London dispersion effects in the coordination and activation of alkanes in -complexes: a local energy decomposition study

Local energy decomposition (LED) analysis decomposes the interaction energy between two fragments calculated at the domain-based local pair natural orbital CCSD(T) (DLPNO-CCSD(T)) level of theory into a number of chemically meaningful contributions. Herein, this scheme is applied to the interaction between the transition metal (TM) and the alkane in σ-complexes. It is demonstrated that the often-neglected London dispersion (LD) energy is a fundamental component of the TM-alkane interaction for a wide range of experimentally characterized σ-complexes. LD effects determine the structure and the thermodynamic stability of σ-complexes and influence the selectivity of CH activation reactions. The magnitude of the LD energy can be modulated by increasing the size of the alkane and of the ancillary ligands on the TM. These results provide further evidence on the fundamental role that London dispersion plays in organometallic chemistry.


Introduction
The London dispersion (LD) energy constitutes the attractive part of the van der Waals potential and it is omnipresent in chemistry. In the Symmetry Adapted Perturbation Theory (SAPT) framework, 2 the leading dipole-dipole term of the LD energy grows with the polarizabilities of the interacting fragments and decays asymptotically as R À6 , with R being the interfragment distance. 3 Hence, as the molecular system size increases, the magnitude of the LD energy increases, often making it stronger than covalent or electrostatic interactions. 4 Due to its size dependence, LD is responsible for the stability of seemingly too crowded molecules, such as diamantyl, 5 all-meta t Bu-triphenylmethane, 6 and all-meta t Bu-hexaphenylethane dimers. 7 Moreover, its importance has been widely recognized in biochemistry, 8 material science, 9 and organic chemistry. 10 On the other hand, LD is typically considered weaker than covalent or other non-covalent interactions, especially for small systems. 11 In particular, although recent studies have underlined its importance in organometallic chemistry, 4,10,12-22 its contribution to coordinate covalent bonds is still often ignored or assumed to be negligible. In fact, the interaction between a transition metal (TM) and a ligand (L) is still often described as a simple donor-acceptor interaction and the structure, the catalytic behaviour and the spectroscopic properties of TM complexes are in most cases discussed using orbital models such as the popular Dewar-Chatt-Duncanson (DCD) bonding model. 23,24 Despite the success of these simple interpretative frameworks, a unified understanding of intermolecular interactions in the various fields of chemical research would require models that explicitly include LD as one of the components of the chemical bond, irrespective of the intermolecular distance or of the interaction strength. Unfortunately, rigorous perturbative approaches like SAPT are not applicable to strong TM-L interactions.
More in general, the LD energy, as well as all the other physical components of the interaction typically discussed in chemical models, is not a quantum mechanical observable. Hence, its theoretical definition is ambiguous, especially when the interacting fragments are close to each other and their electron density significantly overlap. 3 Another major problem is that the computational modeling of organometallic complexes requires the accurate inclusion of electron correlation for systems with many electrons and complicated electronic structures. The most commonly used computational methodology to practically address this problem is Density Functional Theory (DFT). Although it is well-known that DFT does not properly describe LD forces, 25 several strategies have been suggested to deal with this shortcoming, 26,27 thus allowing the calculation of accurate structures and energies for organometallic complexes. In particular, a force-field like correction term (the -D correction) is typically added to the DFT energy to account for long-range LD effects, which are not or poorly described by the exchange-correlation functional. Hence, the magnitude of the -D correction can be used as a qualitative estimate for the LD energy, albeit with caution in those cases when short-to mid-range LD effects are important. 13 Recently, [28][29][30] we proposed a new strategy called Local Energy Decomposition (LED) analysis, which provides a decomposition of interaction energies computed at the domain-based local pair natural orbital coupled cluster DLPNO-CCSD(T) [31][32][33] level into various terms representing the most important chemical components of the interaction. This scheme allows for a simple definition of the LD energy at the DLPNO-CCSD(T) level 29,34,35 and has been already applied in a wide range of different chemical situations. 36 Importantly, the LED analysis can be applied for obtaining a consistent quantification of the LD energy for virtually any system within the range of applicability of DLPNO-CCSD(T) scheme, going from weak noncovalent interactions 29 to strong ionic and covalent bonds. 13,34,37 Herein, the DLPNO-CCSD(T)/LED approach is used to quantify the LD component of TM-L bonds of various strength and to elucidate the factors that contribute to its magnitude in organometallic complexes. As a case study, we consider the bond between the TM and the CH bond of an alkane in s-complexes. These complexes have been suggested as crucial intermediates for alkane C-H activation reactions, [38][39][40][41][42][43][44] which are of paramount importance in various chemical industries. In particular, the development of chemical processes for the conversion of alkanes from fossil sources to more-valuable compounds has attracted great attention over past decades. [44][45][46][47][48] In the simplest mechanistic scenario, the CH bond initially coordinates to the TM to form a s-complex (intermediate 2 in Scheme 1a), eventually leading to the cleavage of the CH bond through oxidative addition to the TM.
