The collision-induced hyper-Rayleigh (CIHR) spectra of the gaseous H 2 – Ar mixture are discussed in the binary regime on the basis of our ab initio computed H 2 – Ar collision-induced (CI) first dipole hyperpolarizability tensor Δ β ( R ) . A method for the computation of the spherical, rotationally adapted components Δ β λ L ( s , K ) ( R ) of Δ β ( R ) needed for spectroscopic line shape analysis is proposed. Both the vector and the septor parts of the H 2 – Ar CIHR spectrum are evaluated at room ( T = 295 K ) temperature. The spectra are calculated assuming the full quantum computations based on the Schrödinger equation of the relative translational motion of H 2 – Ar as well as semiclassical methods (classical trajectory approach and Birnbaum–Cohen model translational profiles). The H 2 – Ar pair CIHR septor spectrum has been found stronger than the vector one.