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Journal of Mathematical Chemistry

, Volume 56, Issue 1, pp 103–119 | Cite as

An efficient algorithm for the determination of force constants and displacements in numerical definitions of a large, general order Taylor series expansion

  • Russell Thackston
  • Ryan C. FortenberryEmail author
Original Paper

Abstract

The venerable Taylor series expansion is a workhorse of computational modeling within the computational and quantum chemistry disciplines. It provides a highly-descriptive means of constructing complicated functions reliably. However, higher-order implementations or broadly defined functions with large numbers of variables can create a significant bottleneck for implementation of the Taylor series model. Most notably, construction of the internuclear potential for anharmonic vibrational frequency computations of large molecules like polycyclic aromatic hydrocarbons can become intractable to setup, much less compute due to the sheer volume of internal coordiantes. This work highlights the use of a lazy cartesian product to generate intelligently the required force constant definitions and subsequent displacements for numerical differentiation schemes with direct application to the definition of vibrational wave functions of molecules such as polycyclic aromatic hydrocarbons which can have dozens to hundreds of atoms. The key feature of the algorithm is its ability to generate directly only the non-zero rows of the Cartesian product (i.e. the Taylor series) while skipping large sections of unnecessary work in computing force constants and the subsequent displacements for unrequested higher-order derivatives with further considerations for symmetry. The savings of this algorithm are orders of magnitude greater than naïvely written nested loops. Furthermore, this algorithm facilitates highly balanced job partitioning, a key element of parallelization and high performance computing necessary for such large molecules. This approach can open the use of the Taylor series to formerly size-forbidden molecules.

Keywords

Taylor series Cartesian product Numerical differentiation Potential energy surfaces Polycyclic aromatic hydrocarbons 

Notes

Acknowledgements

The authors acknowledge the AWS Cloud Credits for Research program for their support of this research, as well as Georgia Southern University for the provision of startup funds and NASA Grant NNX17AH15G for RCF. RCF would also like to thank Cameron Mackie of the Leiden Observatory and Dr. Timothy J. Lee of the NASA Ames Research Center for stimulating conversations relating to computing the spectral features of PAHs.

Compliance with ethical standards

Conflict of interest

The authors claim no conflict of interest or other competing stakes for the work reported herein.

Ethical standard

This research is compliant with all of the standard ethical practices related to scientific research.

