An–imidophosphorane (An = U–Pu) bond covalency and proton-coupled electron transfer thermodynamics driven by orbital energy matching

Abstract

A series of mid-actinide (An = U–Pu) tetrahomoleptic complexes supported by highly electron-donating imidophosphorane ligands, NPC ([NP^tBu(pyrr)₂]⁻, where ^tBu = C(CH₃)₃; pyrr = pyrrolidinyl = N(C₄H₈)), are systematically investigated computationally and experimentally to elucidate the nature of actinide–ligand (An–L) covalency across the An^3+/4+/5+ oxidation states. Trends in An–L bonding and redox properties for these complexes, together with their protonated counterparts, are examined using orbital-, electron density-, and energy-decomposition-based methods. This integrated approach reveals progressively improved energy matching between α-spin An 5f and N_im 2p orbitals with increasing atomic number and oxidation state, becoming particularly pronounced in the ligand-dominant π-bonding orbitals of An⁴⁺ and An⁵⁺. In contrast to the An³⁺ species, the enhanced An 5f_π contributions in the higher-valent counterparts drive the increase in An–N_im covalency for later An, thereby inverting the covalency trend to U < Np < Pu. Redistribution of electron density towards the An and N_im atomic basins due to the growing energy-matching assisted covalency correlates with higher pK_a values and increased N_im–H bond dissociation free energies in protonated An⁴⁺ complexes. Electron density at N_im in An⁴⁺ shows a linear correlation with the pK_a values calculated via the Bordwell equation. Calculations predict a cathodic shift of 0.84–1.00 V in the redox couples upon protonation, a trend validated when experimentally accessible. These findings demonstrate an increasing role of covalency driven by orbital energy matching from U to Pu in tuning the thermodynamic driving force for proton-coupled electron transfer in the An⁵⁺ species.

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Introduction

Understanding the origin of metal–ligand (M–L) bonding interactions and their relationship to the thermodynamics of actinide (An) chemical reactivity is essential for advancing separation techniques.^1–9 Consequently, the study of f-element M–L bond covalency^10–23 has been a cornerstone of coordination chemistry and spectroscopy for both lanthanides and actinides since Seaborg's seminal work in 1954,²⁴ driving the development of contemporary computational methods. A central goal has been to understand and leverage the An 5f orbitals, which are generally more diffuse and higher in energy than lanthanide 4f orbitals. The differences in the participation of the f orbitals in the M–L bonding, which define the degree of covalency, may also lead to differences in chemical reactivity.^23,25–32 Higher-oxidation-state An complexes offer an ideal framework for examining the role of 5f orbitals in chemical bonding because their contributions become more pronounced. Ultimately, developing a fundamental understanding of M–L bond covalency in a systematic fashion across multiple oxidation states and the An series remains a crucial goal.^33–39

Previous studies have focused on applying theoretical methods to interpret spectroscopic and diffraction data, aiming to define the role of covalent bonding in various An complexes. In this context, methods such as X-ray diffraction (XRD),^32,40–44 X-ray absorption spectroscopy,^45–49 resonant inelastic X-ray scattering maps,^45–49 Mössbauer spectroscopy,⁴² UV-vis-nIR spectroscopies,^41,50 nuclear magnetic resonance (NMR),^32,41,44,51 electron paramagnetic resonance,^31,50 and magnetic measurements^43,46,50,52 serve as valuable tools for experimental assessment of bond covalency, 4f/5f orbital contributions, and M oxidation states. These studies, though, are focused on complexes independent of their chemical reactivity. This limitation is largely derived from the absence of comparative reactions that are accessible experimentally across the An series.

Theoretically, covalency can be treated as a perturbation to the ionic limit, arising from orbital overlap and orbital energy matching in heteroatomic M–L bonds.^6,36,53–75 The latter has garnered increasing attention in recent years and has been noted as a driving force for An–L covalency in borate and dipicolinate compounds.^{6,43,54,57,64,76,77} The interplay between orbital overlap and orbital energy matching varies depending on the oxidation state and identity of the An. These differences arise from changes in the radial extent and energy of the An 5f/6d orbitals relative to the L orbitals, which has been noted for a number of systematic studies of An species.^{37,39,60,61,65,78,79} Enhanced orbital mixing arising from improved orbital energy matching has been reported for Ln systems as well.^80,81 For example, the increased 4f contributions in the Ln³⁺ complexes supported by the 3,5-di-tert-butyl-o-semiquinone ligands⁸⁰ was attributed to the greater energetic accessibility of the 4f shell, leading to deviations from expected bonding trends across the Ln series (La–Gd, except Pm).

Understanding the factors that govern the M–L bonding may also be key to determining the thermodynamic aspects of a complex's reactivity. For example, it was suggested previously that the more covalent character of the U–C bonds in (η⁵-C₅Me₅)₂U[η²-C₂(SiMe₃)₂] makes it prone to react with alkynes, in contrast to the Th counterpart exhibiting more ionic bonding.⁸² Likewise, recent theoretical and inductively coupled plasma tandem mass spectrometer studies on the activation of methane by An cations suggest that there is a crossover in reactivity and bonding between Np⁺ and Pu⁺. It is described as a shift from An–L⁺ bonding defined by orbital overlap for the early An to bonding driven by orbital energy matching for the later An.⁸³ These ties between M–L covalency and reactivity have previously been elucidated for other various An and transition metal complexes.^84–91 Likewise, a recent combined theoretical and experimental study of M^IV–cyclopropenyl complexes (M  =  Ti, Zr, Ce, Hf, Th) showed that the 4f-orbital covalency can give rise to the distinct chemical reactivity observed across a series of isostructural and isoelectronic d- and f-block complexes.⁹² Thus, establishing the origin of M–L covalency is indispensable for uncovering and rationalizing reactivity patterns in f-block species.

Proton-coupled electron transfer (PCET) is a fundamental chemical transformation and has been studied extensively across the periodic table.^93–95 Notably, the kinetics and thermodynamics of PCET reactions of transuranic complexes have only been recently defined,^20,21,96 and a few reports of U complexes involved in PCET have been reported.^97,98 Most existing studies of An PCET focus on the ubiquitous actinyl moiety in aqueous solution,^99–101 where the fundamental PCET reaction step is often kinetically obscured by subsequent processes such as disproportionation, oligomerization, and hydrolysis. Recently, we quantified the kinetics and thermodynamics of PCET reactions of transuranic non-actinyl imidophosphorane compounds of Np and Pu^20,21 and examined the redox chemistry of the An^5+/4+ couple in [An(NPC)₄] (An = Np, Pu) (NPC  =   [NP^tBu(pyrr)₂]⁻, ^tBu  =  C(CH₃)₃, and pyrr  =  pyrrolidinyl = N(C₄H₈)). The calculations revealed increasing An–N_im covalent interactions across the U–Pu species (U < Np < Pu) attributed to enhanced energy matching between An 5f and N_im 2p orbitals. This trend correlates with greater electron density accumulation on the N_im atoms within the An⁵⁺ species, aligning with the observed PCET reaction rates: the Pu⁵⁺ complex undergoes PCET ∼5 orders of magnitude faster than Np⁵⁺, while the U⁵⁺ counterpart does not undergo a PCET reaction under the same conditions.

