When does the RHF method fail

The reason for any Hartree-Fock method failure can be only one thing the wave function is approximated as a sin e Slater determinant. All possible catastrophes come from this. And we might even deduce when the Hartree-Fock method is not appropriate for description of a particular real system. First, let us ask when a single determinant would be OK Well, if out of all determinants which could be constructed from a certain spinorbital basis set, only its energy (i.e. the mean value of Hamiltonian for this determinant) were close to the true energy of the molecule. In such a case, only this determinant would matter in the linear combination of 

Electronic Motion in the Mean Field Atoms and Molecules 

In other words, the danger for the RHF method is when the energy difference between HOMO and LUMO is small. For example, RHF completely fails to describe metals properly Always, when the HOMO-LUMO gap is small, expect bad results. 

An example is provided by the H2 molecule at long internuclear distances. 

In the simplest LCAO MO approach, two electrons are described by the bonding orbital xa and Xb are Is orbitals centred on the H nuclei, a and b, respectively) 

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