No Access Submitted: 02 February 2018 Accepted: 23 March 2018 Published Online: 09 April 2018
J. Chem. Phys. 148, 241725 (2018); https://doi.org/10.1063/1.5024577
The accurate representation of multidimensional potential energy surfaces is a necessary requirement for realistic computer simulations of molecular systems. The continued increase in computer power accompanied by advances in correlated electronic structure methods nowadays enables routine calculations of accurate interaction energies for small systems, which can then be used as references for the development of analytical potential energy functions (PEFs) rigorously derived from many-body (MB) expansions. Building on the accuracy of the MB-pol many-body PEF, we investigate here the performance of permutationally invariant polynomials (PIPs), neural networks, and Gaussian approximation potentials (GAPs) in representing water two-body and three-body interaction energies, denoting the resulting potentials PIP-MB-pol, Behler-Parrinello neural network-MB-pol, and GAP-MB-pol, respectively. Our analysis shows that all three analytical representations exhibit similar levels of accuracy in reproducing both two-body and three-body reference data as well as interaction energies of small water clusters obtained from calculations carried out at the coupled cluster level of theory, the current gold standard for chemical accuracy. These results demonstrate the synergy between interatomic potentials formulated in terms of a many-body expansion, such as MB-pol, that are physically sound and transferable, and machine-learning techniques that provide a flexible framework to approximate the short-range interaction energy terms.
This work was supported by the National Science Foundation through Grant No. ACI-1642336 (to F.P. and A.W.G.). This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI-1548562. J.B. is grateful for a Heisenberg professorship funded by the DFG (No. Be3264/11-2). E.Sz. would like to acknowledge the support of the Peterhouse Research Studentship and the support of BP International Centre for Advanced Materials (ICAM). M.C. was supported by the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 677013-HBMAP). G.I. acknowledges funding from the Fondazione Zegna.
  1. 1. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953). https://doi.org/10.1063/1.1699114, Google ScholarScitation, ISI
  2. 2. W. Wood and F. Parker, J. Chem. Phys. 27, 720 (1957). https://doi.org/10.1063/1.1743822, Google ScholarScitation, ISI
  3. 3. B. J. Alder and T. E. Wainwright, J. Chem. Phys. 31, 459 (1959). https://doi.org/10.1063/1.1730376, Google ScholarScitation, ISI
  4. 4. B. Alder and T. Wainwright, J. Chem. Phys. 33, 1439 (1960). https://doi.org/10.1063/1.1731425, Google ScholarScitation, ISI
  5. 5. P. Ball, Chem. Rev. 108, 74 (2008). https://doi.org/10.1021/cr068037a, Google ScholarCrossref, ISI
  6. 6. B. Guillot, J. Mol. Liq. 101, 219 (2002). https://doi.org/10.1016/s0167-7322(02)00094-6, Google ScholarCrossref, ISI
  7. 7. C. Vega and J. L. Abascal, Phys. Chem. Chem. Phys. 13, 19663 (2011). https://doi.org/10.1039/c1cp22168j, Google ScholarCrossref, ISI
  8. 8. I. Shvab and R. J. Sadus, Fluid Phase Equilib. 407, 7 (2016). https://doi.org/10.1016/j.fluid.2015.07.040, Google ScholarCrossref
