Ligand size dependence of UN and UO bond character in a series of uranyl hexaphyrin complexes: quantum chemical simulation and density based analysis

A series of uranyl complexes with hexaphyrin ligands are investigated at the density functional level of theory and analysed using a variety of density-based techniques. A relationship is identified between the size of the ligand and the stability of the complex, controlled by the presence of meso-carbon centres in the porphyrin ring. The complex with the smallest ligand, cyclo[6]pyrrole, is found to have enhanced covalent character in equatorial U–N bonds as defined by the quantum theory of atoms in molecules (QTAIM), as well as enhanced stability, compared to the larger complexes. QTAIM data are supported by electron density difference distributions, integrated electronic properties and analysis of the reduced density gradient (RDG), which all show unambiguous evidence of electron sharing in all U–N bonds. In all complexes, a weakening of the covalent axial U–Oyl interaction in comparison to free uranyl is found, with evidence for a separation of electronic charge resulting in a more ionic interaction. A relationship between covalent character in the U–N bonds and the magnitude of uranyl charge redistribution is identified, where the greater the covalent character of the U–N interaction, the more ionic the U–Oyl interaction appears. The complex with the largest ligand, hexaphyrin(1.1.1.1.1.1), is found to have additional interactions with the uranyl oxygen centres, perturbing the U–Oyl interaction.

Worldwide, 10.9% of energy is now generated by nuclear fission, 21 with a typical reactor producing 20 metric tons of spent fuel per year. 22This consists mainly of a mixture of uranium, plutonium, small amounts of the minor actinides neptunium, americium and curium, as well as fission products including lanthanides and transition metals.The amount of spent fuel generated is only likely to increase as the world becomes more reliant on fission power in order to meet growing energy demands, and how best to manage the long-lived and highly radiotoxic actinides in this spent fuel is an ongoing issue faced by the nuclear power industry.The minor actinides, whose chemistry is dominated by the trivalent oxidation state, 3 present a particular challenge as current practices for their management require efficient separation from the lanthanides, which exhibit very similar chemistry.For this reason, a significant research effort is currently focussed on the identification of ligands which show actinide selectivity.
To this purpose, relatively soft N-donor ligands have been investigated with varying degrees of success. 9Several, such as ligands from the BTP, BTBP and BTPhen families, have exhibited promising selectivity, [23][24][25] however the source of this selectivity is not fully understood, and can be destroyed, or greatly enhanced, with small modifications to the ligand, [26][27][28][29][30] albeit not yet in a predictable way.
The expanded porphyrins are large, flexible, synthetic ligands capable of coordinating actinyl (di)cations in the equatorial plane [53][54][55][56][57] via pyrrolic nitrogen centres.8][59][60] Additionally, expanded porphyrins follow the CHON principle, a desirable criteria of an industrially useful separation ligand, specifically that it consist only of carbon, hydrogen, nitrogen and oxygen, which can be fully combusted after use to form environmentally safe products.
Several uranyl complexes with expanded porphyrin ligands have been experimentally realised.The hexaphyrins are expanded porphyrin macrocycles comprised of six pyrrolic subunits separated by varying numbers of meso-carbon atoms.Sessler et al. reported the synthesis of a uranyl complex of the hexaphyrin ligand isoamethyrin(1.0.1.0.0.0), 54 where the numbers in parentheses denote the number and position of meso-carbons (see Fig. 1), and suggested its possible use as a colorimetric actinide sensor. 54,61he Sessler group has also reported neptunyl and plutonyl complexes with isoamethyrin 54,62 as well as several other expanded porphyrins and similar expanded porphyrin-like macrocycles, 55 and uranyl complexes of the hexaphyrin ligands cyclo [6]pyrrole (comprising no meso-carbons) and amethyrin(1.0.0.1.0.0). 53,56ranyl complexes of the rubyrin(1.1.0.1.1.0),rubyrin(1.1.1.1.0.0) and hexaphyrin(1.1.1.1.1.1)6][77][78][79][80] Due to the existence of many possible substitution sites, the basic hexapyrrolic structure of the ligands may be modified with peripheral or meso-substituents, 63,71,72,[81][82][83] or for example, replacement of a pyrrolic subunit with a different kind of ring, e.g. a furan or pyridine subunit. 84][87][88][89][90][91][92][93] Although the equatorial bonding characteristics of uranyl, with its formally empty 5f-shell, and the trivalent minor actinides with their partially filled f-shells are expected to differ, it is intended in this instance simply to investigate the potential for expanded porphyrin ligands to coordinate a uranyl dication and to quantify the nature of the U-N bonds in such complexes with the aim being to begin to identify the characteristics of a ligand which lead to increased equatorial covalency.In this way, if the selectivity shown by ligands from the BTP/BTBP/ BTPhen families is indeed covalency driven, results obtained here may be useful for informing the design of future selective ligands.Additionally, the presence of the distinctive and experimentally accessible U-O yl stretching modes [94][95][96] may act as probes of equatorial covalency. 97Ultimately it is intended that this work will be extended to include complexes of uranyl and plutonyl, and perhaps eventually the trivalent minor actinides.
Previously, we have performed an in-depth study on the equatorial coordination behaviour of uranyl with a range of monodentate first row ligands. 97Using density based analysis, we determined a strong correlation between covalent character in equatorial bonding and the vibrational frequencies of the U-O yl stretching modes.Subsequently, we compared two complexes of uranyl with multidentate ligands: 98 a complex with the hexadentate macrocyclic expanded porphyrin ligand isoamethyrin and a complex with two tridentate bis-triazinylpyridine (BTP) ligands which have been shown to act selectively for An(III) over Ln(III) in industrial separation processes.Our theoretical study concluded that the U-N bonding in these two complexes was strongly similar: largely ionic equatorial bonds with a comparable degree of covalency and a commensurate weakening of the U-O covalent interaction suggesting a redistribution of charge in the uranyl unit, with the very similar equatorial bonding in the two complexes suggesting that expanded porphyrin ligands may be interesting candidates for future investigations of actinide selectivity. 98n this contribution, we combine quantum chemical calculations at the density functional (DFT) level of theory with density-based analysis techniques to show that complexes of uranyl with hexaphyrin ligands provide an interesting set of systems for investigating covalency in the U-N interaction.We determine the effect of altering the size of the ligand core on the degree of covalent character in equatorial U-N bonds, and quantify the extent of the charge redistribution in the uranyl unit that occurs upon complexation using the experimentally accessible U-O yl stretching frequencies.
Quantum chemical calculations have been used to investigate eight complexes in total, of which three have previously been synthetically realised: UO 2 -cyclo [6]pyrrole, which contains no meso-carbon atoms, as well as which contains six, one bridging each of its pyrrolic subunits.Finally, one system for which both the complex and the ligand are hypothetical, UO 2 -hexaphyrin(1.1.0.0.0.0), has been investigated.These ligands were selected so that the effect of increasing the size of the hexaphyrin core via meso-substitution could be directly investigated.
All complexes have been optimised at the density functional level of theory and, as a representative sample, four complexes were selected for detailed density analysis.For simplicity, the eight complexes will be referred to throughout using the labelling defined in Table 1.
In order to avoid any potential ambiguities that may arise from employing orbital-based analysis techniques, we focus on electron-density based analytical approaches.Such densitybased analyses have previously been successfully applied to f-element complexes 14,[36][37][38][39]42,47,99,100 and, in particular, allow direct comparison of results obtained using DFT with those obtained using multiconfigurational methodologies.
The quantum theory of atoms in molecules 101,102 (QTAIM) has been used to assess covalent character in U-N and U-O bonds.QTAIM relies upon the partitioning of a molecule into atomic basins bound by surfaces satisfied by the condition rr(r)Án(r) = 0. n(r) is the unit vector normal to the basin surface.Points at which the gradient in the electron density vanishes, rr(r) = 0, define the critical points in r(r).Of particular interest in this study are ''bond critical points'' (BCPs), which are found where the line of maximum density defining a bond path between bonded atoms is at its minimum. 103Values of topological indicators at the BCP characterise the bonding interaction.In general, where the BCP has r 4 0.20 a.u. and r 2 r o 0, a bond can be described as covalent.Broadly speaking, the higher the value of r, the higher the covalent character of the bond.The energy density, H, can also be used as a measure of covalency. 101,102For a covalent bond, its value is expected to be negative, with the degree of covalency indicated by the magnitude.In addition, integrated properties such as atomic populations, as well as localisation and delocalisation indices can be obtained by integrating over atomic basins and give insight into both electron sharing and charge donation.
We complement our QTAIM studies by considering regions of weak interaction, 104 by investigating the reduced density gradient (RDG), defined as s(r) = |rr(r)|/2(3p 2 ) 1/3 r(r) 4/3 , and, finally, visualise electron density difference distributions upon complexation to qualitatively analyse the changes that are undergone upon complex formation.We compare these distributions to our quantitative data in order to develop a comprehensive description of the nature of U-N and U-O bonding in these complexes.

