CISD: the natural truncation
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FCI is exact and unaffordable; the obvious compromise is to keep only the determinants that matter most. Brillouin's theorem already told you singles do not couple to the HF ground state directly, and the determinants lesson told you the Hamiltonian connects states differing by at most two orbitals. So the leading correction is doubles, and the natural truncation is configuration interaction with singles and doubles — CISD.
Diagonalize H in this much smaller space and the lowest eigenvalue is the CISD energy — variational, so always above FCI, but recovering the bulk of the correlation a single molecule needs. For one H₂ in a minimal basis, the only excitation available is the double, so CISD is FCI. The trouble only appears when you have more than one thing to correlate at once.
CISD keeps single and double excitations from the HF reference. Given Brillouin's theorem, what role do the singles actually play in the CISD ground-state energy?
That trouble has a name — size consistency — and you are about to compute it: two H₂ molecules pushed apart, where the right answer must be exactly twice one H₂, and CISD will quietly miss.
For TWO non-interacting H₂ molecules, FCI gives exactly 2× one H₂ but CISD gives a higher energy. The missing piece is one specific determinant. Which?