Matta, Boyd. The quantum theory of atoms in molecules

352 13 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards

because of the empty eg orbitals, disposed in the direction of the incoming ligand density). This configuration has been associated with a ‘‘lock and key’’ mechanism [26], but is basically the visual representation of the LFT prediction. The atomic graphs of Fe(CO)5 and Ni(CO)4 have, respectively, trigonal prismatic [6, 9, 5] and octahedral [6, 12, 8] shapes, though somewhat dependent on the density model employed [25]. Atomic charges show that back-donation is in the order Cr > Fe ANi, in agreement with computed and observed stretching frequencies.

Corte´s-Guzma´n and Bader [27] related the atomic quadrupole moments of C and O to the amount of s-donation and p-back-donation. The changes occurring to quadrupole moments of a carbonyl axially coordinated to a metal are expected to address the accumulation of density along the axis (because of s-donation) in contrast with that perpendicular to the axis, occupying a torus around the bound atom (indicative of increased p-density on the carbonyl C). Although the theoretical quadrupole moments were confirmed by the experimental determination [25], some caution is necessary because the atomic volumes of C atoms change quite substantially on coordination.

13.3.1.2 Donor–Acceptor Interactions of Heavy Elements

Many complexes contain a donor–acceptor interaction between a heavy maingroup element and a metal. One might expect L(r) to be a ected by the small amount of charge concentrated in the atomic valence shell of the donor, and questions about the real nature of such interactions could arise. For this reason we studied the AsaCo bond in Co2(CO)6(AsPh3)2 [28]. Although the region of negative Laplacian around As is very small in theoretical maps, and absent from experimental maps, we found many similarities with more classical donor– acceptor bonds. If the whole class of Co2(CO)6(XH3)2 molecules (X ¼ N, P, As) is considered (Fig. 13.4 and Table 13.1) we note they share very similar features of H(r) distribution and that the Laplacian lobe corresponding to the location of the donor electron pair on X decreases for the heavy elements. Despite this, the topological indexes do not depict the CoaAs bond as a weak interaction: the delocalization is larger than in NaCo and only slightly smaller than in PaCo. We can, moreover, appreciate the tight correlation between the Hb/rb and d(Co, X).

13.3.2

Direct Metal–Metal Bonding

The most studied metal–metal bonds are those in homoleptic M2(CO)n dimers or in some homo or heteroleptic small metal clusters. The presence of direct chemical bonding between two metals was the subject of discussion for many years. When dealing with ‘‘unsupported’’ MaM contacts, lack of bridging ligands and the 18-electron rule drive chemists to speak of direct covalent (as in Mn2(CO)10) or even dative (as in CrOs(CO)10) [29] MaM bonds. Some doubts arose, however, when semiempirical calculations were performed to address 1,3 M CO interactions as the major source of attraction [30]. Previous accurate electron-density determinations by X-ray di raction [31, 32] were unable to furnish a solution to the

13.3 Two-center Bonding 353

Fig. 13.4 ‘2r(r) and H(r) distributions in [Co2(CO)6(NH3)2] (a and b), [Co2(CO)6(PH3)2] (c and d), and [Co2(CO)6(AsH3)2] (e and f ). Contours and labeling are as in Fig. 13.2.

354 13 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards

puzzle. In fact, interpretation of rather noisy deformation density maps could not produce a clear picture of the MaM interactions. Theoretical deformation maps, also, could not reveal bonding e ects, because density accumulation in MaM bonds is visible only through fragment deformation maps [33], i.e. using as promolecule the superimposition of computed [MLn] fragments rather than spherical atoms.

QTAIM later o ered, instead, a less ambiguous understanding of the MaM interactions, with clear distinction between unsupported species (in which an MaM bond path is found) and ligand bridged species (in which a MaM bond path is usually not found) [10, 34]. QTAIM was used to interpret the experimental charge density of Mn2(CO)10 [35, 36] and Co2(CO)6(AsPh3)2 [28] that confirmed the presence of a bond path linking the two metals. While Macchi et al. [28] and Farrugia et al. [36] considered the MaM interactions as genuine covalent bonds (based especially on Cremer and Kraka’s criterion), however, [14] Bianchi and coworkers classified the MnaMn bond as metallic with features ‘‘between ionic and covalent’’ [35b]. The main argument was the positive Laplacian found at the MaM bcp. On the basis of a similar reasoning, Uhl et al. classified the NiaNi interaction in CpNi(m-InCH3)2NiCp as closed-shell [37].