Computational details
Density functional theory (DFT) calculations have been performed using version 6.4 of the TURBOMOLE quantum chemistry software package. 105Ahlrichs def2-TZVP basis sets of triplezeta quality have been used for the C, H, O and N atoms 106 and the Ahlrichs def-TZVP basis set of triple-zeta quality, which incorporate a relativistic ECP comprising 60 core electrons 107 has been used for the U atoms.All complexes summarised in Fig. 2 and 3 were optimised using the PBE exchange-correlation xc-functional 108 which employs the generalised gradient approximation (GGA).Subsequent reoptimisation of a representative sample of complexes was also performed using the B3LYP  hybrid-GGA xc-functional 109,110 to approximate the exchange correlation energy.This was done in order to investigate the effects of incorporating exact exchange on properties of the electron density.Both PBE and B3LYP have previously be shown to be suitable for the accurate modelling of actinide-containing systems. 3,14,97,98,111,112Initial optimisations were carried out in the gas phase.Subsequently, the COSMO continuum solvation model, 113 using a relative permittivity of e r = 8.9 was used to simulate solvation in DCM, for which there is experimental precedent. 54Vibrational analysis was performed to ensure that structures represented energetic minima.‡ Due to the closed shell nature of these systems, the effects of spin-orbit coupling were not included in the calculations.All of the hexaphyrin macrocycles reported experimentally feature methyl substituents on the periphery of their pyrrole units (see Fig. 3).These are generally assumed to have little effect on the geometry and electronic structure of the molecule and in order to minimise computational expense, such substituents are often omitted when performing DFT simulations.However it has been shown that for UO 2 -isoamethyrin(1.0.1.0.0.0), this simplification leads to severely overestimated (by B0.1 Å) U-N bond lengths. 114Thus, as with our previous study comparing UO 2isoamethyrin(1.0.1.0.0.0) with [UO 2 (BTP) 2 ] 2+ , optimisations have been performed with and without these substituents so that their effects on geometry could be assessed.U-N and U-O bond lengths have been compared to experimental data where available.For electron density analysis, single point energy calculations were performed at the optimised geometries using the SARC all-electron uranium basis set 115 and the second-order Douglas-Kroll-Hess (DKH2) Hamiltonian to account for scalar relativistic effects. 116,117QTAIM analysis was performed using the Multiwfn 118 and AIMAll 119 codes in order to calculate topological and integrated properties of the electron density.RDG and density difference data were visualised using the VMD code. 120

