

Density Matrix Methods in Linear Scaling Electronic Structure Theory

  • Chapter
  • First Online:
Linear-Scaling Techniques in Computational Chemistry and Physics

Part of the book series: Challenges and Advances in Computational Chemistry and Physics ((COCH,volume 13))

Abstract

We review some recursive Fermi operator expansion techniques for the calculation of the density matrix and its response to perturbations in tight-binding, Hartree-Fock, or density functional theory, at zero or finite electronic temperatures. Thanks to the recursive formulation, the expansion order increases exponentially with the number of iterations and the computational cost scales only linearly with the system size for sufficiently large sparse matrix representations. The methods are illustrated using simple models that are suitable for small numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 299.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 379.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 379.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Roothaan CCJ (1951) Rev Mod Phys 23:69

    Article  CAS  Google Scholar 

  2. Hohenberg P, Kohn W (1964) Phys Rev 136 B:864

    Article  Google Scholar 

  3. Kohn W, Sham LJ (1965) Phys Rev 140:1133

    Article  Google Scholar 

  4. Soler JM, Artacho E, Gale JD, Garcia A, Junquera J, Ordejon P, Sanchez-Portal D (2002) J Phys: Condens Matter 14:2745

    Article  CAS  Google Scholar 

  5. Bowler DR, Choudhury R, Gillan MJ, Miyazaki T (2006) Phys Stat Sol B 243:898

    Article  CAS  Google Scholar 

  6. Bock N, Challacombe M, Gan CK, Henkelman G, Nemeth K, Niklasson AMN, Odell A, Schwegler E, Tymczak CJ, Weber V, FREEON (2008) Los Alamos National Laboratory (LA-CC 01-2; LA-CC-04-086) Copyright University of California., URL http://www.nongnu.org/freeon/

  7. Rudberg E, Rubensson E, Salek P, ERGO (2006) Version 1.1: a quantum chemistry program for large-scale self consistent field calculations

    Google Scholar 

  8. Cawkwell M et al., LATTE (2009) A first principles self-consistent tightbinding program for large-scale atomistic simulations

