Actinide inverse trans influence versus cooperative pushing from below and multi-center bonding

Motta, Laura C.; Autschbach, Jochen

doi:10.1038/s41467-023-39626-8

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Published: 18 July 2023

Actinide inverse trans influence versus cooperative pushing from below and multi-center bonding

Nature Communications volume 14, Article number: 4307 (2023) Cite this article

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Abstract

Actinide-ligand bonds with high multiplicities remain poorly understood. Decades ago, an effect known as 6p pushing from below (PFB) was proposed to enhance actinide covalency. A related effect—also poorly understood—is inverse trans influence (ITI). The present computational study of actinide-ligand covalent interactions with high bond multiplicities quantifies the energetic contributions from PFB and identifies a hitherto overlooked fourth bonding interaction for 2nd-row ligands in the studied organometallic systems. The latter are best described by a terminal O/N ligand exhibiting quadruple bonding interactions with the actinide. The 4th interaction may be characterized as a multi-center or charge-shift bond involving the trans ligand. It is shown in this work that the 4th bonding interaction is a manifestation of ITI, assisted by PFB, and provides a long-sought missing piece in the understanding of actinide chemistry.

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Introduction

Chemical bonding is an essential concept in chemistry. However, the covalent participation of metal 5f atomic orbitals (AOs) in actinide (An) bonding remains comparatively poorly understood. Initial suggestions of An f-shell covalency^1,2 were met with contention³, because the 5f shell was thought to be radially too contracted to be able to participate in covalent bonding. Although f-shell covalency was correctly asserted more than 70 years ago and has important fundamental as well as practical aspects, its role in actinide chemistry remains challenging to interpret. Fundamentally, the actinides show unusual or unique bonding patterns that continue to push the frontiers of chemistry. As a practical matter, actinide chemistry must be understood and mastered as part of a safe and sustainable nuclear fuel cycle. For example, actinide nitrides have great potential as nuclear-accident tolerant fuel⁴.

This article is concerned with a poorly understood—yet extremely important—aspect of An chemistry, viz. unusually high An–ligand bond orders (BOs). It was reported in 2012 that carbon in CUO is triply to quadruply bonded to uranium⁵, with one σ and two π bonds similar to those in uranyl^VI ([OUO]²⁺), and additionally a weaker so-called rearward σ bond. The rearward σ bond has not been identified in the isoelectronic uranyl^VI, although it has been suggested that oxo 2s lone pair interactions with uranium may take place^6,7,8. It was also speculated that N is quadruply bonded to U in NUN⁹.

At the heart of actinide-ligand multiple bonding lie two—also poorly understood—aspects of An chemistry, namely 6p pushing from below (PFB)^{7,8,10,11,12,13,14,15} and the inverse trans influence (ITI)^{7,8,16,17,18,19,20,21}. PFB was initially¹⁰ identified for uranyl^VI, to explain among other aspects the finding that the σ_u HOMO, being a bonding linear combination of ligand AOs with 5f AOs, was calculated higher in energy than the σ_g involving the metal 6d/7s shells, and higher than the π_g/u bonding combinations. This was surprising, given that a particularly strong covalent σ interaction involving 5f is expected to stabilize the σ_u MO^10,22,23, and because the σ_g should be higher in energy since it involves the higher-energy 6d metal AOs instead of 5f. The occurrence of a σ_u HOMO was then rationalized by an electrostatic repulsion between the ligand lone pairs and the semi-core U 6p shell, potentially accompanied by 6p-5f hybridization. In addition, 6p-oxygen repulsion renders the oxygen a stronger σ donor that it would otherwise be.¹⁰. Typically, PFB is assumed for charge-dense ligand anions such as (formal) O²⁻ or N³⁻. It was recently suggested that PFB is also present in certain thorium compounds²⁴, prompting the question how general this effect truly is.

PFB has also been associated with ITI in actinide chemistry. ITI plays a vital role in the successful isolation of terminal organometallic actinide nitrido and oxo multiple bonds. Denning and co-workers^7,8 proposed that a U-ligand bond trans to a strong electron-rich bond (e.g., U≡O), is stabilized by parity-allowed 5f-6p mixing, resulting in ITI. The exact role of PFB in the ITI is unknown, however^12,16,18. The original extended-Hückel theory study of PFB by Tatsumi and Hoffmann¹⁰, and subsequent related theoretical work^11,12,13,24 did not quantify PFB, and by extension ITI, in terms of energy across different types of compounds. Furthermore, the delocalized nature of the valence canonical MOs has made it difficult to assess the contributions of actinide 6p AOs in larger systems, such that PFB has been evaluated indirectly, for instance via calculations excluding or including 6p in frozen cores^13,16,18,24.