Geometrical characterisation and energetic stability
Geometries of the complexes optimised using the PBE xc-functional in the gas phase are shown in Fig. 2 and 3.It is apparent (see Table S1, ESI †) that, in most cases, the inclusion of peripheral alkyl groups has a relatively small effect on the average U-N bond length, with differences of only a few hundredths of an Angstrom, and results in no significant structural variation.The exception is with the C 2 x complexes, in which simplified and substituted forms have differences in average bond lengths of approximately 0.1 Å, nearly an order of magnitude higher than for the C 0 , C 4 and C 6 complexes.All C 2 x complexes exhibit significant structural variation when peripheral substituents are included, with a degree of non-planarity introduced that acts to shorten the U-N bonds.Based on these results, C 0 , C 4 x and C 6 complexes will all be considered in their simplified form for the remainder of this contribution, and substituents will only be considered in the case of the C 2 x complexes, with the substituted system referred to as C 2 x 0 .Table 2 summarises U-N bond lengths for the complexes C 0 , C 2 x 0 , C 4 x and C 6 and U-O bond lengths can be found in Table 3.
An examination of U-N and U-O bond lengths reveals that complex C 0 has the shortest average U-N bond lengths of all eight complexes, 2.532 Å (2.527 Å), when optimised in the gas phase (DCM).C 0 features four longer and two slightly shorter (by B0.01 Å) U-N bonds.This complex also has the longest calculated U-O bond length, 1.799 Å (1.812 Å), when optimised in the gas phase (DCM), thereby exhibiting the strongest perturbation of the uranyl unit due to equatorial complexation.Additionally, a very slight degree of non-planarity is introduced upon solvation (and upon addition of peripheral substituents), although this causes no significant changes to bond lengths.When compared to experimental values, the U-N bond lengths of C 0 are reproduced to within 0.01 Å (0.01 Å) in the gas phase (DCM), an excellent level of agreement, while the U-O bonds  are reproduced to within 0.02 Å (0.03 Å) in the gas phase (DCM), a good level of agreement. 53It is worth mentioning that the crystal structure of C 0 exhibits nearly perfect planarity, in agreement with the calculated gas phase structure.
Moving on to the complexes containing two meso-carbons, the simplified complexes C 2 a, C 2 b and C 2 c share several characteristics.All are perfectly or very nearly planar, with average U-N bond lengths of 2.776-2.804Å (2.770-2.799Å) when optimised in the gas phase (DCM) and U-O bonds of 1.777-1.778Å (1.786-1.793Å) when optimised in the gas phase (DCM).Our calculated U-N and U-O bond lengths for both C 2 b and C 2 b 0 are in good agreement with the previous theoretical values reported by Shamov and Schreckenbach, 114 and, for C 2 b 0 , structural parameters are in good agreement with experimental data. 54Shamov and Schreckenbach's work illustrated the importance of including these substituents when modelling uranyl isoamethyrin, and here we find that substituents have a similar distorting effect on both C 2 a 0 and the hypothetical C 2 c 0 , where a presumably sterically-induced twisting of the ligand results in average U-N bond lengths of 2.673-2.689Å (2.659-2.674Å) when optimised in the gas phase (DCM), B0.1 Å shorter than their simplified analogs.Commensurately, U-O bonds in the C 2 x 0 complexes are slightly longer than their simplified analogs, 1.787-1.789Å (1.799-1.802Å) when optimised in the gas phase (DCM).These differences demonstrate that careful consideration of the effects of substituents and solvation models is essential when modelling hexaphyrin complexes.It may be interesting to investigate the rigidity of these structures using perhaps MD simulations, although this has not been performed as part of this study.
Non-planarity in C 4 a, C 4 b and C 4 c is pronounced.Average U-N bond lengths are typically a few hundredths of an Angstrom longer than for the C 2 x 0 complexes, 2.701-2.707Å (2.692-2.706Å) when optimised in the gas phase (DCM).U-O bonds are commensurately slightly shorter than those in the C 2 x 0 complexes, 1.783-1.785Å (1.792-1.793Å) when optimised in the gas phase (DCM).C 6 , the largest complex, is also highly non-planar, allowing equatorial U-N bonds of comparable length to the C 2 x 0 and C 4 x complexes to be formed.Interestingly, in this complex the ligand is folded almost completely in half (reminiscent of a Pacman-style ligand 16,89,121,122 ), although here the uranyl unit is coordinated at the ligands centre.This folding has the effect of bringing some ligand atoms significantly closer to the -yl oxygen ions than in any of the other complexes.Average U-N bonds are 2.700 Å (2.697 Å) when optimised in the gas phase (DCM).U-O bond lengths are 1.789 Å (1.796 Å) and 1.784 Å (1.791 Å), with an 0.005 Å elongation of one bond due to the aforementioned C-O yl interaction.
Reoptimisations of C 0 , C 2 b 0 , C 4 a, and C 6 were performed using the hybrid B3LYP functional but led to no significant structural changes.U-N bond lengths were calculated to be in the region of 0.005 Å longer and U-O bond lengths were found to be shorter by approximately the same amount.Irrespective of the functional employed, continuum solvation acts to give a slight shortening of the U-N bonds and a corresponding lengthening of U-O bonds.

