Structure and energetics
Structural and energetic data emerging from our ZORA-BLYP/TZ2P computations are collected in Tables 1–1, 2, 3, 4. Most ML2 complexes have a linear LML angle, which leads to either D3h-symmetric complexes M(NH3)2 and M(PH3)2 or D∞h-symmetric complexes M(CO)2. However, numerous significantly smaller angles appear throughout Table 1 as well, where the symmetry of the complexes is lowered to C2v. For instance, the complexes become increasingly bent when the ligands are varied along NH3 (a strong σ donor), PH3 (a σ donor and π acceptor) and CO (a strong π acceptor). This is most clearly seen for the group 9 complexes, where, for example, the angle decreases along Rh(NH3)2−, Rh(PH3)2− and Rh(CO)2− from 180.0° to 141.2° and 130.8° (Figure 2). In a later section, we will show that the π-backbonding properties of the complexes constitute a prominent part of the explanation of why d10-ML2 complexes can adopt nonlinear geometries. The increasingly strong π backbonding along this series also results in stronger metal–ligand bonds (see Table 2 for bond dissociation energies (BDEs) and Table 3 for energy decomposition analyses (EDA) results for ML complexes).
Table 4. Ligand orbital energies ε [eV] and proton affinities [kcal mol−1].[a]
The extent of bending systematically decreases when the π-backbonding capability of the metal center decreases from the group 9 anions, via neutral group 10 atoms, to the group 11 cations. This is clearly displayed by the series of isoelectronic complexes Rh(CO)2−, Pd(CO)2 and Ag(CO)2+ along which the LML angle increases from 130.8° to 155.6° to 180° (Table 1). The data in Table 3 for the corresponding monocoordinate RhCO−, PdCO and AgCO+ nicely show how along this series the distortive π-orbital interactions Δ indeed become weaker, from −120 to −51 to −11 kcal mol−1, respectively. In the case of group 9 metals, both phosphine and carbonyl complexes are bent, whereas, for group 10 metals, only the carbonyl complexes deviate from linearity. Complexes with a metal center from group 11 all have a linear LML configuration. The reduced π backbonding also leads to weaker metal–ligand bonds. For the cationic metal centers, for which π backdonation plays a much smaller role, the metal–ligand BDEs decrease in the order NH3>PH3>CO (see Table 2). This trend originates directly from the σ-donating capabilities of the ligands as reflected by the energy of the lone-pair orbital ε(LP), which decreases in this order (see Table 4). Note that, for the same reason, the basicity of the ligand as measured by the proton affinity (PA) decreases along NH3>PH3>CO.26 For the anionic group 9 metal centers, the opposite order is found, that is, metal–ligand BDEs decrease in the order CO>PH3>NH3, following the π-accepting capabilities of the ligands.
Linearity also increases if one descends in a group. For example, from Ni(CO)2 to Pd(CO)2 to Pt(CO)2, the LML angle increases from 144.5° to 155.6° to 159.0°. Interestingly, this last trend is opposite to what one would expect proceeding from a steric model. If one goes from a larger to a smaller metal center, that is, going up in a group, the ligands are closer to each other and thus experience stronger mutual steric repulsion. But instead of becoming more linear to avoid such repulsion, the complexes bend even further in the case of the smaller metal. For example, when the palladium atom in Pd(CO)2 is replaced by a smaller nickel atom, the LML angle decreases from 155.6° in Pd(CO)2 to 144.5° in Ni(CO)2. Later on, we show that this seemingly counterintuitive trend also originates from enhanced π backbonding which dominates the increased steric repulsion.
General bonding mechanism
Figure 3. A) Schematic MO diagrams for the bonding mechanism between PdCO and CO in linear Pd(CO)2 (left) and at a LML angle of 90° (right): dominant interactions (—), other interactions (- - - -), π backbonding (—). B) Schematic representation of the bonding overlaps of the donating orbital on PdCO (black) with the π-accepting orbital on the second CO ligand (red).