The aim of the present study is therefore two-fold. First, we investigate the extent and energetic contributions from PFB to the electronic structure and its relevance to ITI for a variety of actinide compounds, using modern quantum chemical bonding analyses. The computational strategy harnesses all-electron relativistic Kohn-Sham Density Functional Theory (KS-DFT) and multiconfigurational wavefunction theory (WFT) to identify key orbitals involved in the metal–ligand bonding. The systems are analyzed by the complementary natural bond orbital (NBO) and natural orbitals for chemical valence (NOCV) frameworks, various BO measures, as well as orbital entanglement. Second, PFB is investigated in conjunction with the bonding contributions of the terminal nitrido and oxo 2s lone pairs in the exceptionally covalent [(R^a)₃N–An–Y]ⁿ⁻ systems^{20,24,25,26,27} [R^a = CH₂CH₂NSiⁱPr₃, Y = N or O, and n = 0, 1, or 2; the compounds have also been labeled as [An(TREN^TIPS)]ⁿ⁻]; see Fig. 1a below] and the systems¹⁷ R–U^VI(R^b)₃–O [R = Me = H₃C or R = Ph-C ≡ C (Ph = C₆H₆), \({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{b}}}}}}}}}={{{{{{{\rm{N}}}}}}}}{({{{{{{{{\rm{SiMe}}}}}}}}}_{3})}_{2}\); see Fig. 1b below]. The terminal nitrido and oxo ligands were previously assigned as triple or double bonded to the actinide (Fig. 1). We find that these bonds also have considerable rearward σ contributions, aided by PFB. It is therefore appropriate to assign quadruple bond character to the terminal nitrido ligands (similar to C≣U in CUO), and even the terminal bonds with O have quadruple bond character, which is exceedingly rare in large organometallic complexes, if not unheard of. The terminal quadruple bond includes a 3-center 4-electron (3c4e) or charge-shift interaction involving the trans ligand, and it is facilitated by the covalent, steric, and likely also the electrostatic aspects of PFB, explaining the ITI phenomenon in actinide multiple bonding.

**Fig. 1: Resonance models for the organoactinide compounds.**

Results

Bond orders

NBO-derived BOs from the B3LYP calculations are collected in Table 1. See Supplementary Table 3 for additional BOs according to Mayer²⁸, Nalewajski-Mrozek (N-M set 3)²⁹, and Gopinathan-Jug (G-J)³⁰. All calculated BOs depend on their underlying partitioning of the density (matrix) and the accompanying definition of the promolecule³¹. Mayer BOs for dative (donation) bonds in systems with metal atoms are often too low to conform to chemical intuition^5,9,25,31,32. N-M and NBO BOs have been shown to reproduce expected bond multiplicities successfully for a variety of transition metals complexes^31,33, resulting in overall similar bonding patterns. The G-J BOs for our samples are smaller than N-M and NBO because they exclude valence-bond-style ionic contributions, which is likely problematic when considering dative (donation) bonds that are polarized toward the ligands. We focus primarily the NBO-based analysis. The additionally available BO decomposition into contributions from individual Natural Localized Molecular Orbitals (NLMOs)³³, aids the interpretation of the NBO-based BOs.

Table 1 Calculated (DFT/B3LYP) An–ligand bond orders and An 6p-hole for ionic and organometallic actinide compounds

Full size table

For the organometallic compounds, the terminal An-N/O BOs are considerably larger than 3, therefore indicating the presence of a fourth bonding interaction. Quadruple bonding between an actinide and a C, N, or O ligand requires the participation of the rearward 2s-rich ligand σ lone pair and is therefore almost unheard of. As mentioned, however, the rearward σ bond has been reported for CUO⁵. The present NBO analysis for CUO is in accordance with this previous assignment. We note in passing that the N-M C–UO BO of 4.4 (Supplementary Table 3) should not be interpreted as a quintuple bond, given that the bond is with carbon. Instead, one should view the BO as indicating a particularly covalent quadruple dative bond.

