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