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

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

References

  1. 1.

    H.Z. Hassan, A.A. Mohamad, G.E. Atteia, J. Comput. Appl. Math. 236, 2622 (2012)

    Article  Google Scholar 

  2. 2.

    J. Čížek, J. Chem. Phys. 45, 4256 (1966)

    Article  Google 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)

    Article  Google 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. 33–136

    Google 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)

    Google 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. 115–140

    Google Scholar 

  9. 9.

    J.K.G. Watson, in Vibrational Spectra and Structure, ed. by J.R. During (Elsevier, Amsterdam, 1977), pp. 1–89

    Google 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)

    Article  Google Scholar 

  12. 12.

    J.M.L. Martin, T.J. Lee, P.R. Taylor, J.P. François, J. Chem. Phys. 103(7), 2589 (1995)

    CAS  Article  Google Scholar 

  13. 13.

    X. Huang, T.J. Lee, J. Chem. Phys. 129, 044312 (2008)

    Article  Google Scholar 

  14. 14.

    X. Huang, T.J. Lee, J. Chem. Phys. 131, 104301 (2009)

    Article  Google Scholar 

  15. 15.

    X. Huang, P.R. Taylor, T.J. Lee, J. Phys. Chem. A 115, 5005 (2011)

    CAS  Article  Google Scholar 

  16. 16.

    R.C. Fortenberry, X. Huang, J.S. Francisco, T.D. Crawford, T.J. Lee, J. Chem. Phys. 136, 234309 (2012)

    Article  Google Scholar 

  17. 17.

    R.C. Fortenberry, X. Huang, J.S. Francisco, T.D. Crawford, T.J. Lee, J. Phys. Chem. A. 116, 9582 (2012)

    CAS  Article  Google Scholar 

  18. 18.

    X. Huang, R.C. Fortenberry, T.J. Lee, J. Chem. Phys. 139(8), 084313 (2013)

    Article  Google Scholar 

  19. 19.

    R.C. Fortenberry, X. Huang, T.D. Crawford, T.J. Lee, J. Phys. Chem. A 118, 7034 (2014)

    CAS  Article  Google Scholar 

  20. 20.

    D. Zhao, K.D. Doney, H. Linnartz, Astrophys. J. Lett. 791, L28 (2014)

    Article  Google Scholar 

  21. 21.

    M.J.R. Kitchens, R.C. Fortenberry, Chem. Phys. 472, 119 (2016)

    CAS  Article  Google Scholar 

  22. 22.

    V. Barone, J. Chem. Phys. 122, 014108 (2005)

    Article  Google Scholar 

  23. 23.

    M.R. Hermes, S. Hirata, J. Chem. Phys. 139, 034111 (2013)

    Article  Google Scholar 

  24. 24.

    J. Vázquez, M.E. Harding, J.F. Stanton, J. Gauss, J. Chem. Theory Comput. 7, 1428 (2011)

    Article  Google Scholar 

  25. 25.

    R.A. Theis, R.C. Fortenberry, J. Phys. Chem. A 119, 4915 (2015)

    CAS  Article  Google 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)

    Article  Google Scholar 

  27. 27.

    H.J. Werner, F.R. Manby, P.J. Knowles, J. Chem. Phys. 118, 8149 (2003)

    CAS  Article  Google Scholar 

  28. 28.

    R. Thackston, R.C. Fortenberry, J. Comput. Chem. 36, 926 (2015)

    CAS  Article  Google Scholar 

  29. 29.

    R.C. Fortenberry, R. Thackston, Int. J. Quantum Chem. 115, 1650 (2015)

    CAS  Article  Google Scholar 

  30. 30.

    A.G.G.M. Tielens, Annu. Rev. Astron. Astrophys. 46, 289 (2008)

    CAS  Article  Google 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)

  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)

    Article  Google Scholar 

  33. 33.

    A. Candian, P.J. Sarre, A.G.G.M. Tielens, Astrophys. J. Lett. 791, L10 (2014)

    Article  Google Scholar 

  34. 34.

    E.L.O. Bakes, A.G.G.M. Tielens, Astrophys. J. 499, 258 (1998)

    CAS  Article  Google Scholar 

  35. 35.

    M.G. Wolfire, Astrophys. Space Sci. 336, 229 (2011)

    CAS  Article  Google Scholar 

  36. 36.

    C. Boersma, R.H. Rubin, L.J. Allamandola, Astrophys. J. 753, 168 (2012)

    Article  Google 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)

    Article  Google Scholar 

  38. 38.

    B.A. Croiset, A. Candian, O. Berné, A.G.G.M. Tielens, Astron. Astrophys. 590, A26 (2016)

    Article  Google Scholar 

  39. 39.

    K. Nikolaou, P. Masclet, G. Mouvier, Sci. Total Environ. 32(2), 103 (1984)

    CAS  Article  Google Scholar 

  40. 40.

    R.J. Gray, Org. Geochem. 17(4), 535 (1991)

    CAS  Article  Google Scholar 

  41. 41.

    N.T.K. Oanh, L.B. Reutergårdh, N.T. Dung, Environ. Sci. Technol. 33, 2703 (1999)

    CAS  Article  Google Scholar 

  42. 42.

    H. Takagi, T. Isoda, K. Kusakabe, S. Morooka, Energy Fuels 13, 934 (1999)

    CAS  Article  Google Scholar 

  43. 43.

    C.S. McEnally, L.D. Pfefferle, B. Atakan, K. Kohse-Höinghaus, Prog. Energy Combust. Sci. 32, 247 (2006)

    CAS  Article  Google Scholar 

  44. 44.

    C.S. McEnally, L.D. Pfefferle, Combust. Flame 148, 210 (2007)

    CAS  Article  Google Scholar 

  45. 45.

    C.J. Mackie, A. Candian, X. Huang, T.J. Lee, A.G.G.M. Tielens, J. Chem. Phys. 142, 244107 (2015)

    Article  Google 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)

    Article  Google Scholar 

  47. 47.

    A. Ricca, C.W. Bauschlicher Jr., C. Boersma, A.G.G.M. Tielens, L.J. Allamandola, Astrophys. J. 754, 75 (2012)

    Article  Google 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)

  50. 50.

    M. Gupta, Automatic data partitioning on distributed memory multicomputers. Ph.D. thesis, University of Illinois at Urbana-Champaign (1992)

  51. 51.

    C.W. Bauschlicher, E. Peeters, L.J. Allamandola, Astrophys. J. 678, 316 (2008)

    CAS  Article  Google Scholar 

  52. 52.

    R.C. Fortenberry, X. Huang, A. Yachmenev, W. Thiel, T.J. Lee, Chem. Phys. Lett. 574, 1 (2013)

    CAS  Article  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ryan C. Fortenberry.

Ethics declarations

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Thackston, R., Fortenberry, R.C. An efficient algorithm for the determination of force constants and displacements in numerical definitions of a large, general order Taylor series expansion. J Math Chem 56, 103–119 (2018). https://doi.org/10.1007/s10910-017-0783-3

Download citation

Keywords

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