Theoretical Chemistry Accounts

, Volume 121, Issue 5–6, pp 257–266 | Cite as

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

  • Paweł GrochowskiEmail author
Regular Article


We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.


Potential energy surface Hessian Rotations Cartesian coordinates 



The author would like to thank Grzegorz Bakalarski for his expert help in using Gaussian program, and Joanna Trylska for critical reading of the manuscript. The author acknowledges support from University of Warsaw (115/30/E-343/S/2007/ICM BST 1255) and Polish Ministry of Science and Higher Education (3 T11F 005 30, 2006–2008).


  1. 1.
    Heller EJ (1975) J Chem Phys 62:1544CrossRefGoogle Scholar
  2. 2.
    Prezhdo OV (2002) J Chem Phys 117:2995CrossRefGoogle Scholar
  3. 3.
    Giese K, Petkovic M, Naundorf H, Kuhn O (2006) Phys Rep 430:211CrossRefGoogle Scholar
  4. 4.
    Kapral R (2006) Annu Rev Phys Chem 57:129CrossRefGoogle Scholar
  5. 5.
    Grochowski P, Lesyng B (2003) J Chem Phys 19:11541CrossRefGoogle Scholar
  6. 6.
    Ischtwan J, Collins MA (1994) J Chem Phys 100:8080CrossRefGoogle Scholar
  7. 7.
    Thompson KC, Jordan MJT, Collins MA (1998) J Chem Phys 108:564CrossRefGoogle Scholar
  8. 8.
    Thompson KC, Jordan MJT, Collins MA (1998) J Chem Phys 108:8302CrossRefGoogle Scholar
  9. 9.
    Collins MA (2002) Theor Chem Acc 108:313Google Scholar
  10. 10.
    Trylska J, Grochowski P, Geller M (2001) Int J Quantum Chem 82:86CrossRefGoogle Scholar
  11. 11.
    Burcl R, Carter S, Handy NC (2003) Chem Phys Lett 373:357CrossRefGoogle Scholar
  12. 12.
    Taketsugu T, Watanabe N, Hirao K (1999) J Chem Phys 111:3410CrossRefGoogle Scholar
  13. 13.
    Yagi K, Taketsugu T, Hirao K (2001) J Chem Phys 115:10647CrossRefGoogle Scholar
  14. 14.
    Eckart C (1935) Phys Rev 47:552CrossRefGoogle Scholar
  15. 15.
    Kolossvary I, McMartin C (1992) J Math Chem 9:359CrossRefGoogle Scholar
  16. 16.
    Ghysels A, Van Neck D, Waroquier M (2007) J Chem Phys 127:164108CrossRefGoogle Scholar
  17. 17.
    Miller WH, Handy NC, Adams JE (1980) J Chem Phys 72:99CrossRefGoogle Scholar
  18. 18.
    Pulay P (1977) In: Schaefer HF (ed) Applications of electronic structure theory. Modern electronic structure theory, vol 4. Plenum, New YorkGoogle Scholar
  19. 19.
    Tachibana A, Fukui K (1978) Theor Chim Acta 49:321CrossRefGoogle Scholar
  20. 20.
    Mezey PG (1985) Theor Chim Acta 67:115CrossRefGoogle Scholar
  21. 21.
    Banerjee A, Adams NP (1992) Int J Quant Chem 43:855CrossRefGoogle Scholar
  22. 22.
    Wales DJ (2000) J Chem Phys 113:3926CrossRefGoogle Scholar
  23. 23.
    Adams JE, Stratt RM (1992) J Chem Phys 93:1632CrossRefGoogle Scholar
  24. 24.
    Vansteenkiste P, Van Neck D, Van Speybroeck V, Waroquier M (2006) J Chem Phys 124:044314CrossRefGoogle Scholar
  25. 25.
    Ochterski JW (1999) Vibrational analysis in Gaussian.
  26. 26.
    Li H, Jensen JH (2002) Theor Chem Acc 107:211Google Scholar
  27. 27.
    Ghysels A, Van Neck D, Van Speybroeck V, Verstraelen T, Waroquier M (2007) J Chem Phys 126:224102CrossRefGoogle Scholar
  28. 28.
    Malhiot RJ, Ferigle SM (1954) J Chem Phys 22:717CrossRefGoogle Scholar
  29. 29.
    Waluk J (2007) In: Hynes JT, Klinman JP, Limbach HH, Schowen RL (eds) Hydrogen transfer reactions, Wiley-VCH, WeinheimGoogle Scholar
  30. 30.
    Gaussian 03 Revision D.01 (2004) Gaussian Inc., Wallingford Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Interdisciplinary Centre for Mathematical and Computational ModellingUniversity of WarsawWarsawPoland

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