Decomposition of dand f-Shell Contributions to Uranium Bonding from the Quantum Theory of Atoms in Molecules : Application to Uranium and Uranyl Halides

The electronic structures of a series of uranium hexahalide and uranyl tetrahalide complexes were simulated at the density functional theoretical (DFT) level. The resulting electronic structures were analyzed using a novel application of the Quantum Theory of Atoms in Molecules (QTAIM) by exploiting the high symmetry of the complexes to determine 5fand 6d-shell contributions to bonding via symmetry arguments. This analysis revealed fluoride ligation to result in strong bonds with a significant covalent character while ligation by chloride and bromide species resulted in more ionic interactions with little differentiation between the ligands. Fluoride ligands were also found to be most capable of perturbing an existing electronic structure. 5f contributions to overlap-driven covalency were found to be larger than 6d contributions for all interactions in all complexes studied while degeneracy-driven covalent contributions showed significantly greater variation. σ-contributions to degeneracy-driven covalency were found to be consistently larger than those of individual π-components while the total π-contribution was, in some cases, larger. Strong correlations were found between overlap-driven covalent bond contributions, U–O vibrational frequencies, and energetic stability, which indicates that overlap-driven covalency leads to bond stabilization in these complexes and that uranyl vibrational frequencies can be used to quantitatively probe equatorial bond covalency. For uranium hexahalides, degeneracy-driven covalency was found to anti-correlate with bond stability.


Introduction
The quantification of the covalent contribution to bonding in complexes of the f-elements is an area of great current research being explored via X-ray absorption [1][2][3][4][5][6][7][8][9][10][11], electron paramagnetic [12][13][14][15], nuclear magnetic resonance [16,17], emission [18,19], and photoelectron [20] spectroscopies as well as X-ray diffraction [21] and structural studies [22][23][24][25][26][27][28][29][30].The latter have often been carried out in combination with theoretical studies and a wealth of purely theoretical data also exists .However, covalency is a phenomenon that is open to significant interpretation.It can manifest due to (i) the near-degeneracy of energy levels and/or (ii) the spatial overlap of the electronic wave functions of interacting species and, of these, only the latter leads to the accumulation of electronic charge in the bonding region and, therefore, the potential for energetic stabilization of the bond.Differentiating between these two origins of covalent bond character is, in addition to being of fundamental interest, of significant practical importance.Current European [54] and US [55] strategies for the separation of trivalent lanthanides from the highly radioactive minor actinides (Np, Am, and Cm) in spent nuclear fuel are based on the exploitation of energetic stabilization of complexes of the latter with the origin of this stability believed to be due to the greater chemical availability of the 5f valence shell.
Recently, several groups have adopted Bader's Quantum Theory of Atoms in Molecules (QTAIM) [56] in order to probe covalency from a computational perspective.The QTAIM has the capacity to differentiate between degeneracy-driven and overlap-driven contributions to bonding interactions [57].It is an appealing method of analysis since its results are independent of the orbital basis used to express the electronic structure, which cannot be unambiguously defined.This characteristic means that the QTAIM is equally applicable to electronic structures obtained using any quantum chemical methodology and, since it analyzes the physically observable electron density, it can be applied to experimentally-determined data [21,58].
Complexes of the f-elements typically exhibit strong relativistic effects, substantial dynamical electron correlation, and weak crystal fields and these factors combine to produce electronic structures in which the 5f, 6d, and, to a lesser extent, the 7s valence shells exist at similar energies, which complicates the role that these shells play in covalent interactions.Recently, we have shown that the QTAIM can be exploited in systems of high symmetry to differentiate between the roles played by the 5f and 6d shells [59][60][61].In this study, we further extend this approach to distinguish between σ-components and π-components of covalent bond character.This approach is applied to a series of homo-halide and hetero-halide complexes of the general form [UO 2 X 2 Y 2 ] 2− and UX 2 Y 2 Z 2 where X, Y, Z ∈ {F, Cl, Br}.These complexes have been selected due to their high symmetry (D 2h or higher) when assuming a trans orientation of like ligands and because they include a number of synthetically realized species: [UCl 6 ] 2− and [UF 6 ] − have both been the subject of recent experimental studies of U-X bonding [5,21] while the experimentally determined electronic structure of Cs 2 UO 2 Cl 4 has been previously analyzed using the QTAIM [58].Of the remaining complexes, [UO 2 F 4 ] 2− and [UO 2 Br 4 ] 2− have been observed in the gas and condensed phases, respectively [62,63].[UF 6 ] − and [UCl 6 ] − have both been experimentally characterized [64][65][66] although [UBr 6 ] − remains unreported.

Identification of Model Chemistry
The performance of a number of xc-functionals was assessed in order to determine the most suitable model chemistry for the study.The functionals chosen for this exercise were the GGA functionals PBE [67] and BLYP [68,69] along with the hybrid-GGA functionals PBE0 [70], B3LYP [71,72], and BHLYP [73].A variation of the chosen basis set was not considered since its large size was assumed to give results sufficiently close to the basis set limit.Additionally, it has recently been shown that the def2-QZVP basis set used for the ligand ions is sufficient for modeling anions in the absence of diffuse functions [74].For these test calculations, two representative systems were considered: [UO 2 Cl 4 ] 2− and UF 6 , for which accurate experimental data exists.
Tables 1 and 2 summarize selected structural and vibrational parameters of these systems.In particular, only frequencies of vibrational stretching modes are reported since these most clearly relate to bond strength.As expected, hybrid functionals tend to outperform pure GGA functionals.Of the hybrid functionals, BHLYP (incorporating 50% Hartree-Fock exchange) is outperformed by both B3LYP and PBE0 (incorporating 20% and 25% Hartree-Fock exchange, respectively) especially with respect to vibrational frequencies.B3LYP and PBE0 perform comparably well and are in good to excellent agreement with experimental parameters.B3LYP noticeably outperforms PBE0 in the simulation of U-O bond lengths and vibrational frequencies in [UO 2 Cl 4 ] 2− and was chosen as the xc-functional used in the remainder of this study.The accuracy of the B3LYP functional reported in this study is comparable to that previously reported for other simple uranyl coordination complexes [75] and for f-element hexachlorides (M = U, Ce) [61].

