Two single-reference approaches to singlet biradicaloid problems: Complex, restricted orbitals and approximate spin-projection combined with regularized orbital-optimized Møller-Plesset perturbation theory.

We present a comprehensive study of two single-reference approaches to singlet biradicaloids. These two approaches are based on the recently developed regularized orbital-optimized Møller-Plesset method (κ-OOMP2). The first approach is to combine Yamaguchi's approximate projection (AP) scheme and κ-OOMP2 with unrestricted (U) orbitals (κ-UOOMP2). By capturing only essential symmetry breaking, κ-UOOMP2 can serve as a suitable basis for AP. The second approach is κ-OOMP2 with complex, restricted (cR) orbitals (κ-cROOMP2). Although its applicability is more limited due to the comparative rarity of cR solutions, κ-cROOMP2 offers a simple framework for describing singlet biradicaloids with complex polarization while removing artificial spatial symmetry breaking. We compare the scope of these two methods with numerical studies. We show that AP+κ-UOOMP2 and κ-cROOMP2 can perform similarly well in the TS12 set, a dataset that includes 12 data points for triplet-singlet gaps of several atoms and diatomic molecules with a triplet ground state. This was also found to be true for the barrier height of a reaction involving attack on a cysteine ion by a singlet oxygen molecule. However, we also demonstrate that in highly symmetric systems like C30 (D5h), κ-cROOMP2 is more suitable as it conserves spatial symmetry. Finally, we present an organic biradicaloid that does not have a κ-cROOMP2 solution in which case only AP+κ-UOOMP2 is applicable. We recommend κ-cROOMP2 whenever complex polarization is essential and AP+κ-UOOMP2 for biradicaloids without essential complex polarization but with essential spin-polarization.