Inorganica Chimica Acta 360 (2007) 179–189 www.elsevier.com/locate/ica Metal–ligand bonding in metallocenes: Differentiation between spin state, electrostatic and covalent bonding Marcel Swart Institucio´ Catalana de Recerca i Estudis Avanc¸ats (ICREA), 08010 Barcelona, Spain Institut de Quı´mica Computacional, Universitat de Girona, Campus Montilivi, 17071 Girona, Spain Received 16 June 2006; received in revised form 26 July 2006; accepted 26 July 2006 Available online 5 August 2006 Inorganic Chemistry – The Next Generation. Abstract We have analyzed metal–ligand bonding in metallocenes using density functional theory (DFT) at the OPBE/TZP level. This level of theory was recently shown to be the only DFT method able to correctly predict the spin ground state of iron complexes, and similar accuracy for spin ground states is found here. We considered metallocenes along the first-row transition metals (Sc–Zn) extended with alkaline-earth metals (Mg, Ca) and several second-row transition metals (Ru, Pd, Ag, Cd). Using an energy decomposition analysis, we have studied trends in metal–ligand bonding in these complexes. The OPBE/TZP enthalpy of heterolytic association for ferrocene (À658 kcal/mol) as obtained from the decomposition analysis is in excellent agreement with benchmark CCSD(T) and CASPT2 results. Covalent bonding is shown to vary largely for the different metallocenes and is found in the range from À155 to À635 kcal/mol. Much smaller variation is observed for Pauli repulsion (55–345 kcal/mol) or electrostatic interactions, which are however strong (À480 to À620 kcal/mol). The covalent bonding, and thus the metal–ligand bonding, is larger for low spin states than for higher spin states, due to better suitability of acceptor d-orbitals of the metal in the low spin state. Therefore, spin ground states of transition metal com- plexes can be seen as the result of a delicate interplay between metal–ligand bonding and Hund’s rule of maximum multiplicity. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Metallocenes; Metal–ligand bonding; Density functional theory; Spin state splitting 1. Introduction are still under debate, for instance, in the case of the origin of the rotational barrier in ethane [4–6]. The most straight- Predicting chemical bonding within stable organic com- forward and intuitive approach to understanding chemical pounds is relatively straightforward, with most of these bonding is presented by using an energy decomposition molecules having a closed-shell electronic configuration. analysis (EDA) [7], which partitions the interaction energy This picture changes dramatically when turning to metal into physically meaningful components such as Pauli repul- compounds, especially when dealing with (transition) metal sion, electrostatic interactions and orbital interactions. As atoms having partially filled d-shells [1]. In that case, one such it has been applied (among many others) to hydrogen has to consider more than one possible spin state, which bonding in DNA base pairs [8–10], the origin of the rota- are in many cases close in energy (vide infra) [2,3]. This is tional barrier in ethane, [4] and hydrogen–hydrogen inter- however not the only concern, as our understanding and actions in (non-)planar biphenyl [11,12]. Rayo´n and the interpretation of the nature of chemical bonding within Frenking also used EDA [13] to study the nature of chem- either organic molecules or (transition) metal compounds ical bonding in transition metal compounds, which enabled them for instance to differentiate between the d-bonded bis(benzene)chromium and the p-bonded ferrocene. For E-mail address: [email protected]. understanding the differences between accessible spin states 0020-1693/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ica.2006.07.073 180 M. Swart / Inorganica Chimica Acta 360 (2007) 179–189 in transition metal compounds, it is however not sufficient of iron complexes. For instance, in a recent study [16] we to understand chemical bonding with EDA, but one should have shown that only a small number of DFT functionals also be able to make accurate predictions about the spin are able to correctly predict the sextet ground state of a ground state of these molecules. high-spin iron compound. In that study, we used the crystal Recently, we have shown that the reliability of density structure for all three spin states, and since the crystal functional (DFT) [14,15] methods for giving a proper structure was obtained with for instance the sextet mole- description of relative spin state energies (i.e. spin-state cule in its high spin state, the other spin states (low and splittings) depends largely on the functional form of the intermediate) were disfavored, e.g. they are not in their exchange functional [16]. Standard DFT methods like ‘‘natural’’ geometry. In a follow-up paper [27], we per- BP86 [17,18], BLYP [17,19] and PW91 [20] (that contain formed a more thorough