  9. 9. G. A. Cisneros, K. T. Wikfeldt, L. Ojamäe, J. Lu, Y. Xu, H. Torabifard, A. P. Bartók, G. Csányi, V. Molinero, and F. Paesani, Chem. Rev. 116, 7501 (2016). https://doi.org/10.1021/acs.chemrev.5b00644, Google ScholarCrossref, ISI
  10. 10. J. Barker and R. Watts, Chem. Phys. Lett. 3, 144 (1969). https://doi.org/10.1016/0009-2614(69)80119-3, Google ScholarCrossref, ISI
  11. 11. A. Rahman and F. H. Stillinger, J. Chem. Phys. 55, 3336 (1971). https://doi.org/10.1063/1.1676585, Google ScholarScitation, ISI
  12. 12. J. Mayer and M. Mayer, Statistical Mechanics (John Wiley, New York, 1940). Google Scholar
  13. 13. H. J. Berendsen, J. P. Postma, W. F. van Gunsteren, and J. Hermans, Intermolecular Forces (Springer, 1981), pp. 331–342. Google ScholarCrossref
  14. 14. W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L. Klein, J. Chem. Phys. 79, 926 (1983). https://doi.org/10.1063/1.445869, Google ScholarScitation, ISI
  15. 15. H. Berendsen, J. Grigera, and T. Straatsma, J. Phys. Chem. 91, 6269 (1987). https://doi.org/10.1021/j100308a038, Google ScholarCrossref, ISI
  16. 16. L. X. Dang and B. M. Pettitt, J. Phys. Chem. 91, 3349 (1987). https://doi.org/10.1021/j100296a048, Google ScholarCrossref, ISI
  17. 17. D. M. Ferguson, J. Comput. Chem. 16, 501 (1995). https://doi.org/10.1002/jcc.540160413, Google ScholarCrossref
  18. 18. M. W. Mahoney and W. L. Jorgensen, J. Chem. Phys. 112, 8910 (2000). https://doi.org/10.1063/1.481505, Google ScholarScitation, ISI
  19. 19. H. W. Horn, W. C. Swope, J. W. Pitera, J. D. Madura, T. J. Dick, G. L. Hura, and T. Head-Gordon, J. Chem. Phys. 120, 9665 (2004). https://doi.org/10.1063/1.1683075, Google ScholarScitation, ISI
  20. 20. Y. Wu, H. L. Tepper, and G. A. Voth, J. Chem. Phys. 124, 024503 (2006). https://doi.org/10.1063/1.2136877, Google ScholarScitation, ISI
  21. 21. F. Paesani, W. Zhang, D. A. Case, T. E. Cheatham III, and G. A. Voth, J. Chem. Phys. 125, 184507 (2006). https://doi.org/10.1063/1.2386157, Google ScholarScitation, ISI
  22. 22. S. Habershon, T. E. Markland, and D. E. Manolopoulos, J. Chem. Phys. 131, 024501 (2009). https://doi.org/10.1063/1.3167790, Google ScholarScitation, ISI
  23. 23. K. Park, W. Lin, and F. Paesani, J. Phys. Chem. B 116, 343 (2011). https://doi.org/10.1021/jp208946p, Google ScholarCrossref
  24. 24. I. S. Joung and T. E. Cheatham III, J. Phys. Chem. B 112, 9020 (2008). https://doi.org/10.1021/jp8001614, Google ScholarCrossref, ISI
  25. 25. H. S. Frank and W.-Y. Wen, Discuss. Faraday Soc. 24, 133 (1957). https://doi.org/10.1039/df9572400133, Google ScholarCrossref
  26. 26. T. P. Lybrand and P. A. Kollman, J. Chem. Phys. 83, 2923 (1985). https://doi.org/10.1063/1.449246, Google ScholarScitation, ISI
  27. 27. O. Matsuoka, E. Clementi, and M. Yoshimine, J. Chem. Phys. 64, 1351 (1976). https://doi.org/10.1063/1.432402, Google ScholarScitation, ISI
  28. 28. G. Lie and E. Clementi, Phys. Rev. A 33, 2679 (1986). https://doi.org/10.1103/physreva.33.2679, Google ScholarCrossref
  29. 29. A. Lyubartsev and A. Laaksonen, Chem. Phys. Lett. 325, 15 (2000). https://doi.org/10.1016/s0009-2614(00)00592-3, Google ScholarCrossref
  30. 30. K. Honda and K. Kitaura, Chem. Phys. Lett. 140, 53 (1987). https://doi.org/10.1016/0009-2614(87)80416-5, Google ScholarCrossref