This homoleptic imidophosphorane ligand framework allows for a unique opportunity to study the PCET reaction and An–L covalency in a wide range of oxidation states across the mid-An (U–Pu). Since the An 5f orbitals progressively decrease in energy as the atomic number and oxidation state increase, systematic studies within a conserved ligand field are particularly relevant for understanding the trends in orbital overlap and orbital energy matching. Hence, in this work, a series of mid-An complexes are investigated employing the tetrahomoleptic [NPC]⁻ ligand architecture (1-An^q+; 1-An = [An^q+(NPC)₄]ⁿ, q = 3–5; n = −1, 0, +1) as well as their respective single ligand protonated congeners (2-An^q+; 2-An = [An^q+(HNPC)(NPC)₃]ⁿ, q = 3, 4; n = +1), and higher-oxidation state 1-U⁶⁺, 2-U⁵⁺, 2-U⁶⁺, and 2-Np⁵⁺ complexes (Fig. 1). The origin of covalent interactions and covalency trends of the An–N_im bonds across U, Np, and Pu are evaluated as a function of the An oxidation state through orbital-based, electron density-based, and energy-decomposition approaches. Additionally, the redox chemistry of these complexes is modelled and the N_im–H bond dissociation free energies (BDFEs) and pK_a values are determined. The thermodynamic drive for PCET in the An⁵⁺ species is rationalized through changes in the An–N_im covalency and electron density at the N_im atoms, arising in part from the enhanced An 5f–N_im 2p orbital energy matching and inherent redox properties of each complex. These computational insights are supported by experimental single-crystal XRD (SC-XRD) and electrochemical measurements on selected compounds, enabling a comparative analysis.

Fig. 1 Lewis structures of 1-An^q+ and 2-An^q+ (An = U–Pu). Full value range for each individual variable includes: An charge (q = 3, 4, 5, 6); complex charge (n = −1, 0, +1, +2).

Experimental methods

The complexes of 2-U⁴⁺ and 2-Np⁴⁺ were prepared via protonation of 1-U⁴⁺ and 1-Np⁴⁺, respectively, with [H(OEt₂)₂][B(ArF₅)₄]; a mechanistically distinct preparation of 2-Np⁴⁺ compared to the previously published PCET reaction of 1-Np⁵⁺.²⁰ This new synthetic method for 2-Np⁴⁺ increased the yield from a few crystals, only enough to obtain SC-XRD characterization, to near quantitative yield (99%) facilitating the UV-vis-nIR characterization and electrochemical analysis of this material.²⁰ This high yield also enabled an attempt to reduce 2-Np⁴⁺ with KC₈ to form the putative 2-Np³⁺. A colour change was observed, consistent with the colour change seen during the reduction of 1-Np⁴⁺ to 1-Np³⁺; however, 2-Np³⁺ could not be isolated (see the SI for further synthetic details).

The synthesis of 2-U⁴⁺ was only possible via protonation with [H(OEt₂)₂][B(ArF₅)₄] (90%) as 1-U⁵⁺ is inactive for PCET. All compounds in this series retain a pseudo-tetrahedral core geometry around the An, with those containing a protonated ligand having slightly lower τ₄ values due to the presence of the proton altering the inner coordination sphere (Table S73).¹⁰² In the 1-An^q+ series, all complexes within each oxidation state (q = 3–5) are isomorphic. Similarly, 2-Np⁴⁺ and 2-Pu⁴⁺ are isomorphic, whereas 2-U⁴⁺ is only isostructural with these two, crystallizing in a different space group. The unique protonated ligand in 2-U⁴⁺ is disordered over two positions, H1D and H1E, with 60% and 40% occupancies, respectively (Fig. S63). Due to this disorder and proximity to the heavy atom, U, refinement of the H atom positions was carried out using a riding model. This presents another key distinction between the U complex and its Np and Pu analogues, as in the latter cases the protons were defined in the electron difference map and refined as protons localized on N1D within an applied bond length parameter. Further crystallographic modelling details and selected structural metrics for the full series are highlighted in the SI and Table S73.

The solution NMR of 2-U⁴⁺ (heteronuclear and 2D) agrees with the SC-XRD assignment. The ³¹P{¹H} spectrum shows two signals in a 3 : 1 ratio for their respective intensities at 381.0 ppm (NPC) and –90.9 ppm (HNPC). Each ligand exhibits a substantial paramagnetic shift relative to the free ligand HNPC (47.5 ppm), with shifts greater than those observed for either the Np or Pu analogues. Comparative stacked spectra are presented in Fig. S57.

Computational methods

Density functional theory (DFT) calculations

The geometries of all complexes are optimized in the gas phase using the PBE0 hybrid DFT functional¹⁰³ as implemented in Gaussian 16 (Version A.03).¹⁰⁴ The ECP60MWB¹⁰⁵ small core quasi-relativistic pseudopotential and ECP60MWB_ANO¹⁰⁶ basis sets are employed to describe U–Pu. All remaining atoms are treated with the all-electron Pople basis set 6-311G(d).¹⁰⁷ For clarity, this level of theory will hereafter be referred to as PBE0 ECP. This computational protocol has been successfully applied in our previous studies of An-based imidophosphorane complexes,^20–23 accurately reproducing geometries, electronic structures, and thermodynamic trends, and is therefore adopted in the present work. Geometrical analysis shows good agreement between the calculated and experimentally available structures, with An–N_im and P=N_im bond lengths deviating less than 1.7% and 1.4%, respectively, and all complexes adopting a pseudo-tetrahedral geometry (τ₄ = 0.88–0.99) (Tables S3 and S4).

An–N_im bonding interactions were characterized using the Quantum Theory of Atoms in Molecules (QTAIM)¹⁰⁸ by evaluating the electron density ρ(r) and delocalization index δ(An,N_im) with Multiwfn.¹⁰⁹ Complementary orbital-based insight was obtained from the Adaptive Natural Density Partitioning (AdNDP) analysis¹¹⁰ using the Natural Bond Orbital (NBO7) method.¹¹¹ The interacting quantum atoms (IQA) energy decomposition method,^112,113 as implemented in AIMAll,¹¹⁴ was used to partition the An–N_im interaction energy into classical electrostatic (V_cl) and exchange–correlation (V_xc) terms, with V_xc reflecting electron sharing and treated as the covalent component. IQA is a real-space, topology-based scheme that has proven effective for quantifying chemical bonding in An.^55,115,116

All electrochemical calculations are referenced to the Fc^+/0 couple, which reliably reproduces experimental redox potentials in related An systems.^20,21,23 Gas-phase enthalpic and entropic contributions were omitted because they have little effect on computed redox potentials.¹¹⁷ Covalency metrics and redox potentials were also evaluated using (1) the PBE functional and (2) an all-electron basis set with relativistic DKH corrections. Both approaches yielded qualitatively similar trends to those obtained with the PBE0 ECP protocol (see SI for details).

BDFEs are calculated using total energies from the single point energy calculations in THF using the Douglas–Kroll–Hess fourth order (DKH4) relativistic Hamiltonian including spin–orbit coupling (SOC) and zero-point energy and thermal energy corrections from the corresponding gas phase optimizations as implemented in Gaussian 16. The procedure for calculating N–H BDFEs is based on the experimentally determined free energy of H atom transfer from 1-hydroxy-2,2,6,6-tetramethyl-piperidine (TEMPO-H) in THF ¹¹⁸ (see the SI for details). The pK_a values are calculated from the Bordwell equation, BDFE = 1.37 pK_a + 23.06E⁰ + C_G, where C_G is the solvent-specific H⁺/H˙ standard reduction potential⁹⁴ (66 kcal mol⁻¹ in THF¹¹⁹), E⁰ is the experimentally determined redox potential, and BDFE is the calculated value described above. To minimize uncertainty from computational approximations, pK_a were determined only for species with experimentally measured redox potentials. Molecular electrostatic potentials at the N_im atoms were computed with Multiwfn, given their established correlation with pK_a values.