The chemical bond between the TM and the CH bond in s-complexes is typically described as a donor-acceptor in the framework of the DCD model (Scheme 1b). In this model, the C-H group formally donates electrons from its filled s orbital to the metal center (TM ' HC s-donation) while at the same time the TM back-donates d electrons into the empty C-H s* orbital in the alkane (TM -HC p-backdonation). [49][50][51] This model can be considered as an extension of the three-center two-electron (3c-2e) model initially adopted by Brookhart 52 and later corroborated by energy decomposition schemes at the density functional theory (DFT) level of theory. 42,53 Both s-donation and p-backdonation are expected to decrease the CH bond order, thus favoring the oxidative addition to the TM. 54 Although alkane s-complexes have been proposed for decades, their direct experimental characterization has constituted one of the major challenges for organometallic chemists. 54,55 The few experimental characterized s-complexes can be grouped into three families, differing for the complex structure and their synthetic route. Herein, we base our analysis on a series of s-complexes that includes representative systems of each class.
The first class of s-complexes investigated herein is a series of group VII TM complexes that were previously characterized mainly by means of IR spectroscopy. Their structure is shown in Scheme 2a. The series includes neutral complexes differing for the size of the alkane such as [CpMn(CO) 2 (alkane)] (alkane = C 2 H 6 , n-C 7 H 16 ) [56][57][58][59] and [CpRe(CO) 2 (alkane)] (alkane = CH 4 , n-C 5 H 12 , n-C 7 H 16 ) 56,57,60 as well as the positively charged [(HEB)Re(CO) 2 (n-C 5 H 12 )] + (HEB = hexaethylbenzene). 61 Note that for the longer alkanes many conformers are possible, depending on their relative orientation with the TM. As case studies, two conformers are investigated for each complex, in which the TM interacts with a CH bond located on either the methyl or the methylene carbon. Conformers of the first type are denoted hereafter as ''head-on'' structures, whilst the latter are denoted as ''side-on'' structures.
For all these complexes, the main factors contributing to their formation, the coordination preference of the alkane and the activation of the CH bond are discussed in the framework of the LED analysis and the role played by LD in all these situations is elucidated. Moreover, the factors that contribute to the magnitude of the LD energy are identified and designing principles for s-complexes with tailored bonding features are given.

Method
Geometry optimizations were carried out using the TPSS functional 67 and including the -D3 dispersion correction term 68 with Becke-Johnson damping parameters. 69 The def2-TZVP basis set 70 was used. The def2-ECP effective core potential was used for Re and Rh atoms. Frequencies were calculated at the same level of theory to verify the nature of minima and transition states and to compute free energies. The optimized geometries and vibrational frequencies obtained with these computational settings were found to be in good agreement with available X-ray structures and IR data, as shown in Table S1 in the ESI. † Moreover, these settings were shown to provide geometries and IR frequencies in good agreement with experimental data for the closely related agostic complexes. 12 Single point energies at the DLPNO-CCSD(T) level were computed using TightPNO settings. [31][32][33] The def2-TZVPP basis set was used in conjunction with the def2-TZVPP/C auxiliary basis set. Table S2 (ESI †) shows that the DLPNO-CCSD(T) results are at convergence with respect to the basis set size. The RIJCOSX approximation 71 was used for the calculation of Coulomb and exchange integrals in the HF reference. The def2/J auxiliary basis set was used. The COSX grid was set to GridX5.
Binding energies (DE) at the DLPNO-CCSD(T) level were calculated as the electronic energy difference between the s-complex and its constituting fragments (the alkane and the remaining metallic moiety) at their equilibrium TPSS-D3 geometry. The binding energies were decomposed into dispersive (E disp ) and non-dispersive (DE no-disp ) contributions by means of the LED technique. 28 The dominant term in E disp is the DLPNO-CCSD contribution E C-CCSD disp , which was calculated by summing up all the instantaneous dipole-dipole excitations between the two fragments. It includes contributions from both strong and weak pairs, as detailed in ref. 29. For the sake of simplicity, an effective triples correction contribution to the LD energy was also included. It was estimated by multiplying the intermolecular component of the perturbative triples correction, i.e., the contribution coming from triples of occupied orbitals located on different fragments E 28 by a correction factor g. The g correction was defined as the ratio between the dispersion contribution from the strong pairs and the total intermolecular strong pair contribution. Hence, the total LD energy reads: For the systems studied in this work E C-CCSD disp amounts to about 90% of the overall E disp contribution, as shown in Table S3 (ESI †).
