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Sparse low rank approximation of potential energy surfaces with applications in estimation of anharmonic zero point energies and frequencies

  • Prashant RaiEmail author
  • Khachik Sargsyan
  • Habib Najm
  • So Hirata
Original Paper
  • 17 Downloads

Abstract

We propose a method that exploits sparse representation of potential energy surfaces (PES) on a polynomial basis set selected by compressed sensing. The method is useful for studies involving large numbers of PES evaluations, such as the search for local minima, transition states, or integration. We apply this method for estimating zero point energies and frequencies of molecules using a three step approach. In the first step, we interpret the PES as a sparse tensor on polynomial basis and determine its entries by a compressed sensing based algorithm using only a few PES evaluations. Then, we implement a rank reduction strategy to compress this tensor in a suitable low-rank canonical tensor format using standard tensor compression tools. This allows representing a high dimensional PES as a small sum of products of one dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate the product of sparse canonical low-rank representation of PES and Green’s function in the second-order diagrammatic vibrational many-body Green’s function theory (XVH2) for estimation of zero-point energies and frequencies. Numerical tests on molecules considered in this work suggest a more efficient scaling of computational cost with molecular size as compared to other methods.

Keywords

Potential energy surfaces Tensor decomposition Anharmonic vibrations Green’s function theory Compressed sensing High dimensional integration 

Notes

Acknowledgements

The authors thank Judit Zador at Sandia National Laboratories, Livermore for fruitful discussions. Support for this work was provided through the Scientific Discovery through Advanced Computing (SciDAC) program funded by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences under Award No. DE-FG02-12ER46875. Sandia National Laboratories is a multimission laboratory operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The views expressed in the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Sandia has major research and development responsibilities in nuclear deterrence, global security, defense, energy technologies and economic competitiveness, with main facilities in Albuquerque, New Mexico, and Livermore, California. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

