7.7.3 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, ¹⁵⁸⁰ Ziegler T., Rauk A., Baerends E. J.
Theor. Chem. Acc.
(1977), 43, pp. 261. Link ^{, 998} Noodleman L.
J. Chem. Phys.
(1981), 74, pp. 5737. Link ^{, 997} Noodleman L., Davidson E. R.
Chem. Phys.
(1986), 109, pp. 131. Link ^{, 305} Daul C.
Int. J. Quantum Chem.
(1994), 52, pp. 867. Link ^{, 872} Mak A. M., Lawler K. V., Head-Gordon M.
Chem. Phys. Lett.
(2011), 515, pp. 173. Link ^{, 196} Carter-Fenk K., Herbert J. M.
J. Chem. Theory Comput.
(2020), 16, pp. 5067. Link

which expresses the energy of the open-shell singlet state ($E_{\text{OSS}}$) 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 ($E_{\text{trip}}$). This formula was originally suggested by Ziegler, Rauk, and Baerends, ¹⁵⁸⁰ Ziegler T., Rauk A., Baerends E. J.
Theor. Chem. Acc.
(1977), 43, pp. 261. Link but was popularized by Noodleman in application to metalloenzymes.

Evaluation of Eq. (7.78) requires two separate SCF calculations and these can be performed together, automatically, by specifying SPIN_PROJ = NOODLEMAN in the $rem section; see Example 7.7.2. Analytic gradients have been implemented for this approximate spin-purification approach, which again requires two separate SCF gradient computations. (Hessians are not available, however, even by finite-difference.) For the specific case of open-shell singlets, the ROKS method that is discussed in Section 7.7.4 can be understood as a fully self-consistent or orbital-optimized version of the formula in Eq. (7.78), using a consistent set of orbitals for both terms in Eq. (7.78). In contrast, Noodleman’s approximate spin purification procedure performs two independent SCF calculations with different orbitals in each.