The remaining part of the correlation contribution to the binding energy is denoted as DE C-CCSD(T) no-disp . The sum of this term with the overall HF contribution to the binding energy DE HF gives the overall non-dispersive contribution to the binding energy, i.e., DE no-disp .
In order to provide an in-depth analysis of the factors contributing to the LD energy, the Dispersion Interaction Density (DID) plot 29,72 were employed. They provide a useful spatial analysis of the LD energy and were calculated as detailed in ref. 29.
Finally, the orbital relaxation contribution to the binding energy was calculated as detailed in ref. 29. It provides a useful estimate for the charge transfer and polarization energy at the reference level of theory, consistent with that provided by standard energy decomposition schemes. 73 3 Results and discussion then extended to the whole series of s-complexes shown in Scheme 2 in the next section. The ReÁ Á ÁCH 4 binding energy profile as a function of the ReÁ Á ÁC distance calculated at the DLPNO-CCSD(T) level of theory is shown in the upper panel of Fig. 1. In the same panel, the decay of the LD energy (E disp ) and of the non-dispersive component of the binding energy (DE no-disp ) is also reported. As expected, the binding energy is dominated by LD forces in the long range where no significant charge transfer occurs between the fragments. As the fragments approach each other, the relative contribution of E disp to the binding energy slightly decreases. At the equilibrium geometry, E disp amounts to À46.7 kJ mol À1 , which is about the 67% of the overall binding energy (À69.8 kJ mol À1 ). The remaining 33% comes from DE no-disp . It describes the balance between the orbital relaxation energy at the HF level (À135.2 kJ mol À1 ) and the other components of the interaction (+112.1 kJ mol À1 ), such as permanent electrostatics, electronic preparation and the correlation correction for these terms DE C-CCSD(T) no-disp . 29 For the sake of simplicity, only E disp and DE no-disp will be discussed in the following. Additional information on the various terms of the decomposition can be found in the ESI. † Although ReÁ Á ÁCH 4 is a relatively weak coordination bond, the fact that the E disp amounts to the 67% of the overall binding energy is surprising. These results are consistent with the ones recently 12 found in the context of agostic interactions, i.e. the intramolecular interaction between a C-H bond and a coordinatively unsaturated TM. It was found that short-range LD forces between the agostic C-H bond and the metal center are largely responsible for the stability of a wide range of agostic complexes, consistent with a large number of experimental findings, including the fact that many agostic complexes do not show evidence of significant C-H bond activation. 74 In order to compare different computational methodologies, the binding energy profile obtained with different functionals is compared with the DLPNO-CCSD(T) one in the central panel of Fig. 1. In the bottom panel of the same figure the decay of the -D3 correction for different functionals is compared with the E disp values extracted from the LED scheme.
In the long range, all methods provide similar ReÁ Á ÁCH 4 binding energies. At the equilibrium geometry, the difference between the various functionals increases, with binding energies being À51.6, À55.8, À64.2 and À64.5 kJ mol À1 for BLYP-D3, B3LYP-D3, TPSS-D3 and PBE-D3, respectively. The corresponding -D3 correction follows the opposite trend, being À25.7, À21.6, À16.6 and À12.7 kJ mol À1 . The differences between the -D3 estimates are mostly due to the effect of the damping functions, as also recently discussed by Grimme et al. 13 By comparing these figures with the E disp value (À46.7 kJ mol À1 ), one finds that the functionals that better reproduce the DLPNO-CCSD(T) binding energy (e.g. PBE) are those for which the -D3 correction is smaller and far from the accurate LED values. Hence, in the following only DLPNO-CCSD(T)/LED results will be discussed.
In summary our results clearly show that LD is fundamental for both the formation and the thermodynamic stability of [CpRe(CO) 2 (CH 4 )]. In fact, it dominates the TM-alkane interaction in the long range and significantly contributes to the magnitude of the binding energy at the equilibrium geometry.

