Does DFT+U mimic hybrid density functionals?

  • Pragya Verma
  • Donald G. TruhlarEmail author
Regular Article


This work examines the question of how a Hubbard U correction to a local exchange–correlation functional compares with adding Hartree–Fock exchange to a local functional for both solid-state and molecular properties. We compute a solid-state property, namely the band gap, and thermochemical molecular properties, in particular, main-group bond energies, transition metal–ligand bond energies, and barrier heights, to elucidate whether the DFT+U method mimics hybrid DFT. We find that a calculation with a Hubbard U correction may or may not mimic a hybrid functional—depending on the atom, the subshell, and the property to which it is applied. For band gaps, we find that adding a Hubbard U correction to the valence d orbitals of transition metals increases the band gap, which thereby gets closer to the experimental value, while adding a Hubbard U correction to valence s or p orbitals of main-group elements need not always increase the band gap. For molecular thermochemistry, we find that adding a Hubbard U correction to a local density functional need not have the same effect as adding Hartree–Fock exchange to a local density functional. For example when compared to a DFT calculation with a local exchange-correlation functional, hybrid DFT increases the barrier height in all cases, but DFT+U does not always increase the barrier height. For the band gaps of transition metal monoxides, the Hubbard-corrected results lowered the mean errors significantly and were comparable to what could be achieved with a much more expensive hybrid functional, but for reaction barrier heights and bond energies of molecules, the Hubbard correction was found to lower the mean error by only approximately a kcal/mol. As part of the analysis, we also compare VASP and Gaussian 09 calculations for the same density functional.


Atomization energy Band gap Barrier height Bond energy Density functional theory Hubbard U correction Molecular thermochemistry Solid-state physics 



The authors thank Kaining Duanmu, Shuping Huang, Georg Kresse, and Haoyu Yu for helpful discussions. This work was supported by the Nanoporous Materials Genome Center funded by the US Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, under award DE-FG02-12ER16362.

Compliance with ethical standards

Conflict of interest

The authors declare no competing financial interest.

Supplementary material

214_2016_1927_MOESM1_ESM.pdf (347 kb)
Supplementary material 1 (PDF 348 kb)


