δ and φ back-donation in AnIV metallacycles

In all known examples of metal–ligand (M–L) δ and φ bonds, the metal orbitals are aligned to the ligand orbitals in a “head-to-head” or “side-to-head” fashion. Here, we report two fundamentally new types of M–L δ and φ interactions; “head-to-side” δ and “side-to-side” φ back-bonding, found in complexes of metallacyclopropenes and metallacyclocumulenes of actinides (Pa–Pu) that makes them distinct from their corresponding Group 4 analogues. In addition to the known Th and U complexes, our calculations include complexes of Pa, Np, and Pu. In contrast with conventional An–C bond decreasing, due to the actinide contraction, the An–C distance increases from Pa to Pu. We demonstrate that the direct L–An σ and π donations combined with the An–L δ or φ back-donations are crucial in explaining this non-classical trend of the An–L bond lengths in both series, underscoring the significance of these δ/φ back-donation interactions, and their importance for complexes of Pa and U in particular.

Three 6c-2e p bonds forming p aromatic set over each Cp ring in the propene series S12 SF11 Canonical molecular orbitals for the (η 5 -C5Me5)2U[η 4 -C4(SiMe3)2] complex S13 SF12 AdNDP s bonds found in cumulenes S14 SF13 AdNDP bonding elements found between the C4(SiMe3)2 ligand and the M center in the cumulene series S15 SF14 NPA charges on the M centers in the cumulene series S16 SF15 Composition of the M hybrids in the M-Ca 2c-2e s bonds in the cumulene series S17 SF16 Composition of the Ca hybrids in the M-Ca 2c-2e s bonds in the cumulene series S17 SF17 ON values (in |e|) of the M-Ca s bonds and M-C-C p bonds in the cumulene series S17 SF18 Involvement of the M center in the p bonding with the C4(SiMe3)2 ligand in the cumulene series S19 SF19 Unpaired f-electrons on the M centers in the cumulene series S21 SF20 Involvement of the M center in the φ bonding with the C4(SiMe3)2 ligand in the cumulene series S22 SF21 Calculated UV-Visible-NIR electron absorption spectra for (C5Me5)2An(η 4 -1,2,3,4-PhC4Ph) (   The trend in the NPA charges on the metal center is similar to the propene series, with Th (+1.87) possessing larger charge than U (+1.59) (Supplementary Fig. 15). It shows an approximately 0.7 drop in charges between Th (+1.87) and Pa (+1.17) followed by a consistent increase to Pu (+2.85). Similar to propenes, the NPA charge on the metal center of the Pa complex suggests that its reactivity towards alkynes should be comparable to that of the recently synthesized (η 5 -C5Me5)2U[η 4 -C4(SiMe3)2] as well as Group 4 cumulenes, whereas the reactivity of the corresponding Np and Pu complexes should be similar to the Th complex. As one would expect, the NPA charges of the corresponding Ti and Zr complexes have quite low values (+1.40 and +1.24, respectively), comparable to those of U (+1.59) and Pa (+1.17) complexes. As in the propene series, the cumulene M-C s bonds are primarily formed by hybrid 6d-5f An orbitals with C 2s-2p hybrid orbitals (Supplementary Figs 16, 17), with increasing f-and decreasing d-characters across the series from Th to Pu. Likewise, the ON values of the M-C s bonding does not change significantly in the cumulene series ( Supplementary Fig. 18). The similarity between the cumulenes and propenes extends to their p bonding, though the p interactions are slightly altered in the (η 5 -C5Me5)2An[η 4 -C4(SiMe3)2] (An=Th-Pu) compounds by the presence of additional C atoms. AdNDP found two types of p bonding interactions between the (η 5 -C5Me5)2An and C4(SiMe3)2 fragments associated with one central (Cb-Cb) and two peripheral (Ca-Cb) units ( Supplementary Fig. 19). Since the M-Cb-Cb bonding is due to the L-M donation of the p density of the Cb-Cb fragment to the df-hybrid orbital of the metal center, we call it p interaction. However, the M-Cb-Cb p bonding is indeed formally considered as s interaction with respect to the metal center. The M-Cb-Cb and M-Ca-Cb p bonds are mutually perpendicular, and are responsible for the double bond character within the four C atoms of the cumulene moiety. These 3c-2e bonds can also be viewed as 2c-2e C-C bonds, though with lower ON values (1.62-1.76 |e|). This hints at p electron delocalization over neighboring atoms (Supplementary Fig. 19). Indeed, upon inclusion of the M atom, the ON values increase to 1.86-1.92 |e|. The L-M p donation is found to be the largest for U (0.58 |e|) and the smallest for Th (0.37 |e|) (Fig. 6b). The larger donation to U compared to the Th center is in agreement with the experimental values of the M-C bonds in the U (avg. 2.45 Å) and Th (avg. 2.54 Å) cumulene complexes. However, similar to the propene series, the combined s +p L-M trend does not fully explain the peculiar trend of the M-C distances observed in the An series (Figs. 3a, 6c). Specifically, the Pa-cumulene distances are the shortest in series, while the largest s +p L-M donation is found for the U counterpart.