Multireference calculations

To account for the static correlation arising from the near degeneracy of the 5f orbitals, multireference Complete Active Space Self-consistent Field (CASSCF)¹²⁰ calculations were performed on the DFT-optimized geometries of 1-An³⁺, 1-An⁴⁺, and 1-An⁵⁺ using ORCA 6.1.0.^121,122 The DKH2 Hamiltonian was employed to account for scalar relativistic effects.¹²³ The relativistically contracted SARC-DKH-TZVP basis set was employed for An,¹²⁴ the DKH-def2-TZVP(-f) basis set for N atoms in the first coordination sphere, and the DKH-def2-SVP basis set for the rest of the atoms.^125,126 The RIJCOSX approximation was used to reduce the computational cost. For U, the autoaux¹²⁷ feature was used to generate the auxiliary basis set. Dynamic correlation was added through the N-Electron Valence State Second-Order Perturbation Theory (NEVPT2) method for the calculations of spin-state energetics.¹²⁸ For 1-U⁴⁺ and 1-Np⁵⁺, the ground state is three and two-fold degenerate, respectively. Therefore, the CASSCF calculations were carried out with three triplets for 1-U⁴⁺ and two triplets for 1-Np⁵⁺. For other complexes and other spin states, only one root was considered. Restricted Active Space Self-Consistent Field (RASSCF)¹²⁹ calculations were performed to quantify An orbital populations in the 3c–2e An–N–P π bonds within the 1-An⁴⁺ and 1-An⁵⁺ series (see the SI for details). These multireference results were used to benchmark and contextualize the pronounced An 5f_π contributions obtained at the DFT level for later An.

Results and discussion

Molecular orbitals and density of states

Across all examined oxidation states, the An-dominant orbitals corresponding to the An unpaired 5f electrons in the α-spin manifold shift to progressively lower energies as the An series is traversed, which is captured in both DFT and CASSCF methods (Fig. 2 and Fig. S1–S8). In the 1-An³⁺ series, the 5f orbitals are well separated from the frontier L-based π bonding MOs, allowing for a straightforward qualitative assignment of the 5f³, 5f⁴, and 5f⁵ configurations in the U, Np, and Pu species, respectively. In 1-An⁴⁺, these orbitals gradually approach the L-dominant π bonding MOs, which remain relatively consistent regardless of the An identity (Fig. 2). Consequently, orbitals with considerable 5f contributions in 1-Pu⁴⁺ become ∼0.6–0.9 eV lower in energy than the L-dominant highest occupied molecular orbital, HOMO (Pu 6.3%; L 93.7%). Their average An contribution drops to 51.3% compared to 96.1% in 1-U⁴⁺ and 72.1% in 1-Np⁴⁺ (Table S6). The energy matching of M and L orbitals becomes even more pronounced in 1-An⁵⁺, as all orbitals shift to lower energies, most significantly in 1-Pu⁵⁺ (Fig. 2 and Fig. S3). While the more core-like nature of the 5f atomic orbitals (AOs) in later An is anticipated due to the incomplete screening of the additional nuclear charge,^54,57,67,70 the extensive spread of Pu 5f AOs among L-dominant MOs in 1-Pu⁵⁺ makes assigning the 5f³ configuration more challenging based solely on canonical MOs. As was discussed before,^20,21 in such cases, it is critical to delineate M contributions to the localized non-bonding elements from those describing M–L bonding by means of the electron localization methods, such as NBO/AdNDP (vide infra). However, it is important to note that M/L orbital energies and orbital mixing depend on the amount of Hartree–Fock exchange in the chosen DFT functional. Therefore, only relative orbital energy ordering and composition analyses are meaningful for comparing different An systems, which serves as the basis for the analysis in this study.

Fig. 2 Schematic α-orbital energy diagrams. The light blue shaded regions represent the weighted standard deviation from the average of the 5f orbitals, which is taken from –8.40 eV to +4.00 eV. The grey shaded region represents the energy range of the π bonding orbitals. Averages and standard deviations of the orbital energies are weighted by MO composition. Full α + β MO diagrams for all complexes are shown in Fig. S1–S7. Detailed orbital contributions are shown in Tables S5–S11.

A similar effect is observed upon a single L protonation of 1-An⁴⁺ to form 2-An⁴⁺, e.g., HOMOs of 1-U⁴⁺/1-Np⁴⁺ drop in energy by 0.75–0.81 eV. While the An 5f-dominant orbitals of 2-U⁴⁺/2-Np⁴⁺ can still be identified as the top frontier singly occupied MOs in the α-spin manifold, the Pu 5f atomic orbitals in 2-Pu⁴⁺ evince a considerable intermixing with the L-dominant orbitals. They are spread over ten MOs (HOMO–8 to HOMO–17) with the Pu contributions ranging from 16.6% to 65.8% (Fig. S5 and Table S9). As the overall charge increases due to the protonation of 1-An⁴⁺, the lowest unoccupied molecular orbitals (LUMOs) in 2-An⁴⁺ also stabilize and shift down in energy by 0.78–0.84 eV. It is noteworthy that the energy of the LUMO in 2-U⁴⁺ (–0.82 eV) is substantially lower than its non-protonated counterpart, 1-U⁴⁺ (0.02 eV). Our previous joint experimental and theoretical work²³ revealed that the reduction of 1-U⁴⁺ to 1-U³⁺ could not be achieved either chemically or electrochemically, which was rationalized by the relatively high energy of the LUMO in 1-U⁴⁺. In contrast, 1-Np⁴⁺ could be chemically reduced to 1-Np³⁺, consistent with its lower-energy LUMO (–0.56 eV) calculated at the same level.

To more specifically pinpoint the energy matching between certain orbitals, the density of states¹³⁰ (DOS) was calculated for all the studied complexes (see the SI for details). This analysis provides an alternative perspective to the MO diagrams for comparing the energies of the An 5f and ligand π bonding orbitals as a function of the An identity and oxidation state. The DOS plots have proven to be a valuable tool in the electronic structure analysis of a variety of other An-based molecular systems.^{59,70,75,131–133} The projected density of states (PDOS) plots, shown in Fig. S9–S15, evince a substantial energetic gap between the An 5f and N_im 2p states in the +3-oxidation state, irrespective of the An. This contrasts with the +4-oxidation state, where the Np and Pu complexes display an appreciable An 5f–N_im 2p energetic overlap, with the Pu 5f states being much closer to the N_im 2p states than Np 5f. Meanwhile, the U 5f states remain distinguished from N_im 2p, emphasizing the increased role of energy matching in the later An species. As the An oxidation state increases, the energy of the 5f bands decreases at a faster rate than the N_im 2p bands, leading to even greater energy matching as the An series is traversed. Thus, in 1-Np⁵⁺ and 1-Pu⁵⁺, the occupied An 5f states are completely energetically mixed with the N_im 2p manifold, whereas in 1-U⁵⁺ one 5f state is slightly offset in energy, corresponding to the singly occupied 5f orbital.

Natural atomic orbitals

In addition to the MO diagrams and the PDOS plots, analysis of the natural atomic orbitals (NAOs) can be employed to track how the energy difference between the An 5f and N_im 2p NAOs changes in both α- and β-spin manifolds (Fig. 3). First, the general trends show that the energy difference between the An 5f and N_im 2p NAOs decreases from U to Pu regardless of the An oxidation state, in line with the improved energy matching for later An. The average of the α- and β-NAO energies shows that the An 5f–N_im 2p NAO energy gap in the Np complexes decreases by 0.69–1.00 eV compared to the respective U species. This gap is further reduced by 0.64–0.90 eV in the Pu counterparts relative to their Np counterparts (Table S12). Second, as the oxidation state increases, the orbital energy difference becomes progressively smaller. For instance, the averaged An 5f–N_im 2p NAO gap decreases from 4.81 eV in 1-Pu³⁺ to 2.98 eV in 1-Pu⁴⁺ and further to 2.22 eV in 1-Pu⁵⁺. This result is in accordance with previously reported studies on the An²⁺Cl₂ and An³⁺Cl₃ complexes,^60,61 which showed that the An³⁺Cl₃ species generally had a greater orbital energy matching than An²⁺Cl₂. A similar effect is observed upon ligand protonation, although less abrupt, e.g., the NAO energy gap decreases from 4.81 eV in 1-Pu³⁺ to 4.59 eV in 2-Pu³⁺ and from 2.98 eV in 1-Pu⁴⁺ to 2.88 eV in 2-Pu⁴⁺. Third, in the open shell species, the α-orbitals are found to be the driving force of energy matching as the N_im 2p NAOs energy gap with α-spin An NAOs is always smaller than it is with the respective β-spin An NAOs, owing to the presence of unpaired electron(s) on the An in the α-spin manifold. Across all species, 1-Pu⁵⁺ has the lowest energy α-An 5f NAOs, creating the only negative energy gap from the N_im 2p NAOs (–0.34 eV), with an α/β average energy gap (2.22 eV) even smaller than that of 1-U⁶⁺ (3.22 eV).