    Google Scholar 

  9. Ozaki T, Kino H (2005) Phys Rev B 72:045121

    Article  CAS  Google Scholar 

  10. Tsuchida E, Tsukada M (1998) J Phys Soc Jpn 67:3844

    Article  CAS  Google Scholar 

  11. Hine ND, Haynes PD, Mostofi AA, Skylaris C-K, Payne MC (2009) Comput Phys Comm 180:1041

    Article  CAS  Google Scholar 

  12. Haydock R, Heine V, Kelley MJ (1975) J Phys C 8:2591

    Article  Google Scholar 

  13. Haydock R (1980) Comput Phys Commun 20:11

    Article  Google Scholar 

  14. Haydock R (1980) Solid State Phys 35:3547

    Google Scholar 

  15. Lanczos C (1950) J Res Nat Bur Stan B 45:255

    Article  Google Scholar 

  16. Yang W (1992) Phys Rev Lett 66:1438

    Article  Google Scholar 

  17. Li XP, Nunes RW, Vanderbilt D (1993) Phys Rev B 47:10891

    Article  CAS  Google Scholar 

  18. Daw MS (1993) Phys Rev B 47:10895

    Article  Google Scholar 

  19. Mauri F, Galli G, Car R (1993) Phys Rev B 47:9973

    Article  Google Scholar 

  20. Ordejón P, Drabold DA, Grumbach MP, Martin RM (1993) Phys Rev B 48:14646

    Article  Google Scholar 

  21. Goedecker S, Colombo L (1994) Phys Rev Let 73:122

    Article  CAS  Google Scholar 

  22. Goedecker S (1995) J Comp Phys 118:261

    Article  CAS  Google Scholar 

  23. Kim J, Mauri F, Galli G (1995) Phys Rev B 52:1640

    Article  CAS  Google Scholar 

  24. Wang Y, Stocks GM, Shelton WA, Nicholson DMC, Szotek Z, Temmerman WM (1995) Phys Rev Lett 75:2867

    Article  CAS  Google Scholar 

  25. Abrikosov IA, Niklasson AMN, Simak SI, Johansson B, Ruban AV, Skriver HL (1996) Phys Rev Lett 76:4203

    Article  CAS  Google Scholar 

  26. Silver RN, Roder H (1994) Int J Mod Phys C 5:735

    Article  Google Scholar 

  27. Silver RN, Roder H, Voter AF, Kress JD (1996) Int J Comput Phys 124:115

    Article  CAS  Google Scholar 

  28. Palser AHR, Manolopoulos DE (1998) Phys Rev B 58:12704

    Article  CAS  Google Scholar 

  29. Helgaker T, Larsen H, Olsen J (2000) Chem Phys Lett 327:397

    Article  CAS  Google Scholar 

  30. Niklasson AMN (2002) Phys Rev B 66:155115

    Article  CAS  Google Scholar 

  31. Galli G (1996) Cur Op Sol State Mat Sci 1:864

    Article  CAS  Google Scholar 

  32. Ordejón P (1998) Comput Mater Sci 12:157

    Article  Google Scholar 

  33. Goedecker S (1999) Rev Mod Phys 71:1085

    Article  CAS  Google Scholar 

  34. Scuseria G (1999) J Phys Chem 103:4782

    Article  CAS  Google Scholar 

  35. Wu SY, Jayanthi CS (2002) Phys Rep 358:1

    Article  Google Scholar 

  36. Goedecker S, Scusseria GE (2003) Comput Sci Eng 5:14

    Article  CAS  Google Scholar 

  37. Beylkin G, Coult N, Mohlenkamp MJ (1999) J Comp Phys 152:32

    Article  Google Scholar 

  38. Holas A (2001) Chem Phys Lett 340:552

    Article  CAS  Google Scholar 

  39. Mazziotti DA (2003) Phys Rev E 68:066701

    Article  CAS  Google Scholar 

  40. Niklasson AMN, Tymczak CJ, Challacombe M (2003) J Chem Phys 118:8611

    Article  CAS  Google Scholar 

  41. Niklasson AMN (2003) Phys Rev B 68:233104

