Also, parity is closely related to the concept of true chirality, or reciprocal parity-odd [@Barron1986]. Note that chirality of chiral molecules have been embedded in the magneto-electric coupling term [@Govorov2010], which attributes to the reciprocal parity-odd property. By breaking the reciprocity, it is possible to undergo false chirality, which is nonreciprocal, parity-odd [@Barron1986; @Asadchy2014; @Asadchy2019]. A lossless particle has no intrinsic absorption, so its extinction equals to scattering. For a lossless particle, $\mathbf{T}^{\dag}\mathbf{T}=-\frac{1}{2}(\mathbf{T}^{\dag}+\mathbf{T})$, where the superscript ${\dag}$ denotes Hermitian conjugate [@Waterman1971]. This expression has also been used to check the accuracy of $\mathbf{T}$-matrix for lossless particles in literature [@Mishchenko1996]. $\boldsymbol{\alpha}$-tensor of lossless dipolar particles satisfy: $\frac{k^3}{6\pi} \alpha^\dag \alpha = \frac{1}{2i}(\alpha^\dag - \alpha)$, which reduces to the optical theorem $\frac{k^3}{6\pi}|\alpha|^2 = \mathrm{Im}(\alpha)$ for dipolar scalar $\alpha$ [@Sersic2011]. Studies on properties of higher-order $\boldsymbol{\alpha}$-tensor is currently lacking and remains future research.

Meta-atoms and metaphotonics
============================

In metamaterials and metaphotonics, manipulation of light at the nanoscale utilizes optically resonant subwavelength meta-atoms, whose properties have generally been analyzed by dipolar $\boldsymbol{\alpha}$-tensor [@Arango2013; @Asadchy2014; @Liu2016]. However, recently emerged high-refractive-index particles [@Kuznetsov2012; @Evlyukhin2010; @Babicheva2017; @Babicheva2018; @Terekhov2019] and coupled plasmonic systems [@Fruhnert2017; @Arango2014] often involve higher-order multipole transitions. It should be noted that $\mathbf{T}$-matrix of meta-atoms can 