Inspection of individual natural localized molecular orbital (NLMO) contributions to the BOs for the organometallic compounds reveals participation of the terminal N and O rearward 2s_p lone-pair NLMO in the bonds. The notation indicates a 2s-rich hybrid with secondary 2p contributions. The 2s_p BO contributions are largest for the terminal nitrides (average 0.17 ± 0.02), which are close in magnitude to the corresponding value (0.21) for CUO. The covalent contributions of the oxo 2s_p (average 0.10 ± 0.06) appear weaker because the An-oxo bonds are less covalent overall, but the relative contributions are comparable to those for the terminal N.

The BO analysis also reveals a small but not unimportant participation of the An 6p_σ AOs, that is, a covalent contribution. The An 6p_σ BO contributions are greatest for the small molecules (average 0.015 ± 0.01, lowest for ThO₂) compared to the organometallics (average 0.006 ± 0.002). The widespread participation of the An 6p_σ in the bonding, including the C–U bond in CUO, indicates that PFB is likely to be a general chemical phenomenon, at least among the early actinides. The Th–Cl bond in \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}\)NTh–Cl has a much smaller BO than most of the other bonds listed in Table 1, as may be expected for a chloride ligand. The absence of a notable rearward σ interaction, indicated by the very small 3s_p BO contribution, goes along with a negligible Th 6p_σ contribution for this bond.

Terminal actinide-ligand triple vs. quadruple bond

Although the BO is calculated to be large for all terminal O/N–An bonds in the studied organometallics, and the involvement of the 2s_p ligand lone pair hybrid is evident, further analysis is needed to decide whether these bonds should be assigned as quadruple.

The NBO analysis is designed to determine a best single Lewis structure that can optimally describe the electron density (matrix) upon accounting of some—presumably relatively minor—delocalization. The latter can be described via the contributions from other resonance structures^34,35. The suitability of a given Lewis structure is quantified by a root-mean-square deviation (RMSD) representing a residual non-Lewis electron number^33,34,35. For the closed-shell \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\) compound, the optimal Lewis structure as determined by the NBO algorithms (structure 1, in Fig. 1c) features a terminal U≡N and an N-polarized σ(N_amine−U) bond (83% nitrogen weight). See Fig. 1e, f for NLMO visuals. The analysis (Table 2) shows 1.8 electrons RMSD.

Table 2 NRT weights of the Lewis structures 1 and 2, and the associated non-Lewis RMSD errors in the NRT compared to using only a single resonance structure 1 or 2

Full size table

It has been shown previously for a variety of transition metal complexes that a multi-resonance description based on Natural Resonance Theory (NRT) can be much more appropriate than a description based on a single NBO Lewis structure^34,35,36. In the present case, NRT analysis revealed, for example, for \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\) an important secondary resonance structure featuring a terminal U≣N bond and no N_amine−U bond (Fig. 1c, Lewis structure 2). The fourth bond is an N-polarized covalent interaction between the nitride 2s_p (89%) and a U 5f6d_7s6p (an f-d hybrid with contributions from 7s and 6p, Fig. 1). For \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\), resonance structures 1 and 2 have the same RMSD individually (1.8 electrons), and according to NRT, the system is best described by strong 45/55 percent resonance. In other words, there is a 3c4e N_amine−U−N_nitride interaction. The superiority of the resonance model is confirmed by the small residual RMSD (0.09e) in the NRT, meaning the delocalization in the system is essentially only in the N_amine−U−N_nitride moiety.

Similarly, the NBO-NRT analysis also shows that the resonance model (Table 2) is the best description for the other organometallics, including the \({{{{{{{\rm{PhCC}}}}}}}}/{{{{{{{{\rm{MeU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{b}}}}}}}}})}_{3}{{{{{{{\rm{O}}}}}}}}\) compounds (Fig. 1d–f); this picture is supported by the terminal An-N/O and trans-An BOs in Table 1. However, for the terminal An-oxo compounds, structure 1 is more dominant given the lower RMSD values relative to structure 2. This is also reflected in the lower BOs for these compounds. In comparison, the uranyl ion is described well by a single Lewis structure ([O ≡ U ≡ O]⁺²), whereas CUO is also resonance stabilized by a 3c4e C–U–O interaction, favoring the C≣O bond. Given that the resonance is of the type X–U ≡ Y: \({{\leftrightarrow}}\) X: U≣Y (without or with additional X–U bonds), the interaction produces a pronounced U–Y quadruple bond character.

