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While navigating on Stack Exchange, I was suddenly wondering about a possibility that I never saw described in my textbooks on electrodynamics.

Suppose you have a distribution of electric charges in space with a given total charge $Q$, total dipolar moment $\vec{p}$ and quadrupole components $Q_{ij}$. Neglecting all higher multipoles, the total electric potential is the following: \begin{equation}\tag{1} \phi(x) = \frac{k Q}{r} + \frac{k}{r^3} \, \vec{p} \cdot \vec{r} + \frac{k}{2 r^5} \, Q_{ij} \, x_i \, x_j. \end{equation} To simplify things, suppose that the total charge vanishes: $Q = 0$, and the distribution is discrete (the charges can be counted: $n = 1, 2, 3, \dots, N$). It's then possible to consider pairs of opposite charges as forming a set of simple dipoles, so by the principle of superposition of fields: \begin{equation}\tag{2} \phi(x) = \sum_{n=1}^N \frac{k}{||\vec{r} - \vec{r}_n ||^3} \, \vec{p}_n \cdot (\vec{r} - \vec{r}_n). \end{equation} Of course, the global dipole moment in (1) is \begin{equation}\tag{3} \vec{p} = \sum_{n=1}^N \vec{p}_n. \end{equation} Is there a simple closed form giving the quadrupole cartesian components $Q_{ij}$ in (1) as a function of the dipolar moments $\vec{p}_n$ and the dipoles position $\vec{r}_n$? Also, what constraints can we get on the $\vec{p}_n$ and $\vec{r}_n$ (and their total $Q_{ij}$) so that all other multipolar elements vanish, except $Q_{ij}$?

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