Binding energies
Complex stability was investigated by calculating molecular binding energies as well as ligand deformation energies.Molecular binding energies (DE) were calculated as defined in eqn (1) by subtracting the energies of the optimised uranyl dication and ligand dianion fragments from that of the complex: The highly flexible nature of the macrocyclic ligands considered here is such that it is informative to evaluate the degree by which the fragments deform from their optimal geometries upon complexation.Ligand deformation energies (E DL ) were calculated by subtracting the energy of each ligand in its coordination geometry from that of the optimised free ligand, as in eqn ( 2): And similarly for the uranyl dication: (3) The deformation energies may be considered independently or subtracted from the calculated binding energy to obtain a deformation adjusted binding energy (E DA ).
In this way, complex stabilities can be considered in a manner that allows for the effects of the destabilisation of the uranyl and hexaphyrin fragments to be taken into account.Molecular binding energies and deformation adjusted binding energies calculated from PBE gas phase optimisations are listed in Table 4.The overall trend is for binding energies to fall as the ligands become larger, decreasing by B1.5 eV from C 0 to the C 2 x 0 complexes, then by another B1.5 eV from the C 2 x 0 to the C 4 x complexes, where the effect appears to plateau.There is a decrease of only B0.2 eV from the C 4 x complexes to the C 6 complex.Notably, the complexes predicted to be most stable here, C 0 and C 2 x 0 , are those that have been synthetically realised, while the less stable C 4 x and C 6 complexes have proven, so far, to be experimentally inaccessible.
It is interesting to note that the deformation energy of the ligand increases as the core size is increased.This increase is B1 eV per pair of meso-carbon up to the C 4 x complexes, which appear to represent a maximum.Beyond this, increased flexibility in the ligand presumably reduces the deformation energy penalty.The deformation energy of the uranyl unit decreases slightly as the ligand core size increases: it is comparable, and in fact dominant, to that of the ligand in C 0 , whereas it is of comparable magnitude in all other complexes studied.Combined, the result is a net increase in deformation energy from C 0 to C 6 .Whilst the deformation energy doesn't fully account for the relative stability of the smaller ligands, it does strongly correlate (R 2 = 0.98) as shown in Fig. 4.
When the deformation energy is subtracted from the binding energy to obtain a deformation adjusted binding energy, the relationship between binding energy and ligand size persists and there is a loss of stability for each pair of meso-carbons added, reaching a plateau at the C 4 x and C 6 complexes.

U-O stretching frequencies
Frequencies of the U-O stretching modes calculated using the PBE functional in the gas phase and in the presence of a DCM continuum solvent are presented in Table 5.In previous studies, a degree of U-O bond weakening upon equatorial uranyl complexation has been both spectroscopically observed and theoretically calculated, 42,[94][95][96][97]123,124 with the magnitude of this weakening corresponding to a redshift in the distinctive uranyl stretching modes. Our preious study demonstrated strong correlations between binding energy and the frequency of the uranyl stretching modes in a series of monodentate complexes in which uranyl is coordinated by a first row species.97 Fig. 5 shows both the symmetric and asymmetric stretching frequencies of uranyl in C 0 , C 2 a 0 , C 2 b 0 , C 2 c 0 , C 4 a, C 4 b, C 4 c and C 6 plotted against the deformation adjusted binding energy.It is immediately clear that such a linear relationship is not present here, with only very weak correlation (R 2 r 0.3 in all cases).Plotting these frequencies against the uranyl deformation energy however, as seen in Fig. 6, results in weak correlation    (R 2 = 0.68 and 0.71 for the symmetric and asymmetric modes, respectively) with the C 6 complex being an obvious outlier in what is otherwise an apparent linear relationship.When the C 6 complex is omitted from the linear regression analysis, correlation becomes very strong (R 2 = 0.99 and 0.94 for the symmetric and asymmetric modes, respectively).Returning to the relationship between the symmetric/asymmetric stretching frequencies of uranyl and the binding energy/deformation adjusted binding energy, omitting the C 6 complex results in moderate correlation, with R 2 values of between 0.53 and 0.83 and correlation being slightly stronger when considering the adjusted binding energy.Possible reasons for the anomalous behaviour exhibited by the C 6 complex will be further explored in later sections.For the following density based analysis, a representative set of complexes have been selected.This set comprises C 0 , C 2 b 0 (which is the experimentally best characterised C 2 x 0 complex), C 4 a (the most stable of the three hypothetical C 4 x complexes) and C 6 .All are simplified complexes except for C 2 b 0 since, as discussed above, substitution was found to only impact significantly on C 2 x geometries.The characterisation will focus on PBE-optimised gas phase complexes, with data from B3LYP-optimised and solvated simulations given in ESI † and discussed where relevant.

