Problem of fixing the tensor force part of the nucleon-nucleon potential From parity and rotational invariance of the nucleon-nucleon interaction, the tensor force part of the nucleon-nucleon potential can be fixed to be of the form $$S_{12}=\alpha(r)(\vec\sigma_1\cdot{\vec r})(\vec\sigma_2\cdot{\vec r})+\beta(r)(\vec\sigma_1\cdot\vec\sigma_2)$$ where $\vec\sigma_1,\vec\sigma_2$ are the spin operators of the two nucleons, $r$ is the internucleon separation and $\alpha(r),\beta(r)$ are two unknwon functions of $r$.
Nuclear physics by Krane suggests that the functional forms of $\alpha(r)$ and $\beta(r)$ are fixed to be $\alpha(r)=3/r^2$ and $\beta(r)=-1$, by requiring the average of $S_{12}$ over all angles to be zero.
Why do we require the average of $S_{12}$ to be zero?
 A: Realistic nucleon-nucleon potentials have both a tensor force and a spin exchange force proportional to $\vec \sigma_1\cdot \vec \sigma_2$.
See for example the Argonne $v_{18}$ phenomonological potential described in this link
https://www.phy.anl.gov/theory/research/av18/
as well as more modern effective field theory potentials.
The two parts can then be split to have nice angular momentum properties in terms of their transformations under rotations in spin and space. The $\vec \sigma_1\cdot \vec \sigma_2$ is the the coupling of the two spin rank 1 tensors to a scalar (i.e. rank 0). The tensor force is the coupling of these spins together to give a rank 2 spherical tensor, the coupling of the two $\vec r$ vectors to give another second ranked spherical tensor, and finally these are then coupled together to give a scalar as required to give a rotationally invariant Hamiltonian. This is written
\begin{equation}
\{ \{r \otimes r\}_2 \otimes \{S_{1} \otimes S_{2} \}_2 \}_{0} =
\frac{r^2}{3\sqrt{5}} \left [
3\vec S_1 \cdot \hat r \vec S_2 \cdot \hat r - \vec S_1 \cdot \vec S_2
\right ]\,.
\end{equation}
The notation is described in detail in
D.A. Varshalovich, A.N. Moskalev, and V.K. Khersonskii,
Quantum Theory of Angular Momentum, (World Scientific, Singapore, 1988).
One advantage of this breakup is that with the interaction written in terms of spherical tensors, the Wigner-Eckart theorem can be used to simplify matrix elements of the interaction.