London dispersion in r-complexes
In order to test the generality of our findings for the ReÁ Á ÁCH 4 interaction in [CpRe(CO) 2 (CH 4 )], geometries, binding energies and the corresponding LD contributions were calculated for the whole series of s-complexes investigated in this work. The results obtained are summarized in Fig. 2. Note that, for the sake of simplicity, only electronic energies are discussed in this section, while the free energies at experimental conditions are reported in Table S4 (ESI †). Binding energies span a broad range of values, going from À57.6 to À127.8 kJ mol À1 for [CpMn(CO) 2 (C 2 H 6 )] and [(dcpe)Rh(norbornane)] + , respectively. Similarly, the LD energy ranges from À46.7 to À111.3 kJ mol À1 for [CpRe(CO) 2 (CH 4 )] and [(dcpe)Rh(norbornane)] + , respectively.
Remarkably enough, E disp amounts to at least the 67% of the binding energy for all complexes. For [CpMn(CO) 2 (C 7 H 6 )] (II) and [(dibpe)Rh(norbornane)] + , its magnitude exceeds the binding energy. Hence, without this important component of the interaction, these experimentally characterized complexes would not be stable. These results are consistent with our previous findings on the agostic EtTiCl 3 (dmpe) complex. 12 Having established the fundamental importance of LD for the thermodynamic stability of s-complexes, we aim at identifying the most important factors contributing to its magnitude in a view of aiding to the development of designing principle for stable TM-alkane adducts.
As mentioned in the introduction, LD increases with the polarizability of the interacting fragments and decreases with the intermolecular distance. Hence, the size of the ancillary ligands and of the alkane, the nature and the oxidation state of the TM, and the coordination mode of the alkane (i.e. whether it forms head-on or side-on structures) are all expected to contribute to its magnitude.
A revealing insight into this aspect comes from the analysis of the DID plots (Fig. 3), which provide a simple mean of visualizing the most important contacts contributing to the magnitude of the LD energy of the TMÁ Á Áalkane interaction. 72 The first eye-catching feature of these plots is that the most significant contribution to the LD energy originates from short TMÁ Á ÁH-C contacts, as evidenced by the red color of the density isosurfaces around the C-H groups in close proximity of the TM. In Mn and Re complexes, only one C-H group strongly interacts with the TM, whilst all Rh complexes (Fig. 2b and c) are stabilized by two short TMÁ Á ÁH-C contacts. For this reason, the LD energy in Rh complexes is typically larger than in the other s-complexes investigated in this work.
The size of the alkane also plays an important role in affecting the magnitude of the LD contribution.  Similarly, the use of bulky ligands on the TM also increases the LD contribution to the binding energy. In fact, going from [(dippe)Rh(norbornane)] + to [(dbpe)Rh(norbornane)] + , E disp increases from 103 to 110 kJ mol À1 as a consequence of the substitution on the four i Pr groups on the diphosphine ligand with the more polarizable t Bu groups. However, these figures do not correlate with the corresponding variations in the binding energy. This is probably due to steric effects associated with the inclusion of bulkier t Bu groups, as shown by the fact that r (TM-C) distance slightly increases going from [(dippe)Rh-(norbornane)] + (2.40 Å) to [(dbpe)Rh(norbornane)] + (2.42 Å).
As mentioned above, different coordination modes (e.g. sideon vs. head-on) are possible for the longer alkanes. To investigate the effect of LD on the coordination preference of the alkane, it is useful to compare binding and LD energies for the pairs [CpMn(CO) 2 (n-C 7 H 16  In all cases, LD is larger for side-on than for head-on structures. In fact, the former are more compact and feature many TMÁ Á ÁHC contacts. Consistent with this observation, the TM-alkane binding energy is always slightly larger for side-on structures, with the only exception being the positively charged [(HEB)Re(CO) 2 (n-C 5 H 12 )] + . In this case, the positive charge on the metal probably increases the relative importance of the non-dispersive interaction components, such as polarization and TM ' CH s-donation. Indeed, by subtracting the LD energy from the binding energy, one finds that the nondispersive contributions favour head-on structures over sideon structures in all cases. In fact, due to the less congested environment around the TM center, the head-on structures feature slightly smaller r (TM-C) values (Fig. 2), and charge transfer and polarization effects decay quickly with the intermolecular distance. 29,73 This aspect will be further discussed in Section 3.3 while investigating the physical components affecting the selectivity of CH activation reactions.
These results demonstrate that two opposing effects, i.e., dispersive and non-dispersive energy contributions, determine the coordination preference of s-complexes. Although LD typically dominates in s-complexes, the non-dispersive contributions are also important and particularly significant in charged species. These results are consistent with previous experimental NMR studies showing that [CpRe(CO) 2 (n-C 5 H 12 )] has a slight preference to form side-on structures, 1,75 whilst [(HEB)Re(CO) 2 (n-C 5 H 12 )] + favors head-on structures. 76 In summary, LD is fundamental for the thermodynamic stability of s-complexes. It favors the formation of s-complexes of long alkanes, influences the alkane coordination preference and favors the formation of compact structure with many TMÁ Á ÁCH contacts.

