Interaction-induced pair hyperpolarizabilities by spherical irreducible tensors

Starting from the electrostatic part of the intermolecular multipole interaction energy we derived irreducible spherical tensor formulas for the first-order long-range, interaction-induced first and second pair hyperpolarizabilities Delta beta(LM) and Delta gamma(LM) for arbitrary shape monomers. For atoms a general relation is obtained for the dipole(i)-2(k)-pole hyperpolarizability tensor Z((i+k)) of arbitrary order, between its irreducible spherical components Z(00)[((((11)a(1)1)a(2)...)a(i-2)1)kk] and the Cartesian counterpart Z(zz...z)((i+k)). For isotropic systems the expressions for Delta beta(zzz) and Delta gamma(zzzz) are in full agreement with the Cartesian tensor results of Buckingham, Concannon and Hands [J. Phys. Chem. 98, 10455 (1994)] and Li [J. Chem. Phys. 105, 10954 (1996)]. Our irreducible spherical tensor results for Delta beta(LM) and Delta gamma(LM) are very desirable when dealing with molecular rotations, e.g., in spectral line shape calculations of interaction-induced hyper-Rayleigh and/or hyper-Raman light scattering. Finally we show how our spherical tensor formula for Delta beta(LM) can be used for calculation of, one molecule forbidden, hyper-Rayleigh and hyper-Raman (A(1) vibration) pair hyperpolarizability Delta beta(zzz) for molecules of octahedral symmetry O-h. (C) 1999 American Institute of Physics. [S0021-9606(99)51140-9].