Matta, Boyd. The quantum theory of atoms in molecules
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13.4 Three-center Bonding 363
In the same work, Scherer et al. discussed other interesting features of the bond paths at the metal atom. In a classical DCD complex the acceptor metal d-orbital is empty and usually bisects the CaMaC angle, whereas the orbitals involved in back-donation are exocyclically directed, for example Cr(CO)5(C2H4) in Fig. 13.5 and Table 13.2. Thus, one should expect outward curvature at the metal, as one can appreciate from the ‘2r(r) distribution, which is indicative of charge accumulation by the Cr 3d shell to lie outside the CraC2 ring. This is not observed for late transition metals, however, particularly the d10 elements [28, 49], which have all d orbitals occupied and therefore should have a spherical Laplacian. The back-donation process induces an aspherical density, with charge depletion corresponding with the orbitals involved. Thus, in contrast with Cr, in Ni complexes there is a dominant negative region of the Laplacian associated with an orbital not involved in the back-bonding (and thus completely filled) and pointing toward the center of the MC2 ring. Consequently, the bond paths at Ni (or even at Fe) are slightly endocyclic. Scherer and coworkers also noted that around the midpoints of the NiaC bonds the path curvatures change again (becoming exocyclic), because there the p-back-bonding orbitals dominate. It should be noted that this is not the rule, but an exception for electron-richer metal.
The Laplacian usually shows two distinct charge concentrations, corresponding with the MaC bonds, which become larger as the MaC bond strengthens, in agreement with the shape of the bond paths and with the larger amount of electron density at bcp. Accordingly, MaC and CbC delocalization indexes reproduce quite well the reallocation of the olefin p-density in MaC interactions. Eventually, in coordination mode 3, the distribution of total energy density is characterized by a unique negative region incorporating both the metal and the olefin fragment. All the arguments proposed above can be used to identify the di erent bonding contributions in parallel (||) or perpendicular (?) coordination of an olefin in the equatorial plane of Fe(CO)4(C2H4). From simple orbital reasoning [50] the parallel stereochemistry is favored, because back-donation is more active. Assuming the same coordination geometries for parallel and perpendicular Fe(CO)4(C2H4), we can appreciate from Table 13.2 that the parallel conformer has the features associated with larger back-donation and stronger MaC interactions (i.e. smaller d(CbC), larger d(MaC), and smaller inward curvature of the bond paths).
13.4.2 s-Complexes
Although the donation of p density to a metal was discovered and studied much earlier than that of s density, p and s-complexes share many similarities. As far as hydrogen is concerned, s-complexes span dihydrogen compounds, agostic interactions, and bridging hydrides [51], the bonding of which is usually described in terms of 3c–2e bonds. There is now general consensus that 3c–2e systems may have many di erent geometries, take some 3c–4e character (depending from the relevance of p-donation from the metal), and, eventually, evolve into two 2c–2e bonds.
364 13 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards
13.4.2.1 Dihydrogen and Dihydride Coordination
The first dihydrogen complexes (W(CO)3(PR3)2(H2), R ¼ cyclohexyl, isopropyl) were reported in 1984 by Kubas et al. [52]. Some pioneering QTAIM studies on h2 coordination of H2 to a metal were published in the early nineties [53, 54]. In the dihydrogen complexes the mechanism of coordination is reminiscent of the DCD model, although based on donation of s-density. Accordingly, whereas the metal–olefin coordination is characterized by the p-complex–metallacycle dichotomy, the dihydrogen complexes have similar uncertainty with regard to dissociation into dihydride. Similarly, there are weakly bound complexes, observed when the metal center is quite acidic and characterized by a TM molecular graph (e.g. Cr(CO)5(H2)), and strongly bound complexes, characterized by more substantial back-donation and a genuine ring graph (e.g. Cr(PH3)5(H2)). Because the H atoms lack of directional orbitals, however, the MaH bond paths of the ring structure are quite inwardly curved and the H2 density is only weakly polarized in the direction of the metal. Dapprich and Frenking [53] reported a ring structure (though almost collapsed) for (CO)5M(H2) molecules (M ¼ Cr, Mo, W) based on MP2 and CCSD(T) calculations with e ective core potential basis sets. The overall picture is, nevertheless, not very di erent because in both circumstances the dominant interaction is the HaH bond, as demonstrated by the topological and delocalization indexes (Table 13.2), whereas the metal–ligand interaction is just above a pure closed-shell limit (especially in Cr(CO)5(H2)). On the basis of charge decomposition analysis, Dapprich and Frenking [53] estimated the amount of donation and back-donation, concluding that H2 is a much weaker p-acceptor than CO, thus justifying the molecular graphs observed.