    Article  CAS  Google Scholar 

  42. Xiang HJ, Liang WZ, Yang J, Hou JG, Zhu Q (2005) J Chem Phys 123:124105

    Article  CAS  Google Scholar 

  43. Jordan DK, Mazziotti DA (2005) J Chem Phys 122:084114

    Article  CAS  Google Scholar 

  44. Niklasson AMN (2008) J Chem Phys 129:244107

    Article  CAS  Google Scholar 

  45. McWeeny R (1956) Proc R Soc London Ser A-Math 235:496

    Article  CAS  Google Scholar 

  46. Kenney CS, Laub AJ (1991) SIAM J Matrix Anal Appl 12:273

    Article  Google Scholar 

  47. Kenney CS, Laub AJ (1995) IEEE Trans Auto Cont 40:1330

    Article  Google Scholar 

  48. Goedecker S (1999) Rev Mod Phys 71:1085

    Article  CAS  Google Scholar 

  49. Bernstein N (2001) Europhys Lett 55:52

    Article  CAS  Google Scholar 

  50. Bondesson L, Rudberg E, Lu Y, Salek P (2007) J Phys Chem 111:10320

    CAS  Google Scholar 

  51. Rudberg E, Rubensson EH, Salek P (2008) J Chem Phys 128:184106

    Article  CAS  Google Scholar 

  52. Niklasson AMN, Challacombe M (2004) Phys Rev Lett 92:193001

    Article  CAS  Google Scholar 

  53. Weber V, Niklasson AMN, Challacombe M (2004) Phys Rev Lett 92:193002

    Article  CAS  Google Scholar 

  54. Xiang HJ, Yang J, Hou JG, Zhu Q (2006) Phys Rev Lett 97:266402

    Article  CAS  Google Scholar 

  55. Weber V, Niklasson AMN, Challacombe M (2005) J Chem Phys 123:44106

    Article  CAS  Google Scholar 

  56. Niklasson AMN, Weber V,Challacombe M (2005) J Chem Phys 123:44107

    Article  CAS  Google Scholar 

  57. Niklasson AMN, Tymczak CJ, Challacombe M (2006) Phys Rev Lett 97:123001

    Article  CAS  Google Scholar 

  58. Kohn W (1996) Phys Rev Lett 76:3168

    Article  CAS  Google Scholar 

  59. Haynes PD, Payne MC (1999) Phys Rev B 59:12173

    Article  CAS  Google Scholar 

  60. Shao Y, Saravanan C, Head-Gordon M (2003) J Chem Phys 118:6144

    Article  CAS  Google Scholar 

  61. Mauri F, Galli G (1994) Phys Rev B 50:4316

    Article  CAS  Google Scholar 

  62. Ordejón P, Drabold DA, Martin RM, Grumbach MP (1995) Phys Rev B 51:1456

    Article  Google Scholar 

  63. Liu S, Perez-Jorda JM, Yang W (2000) J Chem Phys 112:1634

    Article  CAS  Google Scholar 

  64. Fattebert JL, Gygi F (2006) Phys Rev B 73:115124

    Article  CAS  Google Scholar 

  65. Weber V, VandeCondele J, Hutter J, Niklasson AMN (2008) J Chem Phys 128:084113

    Article  CAS  Google Scholar 

  66. Weber V and Hutter J (2008) J Chem Phys 128:064107

    Article  CAS  Google Scholar 

  67. Cooley JW, Tukey JW (1965) Math Comput 19:297

    Article  Google Scholar 

  68. Frigo M, Johnson SG (2005) Proc IEEE 93:216

    Article  Google Scholar 

  69. Greengard L, Rokhlin V (1987) J Comp Phys 73:325

    Article  Google Scholar 

  70. Greengard L, Rokhlin V (1990) Comp Phys 4:142

    Google Scholar 

  71. Challacombe M, White C, HeadGordon M (1997) J Chem Phys 107:9708

    Article  Google Scholar 

  72. Rudberg E, Salek P (2006) J Chem Phys 125:084106

    Article  CAS  Google Scholar 

  73. Greengard L (1994) Science 265:909

