Two Single-Reference Approaches to Singlet Biradicaloid Problems: Complex, Restricted Orbitals and Approximate Spin-Projection Combined With Regularized Orbital-Optimized M{o}ller-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 M{\o}ller-Plesset method ($\kappa$-OOMP2). The first approach is to combine the Yamaguchi's approximate projection (AP) scheme and $\kappa$-OOMP2 with unrestricted (U) orbitals ($\kappa$-UOOMP2). By capturing only essential symmetry breaking, $\kappa$-UOOMP2 can serve as a suitable basis for AP. The second approach is $\kappa$-OOMP2 with complex, restricted (cR) orbitals ($\kappa$-cROOMP2). Though its applicability is more limited due to the comparative rarity of cR solutions, $\kappa$-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+$\kappa$-UOOMP2 and $\kappa$-cROOMP2 can perform similarly well in the TS12 set, a data set 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 $\text{C}_{30}$ ($\text{D}_{5h}$) $\kappa$-cROOMP2 is more suitable as it conserves spatial symmetry. Lastly, we present an organic biradicaloid that does not have a $\kappa$-cROOMP2 solution in which case only AP+$\kappa$-UOOMP2 is applicable. We recommend $\kappa$-cROOMP2 whenever complex polarization is essential and AP+$\kappa$-UOOMP2 for biradicaloids without essential complex polarization but with essential spin-polarization.