Topological analysis of the electron density
Topological properties of the U-N and U-O bonds are now considered.The values of the electron density, r, its Laplacian, r 2 r, and the energy density, H, were investigated at the U-N and U-O bond critical point (BCP).Also included in this section is the delocalisation index d(A,B), defined as the number of electrons delocalised between two atomic basins A and B. Table 6 contains average and total values of topological descriptors at the U-N BCPs.As a rule of thumb, a covalent bond is expected to have an electron density at the bond critical point of r Z 0.2 a.u., with a negative Laplacian and an energy density which is negative in sign, with its magnitude commensurate with the degree of covalency. 101,102It is immediately apparent from the data in Table 6 that, as expected, none of the U-N bonds investigated here exhibit pronounced covalency, rather each U-N bond has a small degree of covalent character which can be quantified by the values of these topological properties at the BCP.Average and total properties are given in Table 6 since it is the effect of the ligand as a whole on the uranyl unit which is of greatest interest, however there is a strong relationship between individual U-N bond lengths in C 0 , C 2 b 0 , C 4 a and C 6 and r BCP values, (see Fig. 7) where the shorter the bond, the larger the covalent component of the interaction.
When average values are considered (see Table 7), it can be seen that equatorial covalency decreases and U-N bond length increases in the order C 0 4 C 2 b 0 4 C 6 4 C 4 a, with C 0 having by far the most U-N covalency and the other three complexes Fig. 6 Linear fitting of the relationship between uranyl deformation energies and the frequencies of (a) the symmetric, and (b) the antisymmetric stretching modes of uranyl.Fit lines are given for the entire data set (blue) and omitting the outlying C 6 result (red).Generated from data obtained using the PBE functional in the gas phase.
Table 6 Topological parameters obtained at the bond critical points of the U-N bonds and delocalisation indices between the U-N atomic basins, given as total or average values measured in atomic units (a.u.).Data is from structures calculated using the PBE functional in the gas phase  being broadly similar in their U-N character.It is worth emphasising that the U-N bonding character of C 2 b 0 , C 4 a and C 6 is very similar in comparison to the differences between these complexes and C 0 .When the relationship between the frequencies of the uranyl stretching modes and the sum of r BCP (see Table 6) for the U-N bonds is investigated, linear regression reveals only weak correlations with R 2 = 0.74 and 0.61 for the antisymmetric and symmetric modes, respectively.The values of the energy density, H, at the BCPs of all U-N bonds, support conclusions drawn from the electron density, r, of weak covalent character.
H takes negative, albeit very small, values in all complexes, with C 0 having the largest magnitude, indicating the greatest degree of covalent character.The delocalisation index (see Table 6), summed over all U-N bonds, may be considered a direct measure of electron sharing between the uranyl unit and the ligand.Supporting the assertion based on analysis of r BCP that the U-N bonds in C 0 have significantly more covalent character than any of the other complexes, C 0 exhibits the greatest degree of electron delocalisation in its U-N bonds, with C 2 b 0 , C 4 a and C 6 all exhibiting comparable values, similar to those of the [UO 2 (BTP) 2 ] 2+ complex we considered previously. 98n Table 8, various topological parameters of the U-O bonds are given.We find strong correlation (R 2 = 0.97) between average values of r BCP for the U-N bonds and values of r BCP for the U-O bonds (see Fig. 8).This can be explained in terms of the effect on the uranyl unit due to the bonding in the equatorial plane, which acts to destabilise it, as we have previously reported, 97,98 resulting in U-O covalency being weakest when equatorial covalency is strongest.Two things are of note when these data are considered.Firstly, C 0 is again set apart from the other complexes, with significantly greater equatorial covalency and a commensurately smaller degree of covalency in the U-O interaction.Secondly, C 6 again appears to differ from the other complexes in that its two U-O bonds have noticeably different values of r BCP .This is explained by the fact that C 6 has a characteristic unique amongst the complexes investigated here: QTAIM analysis reveals bond paths between two ligand meso-carbons and one of the uranyl oxygens, which is enclosed by the ligand in a manner reminiscent of a Pacman-style complex. 125Topological properties associated with this interaction are given in Table 8, showing that they are weak non-covalent interactions with r BCP an order of magnitude lower than in the U-O bonds.When only the value of r BCP for the unenclosed U-O bond in C 6 is used, the frequencies of the uranyl stretching modes and values of r BCP for the U-O bonds are found to have an improved linear relationship with R 2 = 0.88 and 0.79 for the antisymmetric and symmetric modes, respectively.When the value of r BCP for the enclosed U-O bond of C 6 is used, linear correlations decrease to R 2 = 0.70 and 0.58 for the antisymmetric and symmetric modes, respectively.Thus the uranyl stretching modes in the C 6 complex are significantly perturbed by these additional interactions.
The effect of solvation is to slightly increase topological parameters in all U-N bonds, with a commensurate small decrease in the values of the topological parameters in the U-O bonds (see Tables S6-S9, ESI †).As in our previous work, 97,98 choice of functional appears to have consistent, small, but non-negligible effects on the QTAIM parameters (see Tables S8 and S9, ESI †).For all complexes, use of the hybrid B3LYP xc-functional results in a small but appreciable increase in r BCP for the U-O bond, and a small reduction in delocalisation.At the U-N BCPs, optimisation with B3LYP results in a small reduction in all properties measured compared to those obtained using PBE, implying that inclusion of a proportion of exact exchange results in increased electron localisation. 14,48,97,98