  31. 31. U. Niesar, G. Corongiu, M.-J. Huang, M. Dupuis, and E. Clementi, Int. J. Quantum Chem. 36, 421 (1989). https://doi.org/10.1002/qua.560360845, Google ScholarCrossref
  32. 32. U. Niesar, G. Corongiu, E. Clementi, G. Kneller, and D. Bhattacharya, J. Phys. Chem. 94, 7949 (1990). https://doi.org/10.1021/j100383a037, Google ScholarCrossref
  33. 33. F. H. Stillinger and C. W. David, J. Chem. Phys. 69, 1473 (1978). https://doi.org/10.1063/1.436773, Google ScholarScitation, ISI
  34. 34. L. X. Dang and T.-M. Chang, J. Chem. Phys. 106, 8149 (1997). https://doi.org/10.1063/1.473820, Google ScholarScitation, ISI
  35. 35. C. J. Burnham and S. S. Xantheas, J. Chem. Phys. 116, 1479 (2002). https://doi.org/10.1063/1.1423940, Google ScholarScitation, ISI
  36. 36. S. S. Xantheas, C. J. Burnham, and R. J. Harrison, J. Chem. Phys. 116, 1493 (2002). https://doi.org/10.1063/1.1423941, Google ScholarScitation, ISI
  37. 37. C. J. Burnham and S. S. Xantheas, J. Chem. Phys. 116, 1500 (2002). https://doi.org/10.1063/1.1423942, Google ScholarScitation, ISI
  38. 38. C. J. Burnham and S. S. Xantheas, J. Chem. Phys. 116, 5115 (2002). https://doi.org/10.1063/1.1447904, Google ScholarScitation, ISI
  39. 39. G. S. Fanourgakis and S. S. Xantheas, J. Chem. Phys. 128, 074506 (2008). https://doi.org/10.1063/1.2837299, Google ScholarScitation, ISI
  40. 40. C. J. Burnham, D. J. Anick, P. K. Mankoo, and G. F. Reiter, J. Chem. Phys. 128, 154519 (2008). https://doi.org/10.1063/1.2895750, Google ScholarScitation, ISI
  41. 41. P. Ren and J. W. Ponder, J. Phys. Chem. B 107, 5933 (2003). https://doi.org/10.1021/jp027815+, Google ScholarCrossref, ISI
  42. 42. L.-P. Wang, T. Head-Gordon, J. W. Ponder, P. Ren, J. D. Chodera, P. K. Eastman, T. J. Martinez, and V. S. Pande, J. Phys. Chem. B 117, 9956 (2013). https://doi.org/10.1021/jp403802c, Google ScholarCrossref, ISI
  43. 43. A. P. Bartók, M. J. Gillan, F. R. Manby, and G. Csányi, Phys. Rev. B 88, 054104 (2013). https://doi.org/10.1103/physrevb.88.054104, Google ScholarCrossref
  44. 44. T. Morawietz, A. Singraber, C. Dellago, and J. Behler, Proc. Natl. Acad. Sci. U. S. A. 113, 8368 (2016). https://doi.org/10.1073/pnas.1602375113, Google ScholarCrossref, ISI
  45. 45. B. J. Braams and J. M. Bowman, Int. Rev. Phys. Chem. 28, 577 (2009). https://doi.org/10.1080/01442350903234923, Google ScholarCrossref, ISI
  46. 46. J. Behler, Angew. Chem., Int. Ed. 56, 12828 (2017). https://doi.org/10.1002/anie.201703114, Google ScholarCrossref, ISI
  47. 47. R. Bukowski, K. Szalewicz, G. C. Groenenboom, and A. van der Avoird, Science 315, 1249 (2007). https://doi.org/10.1126/science.1136371, Google ScholarCrossref, ISI
  48. 48. Y. Wang, X. Huang, B. C. Shepler, B. J. Braams, and J. M. Bowman, J. Chem. Phys. 134, 094509 (2011). https://doi.org/10.1063/1.3554905, Google ScholarScitation, ISI
  49. 49. V. Babin, G. R. Medders, and F. Paesani, J. Phys. Chem. Lett. 3, 3765 (2012). https://doi.org/10.1021/jz3017733, Google ScholarCrossref, ISI
  50. 50. V. Babin, C. Leforestier, and F. Paesani, J. Chem. Theory Comput. 9, 5395 (2013). https://doi.org/10.1021/ct400863t, Google ScholarCrossref, ISI
  51. 51. V. Babin, G. R. Medders, and F. Paesani, J. Chem. Theory Comput. 10, 1599 (2014). https://doi.org/10.1021/ct500079y, Google ScholarCrossref, ISI