Toma´s et al. [55] showed that a dihydrogen complex can further proceed in the oxidative addition of H2, which was associated with a small barrier. The final product is an heptacoordinated bipyramidal complex with two hydride ligands in
the pentagonal plane, but not in the cis configuration. The transition state is a
˚´ structure containing an H2 moiety with H H separation of more than 1.5 A. In
some complexes this structure is actually a stable isomer, because the minimum on the potential energy surface is attained at larger H H separation, whereas in others an equilibrium between dihydrogen and dihydride can be established [55]. The main chemical problem is then associated with the presence of some residual H H bonding.
13.4.2.2 Agostic Interactions
Within the QTAIM formalism, Popelier and Logothetis were the first to address the di erences between agostic interactions (in d0 TiIV complexes) and the 3c– 4e hydrogen bonds [56]. They concluded that an agostic interaction is characterized by a bond path with values of the electron density and the Laplacian at the bcp characteristic of an ionic, closed-shell interaction. They proposed criteria based on the variations of atomic Hb properties which, however, lack generality (Table 13.3). Scherer and McGrady [57], in contrast, showed that b-agostic interactions in d0 metals occur because of delocalization of the MaCa bond over the metal–alkyl moiety (negative hyperconjugation) rather than the presence of a
366 13 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards
Fig. 13.8 ‘2r(r) and H(r) distributions in Ni(dhpe)(C2H5)þ (a and b) and TiCl3(C2H5) exo (c and d).
3c–2e M Hb aCb interaction. In a combined experimental and theoretical study on Ti(dhpe)Cl3(C2H5) (dhpe ¼ H2PCH2CH2PH2) a significantly pronounced bond path for the TiaHb was not found [58]. In support of this theory, the authors addressed the characteristic curvature of the TiaCa bond path (Fig. 13.8) and its ellipticity along the whole path. We also note that delocalization indexes are very informative: if an agostic interaction is activated, both d(MaCa) and d(Cb aHb ) decrease in favor of d(Ca aCb ) and d(MaHb ).
It is worth noting that if the main mechanism is hyperconjugative, exo or endo Hb should both be a ected by the agostic interaction. Indeed, the endo and exo (which is not a minimum on the PES) conformers of TiCl3(C2H5) are characterized by the same features of a weak agostic type interaction. If a TiaHb bond path is found in the exo conformer it is simply because of an additional and, perhaps, negligible source of bonding (i.e. a closed-shell Ti Hb interaction) which does not significantly account for the overall stability of that conformer. The H(r) distribution is in agreement with this view (Fig. 13.8).
13.4 Three-center Bonding 367
Scherer and McGrady also pointed out that the availability of local Lewis acidity sites on the metal is a feature that favors agostic interactions, as revealed by the VSCCs about the metal atom.
In contrast, electron-richer metals a ord real 3c–2e agostic interactions – see the comparison between (dhpe)Cl3Ti(CH2H5) and [(dhpe)Ni(CH2H5)]þ. As suggested for the Cp(CO)2Mn/HSiCl3 adduct, however [44, 59], an agostic interaction may transform into (or consist of ) the more classical oxidative addition product. This process is of paramount interest in catalysis.
On the basis of these observations we can locate two extremes: the weak agostic and incipient oxidation adducts. The former is associated with an MYX graph (Scheme 13.2, X ¼ C, Y ¼ H) close to the breaking of the MaY path, the latter with a ring structure close to the breaking of XaY to a ord an XMY graph. They are also distinguishable on the basis of d(MaH)/d(XaH) which is close to zero in the agostic interaction of early transition metals but could reach unity in ‘‘symmetric agostic’’ interactions (Section 13.4.2.3).