Fig. 3 Difference between average An 5f NAO energy and average N_im 2p NAO energy in the α-spin (blue) and β-spin (yellow) manifolds. The average of the α-spin and β-spin values is shown with the grey circle connected by the dashed line for similar complexes. Exact values for the energy differences are shown in Table S12.

AdNDP localized orbitals: population and composition

NAOs transformed to the natural bonding orbitals (NBOs) are further employed for the detailed orbital analysis of the An–N_im bonding interactions via the AdNDP method.¹¹⁰ This approach enables a detailed evaluation of the An–N_im bonding interactions, providing insight into the An contributions and how the orbital composition evolves with the An identity and oxidation state. In all open-shell species, AdNDP recovered non-bonding one-centre one-electron (1c–1e) element(s), corresponding to the An unpaired electron(s). In such cases, they are very localized, with occupation number (ON) values spanning a narrow range, 0.95–1.00 |e|, and with dominant 5f-orbital character (90.2–100.0%, Table S13). The corresponding 5fⁿ configurations deciphered from the AdNDP analysis are instrumental for the validation of the An oxidation state, which is particularly useful for the higher-oxidation state An species exhibiting a significant spread of the An AOs in the canonical MOs. The assignments of the An oxidation state deduced from AdNDP also fall in line with the computed N_{pyramidalization} (the sum of N_pyrr valence angles vs. P–N_pyrr bond length) for each studied complex (Fig. S16 and S17), supporting its use as a geometric indicator of M oxidation state in imidophosphorane complexes, as was suggested previously.^20,22,23

The covalency of the An–N_im interactions can be tracked through the An orbital contributions in the two-centre two-electron (2c–2e) An–N_im σ and three-centre two-electron (3c–2e) An–N_im–P π bonds shown in Fig. 4A and Fig. S18 and S19. A similar characteristic bonding pattern has previously been found in other An compounds featuring an imido-type ligand framework.^134–136 The dominant 6d and 5f An orbital contributions are analysed for the σ- and π-bonding interactions in both α- and β-spin manifolds (Tables S14 and S15), as some differences are noted between them across the considered oxidation states. For clarity, the α-spin manifold will be discussed first. The 6d orbital contributions slightly diminish or remain relatively the same across the An of the same oxidation state. However, they steadily become larger as the An oxidation state increases, i.e., from ∼0.11–0.12 |e| in 1-An³⁺ to ∼0.19 |e| in 1-An⁵⁺. In contrast, the 5f contributions display a divergent trend in the lower-oxidation state (+3) vs. higher-oxidation states (+4, +5). Specifically, in the An³⁺ complexes, the contributions slightly decrease as the An series is traversed, whereas the An⁴⁺ and An⁵⁺ 5f contributions increase progressively from U to Pu. In fact, their magnitude becomes comparable or even surpasses the 6d contributions in 1-An⁵⁺, e.g., the 6d/5f values change from 0.19/0.18 |e| in 1-U⁵⁺ to 0.19/0.29 |e| in 1-Pu⁵⁺. Separate analyses of the σ (6d_σ/5f_σ) and π (6d_π/5f_π) bonding interactions in 1-An⁵⁺ further show that the growth in the overall An orbital population is driven predominantly by the increasing An 5f_π contributions in the α-spin manifold (Fig. 4B and Fig. S20).

Fig. 4 (A) An–N_im bonding interactions recovered by AdNDP and (B) their average total (6d_σ + 5f_σ + 6d_π + 5f_π) An orbital population in the α- and β-spin manifolds. Only populations greater than 0.03 |e| are labelled. See Tables S14 and S15 for exact %An contribution to the σ and π bonds and Tables S16 and S17 for the detailed An orbital population.

The trend observed for the α-spin An orbital populations sharply contrasts with the β-spin counterparts, where both 6d and 5f contributions slightly decrease within the An series of the same oxidation state, e.g., An 6d/5f: 0.18/0.16 |e| 1-U⁵⁺vs. 0.15/0.13 |e| 1-Pu⁵⁺. The 6d/5f contributions still gradually increase from An³⁺ to An⁵⁺, as is the case for the α-spin manifold, although to a lesser extent. However, in contrast to the α-spin manifold, the β-spin 5f contributions do not exceed the 6d counterparts at higher-oxidation state An, and the An 5f_π contributions do not grow as the An series is traversed within a given oxidation state. It is also worthy to note that the α-spin An 5f_σ+π orbital contributions start to outweigh the β-counterparts in the higher-oxidation state An⁴⁺ and An⁵⁺ species, especially for the later An. Also, the overall (α + β) 6d/5f An contributions echo the trends elucidated across all oxidation states for the α-spin manifold alone (Fig. S21 and Table S16). Hence, the overall (α + β) AdNDP data suggest that the increased An–N_im covalent interactions in the An⁴⁺ and An⁵⁺ species are driven by the greater α-spin An 5f_π orbital contributions, which increase with the growing prominence of orbital energy matching.

The greatest An 5f–N_im 2p energy matching found in 1-An⁵⁺ across all considered series correlates with the most covalent An–N_im interactions. As the orbital energy gap between M and L orbitals shrinks upon oxidation of 1-An⁴⁺ to form 1-An⁵⁺, the overall (α + β) An 5f_σ+π orbital population increases by 0.18–0.23 |e| while the An 6d_σ+π increases by only 0.05–0.08 |e|. The increase of the An 5f_π contributions in the 3c–2e An–N_im–P π bonds (0.13–0.21 |e|) is more pronounced compared to the 5f_σ contributions in the 2c–2e An–N_im σ bonds (0.03–0.05 |e|). In contrast, while the 6d orbitals are more diffuse than 5f, the An 6d_π and 6d_σ contributions increase by only 0.03–0.04 |e| (Fig. S21). The greater increase of the α-spin An 5f_π contributions can be demonstrated by plotting the An–N_im–P π bonds at the same isovalue across various systems. For instance, Fig. S22 shows that the An centres in 1-An⁴⁺ and 1-An⁵⁺ exhibit larger α-spin An 5f_π population when the An oxidation state increases from +4 to +5 as well as when the An series is traversed. This trend aligns with progressively improved An 5f–N_im 2p energy matching for the later An in these high-valent species. A similar picture was reported previously for the heavier An systems: AmCl₃ and FmCl₃ compounds, wherein the π α-spin orbital of the former and the π β-spin orbital of the latter also showed considerable signs of energy matching driven covalency.⁶⁰

