# Can someone explain how Qiskit defines the electronic dipole moments?

The Qiskit API defines the electronic dipole moment as

$$\hat{d} = \sum_{p,q} d_{pq}^x a_p^\dagger a_q ,$$ where $$d_{pq}^x,d_{pq}^y,$$ and $$d_{pq}^z$$ are the Cartesian components of the vector $$\textbf{d}_{pq} = \int d\tau \phi_p \frac{1}{\textbf{r}} \phi_q .$$

Can someone please explain how the dipole moment operator can be expressed in the second quantised form this way? I follow that for a discrete one-dimensional system, the expectation value of the dipole moment is easily evaluated by $$\langle \mu \rangle = \langle \psi | \hat{\mu} | \psi \rangle$$ where $$\hat{\mu} = \sum_i \hat{N}_i x_i$$ but I cannot extend this to a three-dimensional system with continuous wavefunctions.

Specifically, I cannot see how the term $$\frac{1}{\textbf{r}}$$ arises in the equation for $$\textbf{d}_{pq}$$. From dimensional analysis it makes sense to have this form as one needs a factor of $$d \tau = 4 \pi r^2 dr$$ to convert from Cartesian to spherical polar coordinates, but I'd like to know how it's derived and what happens to the $$4 \pi$$ factor.