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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 $$ \textbf{d}_{pq} = \int d\tau \phi_p \frac{1}{\textbf{r}} \phi_q . $$

(Link to the API : https://qiskit.org/ecosystem/nature/stubs/qiskit_nature.second_q.properties.ElectronicDipoleMoment.html#qiskit_nature.second_q.properties.ElectronicDipoleMoment)

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.

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You actually spotted a mistake in the Qiskit Nature documentation. Thanks for raising this here. I have opened a PR to fix this.

To learn more about the internal details, it is probably better to look at classical-computational quantum chemistry codes which compute these integral quantities themselves (e.g. PySCF, Psi4, etc.). In Qiskit Nature we merely extract the values of these integrals from those codes and expose them via this property for a user to easily refer to them.

You can find examples of where we extract these quantities from PySCF here and from Psi4 here, just to give 2 examples.

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