Furthermore, Espinosa et al.26 introduced the concept of covalency degree as the total pressure per electron density unit around the BCP that is easily obtained by dividing H(r) by the electron density at that point. It represents a more balanced way to assess covalency in any pairwise interaction. Our results indicate essentially the same trend observed in total energy densities, though separations are more pronounced between Bk–O(1) Bk–O(2).
Further insight into the topology of these complexes can be obtained by looking at the localization ( λ ) and delocalization (δ) indices (Table S7, ESI). These integrated parameters can be related to the bond orders and the actual oxidation state (Z – λ ) of Bk in the molecule. The calculated values assign the +4 oxidation state to both complexes that deviate significantly from calculations previously reported for Bk(IV) in a hexachloride environment (+3.345).5 This could mirror the way that the ligands in complexes 1 and 2coordinate to Bk(IV), where charge-transfer seems not to be as important as ionic and covalent interactions. The δBCP parameters are similar in magnitude to the Wiberg bond orders (Table S5, ESI) reflecting the consistency of the results around the concept of bond order.
A final important remark regarding the electron density is the bond ellipticity. This parameter reveals the symmetry of the electron density along the bond-path. Thus, for pure σ-bonds a value of zero is expected to represent a cylindrically symmetric electron density; whereas, when π contributions take place in the bond, ellipticity deviates from zero. To put Bk–O bonds into perspective, a C–C bond in butane has an ellipticity of 0.01, while a C=C bond in ethene reaches a value of 0.3. As shown in Table 5, the ellipticity in Bk–CO3 bonds is much more cylindrical than Bk– OH bonds, which confirms and agrees with the results shown in Table 4 based on the σ and π contributions to the bond order.

Interacting Quantum Atom (IQA)

As a final approach to shed lights on covalency, IQA27provides a way to measure covalency in terms of energy decomposition by separating contributions from purely Coulomb/electrostatic and exchange (exchange-correlation in case of DFT) interactions between two topological atoms (basins). It is noteworthy that these energies are evaluated by integrating over the basins and have no direct relation to properties obtained at the BCP. Table 6 (decomposition per bond in Table S8, ESI) shows this decomposition analysis for complexes 1and 2 . The behavior of energy densities in Table 5 can be understood by the role of exchange interactions in the total energy of interaction between Bk and O basins. Thus, within IQA, covalency is seen as the weight that exchange energies have in the overall energy of interaction. The results indicate that complex 1 has the same covalent character in the Bk–O(1) and Bk–O(2) bonds. Thus, the small difference observed for the total energy of interaction EINT in 1 is due to Coulombic interactions, though this difference is negligible. In contrast, a total rearrangement is observed in complex 2 . Exchange interactions are decreased in Bk–O bonds due to the strengthening of the Bk–OH bonds and particularly the Bk–OH(1) bond with the concomitant weakening of the opposed berkelium carbonate bonds. This is reflected in the nearly 3% increase of covalency in the bonds.
Table 6. IQA energy decomposition. EC corresponds to Coulombic energies, EX to exchange energies, and EINT refers to total energy of interaction (EC + EX) between Bk and O atoms. Covalent energy is expressed as the weight that exchange interactions have in the total energy. Average energies are given in kJ mol-1. Detailed table is found in ESI