+ \frac{\eta k^6}{160 \pi} \sum_{\alpha\beta}{|Q^M_{\alpha\beta}|^2}
        + \frac{\eta k^8}{1680 \pi} \sum_{\alpha\beta\gamma}{|O^M_{\alpha\beta\gamma}|^2}
    \end{split}
    \label{eqn:power}$$ Note that the proportionality constants in Eqns. \[eqn:radiationfield\] and \[eqn:power\] are different from previous work [@Evlyukhin2016], but the reconstructed scattered fields satisfying the Maxwell’s equations should be identical regardless of the choice of the notations. Also, the notations of the spherical multipoles and VSWFs differ by publications, and our expression is given in Appendix \[appendix:proportionality\].

Extended point polarizability and T-matrix
==========================================

Many optical phenomena have been successfully resolved from induced multipoles, but they are given at a specific illumination. In general, the induced moments are not invariant under the interaction with other particles or under different external excitation fields. Considering that meta-atoms generally have strong anisotropy and can therefore exhibit different induced multipoles depending on its environment and excitation fields, the induced moments obtained from a specific situation do not consistently represent the inherent properties of the meta-atoms.

Hence, a different quantity is required to consistently describe an identical particle in an isolated state, nearby other particles, or in a lattice, and $\boldsymbol{\alpha}$-tensor serves for this purpose. $\boldsymbol{\alpha}$-tensor is defined as a response tensor linearly relating the local fields to the induced multipoles, so it is irrelevant of 