  52. 52. G. R. Medders, V. Babin, and F. Paesani, J. Chem. Theory Comput. 10, 2906 (2014). https://doi.org/10.1021/ct5004115, Google ScholarCrossref, ISI
  53. 53. F. Paesani, Acc. Chem. Res. 49, 1844 (2016). https://doi.org/10.1021/acs.accounts.6b00285, Google ScholarCrossref
  54. 54. J. O. Richardson, C. Pérez, S. Lobsiger, A. A. Reid, B. Temelso, G. C. Shields, Z. Kisiel, D. J. Wales, B. H. Pate, and S. C. Althorpe, Science 351, 1310 (2016). https://doi.org/10.1126/science.aae0012, Google ScholarCrossref, ISI
  55. 55. W. T. S. Cole, J. D. Farrell, D. J. Wales, and R. J. Saykally, Science 352, 1194 (2016). https://doi.org/10.1126/science.aad8625, Google ScholarCrossref
  56. 56. S. E. Brown, A. W. Götz, X. Cheng, R. P. Steele, V. A. Mandelshtam, and F. Paesani, J. Am. Chem. Soc. 139, 7082 (2017). https://doi.org/10.1021/jacs.7b03143, Google ScholarCrossref
  57. 57. G. R. Medders, V. Babin, and F. Paesani, J. Chem. Theory Comput. 9, 1103 (2013). https://doi.org/10.1021/ct300913g, Google ScholarCrossref
  58. 58. S. K. Reddy, S. C. Straight, P. Bajaj, C. Huy Pham, M. Riera, D. R. Moberg, M. A. Morales, C. Knight, A. W. Götz, and F. Paesani, J. Chem. Phys. 145, 194504 (2016). https://doi.org/10.1063/1.4967719, Google ScholarScitation, ISI
  59. 59. B. Cheng, J. Behler, and M. Ceriotti, J. Phys. Chem. Lett. 7, 2210 (2016). https://doi.org/10.1021/acs.jpclett.6b00729, Google ScholarCrossref
  60. 60. C. H. Pham, S. K. Reddy, K. Chen, C. Knight, and F. Paesani, J. Chem. Theory Comput. 13, 1778 (2017). https://doi.org/10.1021/acs.jctc.6b01248, Google ScholarCrossref
  61. 61. G. R. Medders and F. Paesani, J. Chem. Theory Comput. 11, 1145 (2015). https://doi.org/10.1021/ct501131j, Google ScholarCrossref, ISI
  62. 62. S. C. Straight and F. Paesani, J. Phys. Chem. B 120, 8539 (2016). https://doi.org/10.1021/acs.jpcb.6b02366, Google ScholarCrossref
  63. 63. G. R. Medders and F. Paesani, J. Am. Chem. Soc. 138, 3912 (2016). https://doi.org/10.1021/jacs.6b00893, Google ScholarCrossref, ISI
  64. 64. D. R. Moberg, S. C. Straight, C. Knight, and F. Paesani, J. Phys. Chem. Lett. 8, 2579 (2017). https://doi.org/10.1021/acs.jpclett.7b01106, Google ScholarCrossref, ISI
  65. 65. H. Partridge and D. W. Schwenke, J. Chem. Phys. 106, 4618 (1997). https://doi.org/10.1063/1.473987, Google ScholarScitation, ISI
  66. 66. J. Behler and M. Parrinello, Phys. Rev. Lett. 98, 146401 (2007). https://doi.org/10.1103/physrevlett.98.146401, Google ScholarCrossref, ISI
  67. 67. J. Behler, Int. J. Quantum Chem. 115, 1032 (2015). https://doi.org/10.1002/qua.24890, Google ScholarCrossref, ISI
  68. 68. Z. Xie and J. M. Bowman, J. Chem. Theory Comput. 6, 26 (2010). https://doi.org/10.1021/ct9004917, Google ScholarCrossref, ISI
  69. 69. A. P. Bartók and G. Csányi, Int. J. Quantum Chem. 115, 1051 (2015). https://doi.org/10.1002/qua.24927, Google ScholarCrossref
  70. 70. K. Yao, J. E. Herr, and J. Parkhill, J. Chem. Phys. 146, 014106 (2017). https://doi.org/10.1063/1.4973380, Google ScholarScitation, ISI
  71. 71. A. Kamath, R. A. Vargas-Hernández, R. V. Krems, T. Carrington, Jr., and S. Manzhos, J. Chem. Phys. 148, 241702 (2018). https://doi.org/10.1063/1.5003074, Google ScholarScitation, ISI
  72. 72. D. M. Bates and G. S. Tschumper, J. Phys. Chem. A 113, 3555 (2009). https://doi.org/10.1021/jp8105919, Google ScholarCrossref, ISI
