Article pubs.acs.org/JPCA
On the Mechanism of Dihydrogen Activation by Frustrated Lewis Pairs. Insights from the Analysis of Domain Averaged Fermi Holes and Generalized Population Analysis Robert Ponec*,† and Pavel Beran†,‡ †
Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic v.v.i., Prague 6, Suchdol 2, 165 02 Czech Republic ‡ Department of Physical Chemistry, Institute of Chemical Technology, Prague, Technická 5, 166 28 Prague 6, Czech Republic S Supporting Information *
ABSTRACT: The electron reorganization responsible for the facilitation of heterolytic splitting of H−H bond by frustrated Lewis pair (FLP) catalysts has been studied using the analysis of domain averaged Fermi holes and generalized population analysis. The analysis of electron structures of the species along the reaction path has revealed that the anticipated synchronicity of previously considered electron shifts of electron pair of the σHH bond to a vacant orbital on B and from the lone pair on the basic N site to an antibonding σHH* orbital is associated with the build up of extensive delocalized bonding that can conveniently be characterized in terms of multicenter bond indices. In addition, the detailed scrutiny of the IRC-dependence of the 2-center bond indices of the disappearing H−H bond resulted in the proposal of a simple heuristic measure of the efficiency of the FLP catalysts. Attention was also paid to the evaluation of the presumed facilitating effect on the dissociation of the H−H bond of the electric field in the cavity of FLP catalyst. It has been shown that the strength of this field does not reach the critical values required for the efficient facilitation of the splitting of the H−H bond.
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INTRODUCTION The recently discovered ability of the so-called frustrated Lewis pairs (FLP) to facilitate the activation of molecular hydrogen1 has rapidly been recognized as new challenging paradigm for the design of environmentally friendly metal-free hydrogenation catalysts and in the past few years a lot of effort has been spent at exploring the wide synthetic potential of this new class of catalysts2−9 as well as at revealing of the mechanistic details of the activation of the H−H bond.10−20 It has long been recognized that the catalytic activity of FLP is closely tied with the steric strain preventing the formation of an ordinary donor−acceptor bond between Lewis acid and base components and although there is wide general agreement about the heterolytic nature of the splitting of H−H bond,2 the intimate mechanistic details of the activation process are not still completely clarified.10,18 Thus, e.g., the original mechanistic proposal by Stephan,2 in which a stepwise character of the process has been assumed was reconsidered and based on the results of detailed computational study of the potential energy surfaces, the existence of a more favorable concerted reaction path has been reported,12,13 which finally resulted in the proposal of new mechanistic alternative in which the reaction is initiated by the formation of loosely bound but energetically strained (frustrated) complex whose stability originates from © 2013 American Chemical Society
noncovalent dispersion interactions between the parent Lewis acid and base components such as, e.g., [(tBu)3P]···[B(C6F5)3]. This complex acts as a highly reactive species which activates molecular hydrogen by electron transfer from the σHH bond to the acidic B center synchronously coupled with the donation of electrons from the lone pair of the basic site to the antibonding σ*HH orbital. More recently the conclusions of the study12,13 have been reconsidered by Grimme et al.,14 who also confirmed the crucial role of dispersion interactions but also stressed the importance of another factor that could effectively facilitate the anticipated heterolytic splitting of the H−H bond, namely the effect of electric field inside the FLP cavity. In order to contribute to the elucidation of the role of anticipated electron shifts in the mechanism of hydrogen activation, we report in this study the detailed analysis of electron reorganizations accompanying the heterolytic splitting of H−H bond in the FLP cavity using two original methodologies, namely the analysis of domain averaged Fermi holes21−23 and the generalized population analysis.24−26 Both of the above methodologies recently proved to be useful in revealing the Received: February 20, 2013 Revised: March 1, 2013 Published: March 19, 2013 2656
dx.doi.org/10.1021/jp4017932 | J. Phys. Chem. A 2013, 117, 2656−2663
The Journal of Physical Chemistry A
Article
of eigenvectors and eigenvalues that are in the second step subjected to the so-called isopycnic transformation,42 whose goal is to transform the original eigenvectors, that are usually delocalized over the whole molecule, to more localized functions reminiscent of localized orbitals, that can straightforwardly be associated with the classical concepts of bonds and/ or core and lone pairs in terms of which chemists are used to think of molecules. The structural information is primarily being extracted from the resulting populations of localized DAFH functions that allow to detect the bonding electron pairs (chemical bonds and also lone and/or core electron pairs) as well as broken or dangling valences resulting from the formal splitting of the bonds. The electron pairs are generally associated with populations close to 2, whereas the values significantly deviating from this natural limit usually correspond to broken valences. The interpretation of the above primary numerical data is greatly facilitated by the visual inspection of the associated DAFH functions. Because the size of the systems prevented us from performing the analysis at the level of the “exact” approach where the AOM matrices are determined using the integration over real AIM domains, we have used a simpler alternative form of DAFH analysis in which the integration over AIM domains is replaced by the Mulliken-like approach which assumes an electron to be in the domain of a given atom A if it is in an orbital attached to that atom. The reliability of such an approach that makes the analysis much less time-consuming has recently been addressed38 on other model systems by direct comparison with the “exact” AIM generalized approach, and as it has been shown, the differences between both alternative descriptions are only marginal and have no practical impact on the resulting picture of the bonding. Generalized Population Analysis. Another methodology we used to get the complementary insights into the detailed mechanism of hydrogen activation by FLP is the so-called generalized population analysis (GPA). The term GPA is a generic name denoting the whole family of approaches based on the partitioning of the identity (4) that holds at HF and/or DFT level of the theory, into mono-, bi-, tri-, and generally kcenter contributions.
