Equations are proposed for computing from ab initio wave functions quantities like $〈{{S}}_A{{\cdot S}}_B\mathrm{〉,}$ which appear in the Heisenberg model Hamiltonian of magnetism. These equations are based on projection operators derived from Löwdin orthogonalization. They result in local spin operators ${{S}}_A$ which obey the definition of angular momentum operators and commute with each other. These equations are evaluated for several typical closed and open shell molecules. For closed shells in the single Slater determinant approximation, $〈{{S}}_A{{\cdot S}}_B〉$ is −$\frac{3}{8}$ of the bond-order and $〈{{S}}_A^2〉$ is $+\frac{3}{8}$ of the total number of bonds to center A. For open shells there are additional contributions from the unpaired electrons. In favorable cases, these additional terms have the value assumed as the whole answer in the usual applications of the Heisenberg Hamiltonian.

Recent advances in computational methods have made it possible to calculate the wave functions for a wide variety of simple actinide complexes. Equally important is the ability to analyze the information contained therein and produce a chemically meaningful understanding of the electronic structure. Yet the performance of the most common wave function analyses for the calculation of atomic charge and bond order has not been thoroughly investigated for actinide systems. This is particularly relevant because the calculation of charge and bond order even in transition metal complexes is known to be fraught with difficulty. Here we use Mulliken, Löwdin, natural population analysis, atoms-in-molecules (AIM), and fuzzy atom techniques to determine the charges and bond orders of UO22+, PuO22+, UO2, UO2Cl42−, UO2(CO)52+, UO2(CO)42+, UO2(CN)53−, UO2(CN)42−, UO2(OH)53−, and UO2(OH)42−. This series exhibits a clear experimental and computational trend in bond lengths and vibrational frequencies. The results indicate that Mulliken and Löwdin populations and bond orders are unreliable for the actinyls. Natural population analysis performs well after modification of the partitioning of atomic orbitals to include the 6d in the valence space. The AIM topological partitioning is insensitive to the electron donating ability of the equatorial ligands and the relative atomic volume of the formally U(VI) center is counterintuitively larger than that of O2− in the UO22+ core. Lastly, the calibrated fuzzy atom method yields reasonable bond orders for the actinyls at significantly reduced computational cost relative to the AIM analysis.