Stability and classification of the trans-An-terminal multi-center interaction

The BOs and NBO-NRT analyses strongly support the existence of a fourth bonding interaction in the organometallic complexes. However, the analysis does not reveal information about the strength/stability of the 3c4e trans–U–terminal-ligand hyperbond. BOs do not strictly correlate with interaction energies, in particular when comparing different types of chemical bonds^5,37. To evaluate interaction energies related to the covalency of the 2s_p O and N lone pairs, and the trans ligand, NBO-based second-order perturbation energies (ΔE⁽²⁾) for the donor–acceptor interactions were evaluated and compared to those in [OUO]⁺², CUO, and NUN. To allow direct comparisons of the ΔE⁽²⁾ values³³, the X: U≡Y: Lewis structure (structure 1 in Fig. 1) was used consistently. The dominant non-Lewis delocalization interactions are illustrated in Fig. 1c for \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\).

The ΔE⁽²⁾ stabilization interactions for \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\) are illustrated in Fig. 2a and listed in Table 3 for a subset of compounds. For all tested compounds, the energetically dominant interactions (nos. 4 and 5) correspond to donation from the terminal and trans ligands lone pairs to formally unoccupied actinide AOs, that is, dative bonding. The strongest interactions are found for the An-terminal bond, in particular for the nitrido systems, in which apparently good overlap and a favorable energy match combine, according to the principles of MO theory, to yield substantial energetic stabilization. Stabilization by the rearward donation to An is also strong for CUO and NUN, in which the U is quadruply bonded to the ligand. The NBO ΔE⁽²⁾ data indicate that all of the studied actinide compounds are stabilized by the formation of the trans 3e4c bond from the resonance of structures 1 and 2. Given the symmetry of the actinyl ions, the ΔE⁽²⁾ values are the same for interaction nos. 4 and 5.

**Fig. 2: Energetic contributions to metal–ligand bonding.**

Table 3 NBO second-order perturbation stabilization energies for the dominant non-Lewis interactions, and the orbital interaction energies ΔE_orb for the dominant bonding interactions shown in Fig. 2a

Full size table

The trans–ligand–An-terminal-ligand rearward interaction involves orbitals that are polarized toward the corresponding nitrogen, oxygen, or carbon ligands. There are therefore good reasons to classify the rearward trans–An-terminal interaction as a charge-shift (CS) bond³⁸. Indeed, using modern VB calculations, it was argued previously that 3c4e bonds in stable (not transient or transition state) species can be classified as CS bonds³⁸. CS bonds can also be identified using QTAIM³⁸, by a small negative or positive electron density Laplacian (∇²ρ), densities (ρ) ≥ 0.1, and a negative overall energy density (H), at the bond critical point (BCP). In \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{-}{{{{{{{\rm{N}}}}}}}}\) the N_amine−U BCP has a ρ = 0.1, ∇²ρ = 0.1, and H = −0.03, and for the U–N BCP we find ρ = 0.3, ∇²ρ = 0.04, and H = − 0.4. The data are compatible with a CS bond assignment. It is important to note that the QTAIM analysis reflects all of the interactions present in the quadruple terminal bond, not only 2s_p rearward bonding.

To further explore the covalent contributions to the An-Terminal bonding, and the stability of the fourth 3c4e/CS interaction, we carried out ETS-NOCV analyses for the organometallic compounds, and compared the results to [OUO]²⁺, NUN, and CUO (Table 3). The ETS-NOCV analysis clearly identifies four bonding contributions to the absolute values of ΔE_orb, the stabilization energy that arises from the covalency of the metal–ligand orbital interactions, for most systems: one σ, two π, and the additional (weaker) rearward σ interaction. The corresponding NOCVs are shown in Fig. 2b–c for \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{-}{{{{{{{\rm{N}}}}}}}}\) and \({{{{{{{{\rm{MeU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{b}}}}}}}}})}_{3}{-}{{{{{{{\rm{O}}}}}}}}\). Consistent with the other analyses, there is a σ bond, two π bonds, and the σ 3c4e/CS bond. The fourth NOCV clearly displays the multi-center covalent interactions between the trans and terminal ligand via U. For the organometallics, the energetic bond component is the largest for the σ bond and the smallest for the 3c4e/CS bond, but the latter is not negligible. Namely, for \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NAn}}}}}}}}}^{{{{{{{{\rm{n}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\) compounds, ΔE_orb for the 3c4e/CS bond is larger than that of the known fourth rearward σ bond in CUO, and comparable to the covalent energy of the main \([{({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NTh}}}}}}}}}^{{{{{{{{\rm{IV}}}}}}}}}{-}{{{{{{{\rm{Cl}}}}}}}}]\) σ bond.

