Addressing first derivative discontinuities in orbital-optimised opposite-spin scaled second-order perturbation theory with regularisation

Orbital-optimised opposite-spin scaled second-order perturbation theory (O2) generates a single-reference wave function composed of approximate Brueckner orbitals with fourth-order computational scaling. While O2 provides significantly improved treatment of radicals by reducing spin contamination, it has been shown to suffer from first derivative discontinuities for bond stretching near the unrestriction point. That qualitative failure is resolved in this work by the implementation of regularised O2, which includes a regularisation parameter in the denominator of its second-order term. The value of the regularisation parameter is semi-empirically chosen to qualitatively describe bond stretching energetics of hydrogen, ethane and ethene, while also considering the effect of the regularisation parameter on thermochemical errors for the well-known Gaussian-2 (G2) test set. The generality of the empirical scaling and semi-empirical regularisation parameter is studied by application to the 3dMLBE20, DBH24, RSE43 and W4-11 test sets. We demonstrate that accuracy of O2 is roughly maintained and sometimes even improved by regularisation, with root mean squares of regularised O2 between factors of 1.6 and 0.8 from corresponding root mean squares of O2