Derivatives of the approximated electrostatic potentials in unrestricted Hartree–Fock based on the fragment molecular orbital method and an application to polymer radicals

  • Hiroya NakataEmail author
  • Dmitri G. Fedorov
  • Satoshi Yokojima
  • Kazuo Kitaura
  • Shinichiro Nakamura
Regular Article


The analytic energy gradient for the point charge approximation of the embedding potential is derived in the framework of unrestricted Hartree–Fock based on the fragment molecular orbital method (FMO). For this goal, we derive the necessary coupled-perturbed unrestricted Hartree–Fock equations, describing the response terms arising from the use of embedding atomic charges in dimer calculations. By a comparison to numerical gradients and with the aid of molecular dynamics, we show that the gradients have a high accuracy. A speed-up of the factor 7.3 is obtained for the largest system, when approximated potentials are used relative to the exact two-electron embedding. We apply the FMO method to polymer radicals and show that it has satisfactory accuracy in reproducing the geometries and energies of polymer radical reactions.


Solvation Analytic gradient SCZV HF calculation Macromolecules Polymer Molecular orbital calculations 



We thank late Dr. Takeshi Nagata for helpful discussions about the FMO analytic energy gradient. This work was in part supported by the Next Generation Super Computing Project, Nanoscience Program (MEXT, Japan) and Computational Materials Science Initiative (CMSI, Japan). Most calculations were performed on TSUBAME2.0 at the Global Scientific Information and Computing Center of Tokyo Institute of Technology. We also thank the RIKEN Integrated Cluster of Clusters (RICC) at RIKEN and Research Center for Computational Science (Okazaki, Japan) for the computer resources.

Supplementary material

214_2014_1477_MOESM1_ESM.pdf (140 kb)
Supplementary material 1 (f 140 KB)


  1. 1.
    Goedecker S (1999) Rev Mod Phys 71:1085CrossRefGoogle Scholar
  2. 2.
    Scuseria GE (1999) J Phys Chem A 103:4782CrossRefGoogle Scholar
  3. 3.
    Li X, Milliam JM, Scuseria GE, Frisch MJ, Schlegel HB (2003) J Chem Phys 119:7651CrossRefGoogle Scholar
  4. 4.
    Mezey PG, Leszczynski J (2011) Linear-scaling techniques in computational chemistry and physics. Springer, New YorkGoogle Scholar
  5. 5.
    Reimers JR (2011) Computational methods for large systems: electronic structure approaches for biotechnology and nanotechnology. Wiley, New YorkCrossRefGoogle Scholar
  6. 6.
    Gordon MS, Fedorov DG, Pruitt SR, Slipchenko LV (2012) Chem Rev 112:632CrossRefGoogle Scholar
  7. 7.
    Otto P, Ladik J (1975) Chem Phys 8:192CrossRefGoogle Scholar
  8. 8.
    Yang W (1991) Phys Rev Lett 66:1438CrossRefGoogle Scholar
  9. 9.
    Gao JL (1997) J Phys Chem B 101:657CrossRefGoogle Scholar
  10. 10.
    Wang Y, Sosa CP, Cembran A, Truhlar DG, Gao J (2012) J Phys Chem B 116:6781CrossRefGoogle Scholar
  11. 11.
    Korchowiec J, Gu FL, Aoki Y (2005) Int J Quantum Chem 105:875CrossRefGoogle Scholar
  12. 12.
    Aoki Y, Gu FL (2012) Phys Chem Chem Phys 14:7640CrossRefGoogle Scholar
  13. 13.
    Chen XH, Zhang JZH (2004) J Theor Comput Chem 3:277CrossRefGoogle Scholar
  14. 14.
    Hua S, Li W, Li S (2013) Chem Phys Chem 14:108CrossRefGoogle Scholar
  15. 15.