  73. 73. B. Temelso, K. A. Archer, and G. C. Shields, J. Phys. Chem. A 115, 12034 (2011). https://doi.org/10.1021/jp2069489, Google ScholarCrossref, ISI
  74. 74. U. Góra, R. Podeszwa, W. Cencek, and K. Szalewicz, J. Chem. Phys. 135, 224102 (2011). https://doi.org/10.1063/1.3664730, Google ScholarScitation, ISI
  75. 75. T. B. Adler, G. Knizia, and H. J. Werner, J. Chem. Phys. 127, 221106 (2007). https://doi.org/10.1063/1.2817618, Google ScholarScitation, ISI
  76. 76. K. A. Peterson, T. B. Adler, and H.-J. Werner, J. Chem. Phys. 128, 084102 (2008). https://doi.org/10.1063/1.2831537, Google ScholarScitation, ISI
  77. 77. A. Tikhonov, Soviet Mathematics Doklady (American Mathematical Society, 1963), Vol. 5, pp. 1035–1038. Google Scholar
  78. 78. J. Behler, J. Chem. Phys. 134, 074106 (2011). https://doi.org/10.1063/1.3553717, Google ScholarScitation, ISI
  79. 79. F. Chollet et al., “Keras,” https://github.com/keras-team/keras (2015). Google Scholar
  80. 80. Theano Development Team, e-print arXiv:abs/1605.02688 (2016). Google Scholar
  81. 81. D. Nguyen and B. Widrow, in 1990 IJCNN International Joint Conference on Neural Networks (IEEE, 1990), Vol. 3, pp. 21–26. Google ScholarCrossref
  82. 82. A. P. Bartók, M. C. Payne, R. Kondor, and G. Csányi, Phys. Rev. Lett. 104, 136403 (2010). https://doi.org/10.1103/physrevlett.104.136403, Google ScholarCrossref, ISI
  83. 83. A. P. Bartók, R. Kondor, and G. Csányi, Phys. Rev. B 87, 184115 (2013). https://doi.org/10.1103/physrevb.87.184115, Google ScholarCrossref, ISI
  84. 84. A. Bartók-Pártay, S. Cereda, G. Csányi, J. Kermode, I. Solt, W. Szlachta, C. Várnai, and S. Winfield, http://www.libatoms.org (2017). Google Scholar
  85. 85. S. De, A. P. Bartók, G. Csányi, and M. Ceriotti, Phys. Chem. Chem. Phys. 18, 13754 (2016). https://doi.org/10.1039/c6cp00415f, Google ScholarCrossref, ISI
  86. 86. A. P. Bartok, S. De, C. Poelking, N. Bernstein, J. Kermode, G. Csanyi, and M. Ceriotti, Sci. Adv. 3, e1701816 (2017). https://doi.org/10.1126/sciadv.1701816, Google ScholarCrossref
  87. 87. M. W. Mahoney and P. Drineas, Proc. Natl. Acad. Sci. U. S. A. 106, 697 (2009). https://doi.org/10.1073/pnas.0803205106, Google ScholarCrossref
  88. 88. S. De, F. Musil, T. Ingram, C. Baldauf, and M. Ceriotti, J. Cheminf. 9, 6 (2017). https://doi.org/10.1186/s13321-017-0192-4, Google ScholarCrossref
  89. 89. D. J. Rosenkrantz, R. E. Stearns, and P. M. Lewis II, SIAM J. Comput. 6, 563 (1977). https://doi.org/10.1137/0206041, Google ScholarCrossref
  90. 90. M. Ceriotti, G. A. Tribello, and M. Parrinello, Proc. Natl. Acad. Sci. U. S. A. 108, 13023 (2011). https://doi.org/10.1073/pnas.1108486108, Google ScholarCrossref, ISI
  91. 91. M. Ceriotti, G. A. Tribello, and M. Parrinello, J. Chem. Theory Comput. 9, 1521 (2013). https://doi.org/10.1021/ct3010563, Google ScholarCrossref, ISI
  92. 92. F. Musil, S. De, J. Yang, J. E. Campbell, G. M. Day, and M. Ceriotti, Chem. Sci. 9, 1289 (2018). https://doi.org/10.1039/c7sc04665k, Google ScholarCrossref
  93. 93. G. R. Medders, A. W. Götz, M. A. Morales, and F. Paesani, J. Chem. Phys. 143, 104102 (2015). https://doi.org/10.1063/1.4930194, Google ScholarScitation, ISI
  1. © 2018 Author(s). Published by AIP Publishing.