We carried out a subset of ETS-NOCV analyses also for the bonds indicated by the dash in CU–O and \({{{{{{{{\rm{OU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{b}}}}}}}}})}_{3}{-}{{{{{{{\rm{Me}}}}}}}}\), i.e., to evaluate covalent contributions to the trans–An interaction in the 3c4e/CS bond. This analysis is not possible for the \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NAn}}}}}}}}}^{{{{{{{{\rm{n}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\) compounds because the contributions of the trans N_amine cannot be isolated from the TREN^TIPS ligand (Table 3). There is a fourth ΔE_orb bonding contribution from the O lone pair in CU–O, although weaker relative to the C lone pair contributions, in agreement with the NBO-NRT resonance analysis. Likewise, the corresponding NOCV for the OU^VI−Me single σ bond illustrates a multi-center interaction composed of C, U, and O contributions.

The combined NBO and ETS-NOCV evidence lead to the conclusion that the trans-ligand–An–terminal-ligand interaction in An compounds with high An–ligand bond multiplicities is ITI, facilitated by the terminal and trans 2s_p AOs. The participation of the actinide 6p semi-core shell is discussed next.

Analysis of the pushing-from-below mechanism

As discussed earlier, the individual NLMO contributions to the BOs (Table 1) indicate covalent participation of the An 6p_σ for most of the studied compounds. The ΔE⁽²⁾ NBO analysis (Table 3 and Fig. 2; interactions 1–3), also demonstrates clear and energetically important non-Lewis donor–acceptor interactions involving the semi-core 6p_σ (with the exception of \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NTh}}}}}}}}}^{{{{{{{{\rm{IV}}}}}}}}}{-}{{{{{{{\rm{Cl}}}}}}}}\), as noted already). Delocalization of a (partially) filled atomic orbital in a localized orbital framework is covalency, and it creates a partial electron hole in the 6p shell. We find a lesser, but non-negligible extent of 6s and 6p_π delocalization. The extent of stabilization from 6p_σ delocalization becomes smaller when going from \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\) to \({[{({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{IV}}}}}}}}}{{{{{{{\rm{N}}}}}}}}]}^{2-}\).

In a complementary yet consistent picture, the charge-flow channels³⁹ in \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{-}{{{{{{{\rm{N}}}}}}}}\) and \({{{{{{{{\rm{MeU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{b}}}}}}}}})}_{3}{-}{{{{{{{\rm{O}}}}}}}}\) corresponding to four bonding NOCVs (Fig. 1) indicate an outflow of 6p and 6s density to the formally unoccupied 5f and 6d AOs. As expected from the original extended-Hückel theory study of PFB¹⁰, the contributions from the 6p_σ (~5%) to σ_u HOMO are greater than those of the 6p_π ( ~1%) to the HOMO-1 and HOMO-2. The largest 6p_σ (~28%) contributions arise in the σ trans–U–terminal NOCV, which is not entirely surprising given the comparatively larger radial overlap between the ligand 2s and the 6p_σ vs. 6p_π actinide AOs^5,7,8. Thus, the fourth covalent interaction is a manifestation of PFB facilitating ITI, and it does not require inversion symmetry to be present. As discussed earlier, our results indicate that in addition to 6p-5f hybridization, the covalent participation of the ligand 2s_p lone pair in the 3c4e bond is an essential component of ITI.