    Gordon MS, Mullin JM, Pruitt SR, Roskop LB, Slipchenko LV, Boatz JA (2009) J Phys Chem B 113:9646CrossRefGoogle Scholar
  16. 16.
    Flick JC, Kosenkov D, Hohenstein EG, Sherrill CD, Slipchenko LV (2012) J Chem Theory Comput 8:2835CrossRefGoogle Scholar
  17. 17.
    Kobayashi M, Yoshikawa T, Nakai H (2010) Chem Phys Lett 500:172CrossRefGoogle Scholar
  18. 18.
    He X, Merz KM (2010) J Chem Theory Comput 6:405CrossRefGoogle Scholar
  19. 19.
    Kobayashi M, Nakai H (2012) Phys Chem Chem Phys 14:7629CrossRefGoogle Scholar
  20. 20.
    Collins MA (2012) Phys Chem Chem Phys 14:7744CrossRefGoogle Scholar
  21. 21.
    Huang L, Massa L (2012) Future Med Chem 4:1479CrossRefGoogle Scholar
  22. 22.
    Söderhjelm P, Kongsted J, Ryde U (2010) J Chem Theory Comput 6:1726CrossRefGoogle Scholar
  23. 23.
    Sahu N, Yeole SD, Gadre SR (2013) J Chem Phys 138:104101CrossRefGoogle Scholar
  24. 24.
    Frank A, Möller HM, Exner TE (2012) J Chem Theory Comput 8:1480CrossRefGoogle Scholar
  25. 25.
    Kurbanov EK, Leverentz HR, Truhlar DG, Amin EA (2012) J Chem Theory Comput 8:1CrossRefGoogle Scholar
  26. 26.
    Kitaura K, Ikeo E, Asada T, Nakano T, Uebayasi M (1999) Chem Phys Lett 313:701CrossRefGoogle Scholar
  27. 27.
    Fedorov DG, Kitaura K (2009) The fragment molecular orbital method: practical applications to large molecular systems. CRC Press, Boca Raton, FLGoogle Scholar
  28. 28.
    Fedorov DG, Kitaura K (2007) J Phys Chem A 111:6904CrossRefGoogle Scholar
  29. 29.
    Fedorov DG, Nagata T, Kitaura K (2012) Phys Chem Chem Phys 14:7562CrossRefGoogle Scholar
  30. 30.
    Steinmann C, Fedorov DG, Jensen JH (2013) PLoS One 8:e60602CrossRefGoogle Scholar
  31. 31.
    Sugiki SI, Kurita N, Sengoku Y, Sekino H (2003) Chem Phys Lett 382:611CrossRefGoogle Scholar
  32. 32.
    Fedorov DG, Kitaura K (2004) J Chem Phys 121:2483CrossRefGoogle Scholar
  33. 33.
    Fedorov DG, Kitaura K (2005) J Chem Phys 123:134103CrossRefGoogle Scholar
  34. 34.
    Pruitt SR, Fedorov DG, Kitaura K, Gordon MS (2010) J Chem Theory Comput 6:1CrossRefGoogle Scholar
  35. 35.
    Pruitt SR, Fedorov DG, Gordon MS (2012) J Phys Chem A 116:4965CrossRefGoogle Scholar
  36. 36.
    Fedorov DG, Kitaura K (2005) J Chem Phys 122:0541081CrossRefGoogle Scholar
  37. 37.
    Komeiji Y, Mochizuki Y, Nakano T, Mori H (2012) Recent advances in fragment molecular orbital-based molecular dynamics(FMO-MD) simulations. InTechGoogle Scholar
  38. 38.
    Nakata H, Fedorov DG, Nagata T, Yokojima S, Ogata K, Kitaura K, Nakamura S (2012) J Chem Phys 137:044110CrossRefGoogle Scholar
  39. 39.
    Fedorov DG, Avramov PV, Jensen JH, Kitaura K (2009) Chem Phys Lett 477:169CrossRefGoogle Scholar
  40. 40.