φ bonding
While the strength of the interaction between the An atom and the cumulene fragment is primarily dominated by the strength of their s and p bonding (Fig. 6c), these two interactions alone do not fully explain the peculiar trend of the M-C distances observed in the An series (Fig. 3a). Indeed, excepting Pa, the combined s +p L-M interactions can account for the M-C bond length decrease from Th to Pa and follow-up constant increase from Pa to Pu, thus defying the anticipated trend of ionic radii in actinides. However, there is more one interaction, which is found to be the strongest in the case of the Pa complex, i.e. M-L φ back-bonding (Fig. 6d) that helps to bring the donation trend in full consistency with the M-C bond trend (Figs. 3a, 6e).
Specifically, one of the unpaired f-electrons of the M atom, which has the lowest ON value ( Supplementary Fig. 20), participates in a φ bond with the four C atoms of the C4(SiMe3)2 fragment. Similar to the Group 4 cumulenes, no φ bonding interaction is observed in the case of the Th complex due to the absence of f-electrons. The 5c-1e φ bond is formed due to the promotion of the electron density from the singly occupied 5fxz 2 orbital of the An atom into the p * orbital of the four C atoms of the C4(SiMe3)2 ligand ( Supplementary Fig. 21). It features three nodal planes: one nodal plane going through the plane of the four C atoms in addition to two perpendicular nodal planes, in between the Ca and Cb atoms on both sides of the ligand. This unprecedented 5c-1e φ bond is the first example of the "side-to-side" M-L φ interaction occurring between the 5f orbital of the metal and the p * orbital of the ligand. On one hand, the φ back-donation is expected destabilize the Ca-Cb bonds due to their antibonding characters with respect to one another as well as to stabilize Cb-Cb exhibiting bonding interactions. The degree of the φ back-donation within the series (Fig. 6d) is in agreement with the Ca-Cb bond lengths, confirming the longest Ca-Cb bonds in the Pa complex featuring the largest φ backbonding. In contrast to the Ca-Cb bond length trend, the Cb-Cb bonds stay nearly the same along the series. This in agreement with the fact that the back-donation to the central Cb atoms is significantly smaller (27%) than to the peripheral Ca (73%).
On the other hand, the φ back-donation is expected to stabilize the M-Ca interaction, resulting in shorter M-Ca bonds depending on the degree of such donation. As in propenes, the M-L φ backbonding is strongest in Pa (0.27 |e|); however, the φ interaction of U (0.08 |e|) cumulene is significantly stronger than that of Np (0.02 |e|) or Pu (0.02 |e|). While the U-cumulene φ back donation is still considered as a minor effect since it does not impact the overall donation trend, it does so in the case of Pa. Specifically, accounting for the M-L φ donation results in the highest overall donation value for Pa instead of U (Fig. 6e), in agreement with the shortest Pa-cumulene distance. Although the φ back-donation is small for Np (0.02 |e|) and Pu (0.02 |e|), it grows to 0.08 |e| in U, and to 0.27 |e| in Pa, for which it is even larger than the direct L-M p donation from any C-C p bonds taken alone (0.18 |e|, 0.19 |e|). Overall, presence of the M-L φ interactions in the cumulenes supports the observed M-C bond length trend, thus confirming its indispensable role in explaining of the peculiar geometrical changes found in the cumulene series, particularly for Pa.