The covalent aspect of PFB may be accompanied by an energetic destabilization of the ligand 2s_p hybrid by electrostatics and by Pauli repulsion with the 6sp shell, facilitating covalent interactions of 2s_p with the valence 5f and 6d An orbitals. The numerical analysis indeed shows that Pauli repulsion is also an important component of the PFB mechanism. The NBO pairwise steric exchange energies (ΔE_X) associated with the Pauli repulsion between U 6p/6s and the terminal and trans ligand valence NLMOs are listed in Table 4 for [OUO]²⁺, CUO, and \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{-}{{{{{{{\rm{N}}}}}}}}\). Although the Pauli repulsion is more pronounced between the U 6p_σ and ligand 2s_p and 2p_s interactions, the participation of the U 6s and 6p_π shells is not altogether negligible. Overall, it is clear that electrostatic repulsion, Pauli repulsion, and 6p covalency go together in the PFB mechanism.

Table 4 Selected axial pairwise steric exchange energies^a ΔE_X from NBO analysis for interactions between U 6p/6s and ligand valence NLMOs for the dominant interactions shown in Fig. 2a

Full size table

Orbital entanglement investigation

The KS-DFT NBO-NRT and ETS-NOCV analyses evidence a terminal An≣L bond via a 3c4e/CS trans–An–terminal ligand interaction facilitated by the terminal O or N 2s_p AO. Absence of most of the dynamic correlation at the CASSCF level means that the use of NBO-NRT and ETS-NOCV bonding analyses based on this level of multiconfigurational WFT is presently not particularly helpful. To corroborate the rearward lone pair bonding participation by WFT, we therefore conducted a multiconfigurational orbital entanglement analysis^{40,41,42,43,44,45,46}.

Orbital entanglement measures facilitate the qualitative interpretation of the electronic structure in terms of quantum correlation of MOs and have been successful in elucidating bond formation processes⁴¹ and molecular complexation⁴⁰. In particular, the mutual orbital information^40,47,48 indicates entanglement of an orbital pair, hence represent a measure to assess on a qualitative level, independent from the DFT calculations, whether the ligand 2s_p and actinide orbitals are meant to interact. We compare here the orbital entanglement diagrams for OUO⁺², CUO, and \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\), shown in Fig. 3.

Fig. 3: Ground state orbital entanglement diagrams (from MPS(24e,24o) calculations with 12 occupied and 12 virtual orbitals) for the [OUO]⁺² ion, CUO, and \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\).

In the diagrams, the thickness of the connecting lines indicates the extent of entanglement. The diagrams reveal sizable mutual information between the MOs corresponding to the σ and π U≡L bonds. It is also evident that the ligand lone pair is the least important (least entangled) for [OUO]⁺². Although for CUO, both the C and O 2s_p lone pairs are entangled with formally vacant 5f/6d U orbitals, indicative of the rearward bonding, the C 2s_p orbital is more strongly entangled with the LUMO, in qualitative agreement with the preferred C≣U≡O resonance structure. The \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\) diagram is qualitatively similar to CUO in this respect, and there is additionally substantial entanglement involving the MOs dominated by 2p_s and 2s_p AO contributions. Specifically, the 2s_p lone pair shares mutual information with a low lying virtual orbital. Closer inspection of the 2s_p orbitals shows pronounced mixing with the U 5f, 6d, 6p, and 6s and trans N_amine 2p_s AOs. It appears that the entanglement gives away the multi-center bonding and PFB interactions identified in the DFT calculations.

For the orbital entanglement measurements, we found it necessary to use relatively large active spaces [e.g., MPS(24e, 24o)]. Previous MPS calculations of neptunium organometallic compounds demonstrated that small CASSCF spaces lead to overlocalization of the 5f shell and an underestimation of the ligand donation (dative bonding)^49,50. Although (12e, 12o)⁵¹ and (8e, 8o)⁵ active spaces have been used previously to investigate the electronic structure of CUO, these active spaces appear to be insufficient to bring about a full picture of the bonding. To describe the C–U and U–O sets of bonds, 4 C–U and 4 U–O (2p_s, 2s_p and two π each) occupied orbitals are needed. Thus, it is necessary to include at least 8 doubly occupied orbitals (16 electrons) in the active space for CUO. We therefore speculate that the use of smaller active spaces (see Supplementary Fig. 1) in previous research may have obscured the 3-center bonding and ITI interactions in CUO to some degree.