Homohalide Complexes of Uranium and Uranyl
In this section, the simulation and analysis of complexes of the form [UO 2 X 4 ] 2− and UX 6 (X = F, Cl, Br) is considered.Focusing initially on the uranyl complexes, Table 3 summarizes structural parameters obtained using the B3LYP xc-functional.
A clear trend of increasing equatorial bond length is found as the halide mass increases, which is expected with the increasing size of the ligand.There is also, however, a commensurate reduction in the axial U-O bond length, which suggests a weakening of the equatorial bonds.Although equatorial bond lengths are found to be greater than the sum of the ionic radii, the difference is least pronounced for X = F, which again increases as the halide mass increases.This suggests some degree of covalent bond stabilization for the lighter halides.
As with [UO 2 Cl 4 ] 2− (Table 1), the U-O bond length in [UO 2 Br 4 ] 2− is in excellent agreement with the experimentally reported value of 1.766 Å [63].However, it should be noted that the experimental data are derived from XRD studies of condensed-phase structures.No experimentally determined structural data is available for [UO 2 F 4 ] 2− , which has only been observed in the gas phase [62].However, the U-O bond length reported here lies between the computationally-derived gas-phase literature values of 1.808 and 1.851 Å obtained using PBE and CCSD(T), respectively, [62] and compares well to that previously obtained using the B3LYP functional [37,78].The U-Cl bond length is overestimated by 0.078 Å but is identical to that reported previously using the same functional [37] and the U-Br bond length is overestimated by 0.108 Å.The U-F bond length of 2.229 Å is very similar to that reported from PBE (2.233 Å), B3LYP (2.231 Å), and CCSD(T) (2.232 Å) simulations [37,62].The excellent agreement with the CCSD(T) data, in particular, indicates that the origin of the discrepancies between experimental and calculated values for the chloride and bromide complexes is due to crystal packing effects.
The calculated structural data for UX 6 (Table 4) presents a less clear picture.While the U-X bond lengths again increase as the halides mass increases, a comparison with the sum of ionic radii gives no trend.In contrast to the uranyl halides, bond lengths in these species are shorter than the sum of the ionic radii due to the lack of competition with strongly coordinating terminal oxo ligands.However, this shortening is identical in the fluoride and bromide complexes and less than that found in UCl 6 .The good agreement with the experimental gas-phase U-F bond length (Table 2) is apparently not replicated in UCl 6 where the U-Cl bond is overestimated by 0.050 Å in comparison to the value of 2.42 Å reported experimentally [66].However, this value is again derived from an XRD study of a condensed-phase structure and so the discrepancy is comparable to those found for [UO 2 Cl 4 ] 2− and [UO 2 Br 4 ] 2− .The calculated U-Cl bond length is in excellent agreement with that calculated by Batista et al. (2.472 Å) using the same xc-functional [79].There are no experimentally reported simple bromides of uranium [80], but the value reported in this study is in good agreement with and slightly shorter than those reported from GGA-based simulations (2.650-2.687Å) [81], commensurate with the data presented in Tables 1 and 2. The good agreement with available literature data, therefore, leads us to tentatively suggest that an interplay of electronic and steric effects serve to mask any simple trend in UX 6 bond lengths when compared to ionic radii.Bonding analysis of these complexes was performed using the Quantum Theory of Atoms In Molecules (QTAIM) [56].We have recently discussed the merits of this methodology in analyzing the electronic structure of f-element complexes [57].In this study, advantage is taken of the high symmetry of these complexes in order to perform a novel decomposition of a bonding contribution to 5f and 6d uranium valence shells.
Two QTAIM metrics are commonly used to characterize bonding in molecular systems.The first of these is ρ BCP (A, B), which is the magnitude of the electronic density at the bond critical point (BCP) between atoms A and B where the BCP is the saddle point in the electronic density distribution ρ(r) being a local maximum on the interatomic surface and a local minimum perpendicular to the surface.A commonly employed rule of thumb is that covalent interactions are characterized by ρ BCP > 0.2 a.u. and closed shell interactions by ρ BCP < 0.1 a.u. with intermediate values indicating an increasing degree of a covalent charter.ρ BCP can, therefore, be interpreted as a measure of covalent interaction manifested by orbital overlap.
The second QTAIM metric used to characterize bonding is the delocalization index, δ(A, B).The delocalization index is an integrated property and, for chemical systems described by a single electronic configuration, can be defined by the equation below: where S ij (X) is the overlap between molecular orbitals (MOs) ϕ i (r) and ϕ j (r) integrated over the atomic basin associated with atom X (as defined by QTAIM).δ(A, B) is a measure of the number of electrons shared between two atoms and, in the absence of bond polarisation, correlates strongly with a formal bond order.More generally, δ(A, B) can be interpreted as indicating the degree of covalent interaction manifested by energetic degeneracy since it can be large in the absence of significant overlap between species.While, typically, large values of ρ BCP (A, B) are accompanied by large values of δ(A, B), the converse is not universally true.
The complexes considered in this study all possess D 2h (or higher) symmetry.Since the systems are closed shell in nature, the electron density possesses the same symmetry as the complex it describes and, therefore, the atomic basin associated with the central uranium atom also possesses this symmetry (see Figure 1).The delocalization index between the uranium center and any other atom in the complex can then be decomposed by the equation below.
where Γ indexes the irreducible representations (irreps) of the point group and the summation over MOs is now limited to those orbitals spanning the irrep Γ.This definition is valid since the overlap integral S ij (U) is only non-zero when the MOs ϕ i (r) and ϕ j (r) span the same irreducible representation.
We have used a similar, though less developed, approach in considering the contributions to bonding in a series of actinocenes [60].
Inorganics 2018, 6, x FOR PEER REVIEW 5 of 18 where (X) is the overlap between molecular orbitals (MOs) ( ) and ( ) integrated over the atomic basin associated with atom X (as defined by QTAIM).(A, B) is a measure of the number of electrons shared between two atoms and, in the absence of bond polarisation, correlates strongly with a formal bond order.More generally, (A, B) can be interpreted as indicating the degree of covalent interaction manifested by energetic degeneracy since it can be large in the absence of significant overlap between species.While, typically, large values of (A, B) are accompanied by large values of (A, B), the converse is not universally true.
The complexes considered in this study all possess D2h (or higher) symmetry.Since the systems are closed shell in nature, the electron density possesses the same symmetry as the complex it describes and, therefore, the atomic basin associated with the central uranium atom also possesses this symmetry (see Figure 1).The delocalization index between the uranium center and any other atom in the complex can then be decomposed by the equation below.
where Γ indexes the irreducible representations (irreps) of the point group and the summation over MOs is now limited to those orbitals spanning the irrep Γ.This definition is valid since the overlap integral (U) is only non-zero when the MOs ( ) and ( ) span the same irreducible representation.We have used a similar, though less developed, approach in considering the contributions to bonding in a series of actinocenes [60].In D2h symmetry, the components of the 5f and 6d valence shells that can engage in σ-bonding, π-bonding, δ-bonding, and φ-bonding interactions with ligands lying on the principal axis are summarized in Table 5.In the systems under consideration here, δ-bonding and φ-bonding interactions are not possible (or are, at least, energetically highly unfavorable) and σ-bonding and πbonding contributions from the 5f and 6d shell can be decomposed from the total delocalization index.Similarly, can be decomposed into contributions from the 5f and 6d shells even though further decomposition is not possible since only the σ-bonding contribution is non-zero at the BCP  Table 5. Irreducible representations spanned by components of the 5f and 6d valence shells with respect to the principal molecular axis.

Irrep ( Γ)
In D 2h symmetry, the components of the 5f and 6d valence shells that can engage in σ-bonding, π-bonding, δ-bonding, and ϕ-bonding interactions with ligands lying on the principal axis are summarized in Table 5.In the systems under consideration here, δ-bonding and ϕ-bonding interactions are not possible (or are, at least, energetically highly unfavorable) and σ-bonding and π-bonding contributions from the 5f and 6d shell can be decomposed from the total delocalization index.Similarly, ρ BCP can be decomposed into contributions from the 5f and 6d shells even though further decomposition is not possible since only the σ-bonding contribution is non-zero at the BCP (all other contributions to ρ(r) contain a nodal plane passing through the BCP).Lastly, it should be noted that, even though this decomposition is only valid with respect to bonding contributions aligned with the principal axis of the molecule, in all complexes considered here, the principal axis can be chosen to align with any U-O or U-X bond while maintaining D 2h symmetry and, therefore, the decomposition can be applied to all bonding interactions.
Figure 2 summarizes the relevant QTAIM metrics for the U-O bond in [UO 2 X 4 ] 2− complexes (see Supplementary Materials for numerical data).The magnitude of ρ BCP deviates most strongly from that of free uranyl in the case of the fluoride complex, which is indicative of a weakening of the covalent character of the U-O interaction.As the halide mass increases, so does ρ BCP , which implies that the heavier halides perturb the U-O bond to a lesser extent.This is commensurate with the data presented in Table 3 where covalent stabilization of the equatorial bonds appears most pronounced for the fluoride complex.The values of ρ BCP reported in this study compare well to those previously reported for the fluoride and chloride complexes as well as for the free uranyl ion [37].(all other contributions to ( ) contain a nodal plane passing through the BCP).Lastly, it should be noted that, even though this decomposition is only valid with respect to bonding contributions aligned with the principal axis of the molecule, in all complexes considered here, the principal axis can be chosen to align with any U-O or U-X bond while maintaining D2h symmetry and, therefore, the decomposition can be applied to all bonding interactions.Figure 2 summarizes the relevant QTAIM metrics for the U-O bond in [UO2X4] 2− complexes (see Supplementary Materials for numerical data).The magnitude of deviates most strongly from that of free uranyl in the case of the fluoride complex, which is indicative of a weakening of the covalent character of the U-O interaction.As the halide mass increases, so does , which implies that the heavier halides perturb the U-O bond to a lesser extent.This is commensurate with the data presented in Table 3 where covalent stabilization of the equatorial bonds appears most pronounced for the fluoride complex.The values of reported in this study compare well to those previously reported for the fluoride and chloride complexes as well as for the free uranyl ion [37].The 5f contribution to accounts for ~60% of the total value irrespective of the species under consideration, which indicates that the 5f-shell is more involved in covalent interactions with the oxo ligand than the 6d-shell.The percentage variations in the 5f and 6d contributions to are extremely similar with values in the fluoride complex being 71% and 73% of the free uranyl values, respectively.
The corresponding (U, O) data demonstrate broadly the same trend, but provide further insight.The total number of electrons shared is substantially lower in the halide complexes than in free uranyl in agreement with previous work [37], but again increases as the halide mass increases and the perturbation of the U-O bond is reduced.(U, O) is 78% of the free uranyl value in the fluoride complex but the 5f contribution accounts for 65% of the total, which is greater than the 63% contribution in free uranyl and indicates a stability of this contribution to the U-O bond with respect to that of free uranyl.The ratio of 5f/6d contributions is reasonably constant across the halides.For example, the 5f component is 64% of the total value in the bromide complex.It should be noted that there are two components of the 5f shell (and likewise for the 6d shell) that can engage in πinteractions with the oxo ligand and the contributions reported in this paper are the composite of these two distinct interactions.Therefore, the degree of electron sharing is greater for the σcomponent of each bond than for each of the π-components.
While there is an overall reduction in (U, O) upon completion, this reduction is not distributed The 5f contribution to ρ BCP accounts for ~60% of the total value irrespective of the species under consideration, which indicates that the 5f-shell is more involved in covalent interactions with the oxo ligand than the 6d-shell.The percentage variations in the 5f and 6d contributions to ρ BCP are extremely similar with values in the fluoride complex being 71% and 73% of the free uranyl values, respectively.
The corresponding δ(U, O) data demonstrate broadly the same trend, but provide further insight.The total number of electrons shared is substantially lower in the halide complexes than in free uranyl in agreement with previous work [37], but again increases as the halide mass increases and the perturbation of the U-O bond is reduced.δ(U, O) is 78% of the free uranyl value in the fluoride complex but the 5f contribution accounts for 65% of the total, which is greater than the 63% contribution in free uranyl and indicates a stability of this contribution to the U-O bond with respect to that of free uranyl.The ratio of 5f/6d contributions is reasonably constant across the halides.For example, the 5f component is 64% of the total value in the bromide complex.It should be noted that there are two components of the 5f shell (and likewise for the 6d shell) that can engage in π-interactions with the oxo ligand and the contributions reported in this paper are the composite of these two distinct interactions.Therefore, the degree of electron sharing is greater for the σ-component of each bond than for each of the π-components.
While there is an overall reduction in δ(U, O) upon completion, this reduction is not distributed evenly among the four bonding components.Taking the fluoride complex as an example, the 5f σ contribution is largely unchanged and is reduced to 95% of its free value while the 5f π contribution reduces to 72% of its free value.Corresponding 6d contributions reduce to 80% and 69%, respectively.This variation can be understood in terms of the availability of these shells to equatorial bonding interactions: the 5f σ and 6d σ components are orientated along the U-O bond and are, therefore, relatively unavailable for equatorial interactions while the 5f π and 6d π components are available for equatorial σ-bonding and π-bonding interactions, respectively, and are, therefore, most significantly affected by the presence of equatorial ligands.Qualitatively similar behavior is found for the other halide complexes.
More broadly, the quantitative difference in the reduction of ρ BCP (which is only sensitive to σ-interactions) and the σ-contributions to δ(U, O) demonstrate how these metrics provide independent but related data regarding bonding.This is most obvious for the 5f-shell: the 5f contribution to ρ BCP reduces to 71% of its free value upon equatorial coordination by fluoride while the corresponding 5f σ contribution to δ(U, O) maintains 95% of its free value, i.e., overlap-induced covalency is reduced while degeneracy-induced covalency is maintained.
Figure 3 summarizes QTAIM metrics for equatorial U-X bonds in [UO 2 X 4 ] 2− and UX 6 complexes (see Supplementary Materials for numerical data).This data was obtained by taking a U-X bond to define the principal molecular axis and reference to σ-contributions and π-contributions to bonding is given in this study with respect to the U-X bond.This approach is taken throughout this contribution.
Considering first ρ BCP (U, X), overall magnitudes are larger in the UX 6 complexes due to the lack of competition with the strongly coordinating terminal oxo species and the consequently shorter U-X bonds.The ρ BCP (U, X) values are, however, much lower than those of the U-O bond, which indicates, as expected, significantly reduced covalent contributions to bonding.Nonetheless, ρ BCP (U, X) has a maximum value (0.155 a.u.) in UF 6 , consistent with substantial covalent bond character.There is a clear reduction in ρ BCP (U, X) in both sets of complexes as the halide mass increases, which shows a reduction in overlap-induced equatorial covalency.Interestingly, the reduction is significantly more pronounced for the 5f shell.The 5f contribution to ρ BCP is 34% and 38% larger than the 6d contribution in [UO 2 F 4 ] 2− and UF 6 , respectively, but is just 4% and 7% larger in [UO 2 Br 4 ] 2− and UBr 6 , respectively.There is a larger contribution to bonding from the 5f shell in fluoride complexes while 5f and 6d contributions are comparable for the heavier halides.
The value of ρ BCP in UF 6 can be indirectly compared to a recent X-ray diffraction study of [UF 6 ] − in which topological analysis of the experimentally determined electron density gave ρ BCP ~0.135 a.u.for the U-F bond [21].This is approximately 13% lower than the value reported here, but it should be borne in mind that uranium is in the +5 oxidation state in [UF 6 ] − as opposed to the +6 oxidation state in UF 6 .We have previously investigated the dependence of QTAIM parameters on uranium oxidation state in uranium hexachloride [61], finding a reduction in ρ BCP of 17% when comparing the +6 and +5 oxidation states.These recent experimental findings, therefore, appear to be in accord with the present study, which suggests a significant decrease in the overlap-driven covalent bond character upon the reduction of the uranium center in both hexachloride and hexafluoride species.Variation in (U, X) is markedly different in the two sets of complexes.For UX6 species, an overall increase is found as the halide mass increases, which opposes the trend in and shows the danger of attempting to use as an indicator of bond order in the presence of bond polarization.5f components of (U, X) are consistently higher than 6d contributions with the 5fσ contribution dominating the fluoride complex and all components increasing as the halide mass increases.This behavior again opposes that exhibited by and demonstrates that overlap-induced and degeneracy-induced covalency can (i) manifest independently and (ii) both be identified by the methodology employed here.For [UO2X4] 2− species, the trend in 5fσ, 5fπ, and 6dπ components mirrors that of while the variation in 6dσ strongly opposes this trend and results in the overall magnitude of (U, X) remaining largely unchanged with the 6dσ component overtaking 5fσ as the dominant contribution when the halide mass increases.The qualitative difference in (U, X) between the two sets of complexes presumably reflects the strong influence of the axial oxo species on (formally) uranium-based energy levels.

Heterohalide Complexes
In this section, complexes of the form [UO2X2Y2] 2− and UX2Y2Z2 (where X, Y, Z ∈ {F, Cl, Br}) are considered.Again, these complexes possess D2h symmetry or higher, which implies that like ligands are oriented trans to each other and allows for the equivalent analysis to that reported in Section 2.2 to be performed.In the case of the UX2Y2Z2 complexes, only the "axial" U-X bond is considered since the permutation of the ligands means that each of the bonding interactions in the 10 possible complexes is thus included.Variation in δ(U, X) is markedly different in the two sets of complexes.For UX 6 species, an overall increase is found as the halide mass increases, which opposes the trend in ρ BCP and shows the danger of attempting to use δ as an indicator of bond order in the presence of bond polarization.5f components of δ(U, X) are consistently higher than 6d contributions with the 5f σ contribution dominating the fluoride complex and all components increasing as the halide mass increases.This behavior again opposes that exhibited by ρ BCP and demonstrates that overlap-induced and degeneracy-induced covalency can (i) manifest independently and (ii) both be identified by the methodology employed here.For [UO 2 X 4 ] 2− species, the trend in 5f σ , 5f π , and 6d π components mirrors that of ρ BCP while the variation in 6d σ strongly opposes this trend and results in the overall magnitude of δ(U, X) remaining largely unchanged with the 6d σ component overtaking 5f σ as the dominant contribution when the halide mass increases.The qualitative difference in δ(U, X) between the two sets of complexes presumably reflects the strong influence of the axial oxo species on (formally) uranium-based energy levels.

Heterohalide Complexes
In this section, complexes of the form [UO 2 X 2 Y 2 ] 2− and UX 2 Y 2 Z 2 (where X, Y, Z ∈ {F, Cl, Br}) are considered.Again, these complexes possess D 2h symmetry or higher, which implies that like ligands are oriented trans to each other and allows for the equivalent analysis to that reported in Section 2.2 to be performed.In the case of the UX 2 Y 2 Z 2 complexes, only the "axial" U-X bond is considered since the permutation of the ligands means that each of the bonding interactions in the 10 possible complexes is thus included.
Figure 4 summarizes calculated ρ BCP values of the axial bond (numerical values are presented in the Supplementary Materials).Focusing initially on the uranyl halides, the characteristics identified in Section 2.2 are maintained, i.e., that equatorial ligand sets including the lighter, more electronegative halides perturb the axial U-O bond more with the 5f and 6d contributions being approximately equally affected.The effect of altering the equatorial ligand set in the uranium halides is less pronounced, presumably due to the more ionic nature of the interactions.ρ BCP attains a maximum value 0.160 a.u.for the U-F bond with an equatorial ligand set comprised of Cl and/or Br.This is, however, just 3% larger than the value found in UF 6 .Broadly speaking, there is a modest perturbation to ρ BCP when the equatorial ligand set includes F while calculated metrics are almost completely insensitive to replacing Cl by Br.
Inorganics 2018, 6, x FOR PEER REVIEW 9 of 18 Figure 4 summarizes calculated values of the axial bond (numerical values are presented in the Supplementary Materials).Focusing initially on the uranyl halides, the characteristics identified in Section 2.2 are maintained, i.e., that equatorial ligand sets including the lighter, more electronegative halides perturb the axial U-O bond more with the 5f and 6d contributions being approximately equally affected.The effect of altering the equatorial ligand set in the uranium halides is less pronounced, presumably due to the more ionic nature of the interactions.attains a maximum value 0.160 a.u.for the U-F bond with an equatorial ligand set comprised of Cl and/or Br.This is, however, just 3% larger than the value found in UF6.Broadly speaking, there is a modest perturbation to when the equatorial ligand set includes F while calculated metrics are almost completely insensitive to replacing Cl by Br.Delocalization indices exhibit more variation (See Figure 5: numerical values presented in Supplementary Materials).(U, O) exhibits the same variation as reported in Section 2.2 with larger values found when the halide mass increases.The 5fσ contribution is, in all cases, close to the value for free uranyl and insensitive to the equatorial ligand set.Other contributions vary in accordance with the total value, with the 6dπ component exhibiting the greatest sensitivity.As previously discussed, this component of the 6d shell is also able to engage in equatorial π-bonding interactions and the variation seen here is indicative of the formation of such bonding interactions even though this should be interpreted in the context of degeneracy-driven covalency.
All UX2Y2Z2 complexes exhibit qualitatively similar trends.Total values of (U, X) are largely insensitive to the equatorial environment, which increases slightly as the equatorial ligands become heavier and less electronegative.A trend is also exhibited by the 5fπ and 6dσ contributions.The 5fσ contribution decreases as the halide mass increases, but, in contrast, the 6dπ component increases by a greater magnitude.Since both the 5fσ and 6dπ components are available for equatorial π-bonding interactions, the variation in (U, X) can, therefore, be attributed to increased equatorial πinteractions with the 6d shell, which is partially balanced by a reduction in such interactions with the Delocalization indices exhibit more variation (See Figure 5: numerical values presented in Supplementary Materials).δ(U, O) exhibits the same variation as reported in Section 2.2 with larger values found when the halide mass increases.The 5f σ contribution is, in all cases, close to the value for free uranyl and insensitive to the equatorial ligand set.Other contributions vary in accordance with the total value, with the 6d π component exhibiting the greatest sensitivity.As previously discussed, this component of the 6d shell is also able to engage in equatorial π-bonding interactions and the variation seen here is indicative of the formation of such bonding interactions even though this should be interpreted in the context of degeneracy-driven covalency.
All UX 2 Y 2 Z 2 complexes exhibit qualitatively similar trends.Total values of δ(U, X) are largely insensitive to the equatorial environment, which increases slightly as the equatorial ligands become heavier and less electronegative.A trend is also exhibited by the 5f π and 6d σ contributions.
The 5f σ contribution decreases as the halide mass increases, but, in contrast, the 6d π component increases by a greater magnitude.Since both the 5f σ and 6d π components are available for equatorial π-bonding interactions, the variation in δ(U, X) can, therefore, be attributed to increased equatorial π-interactions with the 6d shell, which is partially balanced by a reduction in such interactions with the 5f shell.This is indicative of closer energetic matching of fluorine 2p and uranium 5f levels and of bromine 4p and uranium 6d levels.

Covalency and Bond Stabilization
Lastly, the relationship between equatorial bond covalency and bond stability is considered.For uranyl, the binding energy of the equatorial ligand set is defined by the equation below.
so that is positive for a stabilizing interaction.

Covalency and Bond Stabilization
Lastly, the relationship between equatorial bond covalency and bond stability is considered.For uranyl, the binding energy of the equatorial ligand set is defined by the equation below.
so that E B is positive for a stabilizing interaction.
Inorganics 2018, 6, x FOR PEER REVIEW 11 of 18  ), and the binding energy ( ) of the equatorial ligand set (see Supplementary Materials for numerical data).Extremely strong correlations are found between vibrational frequencies and both and (U, O) (Figure 6a and 6b, respectively), which indicates that the covalency of the U-O bond is strongly stabilizing.We have previously reported similar strong correlations with a more varied ligand set [75].Perhaps more interestingly, strong anticorrelations are also found between these topological parameters and binding energies, which is defined in Equation ( 3).These anti-correlations (Figure 6c,d) illustrate the perturbation of the U-O bond covalency by the presence of the equatorial ligands, which suggests that U-O bond vibrations can be used as a quantitative probe of equatorial ligand stability.
It might be expected that binding energies would also correlate with QTAIM parameters of the axial bonds.Figure 7 summarizes these relationships (see Supplementary Materials for numerical data).Note that since both homoleptic and heteroleptic complexes are considered, averaged topological values are reported.We have taken this approach previously and found strong correlations between averaged values and binding energies [75].
Figure 7a illustrates an extremely strong correlation between ̅ and , which provides further justification for considering these averaged values.Conversely, no correlation is found between ̅ and (Figure 7b).This can be rationalized if one considers the aspects of covalency that and ̅ probe [57]: identifies overlap-driven covalency, i.e., charge accumulation in the bonding region, which might be expected to have a stabilizing effect on the interaction., on the other hand, can be considered a measure of degeneracy-driven covalency for which no energetic stabilization of the bond is necessarily manifested.The absence of correlation presented in Figure 7b is, therefore, indicative of the fact that, in these complexes, degeneracy-driven covalency has no   6a and 6b, respectively), which indicates that the covalency of the U-O bond is strongly stabilizing.We have previously reported similar strong correlations with a more varied ligand set [75].Perhaps more interestingly, strong anti-correlations are also found between these topological parameters and binding energies, which is defined in Equation ( 3).These anti-correlations (Figure 6c,d) illustrate the perturbation of the U-O bond covalency by the presence of the equatorial ligands, which suggests that U-O bond vibrations can be used as a quantitative probe of equatorial ligand stability.
It might be expected that binding energies would also correlate with QTAIM parameters of the axial bonds.Figure 7 summarizes these relationships (see Supplementary Materials for numerical data).Note that since both homoleptic and heteroleptic complexes are considered, averaged topological values are reported.We have taken this approach previously and found strong correlations between averaged values and binding energies [75].
Figure 7a illustrates an extremely strong correlation between ρ BCP and E B , which provides further justification for considering these averaged values.Conversely, no correlation is found between δ and E B (Figure 7b).This can be rationalized if one considers the aspects of covalency that ρ BCP and δ probe [57]: ρ BCP identifies overlap-driven covalency, i.e., charge accumulation in the bonding region, which might be expected to have a stabilizing effect on the interaction.δ, on the other hand, can be considered a measure of degeneracy-driven covalency for which no energetic stabilization of the bond is necessarily manifested.The absence of correlation presented in Figure 7b is, therefore, indicative of the fact that, in these complexes, degeneracy-driven covalency has no clearly-defined energetic effect.It should also be noted that variation in δ is rather modest with maximum deviations less than 2% from the mean value.In contrast, maximum ρ BCP deviations are ~40% from the mean.clearly-defined energetic effect.It should also be noted that variation in ̅ is rather modest with maximum deviations less than 2% from the mean value.In contrast, maximum deviations are ~40% from the mean.Since correlated strongly with and itself correlated strongly with ̅ for the equatorial bonds, the direct relationship between and equatorial bond parameters was considered (see Figure 7c and 7d).As expected, a very strong (albeit slightly weaker) anticorrelation was found with ̅ while correlation with ̅ was not identified.The anti-correlation with ̅ lends further support to our previous assertion that could be used to probe equatorial bond covalency [75].
The success of correlating the averaged equatorial values with binding energies in uranyl halides suggested that a similar approach might be applicable to the uranium hexahalides considered in this study where the average would be with respect to all coordinating species.The binding energy is defined by the equation below.
Figure 8 presents the results of correlating with ̅ and ̅ (See Supplementary Materials for numerical data).As seen for the uranyl halides, ̅ exhibits an excellent correlation with and again demonstrates that ̅ quantitatively measures stabilizing covalent interactions in these species.Remarkably, ̅ shows a very strong anti-correlation with , i.e., as degeneracy-driven covalency increases, bond stability reduces.Since ν UO correlated strongly with E B and E B itself correlated strongly with ρ BCP for the equatorial bonds, the direct relationship between ν UO and equatorial bond parameters was considered (see Figures 7c and 7d).As expected, a very strong (albeit slightly weaker) anticorrelation was found with ρ BCP while correlation with δ was not identified.The anti-correlation with ρ BCP lends further support to our previous assertion that ν UO could be used to probe equatorial bond covalency [75].
The success of correlating the averaged equatorial ρ BCP values with binding energies in uranyl halides suggested that a similar approach might be applicable to the uranium hexahalides considered in this study where the average would be with respect to all coordinating species.The binding energy is defined by the equation below.
Figure 8 presents the results of correlating E B with ρ BCP and δ (See Supplementary Materials for numerical data).As seen for the uranyl halides, ρ BCP exhibits an excellent correlation with E B and again demonstrates that ρ BCP quantitatively measures stabilizing covalent interactions in these species.
Remarkably, δ shows a very strong anti-correlation with E B , i.e., as degeneracy-driven covalency increases, bond stability reduces.

Computational Details
All calculations were performed using version 6.6 of the TURBOMOLE quantum chemistry code [82,83] using scalar-relativistic density functional theory (DFT).Several exchange-correlation (xc-) functionals were considered in order to identify which was most suitable for these simulations and, as reported above, the hybrid-GGA B3LYP functional [71,72] was found to give the best agreement with experimental data in test systems.There are numerous examples in the literature demonstrating the suitability of B3LYP in the study or U(VI) coordination complexes [34,37,78,75].All simulations were performed using the Ahlrichs def2-QZVP basis sets [84] of polarized quadruple-ζ quality for light atoms (O, F, Cl, Br).The small-core pseudopotential of Dolg and co-workers [85], along with the corresponding (14s13p10d8f6g)/[10s9p5d4f3g] basis set [86], was used to model the uranium center by incorporating scalar relativistic effects.This pseudopotential is constructed from calculations employing the Wood-Boring equation (which may be derived directly from the Dirac equation) and includes both a mass-velocity and a Darwin term.The effects of the spin-orbit coupling have not been considered in these closed shell, formally 5f 0 6d 0 species.De Jong et al. have shown that the effects of spin-orbit coupling are rather moderate in the high energy valence region of uranyl but becomes more pronounced when considering the pseudo core 6p-shell [87].However, we have recently shown that the impact of the 6p-shell on QTAIM metrics is modest [88].We would also expect the degree of SO-coupling to be comparable in analogous systems and, therefore, have little impact on the trends reported here.
Geometry optimizations were performed in the gas-phase using default convergence criteria and local energetic minima were identified using analytical frequency analysis.In order to decompose bonding into contributions from different electronic shells, all optimizations were constrained to the D2h point group.Topological and integrated properties of the electron density were evaluated using version 3.3.9 of the Multiwfn code [89].

Computational Details
All calculations were performed using version 6.6 of the TURBOMOLE quantum chemistry code [82,83] using scalar-relativistic density functional theory (DFT).Several exchange-correlation (xc-) functionals were considered in order to identify which was most suitable for these simulations and, as reported above, the hybrid-GGA B3LYP functional [71,72] was found to give the best agreement with experimental data in test systems.There are numerous examples in the literature demonstrating the suitability of B3LYP in the study or U(VI) coordination complexes [34,37,75,78].All simulations were performed using the Ahlrichs def2-QZVP basis sets [84] of polarized quadruple-ζ quality for light atoms (O, F, Cl, Br).The small-core pseudopotential of Dolg and co-workers [85], along with the corresponding (14s13p10d8f6g)/[10s9p5d4f3g] basis set [86], was used to model the uranium center by incorporating scalar relativistic effects.This pseudopotential is constructed from calculations employing the Wood-Boring equation (which may be derived directly from the Dirac equation) and includes both a mass-velocity and a Darwin term.The effects of the spin-orbit coupling have not been considered in these closed shell, formally 5f 0 6d 0 species.De Jong et al. have shown that the effects of spin-orbit coupling are rather moderate in the high energy valence region of uranyl but becomes more pronounced when considering the pseudo core 6p-shell [87].However, we have recently shown that the impact of the 6p-shell on QTAIM metrics is modest [88].We would also expect the degree of SO-coupling to be comparable in analogous systems and, therefore, have little impact on the trends reported here.
Geometry optimizations were performed in the gas-phase using default convergence criteria and local energetic minima were identified using analytical frequency analysis.In order to decompose

Figure 1 .
Figure 1.QTAIM derived an atomic basin of U in UCl6, which exhibits the full symmetry of the molecule.Figures reproduced from Reference[61].
Figure 1.QTAIM derived an atomic basin of U in UCl6, which exhibits the full symmetry of the molecule.Figures reproduced from Reference[61].

Figure 1 .
Figure 1.QTAIM derived an atomic basin of U in UCl 6 , which exhibits the full symmetry of the molecule.Figures reproduced from Reference[61].
Figure 1.QTAIM derived an atomic basin of U in UCl 6 , which exhibits the full symmetry of the molecule.Figures reproduced from Reference[61].

Figure 2 .
Figure 2. Decomposed QTAIM metrics of the U-O bond in [UO2X4] 2− complexes (X = F, Cl, Br) obtained from B3LYP-derived densities.Values from free uranyl are also provided for comparison.f-shell contributions are given in yellow and d-shell contributions are given in blue.

Figure 2 .
Figure 2. Decomposed QTAIM metrics of the U-O bond in [UO 2 X 4 ] 2− complexes (X = F, Cl, Br) obtained from B3LYP-derived densities.Values from free uranyl are also provided for comparison.f-shell contributions are given in yellow and d-shell contributions are given in blue.

Figure 3 .
Figure 3. Decomposed QTAIM metrics of the U-X bond in [UO2X4] 2− and UX6 complexes (X = F, Cl, Br) obtained from B3LYP-derived densities.f-shell contributions are given in yellow and d-shell contributions are in blue.

Figure 3 .
Figure 3. Decomposed QTAIM metrics of the U-X bond in [UO 2 X 4 ] 2− and UX 6 complexes (X = F, Cl, Br) obtained from B3LYP-derived densities.f-shell contributions are given in yellow and d-shell contributions are in blue.

Figure 4 .
Figure 4. values of axial U-O and U-X (X = F, Cl, Br) bonds in [UO2X2Y2] 2− and UX2Y2Z2 complexes derived from B3LYP-generated densities.All values are in a.u.

Inorganics 2018, 6 ,
x FOR PEER REVIEW 10 of 18 5f shell.This is indicative of closer energetic matching of fluorine 2p and uranium 5f levels and of bromine 4p and uranium 6d levels.

Figure 6 .
Figure 6.Correlation between QTAIM parameters of the U-O bond, U-O stretching vibrational frequencies, and the binding energy of the equatorial ligand set in [UO2X2Y2] 2− complexes.(a) correlates and , (b) correlates and (U, O), (c) correlates and , (d) correlates and (U, O).

Figure 6
Figure 6 reports correlations between QTAIM parameters of the UO bond, U-O stretching vibrational frequencies (), and the binding energy ( ) of the equatorial ligand set (see Supplementary Materials for numerical data).Extremely strong correlations are found between vibrational frequencies and both and (U, O) (Figure6aand 6b, respectively), which indicates that the covalency of the U-O bond is strongly stabilizing.We have previously reported similar strong correlations with a more varied ligand set[75].Perhaps more interestingly, strong anticorrelations are also found between these topological parameters and binding energies, which is defined in Equation (3).These anti-correlations (Figure6c,d) illustrate the perturbation of the U-O bond covalency by the presence of the equatorial ligands, which suggests that U-O bond vibrations can be used as a quantitative probe of equatorial ligand stability.It might be expected that binding energies would also correlate with QTAIM parameters of the axial bonds.Figure7summarizes these relationships (see Supplementary Materials for numerical data).Note that since both homoleptic and heteroleptic complexes are considered, averaged topological values are reported.We have taken this approach previously and found strong correlations between averaged values and binding energies[75].Figure7aillustrates an extremely strong correlation between ̅ and , which provides further justification for considering these averaged values.Conversely, no correlation is found between ̅ and (Figure7b).This can be rationalized if one considers the aspects of covalency that and ̅ probe[57]: identifies overlap-driven covalency, i.e., charge accumulation in the bonding region, which might be expected to have a stabilizing effect on the interaction., on the other hand, can be considered a measure of degeneracy-driven covalency for which no energetic stabilization of the bond is necessarily manifested.The absence of correlation presented in Figure7bis, therefore, indicative of the fact that, in these complexes, degeneracy-driven covalency has no

Figure 6 .
Figure 6.Correlation between QTAIM parameters of the U-O bond, U-O stretching vibrational frequencies, and the binding energy of the equatorial ligand set in [UO 2 X 2 Y 2 ] 2− complexes.(a) correlates ν UO and ρ BCP , (b) correlates ν UO and δ(U, O), (c) correlates E B and ρ BCP , (d) correlates E B and δ(U, O).

Figure 6
Figure 6  reports correlations between QTAIM parameters of the UO bond, U-O stretching vibrational frequencies (ν UO ), and the binding energy (E B ) of the equatorial ligand set (see Supplementary Materials for numerical data).Extremely strong correlations are found between vibrational frequencies and both ρ BCP and δ(U, O) (Figures6a and 6b, respectively), which indicates that the covalency of the U-O bond is strongly stabilizing.We have previously reported similar strong correlations with a more varied ligand set[75].Perhaps more interestingly, strong anti-correlations are also found between these topological parameters and binding energies, which is defined in Equation (3).These anti-correlations (Figure6c,d) illustrate the perturbation of the U-O bond covalency by the presence of the equatorial ligands, which suggests that U-O bond vibrations can be used as a quantitative probe of equatorial ligand stability.It might be expected that binding energies would also correlate with QTAIM parameters of the axial bonds.Figure7summarizes these relationships (see Supplementary Materials for numerical data).Note that since both homoleptic and heteroleptic complexes are considered, averaged topological values are reported.We have taken this approach previously and found strong correlations between averaged values and binding energies[75].Figure7aillustrates an extremely strong correlation between ρ BCP and E B , which provides further justification for considering these averaged values.Conversely, no correlation is found between δ and E B (Figure7b).This can be rationalized if one considers the aspects of covalency that ρ BCP and

Figure 7 .
Figure 7. Correlation between QTAIM parameters of the equatorial bonds, the binding energy of the equatorial ligand set, and the U-O stretching vibrational frequencies in [UO2X2Y2] 2− complexes.(a) correlates and ̅ , (b) correlates and ̅ , (c) correlates and ̅ , (d) correlates and ̅ .

Figure 7 .
Figure 7. Correlation between QTAIM parameters of the equatorial bonds, the binding energy of the equatorial ligand set, and the U-O stretching vibrational frequencies in [UO 2 X 2 Y 2 ] 2− complexes.(a) correlates E B and ρ BCP , (b) correlates E B and δ, (c) correlates ν UO and ρ BCP , (d) correlates ν UO and δ.

Figure 8 .
Figure 8. Correlation between the averaged QTAIM parameters of U-X, U-Y & U-Z bonds and the total binding energy in UX 2 Y 2 Z 2 complexes.(a) correlates E B and ρ BCP , (b) correlates E B and δ.

Table 2 .
Calculated structural and vibrational parameters of UF 6 calculated with various exchange-correlation functionals.Values in parenthesis indicate deviations from experimental values.

Table 5 .
Irreducible representations spanned by components of the 5f and 6d valence shells with respect to the principal molecular axis.