A given one-particle-one-hole (\( 1p1h \)) state can be written as
$$ |\Phi_i^a\rangle = \hat{a}_{a}^{\dagger}\hat{a}_i|\Phi_0\rangle, $$while a \( 2p2h \) state can be written as
$$ |\Phi_{ij}^{ab}\rangle = \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_j\hat{a}_i|\Phi_0\rangle, $$and a general \( ApAh \) state as
$$ |\Phi_{ijk\dots}^{abc\dots}\rangle = \hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_{c}^{\dagger}\dots\hat{a}_k\hat{a}_j\hat{a}_i|\Phi_0\rangle. $$We use letters \( ijkl\dots \) for states below the Fermi level and \( abcd\dots \) for states above the Fermi level. A general single-particle state is given by letters \( pqrs\dots \).