Discussion

Taken together, the NLMO BO decomposition, ΔE⁽²⁾ delocalization energies, and ETS-NOCV bonding analyses paint a clear picture of 6p covalency as well as overlap-driven Pauli repulsion as part of the PFB mechanism. Furthermore, state-of-the-art KS-DFT and multiconfigurational WFT-based analyses uncovered a rearward interaction of the 2s_p hybrid ligand lone pair with the metal, previously implicated in the C–U bond in CUO, among a range of actinide compounds. The rearward 2s_p participation facilitates a fourth covalent interaction with terminal nitrido or oxo ligands in several of the studied systems. Actinide 6p_σ (and 6s) contributions can be identified in this fourth, rearward interaction, which means that PFB assists, if not enables, a resonance stabilization resulting in the ITI noted for actinide compounds. It also appears that the present study is the first or perhaps one of few reports so far of oxo ligands featuring quadruple bonding interactions. Although the bonds in the studied actinyl ions do not qualify as quadruple, the oxo rearward 2s_p bonding contributions also lead to additional stabilization. The ligand 2s_p bonding contributions are most prominent for the exceptionally covalent organoactinide compounds that were selected for this study, in which we are also able to identify a 3c4e or CS bond involving the trans ligand. Overall, the data reveal that the PFB mechanism is a vital component of the bond multiplicity in actinide compounds. It is likely much more prevalent than previously anticipated. (With the latter statement, we echo a conclusion from ref. ²⁴). The present computational study therefore shows that there are still many secrets related to the mystery of 5f covalency that can be uncovered, pushing the boundaries of our understanding of chemical bonding.

Methods

Absent experimental structures for gas-phase actinyl ions, CUO, and NUN, the geometries for these systems were optimized in linear symmetry with KS-DFT using the Amsterdam Density Functional (ADF) program version 2022⁵² with the B3LYP functional⁵³ and the zeroth-order regular approximation (ZORA) all-electron scalar relativistic Hamiltonian^54,55. The N–U distance of 1.75 Å for NU²⁺ was taken from ref. ⁵⁶. For the An(TREN^TIPSN/O) and \({{{{{{{\rm{PhCC}}}}}}}}/{{{{{{{{\rm{MeU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}[{{{{{{{\rm{N}}}}}}}}{({{{{{{{{\rm{SiMe}}}}}}}}}_{3})}_{2}]_3\) compounds we used the available experimental crystal structures^{17,20,24,25,26,27}. Given the comparatively large size of the TREN^TIPS ligand, truncated model structures replacing ⁱPr groups by hydrogen were used. Hydrogen positions were optimized as described above. All KS-DFT calculations employed Slater-type orbital (STO) triple-ζ polarized (TZ2P) all electron basis sets⁵⁷. For the subsequent electronic structure analyses, the eXact two-Component (X2C) Hamiltonian was used^58,59,60.

NBO analyses³³ were carried out with version 6 of the code included in the ADF suite. For the calculation of the NBO second-order stabilization energies, reference Lewis structures were specified using the CHOOSE keyword³⁵ to allow direct comparisons between different molecules. Resonant Lewis structures were generated and evaluated using the natural resonance theory (NRT) module in NBO, with high thresholds (20 kcal mol⁻¹) to avoid minor intruding hyperconjugative interactions^35,36. The bonding covalent interactions and the corresponding energy contributions to the total binding energy were evaluated with the extended transition state (ETS) NOCV approach³⁹ as implemented in ADF. For the ETS-NOCV analyses, the molecules were divided into two ionic fragments by cleaving the terminal bond (e.g., UO⁴⁺ and O²⁻ for \({{{{{{{{\rm{UO}}}}}}}}}_{2}^{2+}\), or \({[{{{{{{{\rm{U}}}}}}}}({{{{{{{{\rm{TREN}}}}}}}}}^{{{{{{{{\rm{TIPS}}}}}}}}})]}^{3+}\) and N³⁻ for \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\)), to facilitate comparison among different molecules⁶¹. Pauli repulsion interactions were evaluated for a subset of molecules using the natural steric analysis in NBO⁶². Selected Quantum Theory of Atoms In Molecules (QTAIM) analyses were performed with the Bader module in ADF⁶³.