Reduced density gradient
The reduced density gradient (RDG), s(r) = |rr(r)|/2(3p 2 ) 1/3 r(r) 4/3 , has very small values in regions of covalent and predominantly noncovalent interactions, 104,126 tending towards zero at critical points in the electron density.In regions of covalent interactions, r(r) is large, and where interactions are largely noncovalent, such as the U-N bonding regions of the complexes under investigation here, r(r), as evidenced by our QTAIM analysis, tends to be   small but non-zero.Visualising the s(r) isosurface therefore allows us to qualitatively examine the spatial regions in which these weakly covalent interactions are taking place.These isosurfaces, which are presented in Fig. 9, are colour-mapped with values of r(r)sgn(l 2 ), where sgn(x) is the signum function, returning a value of 1 where x is positive, and À1 where x is negative.l 2 is the second largest eigenvalue of the Hessian of r(r) and is typically negative for attractive interactions and positive for repulsive interactions. 127This allows us to discriminate between attractive and repulsive interactions.The isosurfaces, plotted at a value of s(r) = 0.35, show regions of weak attraction in each of the U-N bonding regions, with the colouring corresponding to the strength of the interaction, as can be best seen for C 2 b 0 .In all complexes, green regions may be interpreted as either weak steric repulsion or weak attraction between adjacent nitrogen atoms, although a distinction is difficult to make.As the ligands become larger and the complexes more non-planar, we find additional regions of weak interaction, between uranyl oxygen ions and nearby pyrrole units in C 4 a, and between the enclosed oxygen ion and interacting meso-carbons in C 6 .These latter interactions correspond to the bond paths identified between the uranium and meso-carbon centres in our QTAIM analysis and, as discussed above, are strong enough to noticeably perturb other molecular properties.Additionally, in C 6 , regions of weak interaction between two pyrrolic nitrogen centres and the uranium ion are seen to extend towards the unenclosed oxygen ion, suggesting the possible presence of further ligand-oxygen interactions, although these are not identified by bond paths in our QTAIM analysis.Scatter plots of s(r) against r(r)sgn(l 2 ) indicating the presence of largely noncovalent interactions via spikes which occur at low densities are given in Fig. S1 (ESI †).In all complexes, s(r) falls to zero at several points, corresponding to critical points in the electron density.s(r) also falls to zero at some small positive values of r(r)sgn(l 2 ), indicating the presence of weak repulsive interactions.

Integrated properties of the electron density
Using the atomic populations, localisation and delocalisation indices associated with the uranyl unit, the accumulation and depletion of charges which occur upon complexation can be further probed.We define two additional measures, where N(UO 2 ) gives the electronic population of the uranyl unit as a whole, and l(UO 2 ) the number of electrons localised on the uranyl unit.In the case of free UO 2

2+
, N(UO 2 ) = l(UO 2 ) = 106 but, when complexed, deviations from this value allow insight into the nature of the interaction between the uranyl unit and the ligand, as we have previously reported. 97,98Table 9 gives the atomic populations, localisation and delocalisation data for the uranyl units in each complex, as well as those of isolated uranyl.
These data allow us to quantify the effect of equatorial complexation by each of the ligands on the uranyl unit.This effect is broadly similar for each of the four complexes, differing only in magnitude.For all complexes N(UO 2 ) is found to be greater than 106 and, as found in our previous comparison of C 2 b, C 2 b 0 and [UO 2 (BTP) 2 ] 2+ , 98 approximately 0.8-0.9a.u. of electronic charge is donated into the uranyl unit.This additional charge is distributed between the uranium ion and each of the oxygen ions and therefore acts to increase the electrostatic repulsion between the ions.
It might be expected that the localisation index, i.e. the amount of electronic charge density localised on an ion, may be used to estimate the strength of an ionic interaction.In all four complexes, greater electron localisation is present on the  oxygen centre compared to free uranyl, alongside a decrease in localisation on the uranium centre, demonstrating that complexation results in increased ionic interaction.Additionally, for all complexes, a reduction in the delocalisation index of the U-O yl bond, d(U,O), which can be considered an alternative measure of bond covalency, is apparent upon complexation, indicating a reduction in the covalent interaction.This provides evidence that the ionic character of the U-O yl bond is enhanced by equatorial complexation, and the elongation and weakening of the U-O yl bond can thus be understood to originate from the fact that this increased ionic interaction comes at the expense of U-O yl bond covalent interaction.The lengthening and weakening of the U-O bond compared to free uranyl seen in all four complexes investigated here can therefore be attributed to these factors.As might be expected, the effects are most pronounced for C 0 , with N(UO 2 ) being 0.9 a.u.greater and l(UO 2 ) 0.6 a.u.less than in free uranyl.These values, combined, suggest the greatest amount of electron delocalisation between the uranyl and the ligand, commensurate with the topological data which demonstrates that C 0 has the largest U-N covalent bond character.This interpretation is in keeping with the qualitative picture given by electron density differences, which show a clear depletion of charge in the U-O yl bonding region along with an accumulation on the O yl centres.
In terms of charge donation onto the uranyl unit, C 2 b 0 and C 4 a are similar to one another.C 6 exhibits similar donation to these complexes, but less of this charge is actually localised on the uranyl unit, with C 6 having a l(UO 2 ) value more comparable to that of C 0 , suggesting greater uranyl-ligand delocalisation than can be accounted for by considering the values of d(U,N).C 6 also has the smallest amount of electronic charge localised on the oxygen centres and the largest amount localised on the uranium centre.This can be explained in terms of the additional interactions between the uranyl oxygen centres and the ligand in this complex.Table 8 shows that, in total, an additional B0.07 a.u. of charge is delocalised in the interactions between the enclosed oxygen centre with the nearby meso-carbons, which partially accounts for the difference between C 6 and C 2 b 0 /C 4 a.Based on the RDG isosurface of C 6 , further examination of the integrated properties reveals that 0.09 a.u. of charge is delocalised between the unenclosed oxygen ion and each of the two nearby nitrogen atoms, contributing to the lower than expected l(UO 2 ) value found in C 6 and suggesting that, while ligand-uranyl interactions in planar complexes may be fairly straightforward, in larger, less planar complexes, there are potentially many other interactions which need to be considered in order to explain the charge redistribution in the uranyl unit.This O-N electron sharing also exists in the other complexes considered, but its magnitude is typically only B50% of that found in C 6 .
Solvation, irrespective of which functional is used (see Tables S11 and S13, ESI †), results in slightly higher values of N(UO 2 ) for all four complexes, while l(UO 2 ) remains largely unchanged.Reoptimisation with B3LYP (see Tables S12 and  S13, ESI †) results in greater localisation compared to the PBE data, as reported in the topological properties.Also apparent are lower values of N(UO 2 ) (by B0.1 a.u.) and higher values of l(UO 2 ) (by B0.01 a.u.).