  1. 1.
    Kohn W, Sham LJ (1965) Phys Rev 140:A1133CrossRefGoogle Scholar
  2. 2.
    Kohn W, Becke AD, Parr RG (1996) J Phys Chem 100:12974CrossRefGoogle Scholar
  3. 3.
    Kohn W (1999) Rev Mod Phys 71:1253CrossRefGoogle Scholar
  4. 4.
    Becke AD (1993) J Chem Phys 98:5648CrossRefGoogle Scholar
  5. 5.
    Seidl A, Görling A, Vogl P, Majewski JI, Levy M (1996) Phys Rev B 53:3764CrossRefGoogle Scholar
  6. 6.
    Hafner J (2008) J Comput Chem 29:2044CrossRefGoogle Scholar
  7. 7.
    Marsman M, Paier J, Stroppa A, Kresse G (2008) J Phys Condens Matter 20:064201CrossRefGoogle Scholar
  8. 8.
    Stephens PJ, Devlin FJ, Chabalowski CF, Frisch MJ (1994) J Phys Chem 98:11623CrossRefGoogle Scholar
  9. 9.
    Heyd J, Scuseria GE, Ernzerhof M (2003) J Chem Phys 118:8207CrossRefGoogle Scholar
  10. 10.
    Zhao Y, Truhlar DG (2008) Theor Chem Acc 120:215CrossRefGoogle Scholar
  11. 11.
    Anisimov VI, Zaanen J, Andersen OK (1991) Phys Rev B 44:943CrossRefGoogle Scholar
  12. 12.
    Anisimov VI, Aryasetiawan F, Liechtenstein AI (1997) J Phys Condens Matter 9:767CrossRefGoogle Scholar
  13. 13.
    Dudarev SL, Liechtenstein AI, Castell MR, Briggs GAD, Sutton AP (1997) Phys Rev B 56:4900CrossRefGoogle Scholar
  14. 14.
    Rohrbach A, Hafner J, Kresse G (2003) J Phys Condens Matter 15:979CrossRefGoogle Scholar
  15. 15.
    Mosey NJ, Carter EA (2007) Phys Rev B 76:155123CrossRefGoogle Scholar
  16. 16.
    Mosey NJ, Liao P, Carter EA (2008) J Chem Phys 129:014103CrossRefGoogle Scholar
  17. 17.
    Himmetoglu B, Floris A, de Gironcoli S, Cococcioni M (2014) Int J Quantum Chem 114:14CrossRefGoogle Scholar
  18. 18.
    Kulik HJ (2015) J Chem Phys 142:240901CrossRefGoogle Scholar
  19. 19.
    Hubbard J (1964) Proc R Soc Lond Ser A 277:455CrossRefGoogle Scholar
  20. 20.
    Ivády V, Armiento R, Szász K, Janzén E, Gali A, Abrikosov IA (2014) Phys Rev B 90:035146Google Scholar
  21. 21.
    Carter EA (2008) Science 321:800CrossRefGoogle Scholar
  22. 22.
    Cococcioni M, de Gironcoli M (2005) Phys Rev B 71:035105CrossRefGoogle Scholar
  23. 23.
    Adamo C, Barone VJ (1999) J Chem Phys 110:6158CrossRefGoogle Scholar
  24. 24.
    Verma P, Maurice R, Truhlar DG (2016) J Phys Chem C 120:9933CrossRefGoogle Scholar
  25. 25.
    Borycz J, Paier J, Verma P, Darago LE, Xiao DJ, Truhlar DG, Long JR, Gagliardi L (2016) Inorg Chem 55:4924CrossRefGoogle Scholar
  26. 26.
    Huang S, Wilson B, Wang B, Fang Y, Buffington K, Stein A, Truhlar DG (2015) J Am Chem Soc 137:10992CrossRefGoogle Scholar
  27. 27.
    Huang S, Wilson B, Smyrl WH, Truhlar DG, Stein A (2016) Chem Mater 28:746CrossRefGoogle Scholar
  28. 28.
    Heyd J, Scuseria GE (2004) J Chem Phys 120:7274CrossRefGoogle Scholar
  29. 29.
    Heyd J, Peralta JE, Scuseria GE, Martin RL (2005) J Chem Phys 123:174101CrossRefGoogle Scholar
  30. 30.
    Heyd J, Scuseria GE, Ernzerhof M (2006) J Chem Phys 124:219901CrossRefGoogle Scholar
  31. 31.
    Paier J, Marsman M, Hummer K, Kresse G, Gerber IC, Angyan JG (2006) J Chem Phys 125:249901CrossRefGoogle Scholar
  32. 32.
    Yu HS, Zhang W, Verma P, He X, Truhlar DG (2015) Phys Chem Chem Phys 17:12146CrossRefGoogle Scholar
  33. 33.
    Kresse G, Furthmüller J (1996) Comput Mater Sci 6:15CrossRefGoogle Scholar
  34. 34.