References

  1. 1.
    J. Almlöf, K. Faegri, K. Korsell, J. Comput. Chem. 3, 385 (1982)CrossRefGoogle Scholar
  2. 2.
    P.G. Mezey, Potential Energy Hypersurfaces (Elsevier, New York, 1987)Google Scholar
  3. 3.
    P.G. Mezey, Theoretica Chimica Acta 54, 95 (1980)CrossRefGoogle Scholar
  4. 4.
    P.G. Mezey, Theoretica Chimica Acta 58, 309 (1981a)CrossRefGoogle Scholar
  5. 5.
    P.G. Mezey, Int. J. Quantum Chem. 20, 279 (1981b)CrossRefGoogle Scholar
  6. 6.
    P.G. Mezey, Int. J. Quantum Chem. 20, 185 (1981c)CrossRefGoogle Scholar
  7. 7.
    P.G. Mezey, Theoretica Chimica Acta 60, 97 (1981d)CrossRefGoogle Scholar
  8. 8.
    P.G. Mezey, Chem. Phys. Lett. 82, 100 (1981e)CrossRefGoogle Scholar
  9. 9.
    P.G. Mezey, Theoretica Chimica Acta 60, 409 (1982a)CrossRefGoogle Scholar
  10. 10.
    P.G. Mezey, Theoretica Chimica Acta 62, 133 (1982b)CrossRefGoogle Scholar
  11. 11.
    P.G. Mezey, Theoretica Chimica Acta 63, 9 (1983)CrossRefGoogle Scholar
  12. 12.
    S. Manzhos, X.G. Wang, R. Dawes, T. Carrington, J. Phys. Chem. A 110, 5295 (2006)CrossRefGoogle Scholar
  13. 13.
    R. Dawes, D.L. Thompson, Y. Guo, A.F. Wagner, M. Minkoff, J. Chem. Phys. 126, 184108 (2007)CrossRefGoogle Scholar
  14. 14.
    R. Dawes, D.L. Thompson, A.F. Wagner, M. Minkoff, J. Chem. Phys. 128, 084107 (2008)CrossRefGoogle Scholar
  15. 15.
    S. Carter, S.J. Culik, J.M. Bowman, J. Chem. Phys. 107, 10458 (1997)CrossRefGoogle Scholar
  16. 16.
    J.M. Bowman, S. Carter, X.C. Huang, Int. Rev. Phys. Chem. 22, 533 (2003)CrossRefGoogle Scholar
  17. 17.
    B.J. Braams, J.M. Bowman, Int. Rev. Phys. Chem. 28, 577 (2009)CrossRefGoogle Scholar
  18. 18.
    J.M. Bowman, T. Carrington, H.-D. Meyer, Mol. Phys. 106, 2145 (2008)CrossRefGoogle Scholar
  19. 19.
    K. Yagi, C. Oyanagi, T. Taketsugu, K. Hirao, J. Chem. Phys. 118, 1653 (2003)CrossRefGoogle Scholar
  20. 20.
    K. Yagi, S. Hirata, K. Hirao, Theor. Chem. Acc. 118, 681 (2007)CrossRefGoogle Scholar
  21. 21.
    O.F. Alış, H. Rabitz, J. Math. Chem. 29, 127 (2001)CrossRefGoogle Scholar
  22. 22.
    G.Y. Li, S.W. Wang, C. Rosenthal, H. Rabitz, J. Math. Chem. 30, 1 (2001)CrossRefGoogle Scholar
  23. 23.
    G.Y. Li, C. Rosenthal, H. Rabitz, J. Phys. Chem. A 105, 7765 (2001)CrossRefGoogle Scholar
  24. 24.
    A. Jäckle, H.-D. Meyer, J. Chem. Phys. 104, 7974 (1996)CrossRefGoogle Scholar
  25. 25.
    A. Jäckle, H.-D. Meyer, J. Chem. Phys. 109, 3772 (1998)CrossRefGoogle Scholar
  26. 26.
    F. Otto, J. Chem. Phys. 140, 014106 (2014)CrossRefGoogle Scholar
  27. 27.
    S. Manzhos, T. Carrington, J. Chem. Phys. 125, 194105 (2006)CrossRefGoogle Scholar
  28. 28.
    G. Avila, T. Carrington, J. Chem. Phys. 143, 044106 (2015)CrossRefGoogle Scholar
  29. 29.
    B. Ziegler, G. Rauhut, J. Chem. Phys. 144, 114114 (2016)CrossRefGoogle Scholar
  30. 30.
    L. Ostrowski, B. Ziegler, G. Rauhut, J. Chem. Phys. 145, 104103 (2016)CrossRefGoogle Scholar
  31. 31.
    M. Valiev, E.J. Bylaska, N. Govind, K. Kowalski, T.P. Straatsma, H.J. Van Dam, D. Wang, J. Nieplocha, E. Apra, T.L. Windus et al., Comput. Phys. Commun. 181, 1477 (2010)CrossRefGoogle Scholar
  32. 32.
    J.M. Bowman, J.S. Bittman, L.B. Harding, J. Chem. Phys. 85, 911 (1986)CrossRefGoogle Scholar
  33. 33.
    S. Chapman, M. Dupuis, S. Green, Chem. Phys. 78, 93 (1983)CrossRefGoogle Scholar
  34. 34.
    J. Ischtwan, M.A. Collins, J. Chem. Phys. 100, 8080 (1994)CrossRefGoogle Scholar
  35. 35.
    G.G. Maisuradze, D.L. Thompson, A.F. Wagner, M. Minkoff, J. Chem. Phys. 119, 10002 (2003)CrossRefGoogle Scholar
  36. 36.
    Y. Guo, A. Kawano, D.L. Thompson, A.F. Wagner, M. Minkoff, J. Chem. Phys. 121, 5091 (2004)CrossRefGoogle Scholar
  37. 37.
    B.G. Sumpter, D.W. Noid, Chem. Phys. Lett. 192, 455 (1992)CrossRefGoogle Scholar
  38. 38.
    T.B. Blank, S.D. Brown, A.W. Calhoun, D.J. Doren, J. Chem. Phys. 103, 4129 (1995)CrossRefGoogle Scholar
  39. 39.
    D.F. Brown, M.N. Gibbs, D.C. Clary, J. Chem. Phys. 105, 7597 (1996)CrossRefGoogle Scholar
  40. 40.
    F.V. Prudente, J.S. Neto, Chem. Phys. Lett. 287, 585 (1998)CrossRefGoogle Scholar
  41. 41.
    H. Gassner, M. Probst, A. Lauenstein, K. Hermansson, J. Phys. Chem. A 102, 4596 (1998)CrossRefGoogle Scholar
  42. 42.
    S. Lorenz, A. Groß, M. Scheffler, Chem. Phys. Lett. 395, 210 (2004)CrossRefGoogle Scholar
  43. 43.
    T. Hollebeek, T.-S. Ho, H. Rabitz, Ann. Rev. Phys. Chem. 50, 537 (1999)CrossRefGoogle Scholar
  44. 44.