Inspection of the Laplacian distribution along an MaM bond path shows that the bcp is located in an L(r) maximum (although Lb < 0) produced by condensation of the two vanishing N shells (if first period transition metals are concerned) [10]. This is somewhat similar to what occurs in simple metal or semimetal diatomic molecules (for example Na2 or B2) [17]. The potential energy density still dominates at the bcp (thus Hb < 0) and the total amount of kinetic energy density per electron is small (Gb/rb f1). Despite the small rb, the MaM interaction is not necessarily weak; the density integrated over the whole zero-flux surface separating two bonded atoms provides meaningful results when di use electrons contribute to the bond (e.g. Na2 in Table 13.1). This concept was stressed in the analysis of Co2(CO)6(AsPh3)2 [28], in which classification problems based on ‘‘traditional’’ QTAIM rules were first addressed. Since then, many other bonding indicators and classification schemes have been proposed, sometimes leading to controversial interpretation. For example, Gervasio et al. [38] applied, to transition metals, the scheme of Espinosa and coworkers [39], who classified bonding interactions according to the spatial region they occupy: the shortest interatomic separation is characteristic of pure shared-shell bonds (‘2rb < 0 and Hb < 0), the longest is characteristic of pure closed-shell bonds (‘2rb > 0 and Hb > 0), and the central region (‘2rb > 0 and Hb < 0) is called a transit region. All interactions with transition metals fall in the intermediate region [38], as one might expect on the basis of their properties at the isolated atom level. This of course suggests MaM and MaL bonds cannot be pure closed shell bonds, as originally proposed [35a], and that these interactions are weaker than covalent bonds between main group atoms. Gervasio et al. [38] also used the flatness criterion, f ¼ rmin=ðrbcpÞmax, [40] to classify chemical bonds in solids: a large flatness was associated with closed-shell bonds and a low flatness with stronger covalent bonds. This criterion cannot be applied to isolated molecules, because rmin would

13.3 Two-center Bonding 355

simply be zero. This limitation has been tackled [38] by analysis of a supported MaM bond system, Co2(CO)6(m-CO)(m-C4H2O2), with an unexpected MaM bond and a ring-type molecular graph. rrcp was therefore taken as the rmin reference for the flatness; it therefore seemed to be quite large, as in closed-shell bonds. The bias introduced is quite severe and would not enable study of an unsupported system. Accurate analysis performed by Farrugia [34c] also revealed that no MCM ring structure is actually present in Co2(CO)6(m-CO)(m-C4H2O2), because the direct CoaCo bond path is missing; this invalidated the conclusions of Ref. [38].

It is worth remarking that the terms ‘‘closed-shell character’’ or ‘‘open-shell character’’ of a chemical bond refer to the electronic configuration of the ‘‘parent’’ fragments. When the bond is formed, such rigorous partition is no longer possible and the electron density itself cannot provide this information, except by resorting to a phenomenological correspondence with prototype situations. It was in this way that Bader [2] derived, for second and third-row molecules, a simple ‘‘translation’’ of the orbital concepts using ‘2r(r) (Table 13.1). However, the absence of VSCC for some heavier elements undermines this scheme [41]. The concept of ‘‘shared interaction’’ mainly implies the concerted movement of a certain fraction of electrons in two (or more) atomic basins. Within QTAIM, the correct indicator for such a concept is the delocalization index d, whose definition is based on the presence of the same electron pair(s) in two (or more) atomic basins. The delocalization index is defined on the basis of pair-density distribution and is, therefore, not accessible from experimental X-ray di raction experiments, in contrast with the electron density [42]. Correspondence with electron density defined properties is, therefore, necessary for interpretation of results from experimental work.

In Table 13.1 it is apparent that the metal–metal bond in unsupported (XH3)(CO)3CoaCo(CO)3(XH3) molecules is associated with d A0:5, which would violate the assumption that the two metals share one electron pair to satisfy the 18-electron rule. We noted, however, that the electron-sharing process in metal carbonyl dimers is rather complex, because it involves not only the two metals but also the equatorial carbonyls (Scheme 13.1(I)) [17]. This is in agreement with the later finding by Ponec et al. [43] on the basis of application of domainaveraged Fermi holes theory to Mn2(CO)10: the MnaMn bond has the character

Scheme 13.1 MaMaCO and MaCOaM bonding in transition metal carbonyl clusters.