CASSCF calculations
In order to assess the multi-reference character of the electronic wave-function in the Pa, U, Np and Pu derivatives of the cumulene series, CASSCF calculations with an active space containing ligand orbitals of the cumulene ligand were carried out. The calculations confirmed the strong multi-reference character of the wave functions of the ground spin-orbit state for each system. As expected from Hund's rules, the ground state wave function is dominated by determinants of the highest spin state in all cases: doublet for Pa, triplet for U, quartet for Np, and quintet for Pu. Specifically, CASSCF calculations considering the 6 σ-and π-type orbitals that describe the π bonds in the cumulene ligand (3 bonding orbitals + 3 anti-bonding orbitals) were performed. Indeed, these orbitals are more likely to mix with the metal f orbitals than those describing σ bonds within the cumulene ligand.
Since CASSCF calculations rapidly become expensive with increasing active space, not all f orbitals are included in the active space. The number of f orbitals to be included is determined using ab initio ligand field theory (AILFT). Details about this approach can be found in the supplementary information attached to Jung et al. Inorg. Chem. 2017, 56, 8802-8816. In a nutshell, in this approach, the CASSCF energies and wave functions of a CAS(n,7) calculation are fitted against a model Hamiltonian that incorporates the effects of electron-electron repulsion, spin-orbit coupling and ligand field interactions. The fit yields a series of parameters for each of the aforementioned interactions. In the case of ligand field interactions, the fit yields the 7x7 matrix of the ligand field Hamiltonian expressed in the basis of the real f-orbitals. The functions that diagonalize this matrix are called ligand field orbitals. They effectively describe how the ligand field interaction splits energetically the seven f orbitals. The shape and relative energy of these orbitals are shown in Supplementary Figure 30 for the cumulene series. The orbitals included in the active space are shown with a black box for each derivative.
The number of computed roots depends on the number of metal orbitals included in the active space, such that only roots contributing to the ground spin-orbit state are computed. Despite the contribution of lower spin states highlighted in the CAS(n,7) calculations, only roots of the highest spin multiplicity are considered for each system. All roots are equally weighted in the state average calculation. Supplementary Table 9 summarizes the computational settings for each system.

S34
For all systems, the electronic wave function of the ground state has a strong multi-reference character, which stems from distributing the unpaired electrons among the orbitals with dominant f-character, but does not involve the ligand orbitals. Indeed, the occupation number of the ligandbased orbitals in any determinant that makes a contribution larger than 1% is always either 2 or 0, which means that these orbitals do not contribute to the multi-reference character of the wave function.
In addition, from the detailed composition of the active space orbitals, it appears that orbitals with dominant f-character are mostly non-bonding as they always hold close to 99% f character. The only noticeable exception is the Pa derivative for which bonding-type metal-ligand mixing (96% vs. 3%) is observed, as shown on Supplementary Figure 35-a. The f orbital that is involved in this mixing has the same symmetry as the one evidenced from DFT calculations and AdNDP analysis to achieve φ back-donation. Since (1) the metal-based orbitals of the active space are mostly non-bonding, and (2) the multireference character of the ground state wave function stems from distributing the unpaired electrons among these non-bonding orbitals, it is expected that the multi-reference character of the wave function will not modify the bonding picture obtained from the AdNDP analysis based on singlereference DFT calculations for the investigated systems.