We start from the reference state
$$ \begin{equation} \tag{1} \vert\Phi_0\rangle = \prod_{i=1}^H a^\dagger_i \vert 0\rangle \end{equation} $$for the description of a system with \( N \) particles. Usually, this reference is the Hartree-Fock state, but that is not necessary.
Here and in what follows, the indices \( i,j,k,\ldots \) run over hole states, i.e. orbitals occupied in the reference state (1), while \( a,b,c,\ldots \) run over particle states, i.e. unoccupied orbitals. Indices \( p,q,r,s \) can identify any orbital. Let \( n_u \) be the number of unoccupied states, and \( N \) is of course the number of occupied states.