Journal of Mathematical Chemistry

, Volume 50, Issue 6, pp 1570–1581

Excess hyperpolarizabilities: the irreducible tensor approach


    • Nonlinear Optics Division, Faculty of PhysicsAdam Mickiewicz University

DOI: 10.1007/s10910-012-9990-0


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.

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© The Author(s) 2012