In accordance with the MO/NAO and PDOS energy plots, it is found that the protonated species 2-An⁴⁺ exhibit greater An–N orbital energy matching than the 1-An⁴⁺ species, but smaller than 1-An⁵⁺. From the geometric standpoint, with the elongation of one bond, the remaining three bonds slightly shrink in 2-An⁴⁺, becoming intermediate to their lengths in the 1-An⁴⁺ and 1-An⁵⁺ counterparts (e.g., 2.127 Å 2-U⁴⁺vs. 2.178 Å 1-U⁴⁺ and 2.094 Å 1-U⁵⁺). AdNDP analysis suggests that the covalency of the An–N_im interactions in 2-An⁴⁺ (across four An–N_im bonds) is closer to the 1-An⁴⁺ species than 1-An⁵⁺, e.g., An 6d + 5f: 1.85 |e| in 2-U⁴⁺vs. 1.83 |e| in 1-U⁴⁺ and 2.85 |e| in 1-U⁵⁺ (Tables S16 and S17). This is explained by the disruption of one of the π bonds due to the formation of the bond in 2-An⁴⁺ (Fig. S19). The An–N’_im bond of the protonated ligand is much weaker, with a lower contribution from the An centre in the bonding interactions compared to the other An–N_im interactions, e.g., 10.8%/9.6% vs. 6.4%/3.4% in the σ/π components of the α-orbitals of 2-U⁴⁺ (Table S15). A similar effect was previously noted in the computational studies of the terminal U^IV imido complex [U(Tren^TIPS)(NH)], where analysis of the protonated precursor [U(Tren^TIPS)(NH₂)] revealed a marked reduction in the U contribution to the π components of the relevant NBOs, i.e., 17.2% vs. 10.4%.¹³⁴

To corroborate the DFT-predicted increase in the An 5f_π orbital contributions from U to Pu in the An⁴⁺ and An⁵⁺ series, multireference calculations were carried out. CAS(n,7)SCF calculations on the 1-An³⁺, 1-An⁴⁺, and 1-An⁵⁺ complexes reveal pronounced multireference character (except for 1-U⁵⁺) as the leading electronic configuration contributes less than 85% to the total wavefunction (Fig. S23–S31 and Tables S18–S34). More extensive RASSCF calculations including the ligand-dominant An–N_im–P π bonding orbitals and their corresponding antibonding orbitals were performed for the 1-An⁴⁺ and 1-An⁵⁺ series, where the An 5f_π contributions are most pronounced (see the SI for details). Analysis of the RASSCF natural orbitals for these species shows ON values of 1.97–1.98 |e| for the bonding orbitals and 0.02–0.03 |e| for the corresponding antibonding orbitals (Fig. S32–S37 and Tables S35–S40). This suggests that the multireference character originates primarily from distributing the unpaired electrons among the largely nonbonding An 5f orbitals rather than from the N_im-based π system. As these nonbonding 5f orbitals drive the multireference behaviour, the RASSCF treatment is not expected to qualitatively alter the bonding trends obtained from the single-reference DFT-based AdNDP analysis. Consistent with the DFT results, RASSCF combined (6d_π + 5f_π) An contributions increase from 1-An⁴⁺ to 1-An⁵⁺ by ∼0.14 |e| on average, compared with 0.18–0.22 |e| at DFT. Although RASSCF calculations show slightly larger or comparable absolute 6d_π contributions relative to 5f_π in both series (Fig. S38), oxidation from 1-An⁴⁺ to 1-An⁵⁺ also leads to a larger increase in 5f_π (0.09–0.15 |e|) than in 6d_π (0.05–0.08 |e|).

QTAIM topological analysis of electron density

To complement the orbital-based bonding analysis, a topological analysis of the electron density was performed using QTAIM. Within this framework, orbital overlap can be evaluated through the electron density, ρ(r), at the An–N_im bond critical point (BCP). Additionally, the delocalization index, δ(An,N_im), serves as a quantitative measure of bond strength, reflecting the combined effects of orbital overlap and orbital energy matching between the interacting atoms (Fig. 5A and B and Table S41). Across all studied complexes, ρ(r) at the An–N_im BCP decreases from U to Pu in each oxidation state. In contrast, ρ(r) increases upon oxidation, from 1-An³⁺ (0.083–0.089 |e| Bohr⁻³) to 1-An⁴⁺ (0.112–0.115 |e| Bohr⁻³) and 1-An⁵⁺ (0.131–0.136 |e| Bohr⁻³). These trends indicate that the An–N_im orbital overlap is enhanced upon oxidation as the two atoms get closer, but diminishes along the An series from U to Pu. However, the δ(An,N_im) values reveal two distinct trends: one for the An³⁺ species, and another for An⁴⁺ and An⁵⁺. For 1-An³⁺ and 2-An³⁺, δ(An,N_im) decreases across the series in parallel with the ρ(r) trends (e.g., from 0.70 in 1-U³⁺ to 0.65 in 1-Pu³⁺). In contrast, for the higher-valent species the trend reverses, with δ(An,N_im) increasing from 0.90/1.14 in 1-U⁴⁺/1-U⁵⁺ to 0.92/1.19 in 1-Pu⁴⁺/1-Pu⁵⁺. While the absolute magnitude of these changes (i.e., 0.02 for 1-An⁴⁺ and 0.05 for 1-An⁵⁺) is small, it is the relative changes and systematic trends across the series that are chemically significant. Comparable relative variations have been demonstrated to be meaningful in prior QTAIM-based studies of actinide bonding.^26,137–140 The divergent behaviour of ρ(r) and δ(An,N_im) suggests that the An 5f–N_im 2p energy matching becomes increasingly important for the higher-valent and later An complexes, compensating for the reduced orbital overlap along the series. These findings are consistent with high energy resolution fluorescence detected X-ray studies combined with multiplet theory of actinyl An–O_yl bonds,⁴⁵ which revealed decreasing overlap-driven but increasing energy degeneracy-driven covalency from U to Pu without a concomitant build-up of bond-path electron density. A similar effect was reported for Ce(IV) halides, where, despite the smaller orbital overlap, covalent mixing between Ce 4f and X (X = F, Cl) 2p orbitals was found to be ∼25% greater in CeCl₆²⁻ than in CeF₆²⁻ due to the substantially improved orbital energy matching in the former.⁸¹ Overall, the δ(An,N_im) trends obtained from QTAIM across all considered species mirror the evolution of An–N_im covalency obtained from AdNDP, decreasing from U to Pu in 1-An³⁺ and 2-An³⁺, and slightly increasing in 1-An⁴⁺, 2-An⁴⁺, and 1-An⁵⁺.

Fig. 5 QTAIM and IQA metrics for the average An–N_im interaction. (A) Electron density ρ(r) at the An–N_im BCP. (B) Delocalization index, δ(An,N_im). (C) Exchange-correlation term V_xc of the An–N_im interaction energy. (D) Classical electrostatic term V_cl of the An–N_im interaction energy.

Spin decomposition of the delocalization indices into α- and β-components shows that δ_β(An,N_im) decreases from U to Pu in each oxidation state for all species (Fig. S39). In contrast, δ_α(An,N_im) mirrors the behaviour of the total δ(An,N_im), decreasing across the An³⁺ series but progressively increasing in An⁴⁺ and An⁵⁺. Notably, the increase in δ_α(An,N_im) for the tetra- and pentavalent species parallels the increasingly larger α-spin An 5f_π contributions along the An series, identified by AdNDP as the dominant driver of the enhanced An orbital population. Thus, the QTAIM data support the AdNDP results, indicating that the reversal in An–N_im covalency trends in An⁴⁺ and An⁵⁺ compared to An³⁺ is driven by the α-spin manifold. Taken together, the AdNDP and QTAIM analyses suggest that, in the 1-An³⁺ and 2-An³⁺ complexes, An–N_im covalency is governed predominantly by orbital overlap with U > Np > Pu. In contrast, in 1-An⁴⁺, 2-An⁴⁺, and 1-An⁵⁺, although orbital overlap remains the primary factor, improved α-spin An 5f–N_im 2p energy matching leads to larger An 5f_π contributions for later An, thereby inverting the covalency trend to U < Np < Pu. Although this analysis cannot quantitatively disentangle the relative contributions of the enhanced An 5f_π population to orbital overlap vs. orbital energy matching, it nevertheless reveals a shift in their balance that becomes particularly important for the higher-valent species and later An.