References

  1. 1.
    H.Z. Hassan, A.A. Mohamad, G.E. Atteia, J. Comput. Appl. Math. 236, 2622 (2012)CrossRefGoogle Scholar
  2. 2.
    J. Čížek, J. Chem. Phys. 45, 4256 (1966)CrossRefGoogle Scholar
  3. 3.
    J. Čížek, Adv. Chem. Phys. 14, 35 (1969)Google Scholar
  4. 4.
    J. Čížek, J. Paldus, Int. J. Quantum Chem. 5, 359 (1971)CrossRefGoogle Scholar
  5. 5.
    T.D. Crawford, H.F. Schaefer III, in Reviews in Computational Chemistry, vol. 14, ed. by K.B. Lipkowitz, D.B. Boyd (Wiley, New York, 2000), pp. 33–136CrossRefGoogle Scholar
  6. 6.
    I. Shavitt, R.J. Bartlett, Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory (Cambridge University Press, Cambridge, 2009)CrossRefGoogle Scholar
  7. 7.
    Y. Yamaguichi, Y. Osamura, J.D. Goddard, H.F. Schaefer III, Analytic Derivative Methods in Molecular Electronic Structure Theory: A New Dimension to Quantum Chemistry and its Applications to Spectroscopy (Wiley, New York, 1994)Google Scholar
  8. 8.
    I.M. Mills, in Molecular Spectroscopy—Modern Research, ed. by K.N. Rao, C.W. Mathews (Academic Press, New York, 1972), pp. 115–140CrossRefGoogle Scholar
  9. 9.
    J.K.G. Watson, in Vibrational Spectra and Structure, ed. by J.R. During (Elsevier, Amsterdam, 1977), pp. 1–89Google Scholar
  10. 10.
    D. Papousek, M.R. Aliev, Molecular Vibration-Rotation Spectra (Elsevier, Amsterdam, 1982)Google Scholar
  11. 11.
    D. Appelö, N.A. Petersson, SIAM J. Sci. Comput. 34, A2982 (2012)CrossRefGoogle Scholar
  12. 12.
    J.M.L. Martin, T.J. Lee, P.R. Taylor, J.P. François, J. Chem. Phys. 103(7), 2589 (1995)CrossRefGoogle Scholar
  13. 13.
    X. Huang, T.J. Lee, J. Chem. Phys. 129, 044312 (2008)CrossRefGoogle Scholar
  14. 14.
    X. Huang, T.J. Lee, J. Chem. Phys. 131, 104301 (2009)CrossRefGoogle Scholar
  15. 15.
    X. Huang, P.R. Taylor, T.J. Lee, J. Phys. Chem. A 115, 5005 (2011)CrossRefGoogle Scholar
  16. 16.
    R.C. Fortenberry, X. Huang, J.S. Francisco, T.D. Crawford, T.J. Lee, J. Chem. Phys. 136, 234309 (2012)CrossRefGoogle Scholar
  17. 17.
    R.C. Fortenberry, X. Huang, J.S. Francisco, T.D. Crawford, T.J. Lee, J. Phys. Chem. A. 116, 9582 (2012)CrossRefGoogle Scholar
  18. 18.
    X. Huang, R.C. Fortenberry, T.J. Lee, J. Chem. Phys. 139(8), 084313 (2013)CrossRefGoogle Scholar
  19. 19.
    R.C. Fortenberry, X. Huang, T.D. Crawford, T.J. Lee, J. Phys. Chem. A 118, 7034 (2014)CrossRefGoogle Scholar
  20. 20.
    D. Zhao, K.D. Doney, H. Linnartz, Astrophys. J. Lett. 791, L28 (2014)CrossRefGoogle Scholar
  21. 21.
    M.J.R. Kitchens, R.C. Fortenberry, Chem. Phys. 472, 119 (2016)CrossRefGoogle Scholar
  22. 22.
    V. Barone, J. Chem. Phys. 122, 014108 (2005)CrossRefGoogle Scholar
  23. 23.
    M.R. Hermes, S. Hirata, J. Chem. Phys. 139, 034111 (2013)CrossRefGoogle Scholar
  24. 24.
    J. Vázquez, M.E. Harding, J.F. Stanton, J. Gauss, J. Chem. Theory Comput. 7, 1428 (2011)CrossRefGoogle Scholar
  25. 25.
    R.A. Theis, R.C. Fortenberry, J. Phys. Chem. A 119, 4915 (2015)CrossRefGoogle Scholar
  26. 26.
    R.C. Fortenberry, Q. Yu, J.S. Mancini, J.M. Bowman, T.J. Lee, T.D. Crawford, W. Klemperer, J.S. Francisco, J. Chem. Phys. 143, 071102 (2015)CrossRefGoogle Scholar
  27. 27.
    H.J. Werner, F.R. Manby, P.J. Knowles, J. Chem. Phys. 118, 8149 (2003)CrossRefGoogle Scholar
  28. 28.
    R. Thackston, R.C. Fortenberry, J. Comput. Chem. 36, 926 (2015)CrossRefGoogle Scholar
