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 [${\beta }_{\lambda \mu \nu }(-2\omega ;\omega ,\omega )$] as well as of the mixed electric-dipole/magnetic-dipole [${\hspace{0.17em}}^{\alpha }{J}_{\lambda \mu \nu }(-2\omega ;\omega ,\omega )$ and ${\hspace{0.17em}}^{\beta }{J}_{\lambda \mu \nu }(-2\omega ;\omega ,\omega )$] or electric-dipole/electric-quadrupole [${\hspace{0.17em}}^{\alpha }{K}_{\lambda \mu \nu o}(-2\omega ;\omega ,\omega )$ and ${\hspace{0.17em}}^{\beta }{K}_{\lambda \mu \nu o}(-2\omega ;\omega ,\omega )$] 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.

中文翻译：

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纯电偶极子和混合第一超极化率的三维表示：改进的单位球表示


在这项工作中，修正的单位球表示 （mUSR） 理论被提出作为一种适用于纯电偶极子 [${\beta }_{\lambda \mu \nu }（-2\omega ）\; 的三维表示的计算工具;\omega ，\omega ）$] 以及混合电偶极子/磁偶极子 [${}^{\alpha }{J}_{\lambda \mu \nu }（-2\omega ;\mathrm{\omega ，\omega }）$ 和 ${}^{\beta }{J}_{\lambda \mu \nu }（-2\omega ;\omega ，\omega ）$] 或电偶极子/电四极子 [${}^{\alpha }{K}_{\lambda \mu \nu o}（-2\Omega ;\omega ，\omega ）$ 和 ${}^{\beta }{K}_{\lambda \mu \nu o}（-2\Omega ;\mathrm{\omega ，\omega }）$] 第一超极化率。这五个量是笛卡尔张量，它们负责超瑞利散射 （HRS） 光谱的手性信号，即 HRS 光学活动 （HRS-OA） 光谱。第一次，对于每个超极化率，以及整个（即可约）笛卡尔张量的三维表示，为构成它们的每个不可约笛卡尔张量 （ICT） 开发了 mUSR。该方案已应用于一系列包含不同程度的吸电子（醌）基团的三个（手性）六螺旋烯分子，并具有相同的（正）旋性。对于这些分子，mUSR 表明，在取代时，对于纯电偶极子和混合电偶极子/磁偶极子超极化能力，获得了最显着的定性和半定量（分子响应的增强）效应。