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When the second CO ligand coordinates opposite the first one (i.e., in a linear LML arrangement), its π*-acceptor orbitals interact with the dπ orbitals on the PdCO fragment. The latter are already considerably stabilized by π backdonation to the first CO ligand (Figure 3 B, left). When, instead, the second ligand is added at an angle of 90°, its π* orbitals overlap with only one dπ orbital, and with one dδ orbital (Figure 3 B, right). This dδ orbital is essentially a pure metal d orbital that has not yet been stabilized by any coordination bond. Consequently, this orbital has a higher energy and is, therefore, a more capable donor orbital for π backdonation into the π* orbital of the second CO ligand. This results in a stronger, more stabilizing donor–acceptor interaction of this pair of orbitals in the 90° (Figure 3 A, right) than in the 180° ML2 geometry (Figure 3 A, left: cf. red-highlighted π interactions). σ-Donation interactions are affected less by bending. It is therefore π backdonation that favors bending. The more detailed energy decomposition analyses in the following sections consolidate this picture.
Bonding mechanism: Variation of ligands
To understand the trends in nonlinearity of our ML2 complexes (see above and Table 1), we have quantitatively analyzed the metal–ligand bonding between ML and the second ligand L as a function of the LML angle. The results are collected in Table 2 and displayed in Figure 4–4, 5, 6, 7. Most of our model complexes have a d10-type ground-state configuration but not all of them, as indicated in detail in Table 2. Yet, all model systems discussed here have been kept in d10-configuration, to achieve a consistent comparison and because, on the longer term, we are interested in understanding more realistic dicoordinated d10-transition-metal complexes that feature, for example, as catalytically active species in metal-mediated bond activation. We start in all cases from the optimal linear ML2 structure (i.e., the complex optimized in either D∞h or D3h symmetry) and then analyze the bonding between ML and L′ as a function of the LML angle, from 180° to 90°, while keeping all other geometry parameters frozen. The analyses were done in Cs symmetry, bending the complexes in the mirror plane, with the out-of-plane hydrogen atoms of M(NH3)2 and M(PH3)2 towards each other. Thus, we are able to separate the orbital interactions symmetric to the mirror plane (A′ irrep) from the orbital interactions asymmetric to the mirror plane (A“ irrep): ΔEoi(ζ)=
[Eq. (3)]. The use of frozen fragment geometries allows us to study purely how the interaction energy changes as the angle is varied, without any perturbation due to geometrical relaxation. Therefore, any change in ΔE stems exclusively from a change in ΔEint=ΔVelstat+ΔEPauli+Δ+Δ. Note that rigid bending of the linearly optimized LML complexes causes minima on the energy profiles to shift to larger angles than in fully optimized complexes, but this does not alter any relative structural or energy order.
In Figure 4, we show the energy decomposition analyses [Eq. (2)] and how they vary along the palladium complexes Pd(NH3)2, Pd(PH3)2 and Pd(CO)2. Upon bending the LML′ complex from 180° to 90°, the average distance between the electron density on LM and the nuclei of L′ decreases (the PdP distance however remains constant), which results in a more stabilizing electrostatic attraction ΔVelstat. Likewise, the Pauli repulsion ΔEPauli increases because of a larger overlap of the lone pair on L′ with the d-derived dσ orbital on the ML fragment. The latter is the antibonding combination of the metal d orbital and the ligand lone pair, with a fair amount of metal s character admixed in an LM bonding fashion. The resulting hybrid orbital is essentially the d orbital with a relatively large torus. The increase in Pauli repulsion that occurs as the LML′ angle decreases stems largely from the overlap of the lone pair on the second ligand L′ with this torus. For Pd(CO)2 for example, the overlap of the L′ lone pair with the dσ hybrid orbital on ML increases from 0.05 to 0.28 upon bending from 180° to 90°. We note that this repulsion induces a secondary relaxation, showing up as a stabilizing Δ, by which it is largely canceled again. The mechanism through which this relief of Pauli repulsion happens is that, in the antibonding combination with the L′ lone pair, the dσ orbital is effectively pushed up in energy and (through its L′-lone-pair component) interacts in a stabilizing fashion with the metal s-derived LUMO on ML.
