with N ↑ for spin-up electrons, N ↓ for spin-down electrons, S for spin magnitude, and Γ for point-group irrep. These fictitious, fragment-indexed quantum numbers are not conserved by the exact, correlated many-electron wave function of the whole system, but they are (in practice) conserved by the approximate |LAS⟩ wave function. Much like a single determinant, a single LAS state is generally not an eigenstate of Ŝ2 unless at most one fragment has a nonzero S .