Differentiation Between Spin State, Electrostatic and Covalent Bonding

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 (658 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 155 to 635 kcal/mol. Much smaller variation is observed for Pauli repulsion (55–345 kcal/mol) or electrostatic interactions, which are however strong (480 to 620 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. 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 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. Density Functional (ADF) program developed by Baer- M. Swart / Inorganica Chimica Acta 360 (2007) 179–189 181

ends and others [32,33]. Molecular orbitals (MOs) were DH 298 ¼ DEtrans;298 þ DErot;298 þ DEvib;0 þ DðDEvib;0Þ298 expanded using a large, uncontracted set of Slater-type þ DðpV Þð1Þ orbitals: TZP [34]. The TZP basis is of triple-f quality, aug- mented by one set of polarization functions. Core electrons Here, DEtrans,298, DErot,298 and DEvib,0 are the differences be- (e.g. 1s for second period, 1s2s2p for third and fourth per- tween the reactants (i.e. M2+ + 2Cp, the isolated metal-ion iod, 1s2s2p3s3p3d for fifth period) were treated by the fro- and the cyclopentadienyl anion rings) and product (i.e. zen core (FC) approximation [33], which has a negligible MCp2, the metallocene) in translational, rotational and effect on the accuracy of geometries [35,36] and energies zero point vibrational energy, respectively; D(DEvib,0)298 is [37–40]. An auxiliary set of s, p, d, f, and g STOs was used the change in the vibrational energy difference as one goes to fit the molecular density and to represent the Coulomb from 0 to 298.15 K. The vibrational energy corrections and exchange potentials accurately in each SCF cycle. Sca- are based on our frequency calculations. The molar work lar relativistic corrections were included self-consistently term D(pV)is(Dn)RT; Dn = 2 for three reactants 2+ using the Zeroth Order Regular Approximation (ZORA) (M + 2Cp ) associating to one product (MCp2). Thermal [41]. corrections for the electronic energy are neglected. Energies and gradients were calculated using the local The total interaction energy DE for the heterolytic asso- 2+ density approximation (LDA; Slater exchange and VWN ciation reaction M + 2Cp ! MCp2 (see below for dis- [42] correlation) with non-local corrections due to cussion of heterolytic versus homolytic association) Handy–Cohen (OPTX exchange) [26] and Perdew– results directly from the Kohn–Sham molecular orbital Burke–Ernzerhof (PBEc correlation) [43] added self-con- (KS-MO) model [7] and is made up of two major compo- sistently. This is the OPBE [28] density functional, which nents (Eq. (2)): is one of the best DFT functionals for the accuracy of DE ¼ DEprep þ DEint ð2Þ vibrational frequencies [28], reaction barriers [44], spin state splittings [16], and geometries [28]; for the accuracy In this formula, the preparation energy DEprep is the energy of geometries an estimated unsigned error of 0.009 A˚ in needed to prepare the ionic fragments and consists of three combination with the TZP basis set is observed [28]. terms (Eq. (3)): The restricted and unrestricted formalisms were used DEprep ¼ DEdeform þ DEcyc–cyc þ DEvalexc ð3Þ for closed-shell and open-shell species, respectively. Spin contamination was in all cases but one found to be neg- The first is the energy needed to deform the separate molec- ligible (the higher lying doublet state of vanadocene ular fragments (in this case only for the cyclopentadienyl showed spin contamination, i.e. an expectation value of anion (Cp) rings) from their equilibrium structure to the S2 of 1.75; a pure spin doublet has an S2 value of 0.75), geometry that they attain in the overall molecular system with typical values of 0.753 for doublet states, or 2.03 (DEdeform), the second (DEcyc–cyc) is the interaction energy for triplet states. In a previous study [45], we used spin between the two Cp rings, which results from electrostatic projection techniques to correct for the spin contamina- repulsion between the negatively charged Cp rings while tion and showed that these corrections are negligible, making one fragment file that contains both Cp rings. and can safely be ignored, for such a small spin contam- The third term is the valence-excitation energy needed to ination. However, for the severely contaminated doublet prepare the metal from its spin-unrestricted (polarized) io- state of vanadocene, the spin projection technique [45] nic state to the spin-restricted (polarized) ionized form was used (both for the energy and the gradients) to (DEvalexc). The valence-excitation energy consists of two obtain the results for the pure spin doublet. terms: the first (positive) term is the energy difference be- Geometries were optimized using the QUILD (QUan- tween the (spherical/non-spherical) spin-polarized metal tum-regions Interconnected by Local Descriptions) (2+) cation in its ground state (e.g. the 5D state for Fe2+) [31,46] program, which is designed for QM/QM and and the spin-restricted cationic form used for the metal cat- QM/MM applications but can also handle QM-only calcu- ion fragment (the fragments need to be spin-restricted). For lations. The QUILD program uses an improved optimiza- the ground state of the ion, we use the ‘‘average of config- tion scheme [31] with adapted delocalized coordinates to uration’’ approach [47]. The second (negative or zero) term handle weakly bound systems, and serves as a wrapper results from changing the fragment occupations for non- around ADF; QUILD handles the geometry optimization singlet metallocenes to reflect the multiplet character of scheme, while ADF only serves to provide the energy and the metal; for singlets, this term is zero. For instance for analytical gradients. Geometries were considered con- copper(II), the spin-restricted cationic fragment is prepared verged when the maximum component of the delocalized with 4.5a and 4.5b d-electrons; within the molecule calcula- gradient was less than 1.0 · 105 atomic units. Vibrational tion, the occupations of the copper-fragment are changed frequencies were obtained from the analytical Hessian. to make 5a and 4b d-electrons, which will lower the energy. Enthalpies at 298.15 K and 1 atm (DH298) were calculated This energy lowering makes up the second part of the va- from electronic bond energies (DE) and vibrational fre- lence-excitation energy. quencies using standard thermochemistry relations for an The interaction energy DEint is the energy released when ideal gas [23], according to Eq. (1): the prepared fragments (i.e. M2+ + 2Cp) are brought 182 M. Swart / Inorganica Chimica Acta 360 (2007) 179–189 together into the position they have in the overall molecule. It is analyzed for our model systems in the framework of the KS-MO model using a Morokuma-type decomposition [48] into electrostatic interaction, Pauli repulsion (or exchange repulsion), and (attractive) orbital interactions (Eq. (4)).

