Restricted active space spin-flip (RAS-SF) with arbitrary number of spin-flips

The restricted active space spin-flip (RAS-SF) approach is a multistate, spin-complete, variational and size consistent method applicable to systems featuring electronic (near-)degeneracies. In contrast to CASSCF it does not involve orbital optimizations and so avoids issues such as root-flipping and state averaging. This also makes RAS-SF calculations roughly 100-1000 times faster. In this paper RAS-SF method is extended to include variable orbital active spaces and three or more spin-flips, which allows the study of polynuclear metal systems, triple bond dissociations and organic polyradicals featuring more than four unpaired electrons. Benchmark calculations on such systems are carried out and comparison to other wave-function based, multi-reference methods, such as CASSCF and DMRG yield very good agreement, provided that the same active space is employed. Where experimental values are available, RAS-SF is found to substantially underestimate the exchange coupling constants, if the minimal active space is chosen. However, the correct ground state is always obtained. Not surprisingly, inclusion of bridge orbitals into the active space can cause the magnitude of the coupling constants to increase substantially. Importantly, the ratio of exchange couplings in related systems is in much better agreement with experiment than the magnitude of the coupling. Nevertheless, the results indicate the need for the inclusion of dynamic correlation to obtain better accuracy in minimal active spaces.