Multi-d-occupancy as an alternative definition for the double-d-shell effect

Despite of the prevalence of first-row transition metal containing compounds in virtually all areas of chemistry, the accurate modeling of these systems is a known challenge for the theoretical chemistry community. Such a challenge is shown in a myriad of facets, among them: difficulties in defining ground states multiplicities; disagreement in the results from methods considered highly accurate; and convergence problems in calculations for excited states. These problems cause a scarcity of reliable theoretical data for transition metal containing systems. In this work we explore the double d-shell effect, that plagues and makes difficult the application of multireference methods to this type of systems. We propose an alternative definition for this effect based on the mixing among d-occupancy configurations, or the multi-d-occupancy character of the wave function. Moreover, we present a protocol able to include this effect in multireference calculations using an active space smaller than the one usually used in the literature. A molybdenum-copper model system and its copper subsystem are used as example study cases, in particular, the molybdenum-copper charge transfer of the former and the electronic affinity of the latter. We have shown that our alternative definition can be used to analyze their reference wave functions qualitatively. Based on this qualitative description it is possible to optimize an active space without a second d-shell, able to obtain relative energies accurately. Seeing the double d-shell effect through the lens of a multi-d-occupancy character it is possible to correctly describe the wave function, improve the accuracy of the relative energies, and cheapen the multireference calculations. That way, we believe that this alternative definition has the potential to improve the modeling of first-row transition metal containing compounds both for their ground and excited electronic structure.