Activation of alkanes in r-complexes
The finding that LD forces determine to a large extent the coordination preference of the alkane, favoring the formation  of compact side-on structures, is in apparent contrast with the fact that the majority of alkane CH activation reactions take place at the methyl carbon, which is usually qualitatively explained in terms of steric effects. 77,78 To provide an insight into this aspect, we investigated a model oxidative addition step in which n-pentane interacts with CpRe(CO) 2 to form the oxidative addition product via the formation of the s-complex intermediate [CpRe(CO) 2 (n-C 5 H 12 )] (Scheme 2a). This reaction has been recently studied by means of DFT calculations, but the role played by the LD energy in this context was not discussed. 53 The computed mechanism at the DLPNO-CCSD(T) level of theory is shown in Fig. 4.
As [CpRe(CO) 2 (n-C 5 H 12 )] (II) is the absolute minimum in the section of the potential energy surface investigated, the oxidative addition is thermodynamically disfavored. This is consistent with the fact that the [CpRe(CO) 2 (n-C 5 H 12 )] intermediate can be characterized by NMR spectroscopy. 1 Although the side-on [CpRe(CO) 2 (n-C 5 H 12 )] (II) is thermodynamically more stable than the head-on [CpRe(CO) 2 (n-C 5 H 12 )] (I) conformer, consistent with the above mentioned experimental observation, the cleavage of the coordinated CH bond in the former requires a larger free energy barrier (DG ‡ = 40.9 kJ mol À1 ) than in the latter (DG ‡ = 29.3 kJ mol À1 ). These results are consistent with previous experimental findings on closely related systems. 79 These results can be explained by considering that the CH bond has to be in close proximity to the TM in order to be activated towards the oxidative addition. However, the equilibrium TMÁ Á ÁC distance in [CpRe(CO) 2 (n-C 5 H 12 )] (II) is significantly longer than that of [CpRe(CO) 2 (n-C 5 H 12 )] (I) (2.68 vs. 2.63 Å) and this difference is partially retained also in the corresponding transition states (2.43 vs. 2.41 Å) and products (2.30 vs. 2.28 Å). This effect probably originates from significant steric effects in the side-on conformation, due to the congested environment around the TM center.
Consistent with these observations, the non-dispersive contribution to the binding energy amounts to À22.4 and À26.8 kJ mol À1 for [CpRe(CO) 2 (n-C 5 H 12 )] (II) and [CpRe(CO) 2 (n-C 5 H 12 )] (I), respectively, and to +63.4 and +52.6 kJ mol À1 for the corresponding transition states. Hence, polarization and charge transfer effects seem to favor the activation of the CH bond at the methyl carbon.
In summary, even though LD determines to a large extent the thermodynamic stability of s-complexes, the activation barrier associated with the oxidative addition step is determined by an interplay of LD, steric and charge transfer effects.

Conclusions
The recently introduced DLPNO-CCSD(T)/LED scheme was used to compute TMÁ Á Áalkane binding energies for a broad range of experimentally characterized s-complexes and to provide an accurate quantification of the London dispersion component of the coordination bond.
It was found that London dispersion significantly contributes to the TMÁ Á Áalkane interaction and is largely responsible for the thermodynamic stability of s-complexes. Its magnitude can be modulated by modifying the size of the substituents on the TM fragment or by increasing the number of carbons in the alkane.
Moreover, LD influences the coordination preference of the alkane to the TM. As LD increases with the number of TMÁ Á ÁCH contacts, side-on structures are generally favored over head-on structures, consistent with previous experimental findings. On the other hand, due to the steric effects, head-on structures This journal is © the Owner Societies 2019 feature a shorter distance between the TM and the coordinated carbon, leading to larger polarization and charge transfer interactions.
As charge transfer effects weaken the CH bond, activating it towards the cleavage, the oxidative addition of methyl CH bonds to the TM is expected to be kinetically preferred, as shown in a prototype oxidative addition reaction and consistent with previous experimental findings.
As LD is ubiquitous in nature, it is expected that our findings concerning its importance in organometallic chemistry can be applied to other TM-L interactions, such as the wide range of s-complexes that serve as intermediates in CH activation reactions.

Conflicts of interest
There are no conflicts to declare.