Singlet biradicals are an important special case, whose description within a single-determinant SCF formalism is typical
characterized by significant spin contamination, (in atomic units), indicating an approximately
equal mixture of singlet () and triplet () wave functions. This is a result
of the fact that a proper description of two electrons in two half-filled orbitals requires a minimum of two determinants.
A simple means to correct for this is to use the approximate spin-purification
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which expresses the open-shell singlet energy () in terms of the energy of the broken-symmetry solution (, meaning the state with ) and the triplet energy (). Evaluation of Eq. (7.36) requires two separate SCF calculations but these can be performed together automatically by specifying OPSING = TRUE in the $rem section; see Example 7.6.1. Analytic gradients have been implemented for this approximate spin-purification approach, which again requires two separate SCF gradient computations. For the specific case of open-shell singlets, the ROKS method that is discussed below can be understood as a fully self-consistent or orbital-optimized version of the formula in Eq. (7.36), using a consistent set of orbitals for both terms in the formula, whereas the method that is requested using OPSING = TRUE performs two independent SCF calculations with different orbitals in each.