13.4.2.3 Hydride Bridges
An apparently di erent type of s-coordination to a metal is that of hydrides bridging two metal atoms. An experimental and theoretical study of the electron density distribution of [Cr2(m2-H)(CO)10] has shown, however [60], that many similarities with agostic interactions can be drawn for MaHaM systems which, traditionally, were classified on the basis of open or closed 3c bonding (Scheme 13.5). Stereochemical considerations led to the conclusion that a closed system is more adherent to reality, because the direction opposite to the axial ligands point toward the MHM ring center rather than to the hydride, suggesting the presence of some direct MaM bonding. It should be noted that this hypothesis was formulated before the agostic interaction was actually discovered.
Scheme 13.5 Hypothesis for bonding in MaHaM systems.
The potential energy surface of [Cr2(m2-H)(CO)10] is particularly flat [3]: the gas phase molecule has Cs symmetry, a nearly symmetric bridge and staggered carbonyl groups (Fig. 13.9) but in the solid state many other conformers can be found as a function of the perturbation induced by the environment; this enables definition of an ‘‘experimental’’ interconversion path between the Cs staggered conformer and a pseudo-D4h (C2v) conformer with eclipsed carbonyl groups and an almost linear MaHaM system (Fig. 13.9). A CraCr bond path is not present and the electron sharing between the two atoms is small and almost constant as a function of CraHaCr bending. The linear geometry, in which no direct metal–
368 13 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards
Fig. 13.9 Scatter plot of the dependence of CraHaCr angles (triangles) and Ceq aCraCraCeq torsion (circles) on Cr Cr distance, taken from the many crystal structures of [Cr2(m2-H)(CO)10] salts. Empty symbols indicate outlier geometry, the nature of which is discussed elsewhere [60].
metal bonding is possible and which contains almost the same amount of delocalization as in the bent structures, should also be noted. QTAIM analysis did not reveal a ‘‘unique’’ e ect responsible for stabilization of one isomer over the others. One is tempted to conclude that ‘‘packing forces’’ have a significant e ect in determining the solid-state conformation of this anion but some general features emerge. For example, the shift of the H atom out of the idealized (OC)ax aM direction seems to be a consistent in MaHaM systems and cannot be attributed to the increased direct M M interaction. Indeed, an unexpected role played by the nearby equatorial carbonyls has been recognized, because of the features of the Fermi hole density distribution and the corresponding H CO delocalization indexes (d approx. 0.1 [60]).
Accurate comparison of [(CO)5CraHaX] systems (Table 13.4) revealed much similarity between two X-moieties Cr(CO)5 and BH3, classically regarded as isolobal [61]. On reducing the acidity of X the limit of a weak agostic interaction can be reached, for example with X ¼ [CH3]þ. Within this framework it is reasonable that, although asymmetric (M HaC) and symmetric hydride bridge (MaHaM) have undisputedly di erent geometries, they share a common bonding nature (at least as far as s-donation of CaH occurs). In more asymmetric (classical) agostic interactions, there is a kind of schizophrenic behavior of the metal, the binding of which could be directed alternately toward H or X (or even be absent, as shown above). As the MaH and XaH bonds become more similar in strength, the MaH overwhelms M X and the structure is characterized by a small M X delocalization. In the limiting case, X ¼ Cr(CO)5, d(MaH)/d(XaH) approaches unity and we can speak of a symmetry-stabilized agostic interaction [60].
