Spin independent symmetric $\hat{U}$

In this case,

$$U(\xi_i,\xi_j) = U\left(\vert{\bf r}_i -{\bf r}_j\vert\right).$$ (2.7)

We write this explicitly with wave functions which are products of orbital wave functions $\psi_\nu({\bf r})$ and spinors $\vert\sigma\rangle$,

$$\langle \xi\vert\nu\rangle = \psi_\nu({\bf r})\vert\sigma\rangle,$$ (2.8)

and take advantage of the fact that the interaction $U$ does not depend on spin. Then, Eq. (IV.2.6) becomes

$$\langle \Psi \vert\hat{U}\vert \Psi\rangle = \frac{1}{2} \sum_{i\ne j}\left[\langle\nu_{j}\nu_{i}\vert U \vert... ...}\nu_{j}\rangle -\langle\nu_{i}\nu_{j}\vert U \vert\nu_{i}\nu_{j}\rangle\right]$$

$$= \frac{1}{2} \sum_{i\ne j}\int\int d{\bf r} d{\bf r}' \Big[ \psi_{... ...) \langle \sigma_i\vert \sigma_i \rangle \langle \sigma_j\vert \sigma_j \rangle$$

$$- \psi_{\nu_i}^*({\bf r}') \psi_{\nu_j}^*({\bf r}) U\left(\vert{\bf... ...gle \sigma_j\vert \sigma_i \rangle \langle \sigma_i\vert \sigma_j \rangle \Big]$$

$$= \frac{1}{2} \sum_{i\ne j}\int\int d{\bf r} d{\bf r}' \Big[ \vert\... ...t^2 \vert\psi_{\nu_i}({\bf r})\vert^2 U\left(\vert{\bf r} -{\bf r}'\vert\right)$$

$$- \psi_{\nu_i}^*({\bf r}') \psi_{\nu_j}^*({\bf r}) U\left(\vert{\bf... ...t) \psi_{\nu_i}({\bf r}) \psi_{\nu_j}({\bf r}') \delta_{\sigma_i\sigma_j}\Big].$$ (2.9)

Using our direct and exchange term notation, Eq. (III.2.23), we can write this in a very simple form as a sum over direct terms $A_{\nu_i\nu_j}$ and exchange terms $J_{\nu_i\nu_j}$,

$$\langle \Psi \vert\hat{U}\vert \Psi\rangle = \frac{1}{2} \sum_{i,j} \left[ A_{\nu_i\nu_j}-J_{\nu_i\nu_j} \delta_{\sigma_i\sigma_j}\right].$$ (2.10)

Here, we recognize that we actually don't need the restriction $i\ne j$ in the double sum: this term is zero anyway.

Note that Eq. (IV.2.9) refers to states $\vert\Psi\rangle$ which are simple Slater determinants. It cannot be used, e.g., for states like the $M=0$ singlet or triplet which are linear combinations

$$\vert\psi_{1\uparrow}\psi_{2\downarrow}\rangle_A \pm \vert\psi_{1\downarrow}\psi_{2\uparrow}\rangle_A,$$ (2.11)

because these would lead to mixed terms

$$\langle \psi_{2\downarrow} \psi_{1\uparrow}\vert U \vert \psi_{1\downarrow}\psi_{2\uparrow}\rangle_A$$ (2.12)

in the expectation value!