Journal of Mathematical Chemistry

, Volume 50, Issue 6, pp 1570–1581 | Cite as

Excess hyperpolarizabilities: the irreducible tensor approach

Open Access
Original Paper


The irreducible spherical and Cartesian tensors built of the products of two interaction tensors: the second order tensor resulting from the product of two second order tensors \({{\sf T}_{\alpha\,\lambda}\,{\sf T}_{\lambda\,\beta}}\) contracted once with the index λ, third order tensor \({{\sf T}_{\alpha\,\beta\,\lambda} {\sf T}_{\lambda\,\gamma}}\) appearing as a product of the third order interaction tensor \({{\sf T}_{\alpha\,\beta\,\lambda}}\) and the second order one \({{\sf T}_{\lambda\,\gamma}}\) contracted once with the index λ and the fourth order product of two second order tensors \({{\sf T}_{\alpha\,\beta}\,{\sf T}_{\gamma\delta}}\), have been considered. This type of products is encountered, e.g., within the London’s dispersive energy formula, inside the second-order virial coefficients of many physical parameters such as the dielectric constant, the Kerr constant, the induced polarizability and hyperpolarizability of a pair of molecules and in other induced quantities. Our results are applied explicitly to the excess induced first and second pair hyperpolarizability.


Irreducible tensors First hyperpolarizability Second hyperpolarizability Collision-induced phenomena 



This paper has been supported by the research project Nb. N N202 069939 sponsored by The Government of Poland. I would like to thank very much George Maroulis for many illuminating discussions.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


  1. 1.
    Kielich S.: Prog. Opt. XX, 155 (1983)CrossRefGoogle Scholar
  2. 2.
    G. Maroulis, T. Bancewicz, B. Champagne, A.D.Buckingham (eds.), Atomic and Molecular Nonlinear Optics: Theory, Experiment and Computation. A Homage to the Pioneering Work of Stanisaw Kielich (1925–1993)(IOS Press, Amsterdam, 2011)Google Scholar
  3. 3.
    Frommhold L.: Collision-induced Absorption in Gases. Cambridge University Press, Cambridge (1993)Google Scholar
  4. 4.
    T. Bancewicz, Y. Le Duff, J.-L. Godet, Modern Nonlinear Optics, Part 1, Advances in Chemical Physics, Vol. 119, 2nd edn, ed. by M. Evans (Wiley, New York, 2000), pp. 89–126Google Scholar
  5. 5.
    Kaatz P., Shelton D.P.: Mol. Phys. 88, 683 (1996)CrossRefGoogle Scholar
  6. 6.
    Głaz W., Bancewicz T., Godet J.-L.: J. Chem. Phys. 122, 224323 (2005)CrossRefGoogle Scholar
  7. 7.
    Bancewicz T., Głaz W., Godet J.-L., Maroulis G.: J. Chem. Phys. 129, 124306 (2008)CrossRefGoogle Scholar
  8. 8.
    Kielich S.: Nonlinear Molecular Optics. Nauka, Moscow (1981)Google Scholar
  9. 9.
    Maroulis G.: J. Phys. Chem. A 104, 4772 (2000)CrossRefGoogle Scholar
  10. 10.
    Bancewicz T., Maroulis G.: Phys. Rev. A 79, 042704 (2009)CrossRefGoogle Scholar
  11. 11.
    Bancewicz T., Maroulis G.: Chem. Phys. Lett. 498, 349 (2010)Google Scholar
  12. 12.
    Li X., Hunt K.L.C.: J. Chem. Phys. 100, 7875 (1994)CrossRefGoogle Scholar
  13. 13.
    Li X., Hunt K.L.C., Pipin J., Bishop D.M.: J. Chem. Phys. 105, 10954 (1996)CrossRefGoogle Scholar
  14. 14.
    Buckingham A.D., Concannon E.P., Hands I.D.: J. Phys. Chem. 98, 10455 (1994)CrossRefGoogle Scholar
  15. 15.
    Bancewicz T.: J. Chem. Phys. 111, 7440 (1999)CrossRefGoogle Scholar
  16. 16.
    Bancewicz T.: Mol. Phys. 50, 173 (1983)CrossRefGoogle Scholar
  17. 17.
    Gray C.G., Gubbins K.E.: Theory of Molecular Fluids. Vol. 1: Fundamentals. Clarendon Press, Oxford (1984)Google Scholar
  18. 18.
    Varshalovich D.A., Moskaliev A.N., Khersonskii V.K.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1988)Google Scholar
  19. 19.
    Maker P.D.: Phys. Rev. A 1, 923 (1970)CrossRefGoogle Scholar
  20. 20.
    Jerphagnon J., Chemla D., Bonneville R.: Adv. Phys. 27, 609 (1978)CrossRefGoogle Scholar
  21. 21.
    Stone A.J.: Mol. Phys. 29, 1461 (1975)CrossRefGoogle Scholar
  22. 22.
    Andrews D.L., Ghoul W.A.: J. Chem. Phys. 75, 530 (1981)CrossRefGoogle Scholar
  23. 23.
    Hunt K.L.C.: Chem. Phys. Lett. 70, 336 (1980)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Nonlinear Optics Division, Faculty of PhysicsAdam Mickiewicz UniversityPoznańPoland

Personalised recommendations