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 $$ \begin{equation} \hat{v} = \sum_{k} \hat{v}_{k} = \sum_{k} C^{(k)}\cdot Q^{(k)}, \tag{27} \end{equation} $$ 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 \( \tilde{v} \).