their $\boldsymbol{\alpha}$-tensors, whose properties can be intuitively interpreted due to the Cartesian basis. Finally, we show that the MST allows efficient and accurate description of electromagnetically coupled meta-atoms, and that analytic scattering objects can be implemented under the multipole approach.

Exact Cartesian multipoles
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In standard electrodynamic textbooks, the spherical multipoles appear from the multipole decomposition of electromagnetic fields using the vector spherical wave functions (VSWFs) as the basis [@Jackson1999]. Because the VSWFs span the vector fields satisfying the transverse Helmholtz type equations, the electromagnetic fields in a homogeneous media can be exactly reconstructed, and the renowned Mie theory is also based on this expansion. Because of the difficulty in interpreting the spherical basis, the spherical multipoles are not directly analyzed per se, but their associated scattering power or radiation fields are.

Therefore, multipole approach in nanophotonics most frequently utilizes the approximate expressions for localized charge-current density multipoles in the Cartesian basis, which are sufficiently straightforward and resemble the expressions in electrostatics and magnetostatics. Despite their popularity, the approximate Cartesian multipoles cannot *exactly* reconstruct the electrodynamic radiation fields and the related scattering phenomena [@Alaee2018]. Scattering from subwavelength nanoparticles with moderate refractive-index generally shows good agreement, but the error grows for 