    Article  CAS  Google Scholar 

  74. Challacombe M, Schwegler E, Almlöf J (1996) J Chem Phys 104:4685

    Article  CAS  Google Scholar 

  75. Schwegler E, Challacombe M (1996) J Chem Phys 105:2726

    Article  CAS  Google Scholar 

  76. Burant JC, Scuseria GE, Frisch MJ (1996) J Chem Phys 105:8969

    Article  CAS  Google Scholar 

  77. Challacombe M, Schwegler E (1997) J Chem Phys 106:5526

    Article  CAS  Google Scholar 

  78. Ochsenfeld C, White CA, Head-Gordon M (1998) J Chem Phys 109:1663

    Article  CAS  Google Scholar 

  79. Prodan E, Kohn W (2005) Proc Natl Acad Sci USA 102:11635

    Article  CAS  Google Scholar 

  80. Kohn W (1959) Phys Rev 115:809

    Article  Google Scholar 

  81. Kohn W (1964) Phys Rev A 133:A171

    Google Scholar 

  82. Smirnov AV, Johnson DD (2001) Phys Rev B 64:235129

    Article  CAS  Google Scholar 

  83. Zeller R (2008) J Phys: Condens Matter 20:294215

    Article  CAS  Google Scholar 

  84. Cankurtaran BO, Gale JD, Ford MJ (2008) J Phys: Condens Matter 20:294208

    Article  CAS  Google Scholar 

  85. Rubensson EH, Salek P (2005) J Comput Chem 26:1628

    Article  CAS  Google Scholar 

  86. Rubensson EH, Rudberg E, Salek P (2008) J Chem Phys 128:074106

    Article  CAS  Google Scholar 

  87. Rubensson EH, Rudberg E, Salek P (2008) J Comput Chem 30:974

    Article  CAS  Google Scholar 

  88. Gustavson FG (1978) ACM Trans Math Soft 4:250

    Article  Google Scholar 

  89. Parr RG, Yang W (1989) Density-functional theory of atoms and molecules, Oxford University Press, Oxford

    Google Scholar 

  90. Golub G, van Loan CF (1996) Matrix Computations, Johns Hopkins University Press, Baltimore, MD

    Google Scholar 

  91. Benzi M, Meyer CD (1995) SIAM J Sci Comput 16:1159

    Article  Google Scholar 

  92. Millam JM, Scuseria GE (1997) J Chem Phys 106:5569

    Article  CAS  Google Scholar 

  93. Challacombe M (1999) J Chem Phys 110:2332

    Article  CAS  Google Scholar 

  94. Benzi M, Kouhia R, Tuma M (2001) Comp Meth App Mech Eng 190:6533

    Article  Google Scholar 

  95. Ozaki T (2001) Phys Rev B 64:195110

    Article  CAS  Google Scholar 

  96. Schweizer S, Kussmann J, Doser B, Ochsenfeld C (2008) J Comput Chem 29:1004

    Article  CAS  Google Scholar 

  97. Niklasson AMN (2004) Phys Rev B 70:193102

    Article  CAS  Google Scholar 

  98. Rubensson EH, Bock N, Holmström E, Niklasson AMN (2008) J Chem Phys 128:104105

    Article  CAS  Google Scholar 

  99. Baer R, Head-Gordon M (1997) Phys Rev Lett 79:3962

    Article  CAS  Google Scholar 

  100. Goedecker S (1998) Phys Rev B 58:3501

    Article  CAS  Google Scholar 

  101. Ismail-Beigi S, Arias TA (1999) Phys Rev Lett 82:2127

    Article  CAS  Google Scholar 

  102. Hastings MB (2004) Phys Rev Lett 93:126402

    Article  CAS  Google Scholar 

  103. Anderson PW (1958) Phys Rev 109:1492

    Article  CAS  Google Scholar 

  104. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipies in FORTRAN. Cambridge University Press, Port Chester, NY

