induced multipole modes at arbitrary incident fields, and has been used to treat scattering objects in many different fields including optics, acoustics, and astrophysics [@DeVries1998; @Mishchenko2008; @Mishchenko2010]. In nanophotonics, analysis of meta-atoms based on their $\mathbf{T}$-matrix started to become remarked rather recently [@Fruhnert2017; @Suryadharma2017]. It has been pointed out that complicated coupled configurations involving multiple meta-atoms can be efficiently studied by describing the meta-atoms in terms of $\boldsymbol{\alpha}$-tensor (or $\mathbf{T}$-matrix) and using the multiple-scattering theory (MST) [@Fruhnert2017]. Electromagnetically coupled discrete scattering objects can be self-consistently treated to describe for collective responses of multiple particles [@DeAbajo1999; @Stout2008; @Stout2011] and periodic particle arrays [@DeAbajo2007; @Baur2018; @Evlyukhin2010; @Babicheva2018; @Babicheva2019; @MahdiSalary2017; @Watson2017], and this approach significantly reduces the calculation loads for complicated, random [@Rahimzadegan2019; @Jenkins2018], or multi-scale systems [@Pattelli2018; @Govorov2010; @Wu2015].

In the following work, we first discuss induced multipoles in different expressions: approximate Cartesian, exact Cartesian, and spherical multipoles, where exact Cartesian and spherical multipoles are essentially identical with different choice of basis [@Alaee2018]. In the next section, we present expression of local fields and field gradients in terms of spherical multipoles. This naturally leads us to obtain transformation between $\boldsymbol{\alpha}$-tensor and $\mathbf{T}$-matrix. This basis transformation is used to analyze meta-atoms based on 