nature of the bonding interactions in systems with nontrivial bonding pattern,27−36 and we believe that their application could bring new interesting insights also to the process of hydrogen activation by FLP. In addition to the insights provided by these two methodologies, the attention was also paid to the evaluation of the presumed role of the electric field as a further factor faciltiating the activation of the H2 molecule.
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THEORY DAFH Analysis. As the details of this analysis have repeatedly been described in previous studies,21−23 we confine ourselves here to only a brief summary of the basic ideas to the extent necessary for the purpose of this study. The most straightforward way of introducing the concept of domain averaged Fermi hole is via the selective integration of the socalled pair correlation function37 C(r , r′) = 2ρ(r , r′) − ρ(r )ρ(r′) g Ω (r ) = − = ρ (r )
∫Ω C(r , r′) dr′
∫Ω ρ(r′) dr′ − 2 ∫Ω ρ(r , r′) dr′
(1)
where ρ(r) and ρ(r,r′) denote ordinary one-electron and pair density, respectively, and the integration is over the finite domain of the space Ω. [For the sake of mathematical rigor it is possible to reformulate the whole approach in terms of density matrices instead of densities38 although either formulation has no impact on the practical applicability of DAFH analysis.] Although there are in principle no constraints of the form of the domains and constructed and analyzed can be the holes averaged over the domains of arbitrary size and shape, the practical experience from previous studies have shown that especially interesting and chemically relevant information can be extracted from the holes for which the domains Ω coincide with atomic domains resulting from any physically sound partitioning of electron density such as AIM,39 Hirschfeld,40 or Mulliken.41 In such a case the holes provide the information about the valence state of an atom in the molecule. However, the analysis can also be performed for more complex domains formed by the union of several atomic domains corresponding, e.g., to different functional groups, interesting molecular fragments, etc. In such a case the holes provide the information about the electron pairs (chemical bonds, core, and/or lone electron pairs) retained in the domain as well as about broken valences created by the formal splitting of the bonds required for the isolation of a given domain from the rest of the molecule. In the framework of HF and formally also DFT approximation (which is of our concern here) the eq 1 can be rewritten in the form (2)
1 k−1
2
=
∫Ω φi(r)φj(r) dr
A < B < ...K
The individual terms resulting from such a partitioning are known as the so-called bond indices and their usefulness for structural elucidations arises from the interesting nontrivial finding that the corresponding values mimic the presence and/ or absence of the bonding interactions between individual atoms in the molecule. Thus, e.g., the diatomic terms resulting from the partitioning (4) for k = 2 represent the 2-center bond indices identical to previously introduced Wiberg−Mayer bond orders.43,44 Similarly, the three-center bond indices45−47 resulting from the partitioning of the identity (4) for k = 3 have been widely used for the detection and the localization of the 3-center bonding in molecules,27−29 and a similar ability is retained also for the indices characterizing the extended bonding delocalized over more than 3-centers.30,48−54 Computations. In order to contribute to the ongoing discussion of the mechanism of the catalytic activity of FLPs,
(2)
where ⟨φi|φj⟩Ω denotes the elements of the so-called atomic overlap matrix (AOM) ⟨φi|φj⟩Ω =
A