Figure 4. Energy decomposition analysis [Eq. (2)] of the interaction between PdL and L in dicoordinated palladium complexes PdL2 as a function of the LML angle (L=NH3, PH3, CO).
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Thus, the interaction energy is split into two contributions which are both stabilizing along a large part of the energy profiles studied and which vary over a significantly smaller range. Therefore, this decomposition allows us to directly compare the importance of Δ with respect to the combined influence of all other terms, contained in ΔE′int. The latter contains the aforementioned counteracting and largely canceling terms of strong Pauli repulsion between A′ orbitals and the resulting stabilizing relaxation effect Δ.
The results of this alternative decomposition appear in Figure 5, again for the series of palladium complexes Pd(NH3)2, Pd(PH3)2 and Pd(CO)2. In each of these complexes, bending begins at a certain point to weaken the ΔE′int energy term and, at smaller LML angles, makes it eventually repulsive as the Pauli repulsion term becomes dominant (see also Figure 4). Numerical experiments, in which we consider the rigid bending process of a complex in which the metal is removed, show that steric repulsion between ligands does contribute to this repulsion, especially at smaller angles. Thus, direct Pauli repulsion between L and L′ in LML′ goes, upon bending from 180° to 90°, from 0.3 to 4.6 kcal mol−1 for Pd(NH3)2 and from 0.4 to 9.0 kcal mol−1 for Pd(CO)2 (data not shown in Figures). This finding confirms that ligands avoid each other for steric reasons, but it also shows that the effect is small as compared to the overall change in the ΔEint curves (see Figure 5). The dominant term that causes ΔEint to go up in energy upon bending is the increasing Pauli repulsion that occurs as the L′ lone pair overlaps more effectively with the LM dσ orbital.
Figure 5. Energy decomposition analysis [Eq. (4)] of the interaction between PdL and L in dicoordinated palladium complexes PdL2 as a function of the LML angle (L=NH3, PH3, CO).
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In a number of cases, the stabilization upon bending from the asymmetric orbital interactions Δ dominates the destabilization from the ΔE′int term. These cases are the complexes that adopt nonlinear equilibrium geometries. This
term gains stabilization upon bending LML′ because the π*-acceptor orbital on the ligand L′ moves from a position in which it can overlap with a ligand-stabilized LM dπ orbital to a more or less pure metal and, thus, up to 1 eV higher-energy dδ orbital (see Table 3), which leads to a more stabilizing donor–acceptor orbital interaction (see Figure 5). The gain in stabilization of Δ upon bending and, thus, the tendency to bend increases along NH3 to PH3 to CO. The reason is the increasing π-accepting ability of the ligands as reflected by the energy ε(π*) of the ligands′ π* orbital which is lowered from +1.42 to −0.24 to −1.92 eV, respectively (see Table 4). Thus, for Pd(NH3)2, where π backdonation plays essentially no role, the
term is stabilized by less than 0.5 kcal mol−1 if we go from 180° to 90°. For PH3, known as a moderate π-accepting ligand, this energy term is stabilized by 1.5 kcal mol−1 from 180° to 90° and, for CO, this stabilization amounts to 2.5 kcal mol−1. Thus, in the case of palladium complexes, the energy profile for bending the complexes becomes progressively more flat as the ligands are better π acceptors, but only the carbonyl ligand generates sufficient stabilization through increased π-backbonding in Δ to shift the equilibrium geometry to an angle smaller than 180°.