356 13 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards

of an ordinary covalent single s-bond but evidence of Mn (CO) intramolecular interactions is also observed. It has been suggested that the shape of the MaC bond paths (slightly bent toward the other metal) and the Laplacian of the equatorial carbonyls may be evidence of a contribution from CO to the MaM [28]. It should be noted that all these observations transfer the problem of the MaM bond in M2(CO)n to multicentered bonding. We will consider this more explicitly in Section 13.4.3 on supported metal–metal bonds.

Bonds between alkali metals in Na2 or K2, which share with CoaCo of Co2(CO)6(XH3)2 the small rb and the positive ‘2rb, but have delocalization indexes able to reproduce the expected bonding electron pair involved, might be regarded as truly unsupported MaM bonds. Once again, the concept of covalence is apparent from Hb (< 0) and from the small amount of kinetic energy density per electron (Gb/rb a0:5 he 1).

13.4

Three-center Bonding

The interactions among three atoms is associated with a variety of molecular graphs (Scheme 13.2) [44] that might represent di erent modes of bonding, although this commonly accepted interpretation should be treated with caution.

Scheme 13.2 Molecular graphs associated with three-center bonding to a metal atom.

13.4 Three-center Bonding 357

The same molecular graph is, in fact, sometimes associated with di erent coordination modes (recognized on the basis of other indicators) or di erent molecular graphs might represent a very similar bonding situation in which minor geometry changes result in di erent bond path connectivity. In three-center (3c) systems, the ring structure is the reference, but not necessarily associated with three separated and localized bonds. Inward curvature of the paths is often an indication of a 3c delocalized bond, at variance with the ring structure, with outward paths typical of three localized bonds. Depending on the kind of interaction, one can observe ring rupture or the degeneration of bcp and rcp into a T-shape structure, a catastrophe point in the conformational space of XMY systems [2].

In organometallic chemistry 3c bonding is quite relevant to interpreting the donation of p and s-density of a preformed bond. This is at the heart, for example, of metal–olefin coordination, agostic interactions, dihydrogen complexes, bridging hydrides, and carbonyl supported metal–metal interactions, all presented below.

13.4.1 p-Complexes

The coordination of an olefin to a metal is one of the most studied chemical bonds involving transition metals – a prototypical bond with enormous implications, for example in catalysis. Many theoretical investigations have been conducted to understand the features of the Mah2(CbC) bond, leading to the dichotomy between Dewar–Chatt–Duncanson (DCD) donor–acceptor complexes and metallacycles. The total number of electrons involved in the bonding is the same, but their spatial localization (and the hypothetical spin state of the interacting fragments) is quite di erent. In the DCD complex, a bonding electron pair (the p-density of the olefin) is s-donated to the metal which, in turn, back donates d- electrons of p symmetry into the empty p of the olefin. In a metallacycle, instead, two localized 2c–2e MaC bonds are formed at the expense of the CaC double bond character (Scheme 13.3).

Scheme 13.3 MO interaction in DCD and metallacycle coordination.

358 13 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards

Fig. 13.5 ‘2r(r) and H(r) distributions in the metal–olefin plane of Na(C2H4)þ (a and b) and Cr(CO)5(C2H4) (c and d). Contours and labeling as in Fig. 13.2.

We recognize three main modes of bonding:

1.Closed shell interaction: The olefin is coordinated to the metal without significant orbital interaction and binding is achieved by electrostatic interaction between a positively charged metal and the electron density of the CbC bond. One example is olefin coordination to a closed shell cation, as in Na(C2H4)þ. In Fig. 13.5, we can appreciate the features of the molecular graphs associated with this mode of bonding, characterized by a T-shape graph (i.e. a single line of maximum electron density is connecting Na and the CbC bcp). In other words, we are unable to locate separate bonds between the metal and each carbon atom. On the basis of ‘2r(r) distribution (Fig. 13.5a), the two fragments are almost unchanged from the isolated Naþ and C2H4 and the closed shell nature of the interaction is also revealed by H(r), the negative regions of

13.4 Three-center Bonding 359

Fig. 13.6 ‘2r(r) distributions in the metal–olefin plane of: (a) superimposition of Ni(CO)2 and (C2H4) densities in the same geometry of the adduct; (b) Ni(C2H4)(CO)2; (c) Ni(C2H4)(PH3)2; (d) Ni(C2H4)(NH3)2. Contours and labeling as in Fig. 13.2.

which inside the two fragments are separated by a large area of H(r) > 0 (Fig. 13.5b). Despite a very low covalence, was shown by Frenking et al. [45] that this mode of coordination is associated with a large stabilization energy (relative to the constituting fragments).

