## Analyzing our results, decomposing the Hamiltonian

The effective interaction is a scalar two-body operator. A general scalar two-body operator $$\hat{v}$$ can be written as $$$$\hat{v} = \sum_{k} \hat{v}_{k} = \sum_{k} C^{(k)}\cdot Q^{(k)}, \tag{27}$$$$ where the operators $$C^{(k)}$$ and $$Q^{(k)}$$ are irreducible spherical tensor operators of rank $$k$$, acting in spin and coordinate space, respectively. The value of $$k$$ is limited to $$k\le 2$$ since the total eigenspin of the two-nucleon system is either $$0$$ or $$1$$. The term with $$k=0$$ refers to the central component of the two-body operator. The values of $$k=1$$ and $$k=2$$ are called the vector and the tensor components, respectively. The vector term is also called the two-body spin-orbit term, although it also contains the anti-symmetric spin-orbit term. Using standard angular momentum algebra it is rather straightforward to relate the matrix elements $$\hat{v}_{k}$$ to those of say $$\hat{v}$$ or $$\tilde{v}$$.