larger particles and high-refractive-index particles. This error has been corrected by toroidal multipoles, which appear from multipole decomposition of the localized current sources [@Gurvitz2019; @Talebi2018; @Baryshnikova2019; @Evlyukhin2016]. However, it has been pointed out that the radiation fields from toroidal multipoles do not have independent (orthogonal) basis to those from electric and magnetic multipoles [@Alaee2018]. Therefore, it is controversial whether to treat the toroidal multipoles as the third multipole family [@Savinov2019], or as a correction to the basic Cartesian multipoles.

Recently, exact expressions for the localized charge-current density multipoles in the Cartesian basis up to MQ were developed without relying on the toroidal multipoles [@Alaee2018; @Fernandez-Corbaton2015]. For completeness, we present expressions of the exact Cartesian multipoles up to MO as

$$\begin{aligned}
        \begin{split}             d^E_\alpha &= 
            -\frac{1}{i\omega}
            \int{{\mathrm{d}^3\mathbf{r}~}\Bigg\{
            J_\alpha j_0(kr)
            +\frac{k^2}{2}\Big[3 {\mathbf{(r \cdot J)}}r_\alpha-r^2J_\alpha\Big]
            \frac{j_2(kr)}{(kr)^2}
            \Bigg\}}
        \end{split}\\
        \begin{split}             d^M_\alpha &= 
            \frac{3}{2}
            \int{{\mathrm{d}^3\mathbf{r}~}{\mathbf{(r \times J)}}_\alpha \frac{j_1(kr)}{kr}}
        \end{split}\\
        \begin{split}             Q^E_{\alpha\beta} &= 
            -\frac{3}{i\omega}
            \int{\mathrm{d}^3\mathbf{r}~}{\Bigg\{}
            \Big[r_\alpha J_\beta + r_\beta J_\alpha
            -\frac{2}{3}\delta_{\alpha\beta}{\mathbf{(r \cdot J)}}\Big]
            \frac{j_1(kr)}{kr}\\
            &\hspace{5mm}
            +2k^2\Big[5{\mathbf{(r \cdot J)}}r_\alpha r_\beta 
            -r^2(r_\alpha J_\beta+r_\beta J_\alpha)
            -r^2\delta_{\alpha\beta}{\mathbf{(r \cdot J)}}\Big]
            \frac{j_3(kr)}{(kr)^3}
            {\Bigg\}}
        \end{split}\\
        \begin{split}             Q^M_{\alpha\beta} &= 5
            \int{{\mathrm{d}^3\mathbf{r}~}{\Big[}
            r_\alpha {\mathbf{(r \times J)}}_\beta + r_\beta {\mathbf{(r \times J)}}_\alpha
            {\Big]}
            \frac{j_2(kr)}{(kr)^2}
            }
        \end{split}\\
        \begin{split}             O^E_{\alpha\beta\gamma} &= 
            -\frac{10}{i\omega}
            \int{\mathrm{d}^3\mathbf{r}~}{\Bigg\{}
            \Big[r_\alpha r_\beta J_\gamma +r_\beta r_\gamma J_\alpha 