IQA energy decomposition

The delocalization index δ(An,N_im) represents the number of electron pairs shared between the An and N_im basins and is often regarded as a real-space analogue of bond order, but it does not itself provide an energetic description of the An–N_im interaction. However, δ is approximately proportional to the exchange–correlation energy, V_xc, and in a zeroth-order approximation the two quantities are related by V_xc ≈ –δ/2R, where R is the internuclear distance.^141–143 Consistent with this relationship, the overall V_xc(An–N_im) trends (Fig. 5C) qualitatively track those of δ(An,N_im). In 1-An³⁺, V_xc becomes less stabilizing from U (–91.5 kcal mol⁻¹) to Pu (–82.1 kcal mol⁻¹). In 1-An⁴⁺ it is nearly invariant (–117.1 to –117.4 kcal mol⁻¹), whereas in 2-An⁴⁺ it becomes progressively more stabilizing from U (–135.3 kcal mol⁻¹) to Pu (–138.8 kcal mol⁻¹). In 1-An⁵⁺, V_xc is most stabilizing at Np (–152.7 kcal mol⁻¹) and slightly less so at U (–150.2 kcal mol⁻¹) and Pu (–150.7 kcal mol⁻¹). The relative gain in exchange–correlation stabilization upon oxidation tends to be larger for higher-valent or protonated species. For example, ΔV_xc(An–N_im) is 32.3 kcal mol⁻¹ for 1-Np³⁺ → 1-Np⁴⁺ but increases to 35.6 and 38.4 kcal mol⁻¹ for 1-Np⁴⁺ → 1-Np⁵⁺ and 2-Np³⁺ → 2-Np⁴⁺, respectively. More broadly, this gain in ΔV_xc(An–N_im) upon oxidation generally increases along the An series: for 1-An³⁺ → 1-An⁴⁺, ΔV_xc(An–N_im) grows from 25.9 kcal mol⁻¹ (U) to 32.3 kcal mol⁻¹ (Np) and 35.0 kcal mol⁻¹ (Pu).

Within the An³⁺ series, the 9.4–9.7 kcal mol⁻¹ destabilizing change in V_xc(An–N_im) from U to Pu is mainly attributed to decreasing orbital overlap, as An 5f–N_im 2p energy matching is not yet developed. Consistent with this, the An 5f_σ, 6d_σ, 6d_π, and 5f_π populations either slightly decrease or remain constant across the An series in both 1-An³⁺ and 2-An³⁺. In contrast, variations in V_xc(An–N_im) driven by changes in orbital overlap upon oxidation are much larger. This can be assessed cleanly only for the U series, because Np- and Pu-based complexes already display additional stabilization from orbital energy matching in the An⁴⁺ and An⁵⁺ species, which obscures a clear separation of the two contributions. For example, the stabilizing gain in V_xc(An–N_im) is 25.9 kcal mol⁻¹ for 1-U³⁺ → 1-U⁴⁺, 30.2 kcal mol⁻¹ for 2-U³⁺ → 2-U⁴⁺, and 32.8 kcal mol⁻¹ for 1-U⁴⁺ → 1-U⁵⁺.

The 2-An⁴⁺ series provides a particularly clear case for estimating the effect of orbital energy matching on V_xc(An–N_im). 2-U⁴⁺ and 2-Np⁴⁺ have identical ρ(r) values at the An–N_im BCP (0.128 |e| Bohr⁻³), indicating similar overlap, yet the slightly larger δ(An,N_im) in 2-Np⁴⁺ (1.08 vs. 1.05) yields 2.0 kcal mol⁻¹ of additional exchange–correlation stabilization relative to 2-U⁴⁺. This extra stabilization is attributed to improved α-spin Np 5f–N_im 2p energy matching in 2-Np⁴⁺. Thus, the overlap-driven gain in V_xc(An–N_im) upon the 2-U³⁺ → 2-U⁴⁺ oxidation is much larger than the stabilization arising from orbital energy matching in going from 2-U⁴⁺ to 2-Np⁴⁺. It aligns with prior works showing that the orbital energy matching provides only modest stabilization relative to overlap-driven covalency and does not produce a substantial build-up of electron density in the internuclear region.^45,53,55,73 In 2-Pu⁴⁺, where the orbital energy matching is even more favourable, V_xc is further stabilized by 1.5 and 3.5 kcal mol⁻¹ relative to 2-Np⁴⁺ and 2-U⁴⁺, respectively, despite a slightly lower ρ(r) at the Pu–N_im BCP (0.127 |e| Bohr⁻³). Across this series, 5f_σ, 6d_σ, and 6d_π populations remain essentially constant or slightly decrease from U to Pu, whereas only the An 5f_π contributions increase, indicating that this “excess” in 5f_π in the later An is primarily responsible for the additional stabilization. This observation is consistent with the study of hydroxypyridinone complexes of Am–Cf, where the experimentally determined greater stability of An⁴⁺ over An³⁺ was reproduced theoretically and attributed to enhanced orbital mixing arising from improved An 5f–ligand π orbital energy matching.³⁶

A similar picture emerges for the 1-An⁵⁺ series. From 1-U⁵⁺ to 1-Np⁵⁺, ρ(r) at the An–N_im BCP decreases slightly (0.136 → 0.135 |e| Bohr⁻³), yet V_xc is stabilized by 2.5 kcal mol⁻¹, again consistent with enhanced Np 5f–N_im 2p energy matching relative to U. In 1-Pu⁵⁺, V_xc becomes 2.0 kcal mol⁻¹ less stabilizing than in 1-Np⁵⁺, even though the 5f_π contribution continues to increase. This reversal reflects a more pronounced decrease in ρ(r) at the An–N_im BCP (0.004 |e| Bohr⁻³ from 1-Np⁵⁺ to 1-Pu⁵⁺vs. 0.001 |e| Bohr⁻³ from 1-U⁵⁺ to 1-Np⁵⁺), consistent with a slight An–N_im bond elongation from Np to Pu (by 0.007 Å). In going from 1-Np⁵⁺ to 1-Pu⁵⁺, this change is accompanied by reduced populations in 5f_σ (–0.04 |e|), 6d_σ (–0.02 |e|), and 6d_π (–0.01 |e|), and a compensating increase in 5f_π (+0.06 |e|). Taken together, these observations suggest that, although the effects of orbital overlap and energy matching cannot be rigorously disentangled in this case, the loss of overlap in 1-Pu⁵⁺ exerts a slightly stronger destabilizing effect on V_xc than the stabilizing contribution from improved energy matching relative to 1-Np⁵⁺. Conversely, comparison of 1-U⁵⁺ and 1-Pu⁵⁺ shows that the appreciably better orbital energy matching in 1-Pu⁵⁺ slightly outweighs the reduction in orbital overlap, thereby rendering V_xc marginally more negative than in 1-U⁵⁺.

In contrast to the increasing An–N_im covalency in the higher-valent species inferred from AdNDP/QTAIM and its impact on the exchange–correlation energy, the classical electrostatic term V_cl of the An–N_im interaction becomes progressively less stabilizing from U to Pu in each oxidation state (Fig. 5D). On an absolute scale, V_cl is substantially more negative than V_xc for a given oxidation state, indicating that the overall An–N_im interaction is dominated by electrostatics. For example, in 1-An⁵⁺, V_cl ranges from –714.3 to –607.7 kcal mol⁻¹, compared with only –152.7 to –150.6 kcal mol⁻¹ for V_xc. Consequently, the net An–N_im interaction energy becomes less stabilizing along the An series in all oxidation states, despite the nuanced evolution of covalency.

pK_a and BDFE

As demonstrated above, the electronic charge accumulation in the bonding region, described by the electron density ρ(r) at the BCP, must be considered alongside the energy separation between the M and L orbitals, which can modulate the extent of M–L orbital mixing and impact the trends in covalency. The covalent mixing parameter, given as λ = H_ML/ΔE_ML (where H_ML is the Hamiltonian matrix element describing the interaction between the M and L orbitals, which in the extended Hückel theory, is approximated as proportional to the orbital overlap integral, and ΔE_ML is the energy gap between these orbitals), may increase as ΔE_ML decreases with improved orbital energy matching.^{6,53,54,58,74}

