7.8.2 Approximate Spin Purification

Singlet biradicals are an important special case, whose description within a single-determinant SCF formalism is typical characterized by significant spin contamination, $⟨{\widehat{S}}^2⟩\approx 1$ (in atomic units), indicating an approximately equal mixture of singlet ($⟨{\widehat{S}}^2⟩=0$) and triplet ($⟨{\widehat{S}}^2⟩=2$) 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 formula ¹⁴⁶⁰ Ziegler T., Rauk A., Baerends E. J.
Theor. Chem. Acc.
(1977), 43, pp. 261. Link ^{, 287} Daul C.
Int. J. Quantum Chem.
(1994), 52, pp. 867. Link ^{, 179} Carter-Fenk K., Herbert J. M.
J. Chem. Theory Comput.
(2020), 16, pp. 5067. Link

which expresses the open-shell singlet energy ($E_{\text{OpSing}}$) in terms of the energy of the broken-symmetry solution ($E_{\text{BS}}$, meaning the state with $⟨{\widehat{S}}^2⟩\approx 1$) and the triplet energy ($E_{\text{trip}}$). Evaluation of Eq. (7.59) 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. (7.59), using a consistent set of orbitals for both terms in the $E_{\text{OpSing}}$ formula, whereas the method that is requested using OPSING = TRUE performs two independent SCF calculations with different orbitals in each.