To ascertain that the results reported herein are only weakly dependent on the chosen DFT functional, additional bonding analyses were conducted for \({{{{{{{{\rm{UO}}}}}}}}}_{2}^{2+}\) and \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\) with the functionals PBE, PBE0, and TPSSh. Likewise, because several optimized sets of bond lengths for CUO have been reported in the literature^{5,51,64,65,66,67}, we carried out additional analyses for CUO with an X2C/DFT (PBE0 functional) geometry⁶⁶, which has slightly shorter C–U and U–O distances of 1.733 and 1.779 Å, respectively, compared to our ZORA/B3LYP bond lengths (1.746, 1.801 Å). Relevant data are provided in Supplementary Tables 1–5, showing that general conclusions can be drawn based on the B3LYP calculations. Given the multi-center nature of the 4th bonding interaction with the rearward 2s_p AO and the general consensus that high An–ligand bond multiplicities are possible in the types of systems studied herein, inclusion of the spin–orbit interaction in the calculations was not deemed to be essential.

In addition to KS-DFT calculations, scalar X2C multiconfigurational wavefunction calculations were performed for OUO²⁺, CUO, and \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\) with the open-source version of the Molcas program (OpenMolcas)^68,69,70 at the complete active space (CAS) self consistent field (SCF)⁷¹ matrix product state (MPS) DMRG level⁷². These calculations were used to generate orbital entanglement diagrams, among other information, with the QCMaquis extension^73,74,75 of Molcas. These calculations employed Gaussian-type all electron atomic natural orbital relativistic semi-core correlation (ANO-RCC) basis sets⁷⁶ in their valence triple-ζ contractions, except for hydrogen atoms where the double-ζ contraction was used instead for computational efficiency. The on-the-fly generated auxiliary-basis RICD functionality was utilized for the electron repulsion integrals⁷⁷. Symmetry was not specifically imposed in the wavefunction calculations. Initial orbitals for the orbital-optimizing DMRG runs were obtained from CASSCF calculations with relatively large active spaces [e.g., (16e, 13o) for CUO, and (12e, 13o) for \({({{{{{{{{\rm{R}}}}}}}}}^{{{{{{{{\rm{a}}}}}}}}})}_{3}{{{{{{{{\rm{NU}}}}}}}}}^{{{{{{{{\rm{VI}}}}}}}}}{{{{{{{\rm{N}}}}}}}}\)]. Series of DMRG calculations were then conducted with active spaces enlarged by subsequent increments of 2 electrons and 2 orbitals, and the orbitals were carefully monitored to ensure that the intended active space was maintained.

Orbital entanglement diagrams were constructed using the AutoCAS software⁴⁴. For the MPS calculations, m = 1024 and 15 sweeps were used. The ground state orbital entanglement diagrams were calculated for (14e, 14o) up to (24e, 24o) active spaces. Finally, to ascertain that the orbital entanglement diagrams are only weakly dependent on the reported CUO bond lengths, the (24e, 24o) and (12e, 12o) orbital entanglement diagrams based on the B3LYP geometry were compared to a (12e, 12o) diagram based on the PBE0 geometry. The diagrams are provided in Supplementary Fig. 1.

Data availability

All data generated and analyzed in this study are included in this article, its supplementary information, and the publicly available source files at https://doi.org/10.5281/zenodo.7853933. The zenodo repository includes all ADF and OpenMolcas output files from which the data presented in this study were extracted. Other data related to this study can be obtained from the corresponding author upon request.

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Acknowledgements

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Heavy Element Chemistry program, under grant DE-SC0001136 to J.A. The authors acknowledge the Center for Computational Research (CCR) at the University at Buffalo for providing computational resources and thank Prof. W. H. E. Schwarz for constructive comments on the manuscript.

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Laura C. Motta
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Department of Chemistry, University at Buffalo, State University of New York, Buffalo, NY, 14260-3000, USA
Laura C. Motta & Jochen Autschbach

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L.C.M. performed all calculations and prepared figures, tables, and supplementary data. J.A. designed the project and secured external funding. Both authors analyzed the data and co-wrote the manuscript.

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Motta, L.C., Autschbach, J. Actinide inverse trans influence versus cooperative pushing from below and multi-center bonding. Nat Commun 14, 4307 (2023). https://doi.org/10.1038/s41467-023-39626-8

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Received: 19 January 2023
Accepted: 22 June 2023
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DOI: https://doi.org/10.1038/s41467-023-39626-8

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