    Kresse G, Furthmüller J (1996) Phys Rev B Condens Matter Mater Phys 54:11169CrossRefGoogle Scholar
  35. 35.
    Duanmu K, Luo S, Truhlar DG Minnesota-VASP functional module (MN-VFM—version 3.0). See for details
  36. 36.
    Peverati R, Truhlar DG (2012) J Chem Theory Comput 8:2310CrossRefGoogle Scholar
  37. 37.
    Roth WL (1958) Phys Rev 110:1333CrossRefGoogle Scholar
  38. 38.
    Terakura K, Williams AR, Oguchi T, Kübler J (1984) Phys Rev Lett 52:1830CrossRefGoogle Scholar
  39. 39.
    Shih B-C, Abtew TA, Yuan X, Zhang W, Zhang P (2012) Phys Rev B 86:165124CrossRefGoogle Scholar
  40. 40.
    Schrön A, Rödl C, Bechstedt F (2012) Phys Rev B 86:115134CrossRefGoogle Scholar
  41. 41.
    Yan J, Nørskov JK (2013) Phys Rev B 88:245204CrossRefGoogle Scholar
  42. 42.
    Sakuma R, Aryasetiawan F (2013) Phys Rev B 87:165118CrossRefGoogle Scholar
  43. 43.
    Zhao Y, Truhlar DG (2009) J Chem Phys 130:074103CrossRefGoogle Scholar
  44. 44.
    Peverati R, Truhlar DG (2012) J Chem Phys 136:134704CrossRefGoogle Scholar
  45. 45.
    Matz R, Luth H (1979) Appl Phys 18:123CrossRefGoogle Scholar
  46. 46.
    Tran F, Blaha P, Schwarz K, Novák P (2006) Phys Rev B 74:155108CrossRefGoogle Scholar
  47. 47.
    Parmigiani F, Sangaletti L (1999) J Electron Spectrosc Relat Phenom 98–99:287CrossRefGoogle Scholar
  48. 48.
  49. 49.
    Lynch BJ, Truhlar DG (2003) J Phys Chem A 107:8996CrossRefGoogle Scholar
  50. 50.
    Zhao Y, Schultz NE, Truhlar DG (2006) J Chem Theory Comput 2:364CrossRefGoogle Scholar
  51. 51.
    Carlson RK, Li Manni G, Sonnenberger AL, Truhlar DG, Gagliardi L (2015) J Chem Theory Comput 11:82CrossRefGoogle Scholar
  52. 52.
    Schultz NE, Zhao Y, Truhlar DG (2005) J Phys Chem A 109:11127CrossRefGoogle Scholar
  53. 53.
    Peverati R, Truhlar DG (2014) Philos Trans R Soc A 372:20120476CrossRefGoogle Scholar
  54. 54.
    Zhang W, Truhlar DG, Tang M (2013) J Chem Theory Comput 9:3965CrossRefGoogle Scholar
  55. 55.
    Zhao Y, Lynch BJ, Truhlar DG (2005) Phys Chem Chem Phys 7:43CrossRefGoogle Scholar
  56. 56.
    Zhao Y, Gonzaĺez-Garcıá N, Truhlar DG (2005) J Phys Chem A 109:2012CrossRefGoogle Scholar
  57. 57.
    Zheng J, Zhao Y, Truhlar DG (2009) J Chem Theory Comput 5:808CrossRefGoogle Scholar
  58. 58.
    Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson GA, Nakatsuji H, Caricato M, Li X, Hratchian HP, Izmaylov AF, Bloino J, Zheng G, Sonnenberg JL, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Kudin YN, Staroverov VN, Kobayashi R, Normand J, Raghavachari K, Rendell A, Burant JC, Iyengar SS, Tomasi J, Cossi M, Rega N, Millam JM, Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Martin RL, Morokuma K, Zakrzewski VG, Voth GA, Salvador P, Dannenberg JJ, Dapprich S, Daniels AD, Farkas Ö, Foresman JB, Ortiz JV, Cioslowski J, Fox DJ (2010) Gaussian 09 Rev. C01. Gaussian Inc., WallingfordGoogle Scholar
  59. 59.
    Zhao Y, Peverati R, Luo S, Yang KR, He X, Yu HS, Truhlar DG Minnesota-Gaussian functional module (MN-GFM, version 6.5). See for details
  60. 60.
    Perdew JP, Burke K, Ernzerhof M (1996) Phys Rev Lett 77:3865CrossRefGoogle Scholar
  61. 61.
    Kudin K, Scuseria GE (2000) Phys Rev B 61:16440CrossRefGoogle Scholar
  62. 62.
    Lynch BJ, Zhao Y, Truhlar DG (2003) J Phys Chem A 107:1384CrossRefGoogle Scholar