    Sawada T, Fedorov DG, Kitaura K (2010) J Am Chem Soc 132:16862CrossRefGoogle Scholar
  41. 41.
    Alexeev Y, Mazanetz MP, Ichihara O, Fedorov DG (2012) Curr Top Med Chem 12:2013CrossRefGoogle Scholar
  42. 42.
    Watanabe T, Inadomi Y, Fukuzawa K, Nakano T, Tanaka S, Nilsson L, Nagashima U (2007) J Phys Chem B 111:9621CrossRefGoogle Scholar
  43. 43.
    Carlson PJ, Bose S, Armstrong DW, Hawkins T, Gordon MS, Petrich JW (2012) J Phys Chem B 116:503CrossRefGoogle Scholar
  44. 44.
    Fukunaga H, Fedorov DG, Chiba M, Nii K, Kitaura K (2008) J Phys Chem A 112:10887CrossRefGoogle Scholar
  45. 45.
    Avramov PV, Fedorov DG, Sorokin PB, Sakai S, Entani S, Ohtomo M, Matsumoto Y, Naramoto H (2012) J Phys Chem Lett 3:2003Google Scholar
  46. 46.
    Roskop L, Fedorov DG, Gordon MS (2013) Mol Phys 111:1622CrossRefGoogle Scholar
  47. 47.
    Okiyama Y, Tsukamoto T, Watanabe C, Fukuzawa K, Tanaka S, Mochizuki Y (2013) Chem Phys Lett 566:25CrossRefGoogle Scholar
  48. 48.
    Sekino H, Matsumura N, Sengoku Y (2007) Comput Lett 3:423CrossRefGoogle Scholar
  49. 49.
    Gao Q, Yokojima S, Kohno T, Ishida T, Fedorov DG, Kitaura K, Fujihira M, Nakamura S (2007) Chem Phys Lett 445:331CrossRefGoogle Scholar
  50. 50.
    Gao Q, Yokojima S, Fedorov DG, Kitaura K, Sakurai M, Nakamura S (2010) J Chem Theory Comput 6:1428CrossRefGoogle Scholar
  51. 51.
    Fedorov DG, Kitaura K (2007) J Comput Chem 28:222CrossRefGoogle Scholar
  52. 52.
    Fedorov DG, Kitaura K (2012) J Phys Chem A 116:704CrossRefGoogle Scholar
  53. 53.
    Mochizuki Y, Fukuzawa K, Kato A, Tanaka S, Kitaura K, Nakano T (2005) Chem Phys Lett 410:247CrossRefGoogle Scholar
  54. 54.
    Ishikawa T, Mochizuki Y, Amari S, Nakano T, Tokiwa H, Tanaka S, Tanaka K (2007) Theor Chem Acc 118(5–6):937CrossRefGoogle Scholar
  55. 55.
    Green MC, Fedorov DG, Kitaura K, Francisco JS, Slipchenko LV (2013) J Chem Phys 138:074111CrossRefGoogle Scholar
  56. 56.
    Nakano T, Kaminuma T, Sato T, Fukuzawa K, Akiyama Y, Uebayasi M, Kitaura K (2002) Chem Phys Lett 351:475CrossRefGoogle Scholar
  57. 57.
    Kitaura K, Sugiki SI, Nakano T, Komeiji Y, Uebayasi M (2001) Chem Phys Lett 336(1,2):163CrossRefGoogle Scholar
  58. 58.
    Nagata T, Brorsen K, Fedorov DG, Kitaura K, Gordon MS (2011) J Chem Phys 134:124115CrossRefGoogle Scholar
  59. 59.
    Nagata T, Fedorov DG, Kitaura K (2009) Chem Phys Lett 475:124CrossRefGoogle Scholar
  60. 60.
    Nagata T, Fedorov DG, Kitaura K (2012) Chem Phys Lett 544:87CrossRefGoogle Scholar
  61. 61.
