What a confusion, indeed! I think the biggest part of my confusion is coming from the "weird" notation used for the creation and annihilation operators, as you correctly pointed out. I checked the whole HK paper and found no definition for those objects: $\psi$ and $\psi^*$. Hohenberg and Kohn use the Capital $\Psi$ for the ground-state wavefunction. So Eqs. (2), (3), and (4) in Hohenberg-Kohn paper are nothing but the same Hamiltonian three terms (respectively from left to right), which I wrote down, reformulated in the language of second quantization? — Sha 10 hours ago

@Sha Yes! Indeed, they don't introduce these objects (at least I couldn't find it either), which actually is not a good style. But you can see that they must be field operators from the definition of the ground state density, too: They write it as an expectation value of $\psi^*(x)\, \psi(x)$, so this object must (!) be a $N$-particle operator (or at least an operator on Fock space), because it acts on $\Psi$. Comparing this definition with your favorite textbook which properly introduces these objects shows you that $\psi^*$ and $\psi$ must denote field operators. — Jakob 7 hours ago

@Sha Long story short: IMHO, Hohenberg-Kohn should've said, at least in one sentence, that they're working in second quantization or that these objects are field operators etc., just to settle the notation, which I think is mandatory in any text. However, the "experienced" reader quickly sees that, so it is no problem for them, but students and beginners might get confused. Yet, I don't know if, back in the 60s, second quantization was used "everywhere" in the field and if it was so common that it is superfluously to add this?! — Jakob 7 hours ago

The whole thing makes sense as long as you think of $\psi$ and $\psi^*$ as field operators instead of the misleading notion of wavefunctions! Thank you so much. As for them should've defined $\psi$ and $\psi^*$ in their paper, I agree with you, but to this, Kohn would have simply replied,“Young men, I am the Kohn of Kohn and Rostoker, the Kohn of Kohn and Hohenberg, the Kohn of Kohn and Sham, the father of DFT! " :) — Sha 7 hours ago

Dear @Jakob your answer, as it stands now, deserves to be accepted as "the answer"; however, would it be too much for you to give the detailed reformulation of the last term in the Hamiltonian, in terms of the second quantization language? — Sha 7 hours ago

@Sha Glad I could help. What do you mean with reformulation? Do you mean derivation? It's actually quite lengthy and I don't have much time, but I can refer you to Density Functional Theory. An advanced course. Engel and Dreizler. Springer. Appendix B, which if I remember correctly deals with this in detail. But as I said, any book on many-body theory should cover this. If you have problems with this derivation then, I'd say you should open a new question, as a) it is a different topic and b) it might interest further readers. — Jakob 6 hours ago

Yes, I meant its derivation. But you are right I have some familiarity with the formalism of second quantization and this derivation is a lengthy one. I thought having its derivation here would be helpful for anyone else in the future. Anyway, thank you for your reference recommendation. — Sha 6 hours ago