# Symmetry in Fock-space 2-body interaction

The simplest two body interaction term for fermions is

$$H = \sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l$$

and I'm trying to determine the symmetries on $$U$$. Unfortunately I keep getting weird sign errors. The first symmetry comes from Hermiticity. To have $$H$$ be Hermitian, we need

$$H = H^\dagger = \sum_{ijkl} U_{ijkl}^\dagger a_l^\dagger a_k^\dagger a_j a_i$$

Then relabel the indices $$i\leftrightarrow l$$, $$j\leftrightarrow k$$:

$$H = \sum_{ijkl} U_{lkji}^\dagger a_i^\dagger a_j^\dagger a_k a_l$$

This should indicate that $$U_{lkji}^\dagger = U_{ijkl}$$. Along similar lines,

$$H = \frac{H + H}{2} = \frac{\sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l + \sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l}{2}$$

Relabel the indices in the second one as $$k\leftrightarrow l$$:

$$H = \frac{\sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l + \sum_{ijkl} U_{ijlk} a_i^\dagger a_j^\dagger a_l a_k}{2}$$

then apply the anticommutation relation:

$$H = \frac{\sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l - \sum_{ijkl} U_{ijlk} a_i^\dagger a_j^\dagger a_k a_l}{2} =\frac{\sum_{ijkl} (U_{ijkl}-U_{ijlk}) a_i^\dagger a_j^\dagger a_k a_l}{2}$$

Thus suggests that $$U_{ijkl}$$ is antisymmetric under the last two indices. Similarly, it should be antisymmetric under the first two indices. Unfortunately, a number of online sources seem to suggest that it should be symmetric, for instance http://sirius.chem.vt.edu/wiki/doku.php?id=crawdad:programming:project3#step_3two-electron_integrals -- here $$\langle{\mu\sigma|\lambda\rho\rangle} = U_{\mu\sigma\lambda\rho}$$, unless I'm somehow very sorely mistaken. Another reason is that $$U_{ijkl}$$ will often get contract with the density matrix $$D_{kl}$$ which is symmetric, and so would vanish if it wasn't antisymmetric.

Is it correct that the symmetries necessary of $$U$$ are the Hermitian symmetry given above, and antisymmetry in the (12) or (34) pairs? Or is it symmetric? Or neither? Thank you.

$$U_{ijkl}$$ is anti-symmetric in $$(12)$$ and $$(34)$$, your derivation is correct.
We define $$\langle ij|kl\rangle=\int\psi_i^*(\vec{r}_1)\psi_j^*(\vec{r}_2)\hat{h}\psi_k(\vec{r}_1)\psi_l(\vec{r}_2)\,d^3r_1d^3r_2$$
1. For general complex wavefunctions, the symmetry of $$\langle ij|kl\rangle$$ is 4-fold: $$\langle ij|kl\rangle = \langle ji|lk\rangle = \langle kl|ij\rangle = \langle lk|ji\rangle$$
2. For real functions, it's 8-fold due to: $$\langle ij|kl\rangle = \langle ji|kl\rangle$$
3. For $$U_{ijkl}$$, only the anti-symmetric part of $$\langle ij|kl\rangle$$ matters: $$U_{ijkl} = \langle ij||kl\rangle = \langle ij|kl\rangle -\langle ij|lk\rangle$$