    Fedorov DG, Ishida T, Uebayasi M, Kitaura K (2007) J Phys Chem A 111:2722CrossRefGoogle Scholar
  62. 62.
    Fedorov DG, Alexeev Y, Kitaura K (2011) J Phys Chem Lett 2:282CrossRefGoogle Scholar
  63. 63.
    Komeiji Y, Nakano T, Fukuzawa K, Ueno Y, Inadomi Y, Nemoto T, Uebayasi M, Fedorov DG, Kitaura K (2003) Chem Phys Lett 372:342CrossRefGoogle Scholar
  64. 64.
    Komeiji Y, Ishikawa T, Mochizuki Y, Yamataka H, Nakano T (2009) J Comput Chem 30:40CrossRefGoogle Scholar
  65. 65.
    Fujita T, Watanabe H, Tanaka S (2009) J Phys Soc Jpn 78:104723CrossRefGoogle Scholar
  66. 66.
    Fujita T, Nakano T, Tanaka S (2011) Chem Phys Lett 506:112CrossRefGoogle Scholar
  67. 67.
    Brorsen KR, Minezawa N, Xu F, Windus TL, Gordon MS (2012) J Chem Theory Comput 8:5008CrossRefGoogle Scholar
  68. 68.
    Nakata H, Nagata T, Fedorov DG, Yokojima S, Kitaura K, Nakamura S (2013) J Chem Phys 138:164103CrossRefGoogle Scholar
  69. 69.
    Sato M, Yamataka H, Komeiji Y, Mochizuki Y, Ishikawa T, Nakano T (2008) J Am Chem Soc 130:2396CrossRefGoogle Scholar
  70. 70.
    Sato M, Yamataka H, Komeiji Y, Mochizuki Y, Nakano T (2010) Chem Eur J 16:6430CrossRefGoogle Scholar
  71. 71.
    Lange AW, Voth GA (2013) J Chem Theory Comput 9(9):4018. doi: 10.1021/ct400516x CrossRefGoogle Scholar
  72. 72.
    Xie W, Orozco M, Gao J, Truhlar DG (2009) J Chem Theory Comput 5:459Google Scholar
  73. 73.
    Ufimtsev IS, Luehr N, Martinez TJ (2011) J Chem Theory Comput 2:1789Google Scholar
  74. 74.
    Kacar G, Atilgan C, Özen AS (2010) J Phys Chem C 114:370CrossRefGoogle Scholar
  75. 75.
    Nagaoka M, Ohta Y, Hitomi H (2007) Coord Chem Rev 251:2522CrossRefGoogle Scholar
  76. 76.
    Elliott JA, Paddison SJ (2007) Phys Chem Chem Phys 9:2602CrossRefGoogle Scholar
  77. 77.
    Karttunen M, Vattulainen I, Lukkarinen A (2004) Novel methods in soft matter simulations. Springer, BerlinCrossRefGoogle Scholar
  78. 78.
    Morales G, Martinez R (2009) J Phys Chem A 113:8683CrossRefGoogle Scholar
  79. 79.
    Zade SS, Bendikov M (2007) Chem Eur J 13:3688CrossRefGoogle Scholar
  80. 80.
    Suhai S (1980) J Chem Phys 73:3843CrossRefGoogle Scholar
  81. 81.
    Hirata S (1998) Phys Rev B 57:11994CrossRefGoogle Scholar
  82. 82.
    Aoki Y, Imamura A, Sasaki T (1988) Bull Chem Soc Jpn 61:1063CrossRefGoogle Scholar
  83. 83.
    Hirata S (2009) Phys Chem Chem Phys 11:8397CrossRefGoogle Scholar
  84. 84.
    Moscatelli D, Cavallotti C, Morbidelli M (2006) Macromolecules 39:9641CrossRefGoogle Scholar
  85. 85.
    Xie W, Song L, Truhlar DG, Gao J (2008) J Chem Phys 128:234108CrossRefGoogle Scholar
  86. 86.
