There is a dipole-dipole interaction $U_{\rm dip}$ between two dipoles $\vec{\mu}_1$ and $\vec{\mu}_2$
$$ U_{\rm dip} = \frac{\vec{\mu}_1\cdot\vec{\mu}_2 -3(\vec{\mu}_1\cdot\hat{R})(\vec{\mu}_2\cdot\hat{R})}{4\pi\epsilon_0 R^3} $$
where $\vec{R}$ is the displacement vector between the position of the two dipoles $\vec{R}_1$ and $\vec{R}_2$.
$$ r_1 + r_2 \leftrightarrow r_1' + r_2' $$
$$ \left<r'_1r'2\right|U{\rm dip}\left|r_1r_2\right> = \frac{\left<r_1'\right|\vec{\mu}_1\left|r_1\right>\left<r'_2\right|\vec{\mu}_2 \left|r_2\right>-3\left<r_1'\right|\vec{\mu}_1\cdot\hat{R}\left|r_1\right>\left<r_2'\right|\vec{\mu}_1\cdot\hat{R}\left|r_2\right>}{4\pi\epsilon_0 R^3} $$