Dependence of the multipole moments on the origin
Multipole moments of a system are defined with an explicit refrence to the co-ordinate system, e.g. $$\boldsymbol{d}=\int dV\, \rho\,\boldsymbol{r}\ \boldsymbol{\mu}=\frac{1}{2c}\int dV\, [\boldsymbol{r}\times\boldsymbol{j}]\ Q_{ij}=\int dV\, \rho(3r_ir_j-\delta_{ij}r^2)$$ Only the leading multipole moment is independent of this choice. Let's assume it's the dipole moment $\boldsymbol{d}$. Now consider a harmonic system with frequency $\omega$. The system will radiate energy, and the radiation intensity $I$ can be written in terms of the multipole moments (sorry if coefficients are not correct) $$I=\frac{2}{3}\frac{\omega^4\boldsymbol{d}^2}{c^3}+\frac{2}{3}\frac{\omega^4\boldsymbol{\mu}^2}{c^3}+\frac{\omega^6Q_{ij}^2}{180c^5}+\dots$$ Now $\boldsymbol{d},\boldsymbol{\mu}$ and $Q_{ij}$ are the amplitudes of the corresponding oscillating moments. According to textbooks, if the dipole moment is present than the dipole term gives the largest contribution to the radiation intensity and other terms could be considered to be corrections. Suppose we want to increase the accuracy by including the magnetic and the quadrupole contributions. My problem is that these corrections do not seem to be well-defined, because the moments themselvs depend on our choice of the coorditate origin. Take for example the magnetic moment and shift the coordinate system by vector $\boldsymbol{a}$. 649
The corresponding change of the magnetic moment is given by $$\Delta\boldsymbol{\mu}=\frac{1}{2c}\int dV\, [\boldsymbol{a}\times\boldsymbol{j}]$$ It is reasonable to take $\boldsymbol{a}$ to be of the order of the whole system size $L$. Indeed, for a generic system with no special symmetry the uncertanty of where is "the center" is of the order of the the system size. But then, the $\Delta\boldsymbol{\mu}$ and $\boldsymbol{\mu}$ seems to be of the same order $\Delta\boldsymbol{\mu}=\boldsymbol{\mu}\propto L j/c$. So the qestion is whether one really can increase the accuracy of the multipole expansion by including only some of the higher order terms (the full sum should be invariant of course)?
Edit: let's consider a specific example, always a good idea. Take a point electric dipole oscillating with frequency $\omega$ and amplitude $\boldsymbol{d}$, i.e. $\boldsymbol{d}(t)=e^{-i\omega t}\boldsymbol{d}$. The total radiation is given by the well-known expression $$I_0=\frac{2\omega^4\boldsymbol{d}(t)^2}{3c^3}$$ Of course, the same must be true for an electric dipole located not at the origin but at some position $\boldsymbol{a}$. The corresponding current and charge densities are $$\rho = -(\boldsymbol{d}(t)\nabla)\delta(\boldsymbol{r}-\boldsymbol{a}),\qquad \boldsymbol{j}=-i\omega\boldsymbol{d}(t)\delta(\boldsymbol{r}-\boldsymbol{a})$$
The dipole moment is of course $\boldsymbol{d}(t)$. Yet, the magnetic moment is non-vanishing $$\boldsymbol{\mu}=\frac{1}{2c}\int\, dV [\boldsymbol{r}\times -i\omega\boldsymbol{d}(t)\delta(\boldsymbol{r}-\boldsymbol{a})]=-\frac{ik}{2}[\boldsymbol{a}\times \boldsymbol{d}(t)]$$ and for generic $\boldsymbol{a}$ will give a non-vanishing contribution to the radiation intensity according to the general formula. 649