    Hratchian HP, Parandekar PV, Raghavachari K, Frisch MJ, Vreven T (2008) J Chem Phys 128:034107CrossRefGoogle Scholar
  87. 87.
    Mayhall NJ, Raghavachari K, Hratchian HP (2010) J Chem Phys 132:114107CrossRefGoogle Scholar
  88. 88.
    Baker J, Kessi A, Delley B (1996) J Chem Phys 105:192CrossRefGoogle Scholar
  89. 89.
    Nagata T, Fedorov DG, Kitaura K (2010) Chem Phys Lett 492:302CrossRefGoogle Scholar
  90. 90.
    Yamaguchi Y, Schaefer HF III, Osamura Y, Goddard J (1994) A new dimension to quantum chemistry: analytical derivative methods in ab initio molecular electronic structure theory. Oxford University Press, New YorkGoogle Scholar
  91. 91.
    Handy NC, Schaefer HF III (1984) J Chem Phys 81:5031CrossRefGoogle Scholar
  92. 92.
    Ochsenfeld C, Gordon MS (1997) Chem Phys Lett 270:399CrossRefGoogle Scholar
  93. 93.
    Clayden J, Greeves N, Warren S, Wothers P (2001) Organic chemistry. Oxford University Press, New YorkGoogle Scholar
  94. 94.
    Valiev M, Bylaska EJ, Govind N, Kowalski K, Straatsma TP, van Dam HJJ, Wang D, Nieplocha J, Apra E, Windus TL, de Jong WA (2010) Comput Phys Commun 181:1477CrossRefGoogle Scholar
  95. 95.
    Wang J, Cieplak P, Kollman PA (2000) J Comput Chem 21:1049CrossRefGoogle Scholar
  96. 96.
    Schmidt NW, Baldridge KK, Baldridge JA, Boatz JA, Elbert ST, Gordon MS, Jensen JJ, Koseki S, Matsunaga N, Nguyen KA, Su S, Windus TL, Dupuis M, Montgomery JA (1993) J Comput Chem 14:1347CrossRefGoogle Scholar
  97. 97.
    Fedorov DG, Kitaura K (2004) J Chem Phys 120(15):6832CrossRefGoogle Scholar
  98. 98.
    Fedorov DG, Olson RM, Kitaura K, Gordon MS, Koseki S (2004) J Comput Chem 25:872CrossRefGoogle Scholar
  99. 99.
    Andersen HC (1983) J Comput Phys 52:24CrossRefGoogle Scholar
  100. 100.
    Allen MP, Tildesley DJ (1987) Computer simulation of liquids. Oxford University Press, New YorkGoogle Scholar
  101. 101.
    Benoit D, Hawker CJ, Huang EE, Lin Z, Russell TP (2000) Macromolecules 33:1505CrossRefGoogle Scholar
  102. 102.
    Grimme S, Antony J, Ehrlich S, Krieg H (2010) J Chem Phys 132:154104CrossRefGoogle Scholar
  103. 103.
    Fedorov DG, Kitaura K (2006) Theoretical development of the fragment molecular orbital (FMO) method, chap. 1. Elsevier, Amsterdam, pp 3–38Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Hiroya Nakata
    • 1
    • 2
    Email author
  • Dmitri G. Fedorov
    • 3
  • Satoshi Yokojima
    • 2
    • 4
  • Kazuo Kitaura
    • 5
  • Shinichiro Nakamura
    • 2
  1. 1.Department of Biomolecular EngineeringTokyo Institute of TechnologyYokohamaJapan
  2. 2.Research Cluster for InnovationWakoJapan
  3. 3.NRINational Institute of Advanced Industrial Science and Technology (AIST)TsukubaJapan
  4. 4.Tokyo University of Pharmacy and Life SciencesTokyoJapan
  5. 5.Graduate School of System InformaticsKobe UniversityKobeJapan

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