check on the validation of density mainly s2 terms in the formulation of their exchange part, functional methods for spin-state splittings, by letting the with s being the reduced density gradient) systematically structure of each spin state separately relax towards its favor low spin states [16]. Hybrid DFT methods like own equilibrium geometry. This resulted in another dra- B3LYP [21,22] that include a portion of Hartree–Fock matic reduction of the list of reliable DFT methods for (HF) exchange systematically favor high spin states and spin-state splittings, basically leaving OPBE [16,28] as the suffer from spin contamination [16], which result directly only DFT method capable of describing spin-state split- from the inclusion of HF exchange. The tendency of tings of iron complexes correctly. Hartree–Fock to favor high spin states can easily be under- Apart from the electronic structure, knowledge of the stood, most conveniently by looking at a d5 system, i.e. a molecular structure forms the basis from which we can pro- system with five d-electrons that can either be all parallel ceed with attempting to understand chemistry and molecu- (see Fig. 1, left) to give the high spin (sextet) state, or form lar biology. By now, molecular structures can be routinely two electron pairs and one single electron (Fig. 1, right) to obtained through experimental techniques, such as X-ray give the low spin (doublet) state. For simplification, sup- diffraction at crystals or NMR spectroscopy, for molecules pose that all five d-orbitals have the same energy level, as ranging in size from a few atoms to biomolecules of several indicated in the figure. thousands of atoms. There are however limitations on the One of the characteristics of Hartree–Fock [23] is the applicability and accuracy of these experimental methods, absence of (favorable) electron correlation between unlike for example, if crystallization is problematic or if crystalli- spins, leaving only (favorable) electron correlation between zation leads to undesired structural deformations as in the like spins through exchange interactions. Now, if we label case of the ‘‘polymerization’’ of manganocene (vide infra). the five electrons as a, b, c, d and e and only look at unique Moreover, reactive intermediates are often of key interest combinations, the exchange interactions for the sextet state but too short-lived for experimental characterization. A on the left in Fig. 1 are a–b, a–c, a–d, a–e, b–c, b–d, b–e, way out of this problem is provided by quantum chemistry c–d, c–e, d–e, i.e. 10 in total. For the doublet state, there [23] that allows for computing the energy of any geometric are only four exchange interactions, i.e., if electrons a–c configuration of a given set of atoms and, thus, also of all are spin-up and d–e spin-down we find: a–b, a–c, b–c, d–e. stationary points on the energy hypersurface [23].An Therefore, Hartree–Fock will always favor high(er) spin important contribution to the successes of quantum chem- states, and as a result, the larger the portion of HF istry (after the obviously important improvements in the exchange in a hybrid functional, the more the hybrid func- accuracy of the quantum chemical methods themselves) tional will favor high spin states. Almost all hybrid func- comes from the ongoing development of still better, i.e. tionals suffer from this, including the popular B3LYP more efficient and numerical accurate techniques for the functional. This has been recognized before by Reiher optimization of molecular geometries. Essential for per- and co-workers [24,25], which lead them to propose a low- forming the optimization efficiently is to choose an appro- ering of the amount of HF exchange within B3LYP, from priate coordinate system, which should both be easy to 20% to 15%. This new functional was called B3LYP*, and construct and enable the full optimization of any geometric although it is performing better than the original B3LYP configuration of a number of atoms. Baker and co-workers functional for spin-state splittings, it is still not as accurate [29] previously showed that delocalized coordinates, which as hoped for (vide infra). are easily made, work well for molecules containing only More important than focusing on the shortcomings of strong (i.e. intramolecular) coordinates, but the application standard and hybrid functionals is the finding [16] that to weakly bound systems was less successful [30,31]. recent and improved pure functionals that also include s4 Recently, we adapted [31] the original delocalized setup terms, such as the OPTX [26] exchange functional, have to be able to treat both strong (intramolecular) and weak been shown to perform better for the spin-state splittings (intermolecular) coordinates efficiently and accurately. 2. Experimental Fig. 1. Schematic representation of sextet (left) and doublet (right) state of All calculations were performed with the Amsterdam ad5 system.