  63. 63.
    Weigend F, Ahlrichs R (2005) Phys Chem Chem Phys 7:3297CrossRefGoogle Scholar
  64. 64.
    Zheng J, Xu X, Truhlar DG (2011) Theor Chem Acc 128:295CrossRefGoogle Scholar
  65. 65.
    Papajak E, Truhlar DG (2011) J Chem Theory Comput 7:10CrossRefGoogle Scholar
  66. 66.
    Marenich AV, Jerome SV, Cramer CJ, Truhlar DG (2012) J Chem Theory Comput 8:527CrossRefGoogle Scholar
  67. 67.
    Hirshfeld FL (1977) Theor Chim Acta 44:129CrossRefGoogle Scholar
  68. 68.
    Duanmu K, Wang B, Marenich AV, Cramer CJ, Truhlar DG (2015) CM5PAC. University of Minnesota, MinneapolisGoogle Scholar
  69. 69.
    Dudarev SL, Botton GA, Savroasov SY, Humphreys CJ, Sutton AP (1998) Phys Rev B 57:1505CrossRefGoogle Scholar
  70. 70.
    Blöchl PE (1994) Phys Rev B Condens Matter Mater Phys 50:17953CrossRefGoogle Scholar
  71. 71.
    Kresse G, Joubert D (1999) Phys Rev B Condens Matter Mater Phys 59:1758CrossRefGoogle Scholar
  72. 72.
    Manz TA Chargemol program for performing DDEC analysis, version 2.2 beta, May 25, 2013. ddec.sourceforge.netGoogle Scholar
  73. 73.
    Da Silva JLF, Ganduglia-Pirovano MV, Sauer J, Bayer V, Kresse G (2007) Phys Rev B Condens Matter Mater Phys 75:045121CrossRefGoogle Scholar
  74. 74.
    Singh V, Kosa M, Majhi K, Major DT (2015) J Chem Theory Comput 11:64CrossRefGoogle Scholar
  75. 75.
    Bui VQ, Pham T-T, Le DA, Thi CM, Le HM (2015) J Phys Condens Matter 27:305005CrossRefGoogle Scholar
  76. 76.
    Xu Z, Joshi YV, Raman S, Kitchin JR (2015) J Chem Phys 142:144701CrossRefGoogle Scholar
  77. 77.
    Iwaszuk A, Nolan M (2011) J Phys Chem C 115:12995CrossRefGoogle Scholar
  78. 78.
    Zakrzewski T, Boguslawski P (2016) J Alloys Compd 664:565CrossRefGoogle Scholar
  79. 79.
    Yang Y, Sugino O, Ohno T (2012) AIP Adv 2:022172CrossRefGoogle Scholar
  80. 80.
    Li W, Walther CFJ, Kuc A, Heine T (2013) J Chem Theory Comput 9:2950CrossRefGoogle Scholar
  81. 81.
    Chen J, Wu X, Selloni A (2011) Phys Rev B Condens Matter Mater Phys 83:245204CrossRefGoogle Scholar
  82. 82.
    Franchini C, Bayer V, Podloucky R, Paier J, Kresse G (2005) Phys Rev B Condens Matter Mater Phys 72:045132CrossRefGoogle Scholar
  83. 83.
    Finazzi E, Di Valentin C, Pacchioni G, Selloni A (2008) J Chem Phys 129:154113CrossRefGoogle Scholar
  84. 84.
    Perdew JP, Wang Y (1992) Phys Rev B 45:244CrossRefGoogle Scholar
  85. 85.
    Essenberger F, Sharma S, Dewhurst JK, Bersier C, Cricchio F, Nordström L, Gross EKU (2011) Phys Rev B 84:174425CrossRefGoogle Scholar
  86. 86.
    Pickett WE, Erwin SC, Ethridge EC (1998) Phys Rev B 58:1201CrossRefGoogle Scholar
  87. 87.
    Jiang H, Gomez-Abal RI, Rinke P, Scheffler M (2010) Phys Rev B 82:045108CrossRefGoogle Scholar
  88. 88.
    Paier J, Hirschl R, Marsman M, Kresse G (2005) J Chem Phys 122:234102CrossRefGoogle Scholar
  89. 89.
    Woon DE, Dunning TH Jr (1993) J Chem Phys 98:1358CrossRefGoogle Scholar
  90. 90.
    Woon DE, Dunning TH Jr (1994) J Chem Phys 100:2975CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Chemistry, Chemical Theory Center, Nanoporous Materials Genome Center, and Minnesota Supercomputing InstituteUniversity of MinnesotaMinneapolisUSA

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