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 ΔβLM and ΔγLM for arbitrary shape monomers. For atoms a general relation is obtained for the dipolei–2k-pole hyperpolarizability tensor Z(i+k) of arbitrary order, between its irreducible spherical components Z00[(((11)a11)a2…)ai−21)kk] and the Cartesian counterpart Zzz…z(i+k). For isotropic systems the expressions for Δβzzz and Δγzzzz are in full agreement with the Cartesian tensor results of Buckingham, Concannon and Hands [J. Phys. Chem. 98, 10455 (1994)] and Li et al. [J. Chem. Phys. 105, 10954 (1996)]. Our irreducible spherical tensor results for ΔβLM and ΔγLM 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 ΔβLM can be used for calculation of, one molecule forbidden, hyper-Rayleigh and hyper-Raman (A1 vibration) pair hyperpolarizability Δβzzz for molecules of octahedral symmetry Oh.