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I am new to this: Does anyone know to define different electric octupole moments in cartesian coordinates? I am looking for expressions that look like this (for electric dipoles).

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Given \begin{align} Y_1^{\pm 1}(\theta,\varphi)&= \pm C_{1} \left(\frac{x\pm iy}{r}\right) \end{align} why not start from the expression for the $\ell=3$ spherical harmonics in Cartesian coordinates and combine them to get real expressions, v.g. \begin{align} Y_3^3(\theta,\varphi)+ Y_3^{-3}(\theta,\phi)\sim (x-iy)^3+(x+iy)^3 = 2x^3-6 xy^2 \end{align} and so forth?

An even more direct way is to start from the real spherical harmonics although that won't help you to have the Cartesian components in terms of the usual spherical harmonics.

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  • $\begingroup$ Thanks @ZeroTheHero, in that case, how can we obtain "unique solutions" for x, y, and z? $\endgroup$ – user2431228 Dec 3 '19 at 13:43
  • $\begingroup$ I’m not sure what you mean. In general a polynomial like $z^3$ will expand in a sum of multipoles but the expansion is unique. $\endgroup$ – ZeroTheHero Dec 3 '19 at 13:48

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