model, where all aforementioned symmetries of the exact wavefunction are preserved. However, a trade-off arises: by allowing an approximate wavefunction to break some of these symmetries, one may be able to reach a lower variational energy. This "symmetry dilemma" has been well documented in literature. Fukutome 58 and later Paldus & Stuber 59 categorized the different variants of HF based on the symmetries they preserve (Figure ). In this work we shall focus on generalized HF (GHF) (sometimes referred to as real GHF to distinguish it from complex GHF), which breaks the T , Ŝ2 and Ŝz symmetries. Practically this means GHF orbitals are not spin pure; each electron in a GHF wavefunction occupies a linear combination of spin-up and spin-down orbitals. The resulting Ŝz symmetry breaking, known as non-collinearity, allows GHF to better describe spin-frustrated systems. As the RHF determinant and its particle-hole excitations span the entire vector space of an N -electron wavefunction, they may be used to express any wavefunction (exact or approximate) within that space. While all SCF solutions are themselves a single determinant, a Thouless rotation 62 can represent symmetry-broken single determinants as a linear combination of these RHF states. Specifically, a unitary one-particle transformation on an arbitrary N -electron single-determinant wavefunction generates another single determinant: