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

Abstract

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 5f- and 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.

1. 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 [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53]. 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 [UO₂X₂Y₂]²⁻ and UX₂Y₂Z₂ 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: [UCl₆]²⁻ and [UF₆]⁻ have both been the subject of recent experimental studies of U–X bonding [5,21] while the experimentally determined electronic structure of Cs₂UO₂Cl₄ has been previously analyzed using the QTAIM [58]. Of the remaining complexes, [UO₂F₄]²⁻ and [UO₂Br₄]²⁻ have been observed in the gas and condensed phases, respectively [62,63]. [UF₆]⁻ and [UCl₆]⁻ have both been experimentally characterized [64,65,66] although [UBr₆]⁻ remains unreported.

2. Results

2.1. 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: [UO₂Cl₄]²⁻ and UF₆, for which accurate experimental data exists.

Table 1. Calculated structural and vibrational parameters of [UO₂Cl₄]²⁻ calculated with various exchange-correlation functionals. Values in parenthesis indicate deviations from experimental values. ^a ref [76], ^b ref. [77].

Parameter		Exp. a,b	BLYP	PBE	PBE0	B3LYP	BHLYP
rUO (Å)	1.774		1.817 (+0.043)	1.800 (+0.026)	1.756 (−0.018)	1.776 (+0.002)	1.735 (−0.039)
rUCl (Å)	2.671		2.765 (+0.094)	2.721 (+0.050)	2.715 (+0.044)	2.749 (+0.078)	2.749 (+0.078)
νUO (cm−1)	834		776 (−58)	803 (−31)	896 (+62)	855 (+21)	953 (+119)
	922		862 (−60)	887 (−35)	975 (+53)	939 (+17)	1023 (+101)
νUCl (cm−1)	206		179 (−27)	192 (−14)	199 (−7)	188 (−18)	196 (−10)
	236		197 (39)	238 (+2)	216 (−20)	206 (−30)	213 (−23)
	267		213 (−54)	226 (−41)	234 (−33)	224 (−43)	232 (−25)

and

Table 2. Calculated structural and vibrational parameters of UF₆ calculated with various exchange-correlation functionals. Values in parenthesis indicate deviations from experimental values. ^a ref [64], ^b ref. [65].

Parameter	Exp. a,b	BLYP	PBE	PBE0	B3LYP	BHLYP
rUF (Å)	1.999	2.036 (+0.037)	2.018 (+0.019)	1.990 (−0.009)	2.009 (+0.010)	1.984 (−0.055)
νUF (cm−1)	534	511 (−23)	521 (−13)	543 (+9)	532 (−2)	548 (+14)
	626	581 (−45)	596 (−30)	633 (+7)	615 (−11)	648 (+22)
	667	613 (−54)	630 (−37)	682 (+15)	660 (+7)	710 (+43)

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 [UO₂Cl₄]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].

2.2. Homohalide Complexes of Uranium and Uranyl

In this section, the simulation and analysis of complexes of the form [UO₂X₄]²⁻ and UX₆ (X = F, Cl, Br) is considered. Focusing initially on the uranyl complexes,

Table 3. B3LYP-calculated structural parameters [UO₂X₄]²⁻. R = 6-coordinated Shannon ionic radius.

Complex	rUO (Å)	rUX (Å)	R = RU + RX (Å)	rUX – R (Å)
[UO2F4]2−	1.824	2.229	0.73 + 1.33 = 2.06	0.169
[UO2Cl4]2−	1.776	2.749	0.73 + 1.81 = 2.54	0.209
[UO2Br4]2−	1.769	2.922	0.73 + 1.96 = 2.69	0.232
UO22+	1.695	-	-	-

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 [UO₂Cl₄]²⁻ (Table 1), the U–O bond length in [UO₂Br₄]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 [UO₂F₄]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 UX₆ (

Table 4. B3LYP-calculated structural parameters UX₆. R = 6-coordinated Shannon ionic radius.

Complex	rUX (Å)	R = RU + RX (Å)	rUX – R (Å)
UF6	2.009	0.73 + 1.33 = 2.06	−0.051
UCl6	2.470	0.73 + 1.81 = 2.54	−0.070
UBr6	2.639	0.73 + 1.96 = 2.69	−0.051

) 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 UCl₆. The good agreement with the experimental gas-phase U–F bond length (Table 2) is apparently not replicated in UCl₆ 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 [UO₂Cl₄]²⁻ and [UO₂Br₄]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 (2.650–2.687 Å) [81], commensurate with the data presented in Table 1 and Table 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 UX₆ 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 ${\rho }_{\mathrm{BCP}}\left(\mathrm{A},\mathrm{B}\right)$, 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 $\rho \left(r\right)$ 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 ${\rho }_{\mathrm{BCP}}$ > 0.2 a.u. and closed shell interactions by ${\rho }_{\mathrm{BCP}}$ < 0.1 a.u. with intermediate values indicating an increasing degree of a covalent charter. ${\rho }_{\mathrm{BCP}}$ 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, $\delta \left(\mathrm{A},\mathrm{B}\right)$. The delocalization index is an integrated property and, for chemical systems described by a single electronic configuration, can be defined by the equation below:

$$\delta \left(\mathrm{A},\mathrm{B}\right)=2{\displaystyle \sum }_{i,j}{S}_{ij}\left(\mathrm{A}\right)S_{ij}\left(\mathrm{B}\right)$$

(1)

where $S_{ij}\left(\mathrm{X}\right)$ is the overlap between molecular orbitals (MOs) ${\phi }_i\left(r\right)$ and ${\phi }_j\left(r\right)$ integrated over the atomic basin associated with atom X (as defined by QTAIM). $\delta \left(\mathrm{A},\mathrm{B}\right)$ 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, $\delta \left(\mathrm{A},\mathrm{B}\right)$ 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 ${\rho }_{\mathrm{BCP}}\left(\mathrm{A},\mathrm{B}\right)$ are accompanied by large values of $\delta \left(\mathrm{A},\mathrm{B}\right)$, 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.

$$\delta \left(\mathrm{U},\mathrm{B}\right)=2{\displaystyle \sum }_{\mathsf{\Gamma }}{\displaystyle \sum }_{i,j}^{\mathsf{\Gamma }}{S}_{ij}\left(\mathrm{U}\right)S_{ij}\left(\mathrm{B}\right)$$

(2)

where $\mathsf{\Gamma }$ indexes the irreducible representations (irreps) of the point group and the summation over MOs is now limited to those orbitals spanning the irrep $\mathsf{\Gamma }$. This definition is valid since the overlap integral $S_{ij}\left(\mathrm{U}\right)$ is only non-zero when the MOs ${\phi }_i\left(r\right)$ and ${\phi }_j\left(r\right)$ 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 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. Irreducible representations spanned by components of the 5f and 6d valence shells with respect to the principal molecular axis.

Irrep ( Γ)	Ag	B1g	B2g	B3g	Au	B1u	B2u	B3u
Component	6dσ, 6dδ	6dδ	6dπ	6dπ	5fδ	5fσ, 5fδ	5fπ, 5fφ	5fπ, 5fφ

. 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, ${\rho }_{\mathrm{BCP}}$ 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 $\rho \left(r\right)$ 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 [UO₂X₄]²⁻ complexes (see Supplementary Materials for numerical data). The magnitude of ${\rho }_{\mathrm{BCP}}$ 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 ${\rho }_{\mathrm{BCP}}$, 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 ${\rho }_{\mathrm{BCP}}$ 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].

The 5f contribution to ${\rho }_{\mathrm{BCP}}$ 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 ${\rho }_{\mathrm{BCP}}$ are extremely similar with values in the fluoride complex being 71% and 73% of the free uranyl values, respectively.

The corresponding $\delta \left(\mathrm{U},\mathrm{O}\right)$ 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. $\delta \left(\mathrm{U},\mathrm{O}\right)$ 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 $\delta \left(\mathrm{U},\mathrm{O}\right)$ 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 ${\rho }_{\mathrm{BCP}}$ (which is only sensitive to σ-interactions) and the σ-contributions to $\delta \left(\mathrm{U},\mathrm{O}\right)$ demonstrate how these metrics provide independent but related data regarding bonding. This is most obvious for the 5f-shell: the 5f contribution to ${\rho }_{\mathrm{BCP}}$ reduces to 71% of its free value upon equatorial coordination by fluoride while the corresponding 5f_σ contribution to $\delta \left(\mathrm{U},\mathrm{O}\right)$ 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 [UO₂X₄]²⁻ and UX₆ 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 ${\rho }_{\mathrm{BCP}}\left(\mathrm{U},\mathrm{X}\right)$, overall magnitudes are larger in the UX₆ complexes due to the lack of competition with the strongly coordinating terminal oxo species and the consequently shorter U–X bonds. The ${\rho }_{\mathrm{BCP}}\left(\mathrm{U},\mathrm{X}\right)$ values are, however, much lower than those of the U–O bond, which indicates, as expected, significantly reduced covalent contributions to bonding. Nonetheless, ${\rho }_{\mathrm{BCP}}\left(\mathrm{U},\mathrm{X}\right)$ has a maximum value (0.155 a.u.) in UF₆, consistent with substantial covalent bond character. There is a clear reduction in ${\rho }_{\mathrm{BCP}}\left(\mathrm{U},\mathrm{X}\right)$ 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 ${\rho }_{\mathrm{BCP}}$ is 34% and 38% larger than the 6d contribution in [UO₂F₄]²⁻ and UF₆, respectively, but is just 4% and 7% larger in [UO₂Br₄]²⁻ and UBr₆, 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 ${\rho }_{\mathrm{BCP}}$ in UF₆ can be indirectly compared to a recent X-ray diffraction study of [UF₆]⁻ in which topological analysis of the experimentally determined electron density gave ${\rho }_{\mathrm{BCP}}$ ~ 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 [UF₆]⁻ as opposed to the +6 oxidation state in UF₆. We have previously investigated the dependence of QTAIM parameters on uranium oxidation state in uranium hexachloride [61], finding a reduction in ${\rho }_{\mathrm{BCP}}$ 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 $\delta \left(\mathrm{U},\mathrm{X}\right)$ 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 ${\rho }_{\mathrm{BCP}}$ and shows the danger of attempting to use $\delta$ as an indicator of bond order in the presence of bond polarization. 5f components of $\delta \left(\mathrm{U},\mathrm{X}\right)$ 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 ${\rho }_{\mathrm{BCP}}$ and demonstrates that overlap-induced and degeneracy-induced covalency can (i) manifest independently and (ii) both be identified by the methodology employed here. For [UO₂X₄]²⁻ species, the trend in 5f_σ, 5f_π, and 6d_π components mirrors that of ${\rho }_{\mathrm{BCP}}$ while the variation in 6d_σ strongly opposes this trend and results in the overall magnitude of $\delta \left(\mathrm{U},\mathrm{X}\right)$ remaining largely unchanged with the 6d_σ component overtaking 5f_σ as the dominant contribution when the halide mass increases. The qualitative difference in $\delta \left(\mathrm{U},\mathrm{X}\right)$ between the two sets of complexes presumably reflects the strong influence of the axial oxo species on (formally) uranium-based energy levels.

2.3. 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 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 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.

Figure 4 summarizes calculated ${\rho }_{\mathrm{BCP}}$ 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. ${\rho }_{\mathrm{BCP}}$ 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 UF₆. Broadly speaking, there is a modest perturbation to ${\rho }_{\mathrm{BCP}}$ 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). $\delta \left(\mathrm{U},\mathrm{O}\right)$ 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 UX₂Y₂Z₂ complexes exhibit qualitatively similar trends. Total values of $\delta \left(\mathrm{U},\mathrm{X}\right)$ 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 $\delta \left(\mathrm{U},\mathrm{X}\right)$ 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.

2.4. 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.

$$E_{\mathrm{B}}=E\left({\mathrm{UO}}_2{}^{2+}\right)+2E\left({\mathrm{X}}^{-}\right)+2E\left({\mathrm{Y}}^{-}\right)-E({\left[{\mathrm{UO}}_2{\mathrm{X}}_2{\mathrm{Y}}_2\right]}^{2-})$$

(3)

so that $E_{\mathrm{B}}$ is positive for a stabilizing interaction.

Figure 6 reports correlations between QTAIM parameters of the UO bond, U–O stretching vibrational frequencies (${\nu }_{\mathrm{UO}}$), and the binding energy ($E_{\mathrm{B}}$) of the equatorial ligand set (see Supplementary Materials for numerical data). Extremely strong correlations are found between vibrational frequencies and both ${\rho }_{\mathrm{BCP}}$ and $\delta \left(\mathrm{U},\mathrm{O}\right)$ (Figure 6a and Figure 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 ${\overline{\rho }}_{\mathrm{BCP}}$ and $E_{\mathrm{B}}$, which provides further justification for considering these averaged values. Conversely, no correlation is found between $\overline{\delta }$ and $E_{\mathrm{B}}$ (Figure 7b). This can be rationalized if one considers the aspects of covalency that ${\rho }_{\mathrm{BCP}}$ and $\overline{\delta }$ probe [57]: ${\rho }_{\mathrm{BCP}}$ identifies overlap-driven covalency, i.e., charge accumulation in the bonding region, which might be expected to have a stabilizing effect on the interaction. $\delta$, 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 $\overline{\delta }$ is rather modest with maximum deviations less than 2% from the mean value. In contrast, maximum ${\rho }_{\mathrm{BCP}}$ deviations are ~40% from the mean.

Since ${\nu }_{\mathrm{UO}}$ correlated strongly with $E_{\mathrm{B}}$ and $E_{\mathrm{B}}$ itself correlated strongly with ${\overline{\rho }}_{\mathrm{BCP}}$ for the equatorial bonds, the direct relationship between ${\nu }_{\mathrm{UO}}$ and equatorial bond parameters was considered (see Figure 7c and Figure 7d). As expected, a very strong (albeit slightly weaker) anticorrelation was found with ${\overline{\rho }}_{\mathrm{BCP}}$ while correlation with $\overline{\delta }$ was not identified. The anti-correlation with ${\overline{\rho }}_{\mathrm{BCP}}$ lends further support to our previous assertion that ${\nu }_{\mathrm{UO}}$ could be used to probe equatorial bond covalency [75].

The success of correlating the averaged equatorial ${\rho }_{\mathrm{BCP}}$ 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.

$$E_{\mathrm{B}}=E\left({\mathrm{U}}^{6+}\right)+2E\left({\mathrm{X}}^{-}\right)+2E\left({\mathrm{Y}}^{-}\right)+2E\left({\mathrm{Z}}^{-}\right)-E\left({\mathrm{UX}}_2{\mathrm{Y}}_2{\mathrm{Z}}_2\right).$$

(4)

Figure 8 presents the results of correlating $E_{\mathrm{B}}$ with ${\overline{\rho }}_{\mathrm{BCP}}$ and $\overline{\delta }$ (See Supplementary Materials for numerical data). As seen for the uranyl halides, ${\overline{\rho }}_{\mathrm{BCP}}$ exhibits an excellent correlation with $E_{\mathrm{B}}$ and again demonstrates that ${\overline{\rho }}_{\mathrm{BCP}}$ quantitatively measures stabilizing covalent interactions in these species. Remarkably, $\overline{\delta }$ shows a very strong anti-correlation with $E_{\mathrm{B}}$, i.e., as degeneracy-driven covalency increases, bond stability reduces.

3. 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 5f⁰6d⁰ 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 D_2h point group. Topological and integrated properties of the electron density were evaluated using version 3.3.9 of the Multiwfn code [89].

4. Summary and Conclusions

The electronic structures of a series of uranium hexahalide and uranyl tetrahalide complexes were simulated at the density functional theoretical (DFT) level. A comparison with previous experimental structural and vibrational data indicated that the B3LYP exchange-correlation functional was best able to accurately simulate these complexes. B3LYP-derived electronic structures were subsequently analyzed using a novel application of the Quantum Theory of Atoms in Molecules (QTAIM). This approach exploited the high (D_2h or greater) symmetry of the complexes to determine 5f-shell and 6d-shell contributions to bonding through symmetry arguments. The analysis also allowed differentiation between overlap-driven and degeneracy-driven contributions to covalency and, for the latter, to further differentiate between σ-type and π-type interactions. This analysis revealed that, of the halogen ligands considered (X = F, Cl, Br), fluoride ligation resulted in the strongest bonds with the most significant covalent character and with values of the density at the U-F bond critical point, ${\rho }_{\mathrm{BCP}}$, as high as 0.16 a.u. As a reference, ${\rho }_{\mathrm{BCP}}\ge$ 0.20 a.u. is often taken to be an indicator of a predominantly covalent interaction. 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 the electronic structure of other bonds in the complex. Taking [UO₂F₄]²⁻ as an example, QTAIM metrics of the U–O bond were reduced to ~75% of the values for free uranyl.

The QTAIM analysis allows for the distinction to be made between covalent bond contributions whose origin lies in either orbital overlap or energy degeneracy. Analysis of the complexes considered here showed that 5f contributions to overlap-driven covalency were 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 each individual π-component even though the total π-contribution was, in some cases, larger.

We have previously reported on strong correlations between uranyl stretching vibrational frequencies and equatorial bond covalency [75]. In this study, 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, overlap-driven covalency was shown to strongly correlate with bond stability while degeneracy-driven covalency was found to strongly anti-correlate. This anti-correlation illustrates the care that must be taken to differentiate between overlap-driven and degeneracy-driven covalent contributions to bond character when employing such analysis to rationalize experimental data. This work demonstrates that the QTAIM can be used to provide a highly detailed characterization of bonding interactions, which distinguishes between the different origins of covalent contributions and provides quantitative data with which to rationalize observable phenomena.