Bonding mechanism: Variation of metals
Applying the same analysis along the series Rh(CO)2−, Pd(CO)2 and Ag(CO)2+, reveals a similar but clearer picture (Figure 6). Along this series of isoelectronic complexes, the equilibrium geometries have LML angles of 130.8°, 155.6° and 180.0°. Similar to the results obtained for the series discussed above, we again find a ΔE′int term that is relatively shallow and eventually, at small angles, dominated by the Pauli repulsion. The ΔE′int term does not provide additional stabilization upon bending the complex. We do observe, however, a Δ component that, from Rh(CO)2− to Pd(CO)2 to Ag(CO)2+, becomes more stabilizing and also gains more stabilization upon bending from 180° to 90°. That is, whereas for Ag(CO)2+ the
remains constant at a value of −5.4 kcal mol−1 as the complex is bent from 180° to 90°; the same component for Pd(CO)2 starts already at a more stabilizing value of −15.1 kcal mol−1 at 180° and is stabilized more than 2.5 kcal mol−1 as the complex is bent to 90°. For Rh(CO)2−, the effect of the additional stabilization upon bending is strongest, almost 10 kcal mol−1, as Δ goes from −28.4 kcal mol−1 at 180° to −37.3 kcal mol−1 at 90°. The mechanism behind this trend is that the donor capability of the metal d orbitals increases as they are pushed up in energy from the cationic AgCO+ to the neutral PdCO to the negative RhCO− (see Table 3). This trend of increasing d orbital energies leads to a concomitant strengthening π backdonation and, thus, an increasing energy difference in the LM fragment between the pure metal dδ and the ligand-stabilized dπ orbitals. Thus, the ”fresh“ dδ orbitals are higher in energy than the ligand-stabilized dπ orbitals by 0.21 to 0.96 to 1.65 eV along AgCO+, PdCO and RhCO−, respectively (see Table 3). Consequently, the LML′ complexes benefit progressively along this series from increasing the overlap of L′ π* with the higher-energy dδ orbitals in the bent geometry.
Figure 6. Energy decomposition analysis [Eq. (4)] of the interaction between MCO and CO in dicarbonyl-transition-metal complexes M(CO)2 as a function of the LML angle (M=Rh−, Pd, Ag+).
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Figure 7. Energy decomposition analysis [Eq. (4)] of the interaction between MCO and CO in dicarbonyl-transition-metal complexes M(CO)2 as a function of the LML angle (M=Ni, Pd, Pt).
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Based on detailed Kohn–Sham MO analyses of individual complexes, we have constructed generalized Walsh diagrams corresponding to bending the ML2 complexes from 180° to 90°. This choice comes down to an alternative perspective on the same problem, and the emerging electronic mechanism, why bending may occur, is fully equivalent to the one obtained in the above analyses based on two interacting fragments LM+L′, namely: Bending ML2 to a nonlinear geometry enables ligand π* orbitals (if they are available on L) to overlap with and stabilize metal d orbitals that are not stabilized in the linear arrangement. The spectrum of different bonding situations has been summarized in two simplified diagrams that correspond to two extreme situations: weakly π-accepting ligands (Figure 8 A) and strongly π-accepting ligands (Figure 8 B). In these diagrams, we position the d orbital in linear ML2 above the other d orbitals, a situation that occurs, for example, for Pd(PH3)2. The relative position of the d may change, and in some complexes, such as, Rh(NH3)2−, it is located below the other d orbitals. These variations do not affect the essential property of the orbitals, namely, their change in energy upon bending the ML2 complex. Furthermore, we speak about weakly π-accepting ligands, not just about (purely) σ-donating ligands, because it turns out that none of our model ligands has negligible π-accepting capability. The resulting Walsh diagrams summarize our results in a more easy to use pictorial manner which, in particular for the situation with strongly π-accepting ligands, is novel.
Figure 8. Simplified Walsh diagrams for bending ML2 complexes that emerge from our Kohn–Sham MO analyses (+/− indicate bonding/antibonding) A) without and B) with π backbonding.
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