Based on the simultaneous decrease in ρ(r) and increase in δ(An,N_im) along the An series for 1-An⁴⁺, 2-An⁴⁺, and 1-An⁵⁺, the enhanced M–L covalency associated with increasingly larger An 5f_π contributions due to the better An 5f–N_im 2p energy matching appears to redistribute electron density toward the atomic centres rather than building up between the nuclei.^45,53,73 This behaviour parallels that reported for An(η-C₅H₅)₃ and An(η-C₅H₅)₄, where stronger 5f mixing across the series was not accompanied by increased charge at the bond midpoint.^67,68 Consistently, QTAIM reveals progressively increasing spin density at the M centre as the An series is traversed in 1-An⁴⁺, 2-An⁴⁺, and 1-An⁵⁺, counterbalanced by the spin density delocalized over the four N_im atoms (Tables S42 and S43). Within the 1-An⁵⁺ series, 1-Np⁵⁺ and 1-Pu⁵⁺ stand out, as their N_im spin densities (–0.27 and –0.65) are substantially larger in magnitude than that of 1-U⁵⁺ (–0.09) (Fig. S40). Similar conclusions are drawn from the NPA data for all the An⁴⁺ and An⁵⁺ complexes, while the An³⁺ counterparts show only minor changes in spin distribution (Tables S44 and S45).

The increase in the N_im spin density along the 1-An⁴⁺ and 1-An⁵⁺ series is also reflected in the increase in the N_imρ(r) at the nuclear critical points (NCPs) (Table S46). In 1-An⁴⁺, the average ρ(r) values at the N_im NCPs increase from 194.522 |e| Bohr⁻³ in 1-U⁴⁺ to 194.531 |e| Bohr⁻³ in 1-Np⁴⁺, and further to 194.551 |e| Bohr⁻³ in 1-Pu⁴⁺ (Fig. 6A). The calculated pK_a values for 2-An⁴⁺ describing a heterolytic cleavage of the bond of the protonated ligand grow in tandem with the increasing electron density at the N_im NCPs in 1-An⁴⁺, from 23.7 for 1-U⁴⁺/2-U⁴⁺ to 27.4 for 1-Pu⁴⁺/2-Pu⁴⁺ (Table S49), evincing a linear correlation with R² = 0.9986 (Fig. 6B). However, this correlation does not hold across multiple oxidation states. For a broader analysis of the trends in pK_a across multiple oxidation states, molecular electrostatic potentials (MEPs) and the sum of 2p NAOs on the N_im atoms (Tables S47 and S48) were used, which were previously employed as descriptors for molecular acidity and basicity.^144–147 In general, more negative MEP values and more positive energy valence NAOs are associated with a higher proton affinity.¹⁴⁸ It is important to note that pK_a was only calculated for the species with experimentally determined E_1/2 values; therefore, the data in Fig. 6C and D only depict the 1-An⁴⁺ species, 1-Pu³⁺, and 1-U⁵⁺. The MEP on the N_im atoms shows an inverse linear correlation (R² = 0.9723) with the calculated pK_a values, becoming more negative in going from 1-U⁵⁺ (–18.3969 E_h/|e|) to 1-Pu³⁺ (–18.6125 E_h/|e|) (Fig. 6C). The sum of the energy of the N_im 2p NAOs computed for these complexes also linearly correlate (R² = 0.9792) with the calculated pK_a (Fig. 6D). Overall, the correlation of ρ(r) at the N_im NCPs, MEP on the N_im atoms, and the N_im 2p NAO values with pK_a suggest their use for the assessment of acidity/basicity.

Fig. 6 (A) ρ(r) at the N_im NCPs as a function of the An identity for the 1-An⁴⁺ and 1-An⁵⁺ species and (B) its correlation with the pK_a values calculated for the protonation of 1-An⁴⁺ complexes. The correlation of the (C) MEP and (D) the sum of energy of the N_im 2p NAOs with pK_a values computed for the protonation of 1-An complexes with experimentally measured E_1/2. See Tables S46–S49 for the detailed information on the ρ(r), MEP, N_im 2p NAO, and pK_a values for the considered complexes.

The computed SOC-corrected BDFE describing the homolytic cleavage of the bond in 2-An⁴⁺ increases across the An series, e.g., from 62.3 kcal mol⁻¹ in 2-U⁴⁺ to 96.1 kcal mol⁻¹ in 2-Pu⁴⁺ (Table S49). In accordance, the smallest pK_a calculated from the Bordwell equation for 2-U⁴⁺ indicates that it has the most acidic proton in the bond. This enhanced acidity correlates with the smallest accumulation of the electron density on N_im in 1-U⁴⁺ (194.522 |e| Bohr⁻³) compared to 1-Np⁴⁺ (194.531 |e| Bohr⁻³) and 1-Pu⁴⁺ (194.551 |e| Bohr⁻³). Likewise, in 1-An⁵⁺, ρ(r) at the N_im NCPs also increases across An, but more steeply: from 1-U⁵⁺ (194.547 |e| Bohr⁻³) to 1-Np⁵⁺ (194.574 |e| Bohr⁻³) and 1-Pu⁵⁺ (194.623 |e| Bohr⁻³). Notably, the value for 1-U⁵⁺ is even smaller than that in the lower-valent 1-Pu⁴⁺ (194.551 |e| Bohr⁻³), which may be related to the divergent PCET reactivity in these complexes: only 1-Np⁵⁺ and 1-Pu⁵⁺ were experimentally shown to undergo PCET,^20,21 whereas 1-U⁵⁺ remained inactive under the same conditions. However, since this PCET reaction is a M–L-based process involving two centres (reduction of An and protonation of N_im) and is highly dependent on the ligand architecture, this correlation requires additional data points to be verified. It would be necessary to consider the net thermodynamics of the reaction including the fate of the THF radical to completely understand the reaction pathway. Combined experimental and computational studies focusing on the covalency and reactivity relations for other high-valent An systems employing modified imidophosphorane ligands are underway.

Redox properties

While trends in the N_im spin and electron densities can be instrumental for predicting thermodynamics of the hetero- and homolytic bond cleavage in these imidophosphorane ligands bound to An⁴⁺, it is the M centre that loses/gains electrons during the oxidation/reduction in these PCET reactions.^{10–12,20,21,23} However, as the M and L orbitals become closer in energy, the likelihood for a ligand-based oxidation increases. As was shown previously for the similar [Ln³⁺(NP*)₄]⁻ (Ln = Pr, Nd, Dy) complexes,¹² oxidation of the Ln centre becomes unattainable for the Nd and Dy counterparts since their M-dominant MOs reside much lower in energy than the L-dominant orbitals compared to Pr, which can be oxidized to the 4+ state. In the present work, we also find that as the N_im 2p and An 5f orbitals get closer in energy in the studied An⁴⁺ and An⁵⁺ complexes, the relative thermodynamic energy difference between the structures resulting from An- and N_im-based oxidations becomes progressively smaller, e.g., from 75.4 kcal mol⁻¹ for 1-U⁴⁺ to 43.7 kcal mol⁻¹ in 1-U⁵⁺, and from 43.2 kcal mol⁻¹ for 1-Pu⁴⁺ to 5.7 kcal mol⁻¹ in 1-Pu⁵⁺ (Fig. S41). However, in all the considered cases, the calculated redox potentials for the thermodynamically favourable M-based oxidations align much more closely with experimental values and are indicative of An-based oxidations (Table S50).

