Full CI: the exact answer in a basis

Hartree-Fock uses one determinant and misses correlation. Full Configuration Interaction (FCI) uses all of them — every way to distribute the electrons among the orbitals — combined as |Ψ⟩ = Σ c_I |Φ_I⟩.

Finding the coefficients is a single (very large) matrix diagonalization Hc = Ec, and the lowest eigenvalue is the exact energy within the basis. FCI recovers every bit of correlation the basis can hold — but the determinant count explodes combinatorially, so it is confined to small systems and serves as the benchmark everything else is judged against.