    P. Rai, Sparse Low Rank Approximation of Multivariate Functions–Applications in Uncertainty Quantification. Ph.D. thesis, Ecole Centrale Nantes (2014)Google Scholar
  45. 45.
    E. Candes, J. Romberg, T. Tao, I.E.E.E. Trans, Inf. Theory 52(2), 489 (2006a)CrossRefGoogle Scholar
  46. 46.
    E. Candes, J. Romberg, T. Tao, IEEE Trans. Inf. Theory 52(12), 5406 (2006b)CrossRefGoogle Scholar
  47. 47.
    K. Kazimierczuk, V.Y. Orekhov, Angew. Chem. Int. Ed. 50, 5556 (2011)CrossRefGoogle Scholar
  48. 48.
    D.J. Holland, M.J. Bostock, L.F. Gladden, D. Nietlispach, Angew. Chem. Int. Ed. 50, 6548 (2011)CrossRefGoogle Scholar
  49. 49.
    L. Zhu, W. Zhang, D. Elnatan, B. Huang, Nat. Methods 9, 721 (2012)CrossRefGoogle Scholar
  50. 50.
    D. Gross, Y.-K. Liu, S.T. Flammia, S. Becker, J. Eisert, Phys. Rev. Lett. 105, 150401 (2010)CrossRefGoogle Scholar
  51. 51.
    A. Shabani, R.L. Kosut, M. Mohseni, H. Rabitz, M.A. Broome, M.P. Almeida, A. Fedrizzi, A.G. White, Phys. Rev. Lett. 106, 100401 (2011)CrossRefGoogle Scholar
  52. 52.
    J.N. Sanders, S.K. Saikin, S. Mostame, X. Andrade, J.R. Widom, A.H. Marcus, A. Aspuru-Guzik, J. Phys. Chem. Lett. 3, 2697 (2012)CrossRefGoogle Scholar
  53. 53.
    Y. August, A. Stern, Opt. Lett. 38, 4996 (2013)CrossRefGoogle Scholar
  54. 54.
    D. Xu, Y. Huang, J.U. Kang, Opt. Lett. 39, 76 (2014)CrossRefGoogle Scholar
  55. 55.
    X. Andrade, J.N. Sanders, A. Aspuru-Guzik, Proc. Natl. Acad. Sci. 109, 13928 (2012)CrossRefGoogle Scholar
  56. 56.
    J. Almeida, J. Prior, M. Plenio, J. Phys. Chem. Lett. 3, 2692 (2012)CrossRefGoogle Scholar
  57. 57.
    H. Schaeffer, R. Caflisch, C.D. Hauck, S. Osher, Proc. Natl. Acad. Sci. 110, 6634 (2013)CrossRefGoogle Scholar
  58. 58.
    L.J. Nelson, G.L. Hart, F. Zhou, V. Ozolins et al., Phys. Rev. B 87, 035125 (2013)CrossRefGoogle Scholar
  59. 59.
    J.N. Sanders, X. Andrade, A. Aspuru-Guzik, ACS Central Sci. 1, 24 (2015)CrossRefGoogle Scholar
  60. 60.
    G. Blatman, B. Sudret, J. Comput. Phys. 230, 2345 (2011)CrossRefGoogle Scholar
  61. 61.
    A. Doostan, H. Owhadi, J. Comput. Phys. 230, 3015 (2011)CrossRefGoogle Scholar
  62. 62.
    W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus, vol. 42 (Springer, New York, 2012)CrossRefGoogle Scholar
  63. 63.
    B.N. Khoromskij, Chemom. Intell. Lab. Syst. 110, 1 (2012)CrossRefGoogle Scholar
  64. 64.
    L. Grasedyck, D. Kressner, C. Tobler, GAMM Mitt. 36, 53 (2013)CrossRefGoogle Scholar
  65. 65.
    V. Khoromskaia, B.N. Khoromskij, Phys. Chem. Chem. Phys. 17, 31491 (2015)CrossRefGoogle Scholar
  66. 66.
    U. Benedikt, A.A. Auer, M. Espig, W. Hackbusch, J. Chem. Phys. 134, 054118 (2011)CrossRefGoogle Scholar
  67. 67.
    P. Rai, K. Sargsyan, H. Najm, M.R. Hermes, S. Hirata, Mol. Phys. 115, 2120 (2017)CrossRefGoogle Scholar
  68. 68.
    F. Bach, R. Jenatton, J. Mairal, G. Obozinski et al., Found. Trends Mach. Learn. 4, 1 (2012)CrossRefGoogle Scholar
  69. 69.
    D.L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006)CrossRefGoogle Scholar
  70. 70.
    K. Sargsyan, C. Safta, H.N. Najm, B.J. Debusschere, D. Ricciuto, P. Thornton, Int. J. Uncertain. Quantif. 4, 63–93 (2014)CrossRefGoogle Scholar
  71. 71.
    B.J. Debusschere, H.N. Najm, P.P. Pebay, O.M. Knio, R.G. Ghanem, O.P. Le Maittre, SIAM J. Sci. Comput. 26, 698 (2004)CrossRefGoogle Scholar
  72. 72.
    W. Bader, T.G. Kolda, et al., MATLAB Tensor Toolbox Version 2.6 (2015). http://www.sandia.gov/~tgkolda/TensorToolbox
  73. 73.
    M.R. Hermes, S. Hirata, J. Chem. Phys. 139, 034111 (2013)CrossRefGoogle Scholar
  74. 74.
    M.R. Hermes, S. Hirata, J. Chem. Phys. 141, 084105 (2014), ibid 143, 129903(E) (2015)Google Scholar
  75. 75.
    L.B. Harding, Y. Georgievskii, S.J. Klippenstein, J. Phys. Chem. A 121, 4334 (2017)CrossRefGoogle Scholar
  76. 76.
    M.R. Hermes, S. Hirata, Int. Rev. Phys. Chem. 34, 71 (2015)CrossRefGoogle Scholar
  77. 77.
    C. Battaglino, G. Ballard, T.G. Kolda, SIAM Journal on Matrix Analysis and Applications 39, 876 (2018)CrossRefGoogle Scholar

Copyright information

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2019

Authors and Affiliations

  • Prashant Rai
    • 1
    Email author
  • Khachik Sargsyan
    • 1
  • Habib Najm
    • 1
  • So Hirata
    • 2
  1. 1.Sandia National LaboratoriesLivermoreUSA
  2. 2.Department of ChemistryUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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