As the An 5f α-orbitals are stabilized across the An series, the calculated redox potentials exhibit a cathodic shift, e.g., from –1.72 V in 1-U^5+/4+ to –0.16 V in 1-Pu^5+/4+. Protonation induces a similar effect by substantially stabilizing their orbitals, leading to cathodic shifts of 0.84–1.00 V relative to the corresponding non-protonated species, e.g., the potential of the U^5+/4+ couple shifts from –1.72 V in 1-U^5+/4+ to –0.72 V in 2-U^5+/4+. This result opens an avenue for achieving lower M oxidation states via chemical reduction of the protonated species. The substantially stabilized LUMO in 2-U⁴⁺ (–0.82 eV) compared to the one in the non-protonated counterpart, 1-U⁴⁺ (0.02 eV), suggests a possible route to access the 2-U³⁺ complex. The calculated 2-U^4+/3+ potential of –2.96 V is at the border of the THF electrochemical window but is more positive than for the 1-Np^4+/3+ couple (–3.23 V), which was only able to be observed chemically and not electrochemically. Encouragingly, the calculated 2-Np^4+/3+ potential is even more positive, at –2.32 V, suggesting that this couple may be experimentally accessible, as was observed visually but not chemically isolable (vide supra). Conversely, this cathodic shift afforded by the ligand protonation hinders the ability for M oxidation, which precludes the observation of the 2-Pu^5+/4+ couple experimentally.

The experimentally available redox potentials subsequently validate what is predicted by the DFT calculations for several complexes (Fig. 7 and Fig. S72). Cyclic voltammetry (CV) studies of 2-U⁴⁺ and 2-Np⁴⁺ were performed in 0.05 M [N(ⁿBu)₄][BPh₄] in THF and referenced to Fc^+/0 (see the SI for full details). The CV of 2-Np⁴⁺ shows a quasi-reversible wave with E_1/2 = −2.60 V, assigned to the 2-Np^4+/3+ couple (–2.32 V theor.), an irreversible wave with E_1/2 = −0.16 V, assigned to 2-Np^5+/4+ couple (0.10 V theor.), and an additional small third feature present with E_1/2 = −0.70 V, identical to the 1-Np^5+/4+ couple (−0.75 V theor.). A CV of 2-U⁴⁺ also boasts three features, two mirroring the waves of the 5+/4+ and 6+/5+ couples seen in 1-U⁴⁺, and a third new feature with E_1/2 = −0.92 V assigned to 2-U^5+/4+ (−0.72 V theor.) via an electrochemical titration. During this titration, both features assigned to 1-U⁴⁺ gradually disappear with higher acid concentrations which is followed by a notable increase in reversibility of the 2-U^5+/4+ couple, confirming these assignments. Details on the electrochemical titration of 2-U⁴⁺ with sequential equivalents of [H(OEt₂)₂][B(ArF₅)₄] are shown in Fig. S73.

Fig. 7 Cyclic voltammograms at 200 mV/s of 1-U⁴⁺ (top), 2-U⁴⁺, 1-Np⁴⁺, 2-Np⁴⁺, 1-Pu⁴⁺, and 2-Pu⁴⁺ (bottom) (∼3 mM; 0.05 M [N(ⁿBu)₄][BPh₄] in THF). All CVs in this work were recorded under a N₂ atmosphere with WE: glassy carbon; RE: polished Ag wire pseudoreference, fritted; CE: Pt wire. CVs are plotted in IUPAC convention and individually referenced to an internal Fc^+/0 couple, with arrows indicating direction and starting point of scan. Applied voltage was corrected for Ohmic drop using positive feedback R_u. Additional details can be found in the SI.

Conclusions

The highly versatile NPC ligand provides a unique platform to interrogate M–L bond covalency over multiple oxidation states for the mid-actinides (U–Pu). Our results demonstrate that the relative energies of the An 5f and N_im 2p orbitals control not only the magnitude, but also the mechanism of An–N_im covalency and thereby impact molecular properties such as pK_a, BDFEs, and redox potentials. While An–N_im covalency decreases from U to Pu in the +3 oxidation state, it increases across the An series in the +4 and +5 oxidation states, where improved orbital energy matching plays an increasingly important role. In the higher-valent 1-An⁴⁺, 2-An⁴⁺, and 1-An⁵⁺ complexes, MO/PDOS and NAO analyses, together with AdNDP/QTAIM data, reveal that enhanced α-spin An 5f–N_im 2p energy matching drives the growth in An–N_im covalency for later An. This is manifested by increasing An 5f_π contributions and larger spin densities at both the An and N_im centres due to the redistribution of electron density toward the atomic basins rather than the build-up of density in the internuclear region. These changes in the nature of covalency correlate with the divergent PCET behaviour observed for the An⁵⁺ complexes, where enhanced An 5f–N_im 2p mixing and growing density at N_im modulate the thermodynamic driving force for the PCET reactivity. IQA energy decomposition analysis further sheds light on the shift in balance between overlap-driven and energy-matching-driven covalency by evaluating exchange–correlation energy V_xc of the An–N_im bond in the An⁴⁺ and An⁵⁺ species. While changes in orbital overlap primarily control covalency upon oxidation from An³⁺ to An⁴⁺ and from An⁴⁺ to An⁵⁺, progressively improved An 5f–N_im 2p energy matching within the An series imparts an additional, subtler stabilization of the An–N_im interaction in the later, higher-valent An.

Protonation of the 1-An⁴⁺ and 1-An⁵⁺ complexes substantially stabilizes the An 5f α-orbitals in 2-An⁴⁺ and 2-An⁵⁺, leading to pronounced cathodic shifts in the An^4+/3+ and An^5+/4+ redox potentials and thereby altering the accessibility of these couples. The calculated pK_a values of the bond in 2-An⁴⁺ strongly correlate with the electron density at N_im in the corresponding 1-An⁴⁺ complexes, suggesting that this QTAIM descriptor may serve as a predictive proxy for protonation in other tetravalent systems. More broadly, both the MEP at N_im and the summed energies of N_im 2p NAOs correlate with pK_a across multiple oxidation states.

Overall, tuning the energies of An 5f orbitals through ligand design is shown to be vital for controlling An covalency, PCET thermodynamics, and redox accessibility. The strongly electron-donating NPC ligand can support An in the pentavalent state up to Pu. Modifying the NPC framework to systematically adjust electronic and steric properties, while correlating these changes with the An–L orbital overlap and energy matching, offers a promising route to rationally control An oxidation states and reactivity. Future combined theoretical and experimental studies employing tailored NPC-type ligands capable of stabilizing even higher An oxidation states within this framework are underway.

Author contributions

The study was conceived by I. A. P. and H. S. L. The theoretical calculations and analysis of the results were performed by C. M. S., S. D., J. W. S., and I. A. P. The experimental portion of the work was performed by K. S. E., J. E. N., and H. S. L. C. M. S. wrote the first draft and all the authors contributed to the writing, editing, and proofreading of the manuscript.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this article have been included as part of the supplementary information (SI). Supplementary information: computed structures, extended MO diagrams, orbital composition data, extended PDOS plots, AdNDP/QTAIM results, detailed redox potentials, complete experimental and methodological details, spectroscopic data (NMR, UV/vis, IR), selected crystallographic tables, and electrochemical characterization. See DOI: https://doi.org/10.1039/d6cp00281a.

Figshare repository contains the XYZ coordinates of the optimized complexes at https://doi.org/10.6084/m9.figshare.29042807.

CCDC 2445923 contains the supplementary crystallographic data for this paper.¹⁴⁹

Acknowledgements

I. A. P., C. M. S., and S. D. acknowledge the NNSA grant DE-NA0004151 as part of the TRUCoRE consortium funded by the NNSA Stewardship Science Academic Alliance program. Also, I. A. P. acknowledges startup funds and the Meyer Early Career Launch Fellowship at Washington State University (WSU). Computational resources at the ARCC HPC cluster at the University of Akron and at the Kamiak HPC cluster at the Center for Institutional Research Computing at WSU are gratefully acknowledged. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Heavy Element Chemistry program under Award Number DE-SC0019385 (K. S. E., J. E. N., H. S. L.).

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