Coulomb interaction and conservation laws In many-body solid-state physics, the Coulomb interaction term in the Hamiltonian usually implies the momentum conservation law in indicies:
$$H_c=\frac{1}{2} \sum_{\mathbf{k},\mathbf{k}',\mathbf{q} \neq 0} V_{\mathbf{q}} a^{\dagger}_{\mathbf{k}'-\mathbf{q}} a^{\dagger}_{\mathbf{k}+\mathbf{q}}a_{\mathbf{k}'} a_{\mathbf{k}},$$ 
where $\mathbf{k},\mathbf{k}',\mathbf{q}$ are quasi-momenta and quantum numbers for the continuum spectrum of electron gas simultaneously.
In quantum chemistry textbooks, the Coulomb term usually looks like:
$$H_c=\sum_{i,j,k,l} V_{i,j,k,l} a^{\dagger}_{i} a^{\dagger}_{j}a_{k} a_{l}$$ 
Numbers $i,j,k,$ and $l$ are running over some discrete energy spectrum. Is it possible to state any conservation laws for quantum numbers $i,j,k,$ and $l$ in the expression above? Should not they obey any conservation law, selection rules or any additional restrictions? I will appreciate any references to textbooks or papers.
 A: Quantum chemistry treats localized systems for which the conservation of total momentum is a true but not-very-useful fact; it therefore doesn't make much sense to incorporate it very explicitly into the notation. (Solid-state systems, on the other hand, are infinite lattices that are invariant under discrete translations, so that the total electron momentum will be a constant of the motion. The formalism is then adapted to this.)
In the quantum chemical formalism, the interaction coefficients are
$$V_{ijkl}=\langle\phi_i\phi_j|\hat{v}|\phi_k\phi_l\rangle,$$
where the creation operator $a_i^\dagger$ creates an electron in the state $|\phi_i\rangle$. The pairwise Coulomb repulsion $\hat{v}$ is indeed translation invariant, in that it commutes with the total translation $\hat{U}=e^{i(\hat{\mathbf{p}}_1+\hat{\mathbf{p}}_2) \cdot \mathbf{r}}$ by any displacement $\mathbf{r}$. Thus the coefficient is also equal to
$$V_{ijkl}=\langle\phi_i|e^{i\hat{\mathbf{p}} \cdot \mathbf{r}} \otimes\langle\phi_j|e^{i\hat{\mathbf{p}} \cdot \mathbf{r}} \cdot\hat{v}\cdot e^{-i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi_k\rangle\otimes e^{-i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi_l\rangle.$$
Unlike in solid-state systems, though, orbitals like $e^{-i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi_k\rangle$ are not related in any way to the rest of the basis, other than in the single necessary expansion
$$e^{-i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi_k\rangle=
\sum_j |\phi_j\rangle\langle\phi_j|e^{-i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi_k\rangle.$$
With this you can formulate the global translation invariance as a condition on the $V_{ijkl}$:
$$V_{ijkl}=\sum_{i',j',k',l'}
\langle\phi_i|e^{i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi_{i'}\rangle
\langle\phi_j|e^{i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi{j'}\rangle
\langle\phi_{k'}|e^{-i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi_k\rangle
\langle\phi_{l'}| e^{-i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi_l\rangle
V_{i'j'k'l'}.$$
The reason this looks so ugly is that there is as yet no selection rule on the matrix elements of the translation between the different basis functions, such as $\langle\phi_i|e^{i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi_{i'}\rangle$. In quantum chemical applications, the basis is localized around the nuclei and there will not be any such selection rule, so the above is the best you'll get. (In practice this is not a problem as you know $\hat{v}$ beforehand and use it to calculate the $V_{ijkl}$. If you want to postulate some coefficients then you do need to check the above relation for all displacements $\mathbf{r}$ or your hamiltonian will not be translation invariant.)
Note, though, that since the above formalism is completely general, you still have the option to choose a translation-invariant basis, for which $e^{i\hat{\mathbf{p}} \cdot \mathbf{r}} |\phi_{i}\rangle=e^{i{\mathbf{p_i}} \cdot \mathbf{r}} |\phi_{i}\rangle$, as in solid-state applications. In this case the matrix elements will simplify to delta functions and the coefficients will be forced into the first form you give.
