Consistent spin decontamination of broken-symmetry calculations of diradicals

Broken-symmetry calculations of diradicals exploit the mean-field energies of determinants that are not eigenfunctions of the Ŝ² operator, the mean value of which is close to 1 for the m_s = 0 solution. This spin contamination must be corrected. Two different contributions affect ⟨Ŝ²⟩, namely, the mixing between neutral and ionic valence bond components, the so-called kinetic exchange, which decreases ⟨Ŝ²⟩, and the spin polarization of the supposedly closed shell orbitals, which increases ⟨Ŝ²⟩. The popular Yamaguchi formula is valid for the first effect but irrelevant for the second one. From a few constrained broken-symmetry calculations, one may treat separately the two contributions and apply their specific spin decontamination correction. This work proposes a consistent spin-decontaminated procedure for the evaluation of singlet-triplet gaps in diradicals.