mode (Fig. \[fig:HDQ\]b,d). Although the MQ mode shows a broad response, it originates from the sum of many resonances (Fig. \[fig:HDQ\]d). This example shows that deeply fundamental properties of meta-atoms may be analyzed from their higher-order $\boldsymbol{\alpha}$-tensors.

Multiple-scattering theory and electromagnetically coupled systems
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In the previous section, we have shown that higher-order $\boldsymbol{\alpha}$-tensor is necessary for analyzing isolated meta-atoms and interpreting their interaction with light. Additionally, $\boldsymbol{\alpha}$-tensor can be used to model interacting meta-atoms for further research. In literature, self-consistent coupled multipole equations have been formulated using the Green’s tensor to illustrate periodic 2D arrays of plasmonic [@DeAbajo2007; @Auguie2008; @Swiecicki2017; @Babicheva2018] and dielectric [@Evlyukhin2010; @Babicheva2019] spheres, and finite [@Martikainen2017; @Draine1994] or random [@Watson2017] systems. In this approach, electromagnetic interactions between scattering objects are taken into account without any approximation, while single scattering objects are described by $\boldsymbol{\alpha}$-tensors approximated to low-order multipole orders. This approach is simplified version of superposition $\mathbf{T}$-matrix method (STMM), which has been extensively studied for coupled spheres [@DeAbajo1999; @Stout2008; @Stout2011]. Earlier studies have usually incorporated small spheres, whose $\boldsymbol{\alpha}$ can be easily obtained using the quasistatic approximation (see Appendix \[appendix:quasistatic\] for more details), to study their interaction with light and with nearby scattering objects or molecules 