The implementation of the Pauli principle looks different in the \( J \)-scheme compared with the \( m \)-scheme. In the latter, no two fermions or more can have the same set of quantum numbers. In the \( J \)-scheme, when we write a state with the shorthand $$ |^{18}\mathrm{O}\rangle_J =|(ab)JM\rangle, $$ we do refer to the angular momenta only. This means that another way of writing the last state is $$ |^{18}\mathrm{O}\rangle_J =|(j_aj_b)JM\rangle. $$ We will use this notation throughout when we refer to a two-body state in \( J \)-scheme. The Kronecker \( \delta \) function in the normalization factor refers thus to the values of \( j_a \) and \( j_b \). If two identical particles are in a state with the same \( j \)-value, then only even values of the total angular momentum apply. In the notation below, when we label a state as \( j_p \) it will actually represent all quantum numbers except \( m_p \).