traditional numerical methods. Dual particles at dipole approximation have $\alpha_{e1}^{e1}$ = $\alpha_{m1}^{m1}$ = $\pm\alpha_{e1}^{m1}$ = $\pm\alpha_{m1}^{e1}$, where $\pm$ corresponds to left- (LDP) and right-dual particles (RDP), which interact with left- (LCP) and right-circularly-polarized lights (RCP), respectively.

In this section, we implemented dual particles using isotropic (scalar), dipolar $\boldsymbol{\alpha}$-tensors with Lorentzian resonance. We confirmed that two oppositely dual particles are completely uncoupled with each other, and only LDP is excited at LCP incidence (Fig. \[fig:dual\]a), and RDP is excited at RCP incidence (Fig. \[fig:dual\]b). This is because LDP is excited by LCP and re-radiates LCP, which cannot excite RDP. In a mixture of LDPs and RDPs, RDPs are completely transparent upon LCP incidence, and LDPs are electromagnetically coupled.

This section shows $\boldsymbol{\alpha}$-tensors as a powerful method to treat symmetries and conservation laws at the microscopic level (single scattering object) without dealing with the geometrical parameters. In fact, implementation of $\boldsymbol{\alpha}$-tensors without their relevant geometrical structures has been performed in literature to treat realistic molecules at weak excitations [@Novotny2009], where the molecules were approximated as $\boldsymbol{\alpha}$-tensors and implemented in the framework of MST to study plasmon-enhanced circular dichroism [@Govorov2010] and helical dichroism [@Wu2015].

Conclusions
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In summary, we have introduced the systematic 