37013 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards
13.4.3
Carbonyl-supported Metal–Metal Interactions
Metal cluster scientists have extensively debated the role of direct MaM interaction in carbonyl-bridged 3c–4e metal–metal bonds. This has led to several papers reporting studies of the two most representative molecules, Fe2(CO)9 and Co2(CO)8 [17]. The QTAIM shows that the most relevant di erence between bridged and unbridged isomers is, respectively, the absence or presence of a MaM bond path (Scheme 13.1(I) and (IIIb)). The topology of a bridged system
was first reported for Co2(CO)8 by MacDougall [10, 33a]. Fe2(CO)9 was the topic of a long theoretical debate. Whereas empirical rules predict direct FeaFe bond-
ing, many MO calculations (semiempirical or ab initio) concluded there was no interaction between the two metals given the small d overlap [62], though VB calculations provided more evidence of an interaction [63]. Mealli and Proserpio [64] found that an FeaFe bond is formally present even if the through-bond intermetal repulsion overcomes the attractive through-space FeaFe interaction. QTAIM analyses have been reported by MacDougall [10] and by Bo et al. [34], who did not find a direct MaM bond path and concluded that ‘‘Fe2(CO)9 is built up by the bridging carbonyls’’ [34], in agreement with the earlier suggestion by Summerville and Ho mann [65]. Bo et al. also noted many features of the m2-coordination:
the larger envelope of the valence shell surrounding the carbon, indicative of more delocalized bonding through the metals;
a larger electronic population on the carbonyl carbon, as a consequence of the better metal-to-ligand charge transfer in the bi-coordinative mode; and
the presence of two nonbonded VSCCs on the bridging oxygen, indicative of an incipient rehybridization.
Delocalization of the bond was confirmed by analysis of the Fermi hole density maps [34].
The first experimental validation of the electron-density distribution in m-CO systems came from analysis of the tetrahedral cluster Co4(CO)11(PPh3) [66]. In agreement with all-electron HF calculations on the C3v Co4(CO)12 isomer, no direct CoaCo bond path was found for the three bridged edges (the overall topological features of which resembled those of Co2(CO)8).
To complete the analysis of the hypothetical conversion path from terminal to symmetrically bridging M2(CO) systems, analysis of a semibridging conformation was undertaken [67]. This study proved that the terminal-to-bridge metamorphosis of a carbonyl, although accompanied by an abrupt change in the molecular graph, actually lies on a type of continuum, especially if the electron-sharing process is considered. The delocalization indexes of MaM and MaC interactions change along the interconversion path in such a way that the overall sharing index is almost constant. This explains both the carbonyl fluxionality in transition metal clusters and the observed continuity of conformations in known MaCOaM
13.5 Concluding Remarks 371
fragments [17]. Thus, even when the carbonyl is terminally bound to just one of the two metal atoms, an MaCOaM system is characterized by interplay of direct and indirect MaM and MaC interactions that results in substantial delocalization through the system. This interplay generates di erent molecular graphs and hampers formation of any truly localized bond, which explains why the MaM unsupported bond has less delocalization than expected (in favor of 1,3 M CO interactions). In symmetrically bridged metal dimers the view proposed is in agreement with the many indications of MaM bonding (first and foremost the 18-electron rule) because substantial metal–metal electron delocalization is actually present, even in the absence of a direct bond path.
On the basis of H(r) distribution and analysis of the contribution of each molecular orbital, Rehinhold et al. [68] came to the conclusion that some direct MaM bonding is actually visible in supported MaM bonds. We note that this view does not contradict the interplay of interactions introduced above and made more visible by the delocalization indexes.
13.5
Concluding Remarks
In the last few years QTAIM has become the model for interpreting theoretical and experimental electron density distributions. Within this framework, the link between bonding modes and topological properties has been fully achieved for molecules of main group atoms. In contrast, the correspondence rules derived cannot be extended in a straightforward manner to organometallic compounds, because bonds to a transition metal have di erent and much narrower spectrum of topological indexes. Metals are always characterized by di use ns valence density lacking concentration of charge in the bonding region, even when electron sharing is important. Thus, classifications based on ‘2r(r) might be misleading or at least incomplete.
On moving from 2c to 3c interactions the situation becomes even more complicated, because there are sudden changes in the molecular graph produced by small perturbations of the relative ‘‘weight’’ of each bonding contribution. When a system is close to a catastrophe point, it is the shape rather than the topology of a given molecular graph that is informative, and several examples have been discussed in which the presence of a bond path was actually misleading because it could not completely explain the molecular geometry observed.
We have shown that d(A, B) and H(r) are the most reasonable indicators of the covalent contribution to the total bonding between two (or more) atoms, not least because they are rather insensitive to abrupt changes of the molecular graphs. They are a bridge between traditional chemical categories (mainly formed on the basis of simple MO schemes) and quantum mechanics of the electron density; much work is, however, still required to derive the appropriate correspondence rules, which must take into account subtler details of the molecular graphs.