Restricted and unrestricted Hartree-Fock theory

In Fock space, the exact solution to the SchrOdinger equation is an eigenfunction of the total and projected spins. Often, we would like the approximate solutions to exhibit the same spin symmetries. However, according to the discussion in Section 4.4, it does not follow that the Hartree-Fock wave function - which is not an eigenfunction of the exact Hamiltonian - possesses the same symmetries as the exact solution. In general, therefore, these and other symmetries of the exact state must be imposed on the Hartree-Fock solution. 

In the unrestricted Hartree-Fock (UHF) approximation, on the other hand, the wave function is not required to be a spin eigenfunction and different spatial orbitals are used for different spins. Occasionally, the spatial symmetry restrictions are also lifted, and the MOs are then no longer required to transform as irreducible representations of the molecular point group. It should be noted that, in particular for systems close to the equilibrium geometry, the symmetries of the exact state are sometimes present in the UHF state as well even though they have not been imposed during its optimization. In such cases, the UHF and RHF states will coincide. 

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