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I can find two different ways of writing the quadrupole moment tensor

$$Q = \int \mathrm{d}^3r \rho(r) \left(3 r\otimes r - |r|^2I\right)$$ or

$$Q = \int \mathrm{d}^3r \rho(r) \left(r\otimes r - \frac{|r|^2}{3}I\right).$$

I am confused. Which one is it?

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There is no reason that physicists have to agree on how to normalize a quadrupole moment. Both conventions are in use, and equations involving $Q$ differ depending on the choice, so ultimately everything you actually measure, such as an electrostatic force, is the same regardless of the convention. In my experience, the second choice is in more common use.

Sometimes $Q$ isn’t defined as trace-free, so the second term is missing. So that’s another choice.

Sometimes people prefer to define a quadrupole moment $Q_{lm}$ based on spherical harmonics rather than a cartesian one. Yet another choice!

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