\psi_i(x') \psi_j(x)
\label{rho1_spin}$$ after [[**\[expansion\]**]{}]{} and [[**\[SDs\]**]{}]{} are inserted into [[**\[rho\]**]{}]{}. In [[**\[rho1\_spin\]**]{}]{} the subindex ${{\bf{k}}}$ specifies the $N-1$ NSOs present in $\Phi_{i,{{\bf{k}}}}$ and $\Phi_{j,{{\bf{k}}}}$, excluding $\psi_i$ and $\psi_j$. As the $\psi$ are the NSOs of $\Psi$, the eigenfunctions of $\rho(x',x)$, the bracketed quantity in [[**\[rho1\_spin\]**]{}]{} must vanish for $i\neq
j$. Regard the coefficients $C_{i,{{\bf{k}}}}$ and $C_{j,{{\bf{k}}}}$ as the components of vectors ${{\bf{C}}}_i$ and ${{\bf{C}}}_j$ and the bracket as their scalar product ${{\bf{C}}}_i \cdot {{\bf{C}}}_j$, which must vanish. There are two realizations of this orthogonality constraint, a condition of consistency between $\Psi$ and the $\psi$. In the first, an [*inclusive*]{} and most general form (OC), the presence of $\Phi_{i,{{\bf{k}}}}$ in $\Psi$ does not exclude the presence of $\Phi_{j,{{\bf{k}}}}$. The individual terms in the scalar product need not vanish, only their sum must. In the second, an [*exclusive*]{} form, the pair-difference constraint (PDC) is a special case of the OC, in which the presence of $\Phi_{i,{{\bf{k}}}}$ excludes $\Phi_{j,{{\bf{k}}}}$ so that either $C_{i,{{\bf{k}}}}$ or $C_{j,{{\bf{k}}}}$ is zero for each ${{\bf{k}}}$, and the sum vanishes term by term. Under the PDC, those $\Phi$ present in the expansion of $\Psi$ must differ from one another by at least two NSOs, forming a subspace $\{\Phi\}_{\text{PDC}}$ 