    Google Scholar 

  105. Challacombe M (2000) Comput Phys Commun 128:93

    Article  CAS  Google Scholar 

  106. Saravanan C, Shao C, Baer R, Ross PN, Head-Gordon M (2003) J Comput Chem 24:618

    Article  CAS  Google Scholar 

  107. Goedecker S, Teter M (1995) Phys Rev B 51:9455

    Article  CAS  Google Scholar 

  108. Voter AF, Kress JD, Silver RN (1996) Phys Rev B 53:12733

    Article  CAS  Google Scholar 

  109. Liang WZ, Baer R, Saravanan C, Shao YH, Bell AT, Head-Gordon M (2004) J Comp Phys 194:575

    Article  Google Scholar 

  110. Weisse A, Wellein G, Alvermann A, Fehske H (2006) Rev Mod Phys 78:275

    Article  Google Scholar 

  111. Pessoa S (2002) Phys Rev B 46:14570

    Article  Google Scholar 

  112. Bergman A, Holmström E, Niklasson AMN, Nordström L, Frota-Pessoa S, Eriksson O (2004) Phys Rev B 70:174446

    Article  CAS  Google Scholar 

  113. Matsubara T (1955) Prog Theor Phys 14:351

    Article  Google Scholar 

  114. Goedecker S (1993) Phys Rev B 48:17573

    Article  CAS  Google Scholar 

  115. Liang WZ, Saravanan C, Shao Y, Baer R, Bell AT, Head-Gordon M (2003) J Chem Phys 119:4117

    Article  CAS  Google Scholar 

  116. Ozaki T (2007) Phys Rev B 75:035123

    Article  CAS  Google Scholar 

  117. Ceriotti M, Kühne TD, Parrinello M (2008) J Chem Phys 129:024707

    Article  CAS  Google Scholar 

  118. Moler C, Loan CV (2003) SIAM Rev 45:3

    Article  Google Scholar 

  119. Nemeth K, Scuseria GE (2000) J Chem Phys 113:6035

    Article  CAS  Google Scholar 

  120. Daniels AD, Scuseria GE (1999) J Chem Phys 110:1321

    Article  CAS  Google Scholar 

  121. Mazziotti DA (2001) J Chem Phys 115:8305

    Article  CAS  Google Scholar 

  122. Niklasson AMN, Tymczak CJ, Roder H (2002) Phys Rev B 66:155120

    Article  CAS  Google Scholar 

  123. Larsen H, Helgaker T, Olsen J, Jorgensen P (2001) J Chem Phys 115:9685

    Article  CAS  Google Scholar 

  124. Weber V, Niklasson AMN, Challacombe M (2004) Phys Rev Lett 92:193002

    Article  CAS  Google Scholar 

  125. Niklasson AMN, Weber V (1996) J Chem Phys 127:44107

    Google Scholar 

  126. Ochsenfeld C, Head-Gordon M (1997) Chem Phys Lett 270:399

    Article  CAS  Google Scholar 

  127. Ochsenfeld C, Kussmann J, Koziol F (2004) Angewandte Chemie 43:4485

    Article  CAS  Google Scholar 

  128. Izmaylov AF, Brothers EN, Scuseria GE (2006) J Chem Phys 125:224105

    Article  CAS  Google Scholar 

  129. Coriani S, Host S, Jasnik B, Thogersen L, Olsen J, Jorgensen P, Reine S, Pawlowski F, Helgaker T, Salek P (2007) J Chem Phys 126:154108

    Article  CAS  Google Scholar 

  130. Lucero MJ, Niklasson AMN, Tretiak S, Challacombe M (2008) J Chem Phys 129:064114

    Article  CAS  Google Scholar 

  131. Tretiak S, Isborn CM, Niklasson AMN, Challacombe M (2009) J Chem Phys 130:054111

    Article  CAS  Google Scholar 

  132. Yokojima S, Chen GH (1998) Chem Phys Lett 292:379

    Article  CAS  Google Scholar 

  133. Yabana K, Bertsch GF (1999) Int J Quantum Chem 75:55

    Article  CAS  Google Scholar 

  134. Tsolakidis A, Sanches-Portal D, Martin R (2002) Phys Rev B 66:235416

    Article  CAS  Google Scholar 

  135. Wang F, Yam CY, Chen GH, Fan K (2007) J Chem Phys 126:134104

    Article  CAS  Google Scholar 

  136. Niklasson AMN et al. Unpublished

    Google Scholar 

  137. Helgaker T, Jorgensen P, Olsen J (2002) Molecular electronic-structure theory, 1st ed. Wiley, New York, NY

    Google Scholar 

  138. McWeeny R (1962) Phys Rev 126:1028

    Article  Google Scholar 

  139. Kussmann J, Ochsenfeld C (2007) J Chem Phys 127:204103

    Article  CAS  Google Scholar 

Download references

Acknowledgments

I would like to thank my collaborators that have contributed, directly or indirectly to this work, including Igor Abrikosov, Anders Bergman, Nicolas Bock, Marc Cawkwell, Matt Challacombe, Eric Chisolm, Erik Holmström, Karoly Nemeth, Anders Odell, Travis Peery, Emanuel Rubensson, Heinrich Röder, Pawel Salek, CJ Tymczak, Valery Weber, and John Wills. In particular, I would like to thank the pleasant atmosphere at the Ten Bar International Science Cafe and Travis Peery for many stimulating discussions. We gratefully acknowledge the support of the U.S. Department of Energy through the LANL LDRD/ER program for this work.

Author information

Authors and Affiliations

  1. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Anders M. N. Niklasson

  2. Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, SE-100 44, Stockholm, Sweden

    Anders M. N. Niklasson

Authors
  1. Anders M. N. Niklasson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anders M. N. Niklasson .

Editor information

Editors and Affiliations

  1. Inst. Physical & Theoretical Chemistry, Wroclaw University of Technology, Wyb. Wyspianskiego 27, Warszawa, 50-370, Poland

    Robert Zalesny

  2. Institute of Organic &, Pharmaceutical Chemistry, National Hellenic Research Foundation, Vas. Constantinou Ave. 48, Athens, 116 35, Greece

    Manthos G. Papadopoulos

  3. Dept. Chemistry, University of Saskatchewan, Saskatoon, S7N 0W0, Saskatchewan, Canada

    Paul G. Mezey

  4. Jackson State University, J. R. Lynch St. 1325, Jackson, 39217, Mississippi, USA

    Jerzy Leszczynski

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Niklasson, A.M.N. (2011). Density Matrix Methods in Linear Scaling Electronic Structure Theory. In: Zalesny, R., Papadopoulos, M., Mezey, P., Leszczynski, J. (eds) Linear-Scaling Techniques in Computational Chemistry and Physics. Challenges and Advances in Computational Chemistry and Physics, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2853-2_16

Download citation

Publish with us

Policies and ethics

Search

Navigation