Density difference distributions upon complexation
Finally, we use electron density difference distributions to qualitatively examine the changes undergone by the system upon complexation.The density difference distributions in Fig. 10 are generated by subtracting the electron density of uranyl and ligand fragments held at the coordination geometry from the electron density of the complex.This leaves a map of the changes that occur when a complex is formed, with regions of electron density accumulation coloured blue and depletion coloured in green.Regions of depletion on the ligands are evidence of electron donation from the ligand, and the teardrop shaped regions of accumulation in each U-N bonding region may be interpreted as evidence of covalent interactions.It is possible, particularly in C 0 and C 2 b 0 , to see that the size of these regions of accumulation varies between the different U-N bonds.The regions of accumulation in C 0 are large and well-focused on the bonds whereas C 2 b 0 , exhibits smaller regions for the longest, least covalent bonds and larger regions for the shorter, more covalent interactions.Our previous study 97 has shown that for more ionic uranyl-ligand interactions, these regions of charge accumulation are more diffuse.The striking changes undergone by the uranyl unit lend qualitative support to our assertion that there is a significant redistribution of charge in the uranyl unit upon complexation, based on the calculated increased U-O bond lengths, redshifted frequencies of the uranyl stretching modes, uranyl deformation energies and decreased U-O delocalisation, all compared to free UO 2

2+
. The charge accumulation on the oxygen ions and depletion in the U-O interaction region upon complexation potentially signifies the involvement of density formally associated with the uranium centre with bonding in the U-N region.The depletion in the U-O bond regions is also consistent with a reduction in the covalent character of these bonds, while accumulation on the uranyl oxygen ions and uranium ion suggests that the U-O interaction is, in accord with our other analyses, becoming more ionic upon complexation.In addition to this, charge accumulation around the uranium centre appears to have some f-like character (see Fig. S2, ESI †).The size of the regions of accumulation and depletion on the uranyl unit appear to be related to the interactions in the U-N region, i.e. as the amount of electron sharing in the U-N bonds is increased, the effects on the uranyl unit become more pronounced.

Summary and conclusions
We have investigated by means of quantum chemical calculations and a range of density based analyses the U-N and U-O interactions of several uranyl hexaphyrin complexes.It has been determined through a comparison of the geometries of simplified and peripherally substituted complexes that it is of great importance when dealing with systems such as these to ensure that the effects of common simplifications such as the removal of alkyl groups are indeed minimal.This is especially important when dealing with hypothetical complexes.It was found that in most, but not all, cases that removal of the alkyl groups had little geometrical impact.
We initially established a relationship between complex stability and ligand size, as well as a weak relationship between stability and uranyl stretching mode frequencies, when a set of eight complexes were considered.This relationship was significantly weaker than that found by us in a previous study, 97 but could be strengthened by omitting the anomalous data associated with the C 6 complex.Subsequently, justification for considering the C 6 complex as being qualitatively different from the others was found in the density based analysis.
A representative set of four complexes was selected for further analysis.We found a relationship between complex stability and the degree of covalent character as defined by the total value of r BCP in the U-N bonds in these complexes.The magnitude of the energy density, H, is also higher for these U-N BCPs than in any other complex.Additionally, there is unambiguous evidence of electron sharing in all U-N bonds, and this is at a maximum for C 0 .This complex has short, strong U-N bonds and, commensurately, we see the largest effects on the uranyl unit here: complex C 0 has the most significant reduction in U-O bond covalency when compared to free uranyl, the most significant U-O bond lengthening, and pays the greatest energy penalty in terms deformation of the uranyl unit.Despite this, it is the most stable complex of any we have investigated here, which may, in part, be due to covalent stabilisation from the relatively large amount of electron sharing in the U-N bonds.
Electron delocalisation and covalent character as defined by values of r BCP and H in the U-N bonds is reduced for C 2 b 0 , C 4 a, and C 6 , although not drastically so, and these all have bond lengths which are comparable due to the flexibility of the ligands, which contort to better fit the uranyl dication in the cavity.There is an energy penalty for this, however and there is a decrease in stability in the order C 0 4 C 2 b 0 4 C 4 a 4 C 6 , a trend which appears to be replicated experimentally: the C 0 and C 2 b 0 complexes are known synthetically, while C 4 a and C 6 are not.This energy penalty can be seen using both the molecular binding energies and the ligand deformation energies which decrease and increase, respectively, with increasing ligand size.There is also a limit as to how much this ligand distortion can act to increase the covalent character of the U-N bonds; a notable plateau is found in terms of both stability and U-N covalent bond character for C 4 a and C 6 .When the frequencies of the uranyl stretching modes are considered, the C 6 data is anomalous, falling somewhere between that of the C 2 b 0 and C 4 a complexes.With all complexes but C 6 , as in our previous study of complexes involving coordination by 1st row species, 97 there is a degree of equatorial planarity.The interactions between the nitrogen ligands and the uranium ion are more or less perpendicular to the U-O yl axis and, as such, the ligands only interact directly with the U atom and this in turn affects the U-O interaction.However in the C 6 complex we see additional interactions between the uranyl oxygen ion enclosed by the ligand and two of the ligand meso-carbons, confirmed by QTAIM analysis and RDG isosurface plots, which also indicate the presence of weak interactions between two pyrrolic nitrogens with the other oxygen ion.These interactions, although weak, act to perturb the frequencies of the uranyl stretching modes as well as the topological and integrated properties of the uranyl unit.
We see dramatic and consistent changes to the uranyl unit upon complexation which are related to the covalent character of interactions in the U-N bonding region as well as the stability of the complex.This is evident in U-O bond lengths, the frequencies of the distinctive uranyl stretching modes (with the exception of C 6 , as discussed above), values of QTAIM descriptors and integrated charges, and can be visualised via electron density difference distributions.These add qualitative support to our assertions that electron density is redistributed in the uranyl unit upon equatorial complexation, with the magnitude of this redistribution related to the magnitude of the equatorial covalent interaction.Density difference distributions also clearly show charge accumulation in the U-N bonding regions, providing qualitative evidence of electron sharing in these interactions.
Analysis of the reduced density gradient allowed visualisation of the regions of weakly covalent interaction in all complexes, with weak attractive regions corresponding to each U-N bond found in all cases and, in the C 6 complex, additional interactions between the uranyl oxygen ions and the ligand identified.
Ultimately, we conclude that hexaphyrin ligands coordinate uranyl in a broadly similar way, with the uranium atom coordinated via largely ionic interactions with small but measurable amounts of covalent character, to six pyrrolic nitrogen atoms.However, the size of the ligand core has pronounced effects on complex stability.U-N covalent character is found to correlate strongly with bond length, however ligand flexibility and its effects (shortened and thus more covalent U-N bonds, but a loss of stability) mean that no clear relationship can be identified between equatorial covalency and stability in these complexes.However, the interesting changes to the electronic structure of the uranyl unit upon complexation suggest that expanded porphyrins are useful systems for investigating the effects of complexation on the uranyl bond.The complicating factor of additional interactions caused by the proximity of the ligand in C 6 suggest that investigation of complexes which have well-defined planarity may be preferable.If selectivity is indeed driven by covalency, it stands to reason that design of ligands to maximise selective behaviour ought to aim to maximise covalent character.The fact that the most stable complex with the greatest equatorial covalency, C 0 , is obtained using the ligand with the smallest core suggests that a fruitful avenue of future research into maximising U-N covalency may be pentapyrrolic complexes of uranyl, of which there are several experimentally realised examples including uranyl pentaphyrin 7 and uranyl superphthalocyanine. 128 Additionally, other actinide complexes of these ligands may be explored in order to assess whether a particular ligand has affinity for a particular actinide.

