Thanks for the answers!
Regarding the {} notation: I completely missed that, but I guess you are indeed right with the integral shorthand. Thanks Stijn!
For the first question: that could be true, but it still seems a little bit weird to me. After all, it should be a scalar since it’s an energy, and we also compute additional corrections for other multipole modes, e.g. there is also a correction coming from the dipole. So that would imply all corrections are monopole corrections. Perhaps it’s just definition, I would just expect it to be called a ‘quadrupole’ correction since it comes from quadrupole objects…