  29. 29.
    R.C. Fortenberry, R. Thackston, Int. J. Quantum Chem. 115, 1650 (2015)CrossRefGoogle Scholar
  30. 30.
    A.G.G.M. Tielens, Annu. Rev. Astron. Astrophys. 46, 289 (2008)CrossRefGoogle Scholar
  31. 31.
    L.J. Allamandola, in PAHs and the Universe: A Symposium to Celebrate the 25th Anniversary of the PAH Hypothesis, ed. by C. Joblin, A.G.G.M. Tielens (EAS Publication Series, Cambridge, UK, 2011)Google Scholar
  32. 32.
    C. Boersma, C. Bauschlicher Jr., A. Ricca, A. Mattioda, J. Cami, E. Peeters, F.S. de Armas, G.P. Saborido, D. Hudgins, L. Allamandola, Astrophys. J. Suppl. Ser. 211, 8 (2014)CrossRefGoogle Scholar
  33. 33.
    A. Candian, P.J. Sarre, A.G.G.M. Tielens, Astrophys. J. Lett. 791, L10 (2014)CrossRefGoogle Scholar
  34. 34.
    E.L.O. Bakes, A.G.G.M. Tielens, Astrophys. J. 499, 258 (1998)CrossRefGoogle Scholar
  35. 35.
    M.G. Wolfire, Astrophys. Space Sci. 336, 229 (2011)CrossRefGoogle Scholar
  36. 36.
    C. Boersma, R.H. Rubin, L.J. Allamandola, Astrophys. J. 753, 168 (2012)CrossRefGoogle Scholar
  37. 37.
    H. Andrews, C. Boersma, M.W. Werner, J. Livingston, L.J. Allamandola, A.G.G.M. Tielens, Astrophys. J. 807, 99 (2015)CrossRefGoogle Scholar
  38. 38.
    B.A. Croiset, A. Candian, O. Berné, A.G.G.M. Tielens, Astron. Astrophys. 590, A26 (2016)CrossRefGoogle Scholar
  39. 39.
    K. Nikolaou, P. Masclet, G. Mouvier, Sci. Total Environ. 32(2), 103 (1984)CrossRefGoogle Scholar
  40. 40.
    R.J. Gray, Org. Geochem. 17(4), 535 (1991)CrossRefGoogle Scholar
  41. 41.
    N.T.K. Oanh, L.B. Reutergårdh, N.T. Dung, Environ. Sci. Technol. 33, 2703 (1999)CrossRefGoogle Scholar
  42. 42.
    H. Takagi, T. Isoda, K. Kusakabe, S. Morooka, Energy Fuels 13, 934 (1999)CrossRefGoogle Scholar
  43. 43.
    C.S. McEnally, L.D. Pfefferle, B. Atakan, K. Kohse-Höinghaus, Prog. Energy Combust. Sci. 32, 247 (2006)CrossRefGoogle Scholar
  44. 44.
    C.S. McEnally, L.D. Pfefferle, Combust. Flame 148, 210 (2007)CrossRefGoogle Scholar
  45. 45.
    C.J. Mackie, A. Candian, X. Huang, T.J. Lee, A.G.G.M. Tielens, J. Chem. Phys. 142, 244107 (2015)CrossRefGoogle Scholar
  46. 46.
    C.J. Mackie, A. Candian, X. Huang, E. Maltseva, A. Petrignani, J. Oomens, W.J. Buma, T.J. Lee, A.G.G.M. Tielens, J. Chem. Phys. 143, 224314 (2015)CrossRefGoogle Scholar
  47. 47.
    A. Ricca, C.W. Bauschlicher Jr., C. Boersma, A.G.G.M. Tielens, L.J. Allamandola, Astrophys. J. 754, 75 (2012)CrossRefGoogle Scholar
  48. 48.
    I. Amazon, Web Services. EC2 instance types -amazon web services (AWS) (2016), https://aws.amazon.com/ec2/instance-types/. Accessed 24 Apr 2016
  49. 49.
    B. Barney, Others, Lawrence Livermore National Laboratory 6(13), 10 (2010)Google Scholar
  50. 50.
    M. Gupta, Automatic data partitioning on distributed memory multicomputers. Ph.D. thesis, University of Illinois at Urbana-Champaign (1992)Google Scholar
  51. 51.
    C.W. Bauschlicher, E. Peeters, L.J. Allamandola, Astrophys. J. 678, 316 (2008)CrossRefGoogle Scholar
  52. 52.
    R.C. Fortenberry, X. Huang, A. Yachmenev, W. Thiel, T.J. Lee, Chem. Phys. Lett. 574, 1 (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Information TechnologyGeorgia Southern UniversityStatesboroUSA
  2. 2.Department of Chemistry and BiochemistryGeorgia Southern UniversityStatesboroUSA

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