Fig. 1
Fig. 1 Molecular structure of isoamethyrin dianion omitting peripheral groups for clarity.Symmetry-distinct coordinating nitrogens are labelled N A , N B and N C .meso-Carbon atoms are labelled C m .

Fig. 3
Fig.3Optimised peripherally substituted C 2 structures, optimised in the gas phase using the PBE exchange-correlation functional.

Fig. 4
Fig. 4 Molecular binding energy plotted against total deformation energy for all eight complexes.Energies are taken from the structures optimised in the gas phase.

Fig. 5
Fig.5Linear fitting of the relationship between deformation adjusted binding energies and (a) antisymmetric and (b) symmetric stretching modes of uranyl; fits are given for the entire data sets (blue) and omitting the outlying C 6 results (red).Generated from data obtained using the PBE xc-functional in the gas phase.

Fig. 7
Fig. 7 Values of U-N r BCP plotted against individual U-N bond lengths for C 0 , C 2 b 0 , C 4 a and C 6 , for complexes optimised using the PBE xc-functional in the gas phase.

Fig. 8
Fig. 8 Average values of r BCP for the U-N bonds plotted against values of r BCP for the U-O bonds.

Fig. 9
Fig. 9 Isosurfaces of the reduced density gradient, s(r), mapped with values of r(r)sgn(l 2 ).Red regions indicate attractive interactions with weakly covalent character.Green areas indicate regions of weak interaction which may be attractive or repulsive.Isosurfaces are rendered at s(r) = 0.35 a.u.

Fig. 10
Fig. 10 Electron density differences upon complexation viewed from above, and in the plane of, the ligand.Blue regions indicate charge accumulation and green areas charge depletion.All densities visualised using an isosurface of r = 0.005 a.u.

Table 1
Complex naming convention used in this study

Table 2
Ranges of U-N bond lengths in Å for complexes optimised using the PBE exchange-correlation functional in the gas phase/DCM

Table 3 U
-O bond lengths in Å for complexes optimised using the PBE exchange-correlation functional in the gas phase/DCM

Table 4
Molecular binding energies in (D E) and deformation adjusted binding energies (DE DA ), with deformation energies of the UO 2 2+ unit (E DL ) and the ligands (E DU ), all given in eV.Data was obtained using the PBE xc-functional, and due to the simple COSMO solvation model being a rather poor approximation for solvated uncoordinated UO 2 2+ , are given in

Table 5
Uranyl stretching frequencies from structures calculated using the PBE functional in the gas phase/DCM n U-Os (cm À1 ) n U-Oas (cm À1 )

Table 7
Average values of r BCP and bond lengths for the U-N bonds in C 0 , C 2 b 0 , C 4 a, and C 6 , given in Angstrom and a.u., respectively.Data is from structures calculated using the PBE functional in the gas phase

Table 9
Integrated properties associated with the uranyl ions of each complex.Data is from structures calculated using the PBE xc-functional in the gas phase