where M is the number of single-particle orbitals. The HF energy is simply defined as the variational minimum within the span of allowed orbital rotations, satisfying Brillouin's theorem. The UHF determinant, | ΦUHF ⟩, contains rotations of the form c † a,↓ c i,↓ and c † a,↑ c i,↑ , which generate a linear combination of single excitations from RHF which is not itself an eigenfunction of ⟨ Ŝ2 ⟩, but may nonetheless be written as a single unrestricted determinant. This is most straightforwardly seen by expressing the rotation as an exponential and decoupling from t a i the totally symmetric,