Original Paper

Journal of Mathematical Chemistry

, Volume 50, Issue 6, pp 1570-1581

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Excess hyperpolarizabilities: the irreducible tensor approach

  • Tadeusz BancewiczAffiliated withNonlinear Optics Division, Faculty of Physics, Adam Mickiewicz University Email author 


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