Three‐dimensional representation of the pure electric‐dipole and the mixed first hyperpolarizabilities: The modified unit sphere representation.

In this work, the theory of the modified unit sphere representation (mUSR) has been proposed as a computational tool suitable for the three‐dimensional representation of the pure electric‐dipole [βλμν(−2ω;ω,ω)] as well as of the mixed electric‐dipole/magnetic‐dipole [ αJλμν(−2ω;ω,ω) and βJλμν(−2ω;ω,ω)] or electric‐dipole/electric‐quadrupole [ αKλμνo(−2ω;ω,ω) and βKλμνo(−2ω;ω,ω)] first hyperpolarizabilities. These five quantities are Cartesian tensors and they are responsible for the chiral signal in the chiroptical version of the hyper‐Rayleigh scattering (HRS) spectroscopy, namely the HRS optical activity (HRS‐OA) spectroscopy. For the first time, for each hyperpolarizability, alongside with the three‐dimensional representation of the whole (i.e., reducible) Cartesian tensors, the mUSRs are developed for each of the irreducible Cartesian tensors (ICTs) that constitute them. This scheme has been applied to a series of three (chiral) hexahelicene molecules containing different degrees of electron‐withdrawing (quinone) groups and characterized by the same (positive) handedness. For these molecules, the mUSR shows that, upon substitution, the most remarkable qualitative and semi‐quantitative (enhancement of the molecular responses) effects are obtained for the pure electric‐dipole and for the mixed electric‐dipole/magnetic‐dipole hyperpolarizabilities. [ABSTRACT FROM AUTHOR]