2.Dewar Chatt Duncanson (DCD) ring complex: If back-donation and donation are active but relatively small, the p electron density is still primarily involved in the CbC bond but the p orbitals are slightly rehybridized and some electron density gradients of the C atoms join the interatomic surface of Ni forming two separate inwardly curved or ‘‘endocyclic’’ MaC bond paths. The more abundant the donation, the straighter the bond paths will appear (Fig. 13.6). This can be emphasized by drawing the contribution to ‘r(r) of the p- orbital of the free olefin as reported in Fig. 13.7. In free ethylene, at short range around each C atom j‘rðrÞj has a mirror-symmetric four-lobe shape. If the unperturbed olefin density is superimposed to the MLn density, a T-shape

360 13 Interactions Involving Metals: From ‘‘Chemical Categories’’ to QTAIM, and Backwards

Fig. 13.7 Side view of j‘rðrÞj generated by the p density in C2H4 in di erent electronic states and geometries: (a) ground state (1Ag) in an isolated molecule; (b) 1A1, as in (CO)2Ni(C2H4); (c) 1A1, as in

(NH3)2Ni(C2H4); (d) 3B2 as in (NH3)2Ni(C2H4).

molecular graph appears, but when the DCD mechanism is switched on, the mirror symmetry of j‘rðrÞj is lost, the j‘rðrÞj lobes undergo a ‘‘disrotation’’ (Fig. 13.7b), and the molecular graph assumes a ring shape (Fig. 13.6).

3.The metallacycle: On increasing the metal–olefin interaction, two localized MaC bonds are formed and the CbC p-density is broken. The benchmark molecular graph is cyclopropane, characterized by exocyclic bond paths [46], although the di erent electronegativity and stereochemical requirements of C and M do not enable this limit to be reached. The topology of metallacycles is substantially similar to that depicted in point 2, but the charge concentrations along the MaC bonds are more pronounced and the concavities of the bond paths at C are reduced. The DCD model no longer works, because the perturbation is now too large and a complete spin reallocation has produced two MaC covalent interactions. Indeed, a proper ring shape structure is retrieved from the superposition of C2H4 density in the

13.4 Three-center Bonding 361

triplet state (3B2) on to the MLn density. The olefin HOMO orbital is associated with more exocyclical gradients (Fig. 13.7c) and the bond order of MaC and CaC bonds becomes much closer in agreement with the delocalization indexes.

Based on QTAIM and charge decomposition analysis, Frenking [47] concluded that the DCD ring complex structure is quite typical of metals in lower oxidation states. It is apparent from Table 13.2, however, that for some 16 or 18-electron complexes of zero-valent metals the range of CaC distances span from shorter (<1.40, hence closer to the classical DCD complex) to larger (>1.44). The main rationale seems to be the acidity of the metal fragment: if the metal is coordinated to electron-withdrawing ligands (for example acidic CO) metal backdonation is poor and the olefin coordination is weaker; in contrast, electrondonating groups opposed to the olefin induce larger back-donation. Accordingly we see that CaCaM bond path angles approach the geometrical angles (see

Scheme 13.4 Definition of bond path di erences in MX2 systems.

Scheme 13.4 for definitions).

The QTAIM data for a series of metal–olefin complexes emphasize the tight correlation between CbC elongation (hence C rehybridization) and the tendency to produce more separated MaC bond paths (Table 13.2). This was previously formulated in experimental determination of r(r) in a typical DCD complex, (Ni(COD)2) [48] (COD ¼ 1,5-cyclooctadiene). Scherer and coworkers [49], on the other hand, have recently drawn, for some 16-electron Ni(XR3)2(C2H4) complexes (X ¼ N, P; R ¼ alkyl, H), the density (and Laplacian) of the molecular orbital mainly responsible of the olefin-to metal-donation. Their outward pointing shapes would suggest that endocyclic bond paths are not the correct markers of donation. Donation of the paired p-density (point 2 above) is, however, always mixed with that of the unpaired sp3-like density (point 3) thus any molecular orbital cannot be representative of the pure DCD bonding mode, especially because the examples chosen are ahead in the metallacycle conversion (Table 13.2). The reasoning in point 2 is based on the expected behavior of the olefin not contaminated by its triplet configuration and it is valid if the objective of molecular graph analysis is to retrieve a given electronic configuration out of a multiconfigurational system.