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.

The page you're referring to (as well as many others, indeed) contains a mistake. See the correct definitions here.

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: $$ U_{ijkl} = \langle ij||kl\rangle = \langle ij|kl\rangle -\langle ij|lk\rangle $$
  • $\begingroup$ I figured out where my confusion was coming from. The sources that claim it is symmetric are working "2-electron orbitals", that is, treating them as bosons with a Fermi exclusion principle. Exchanging bosons will lead to it being even, obviously. $\endgroup$ – Alex Meiburg Mar 1 at 22:45

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