DEint ¼ DV elstat þ DEPauli þ DEorbint ð4Þ

The term DVelstat corresponds to the classical electrostatic interaction between the unperturbed charge distributions of the prepared (i.e. deformed) fragments and is usually eclipsed staggered attractive. The Pauli-repulsion, DEPauli, comprises the destabilizing interactions between occupied orbitals and is Fig. 2. Eclipsed and staggered conformation of metallocenes. responsible for the steric repulsion. The orbital interaction DEorbint in any MO model, and therefore also in Kohn– considered both conformations and found indeed the Sham theory, accounts for electron-pair bonding, charge eclipsed conformation to be the most stable conformation. transfer (i.e. donor–acceptor interactions between occupied Because we are interested in the differences in bonding orbitals on one fragment with unoccupied orbitals of the between different metals, and inasmuch not the conforma- other, including the HOMO–LUMO interactions), and tion, we will discuss only the results for the eclipsed confor- polarization (empty-occupied orbital mixing on one frag- mations. Likewise, although some compounds may be ment due to the presence of another fragment). The orbital expected [55] to exhibit Jahn–Teller distortions to lower interaction energy can be decomposed into the contribu- the symmetry and thus the energy, we explicitly considered tions from each irreducible representation C of the interact- the molecules only within D5h symmetry, to be able to ing system (Eq. (5)) using the extended transition state make a fair comparison of the bonding in the different (ETS) scheme developed by Ziegler and Rauk [49,50]. X metallocenes. DEorbint ¼ DEC ð5Þ C 3.1. Geometric parameters of metallocenes The choice of either neutral (M + 2Cp, i.e. homolytic asso- We begin by looking at the OPBE/TZP [28] optimized ciation) or ionic (M2+ + 2Cp, i.e. heterolytic association) geometries of the metallocene molecules. The metal–car- fragments will influence the absolute value of the bonding bon, carbon–carbon, carbon–hydrogen distances for all energy significantly, and thus also the energy components; spin states of all metallocenes are reported in Table 1, however, Rayo´n and Frenking [13] recently showed that which also contains the bending angle of the hydrogens the contribution of covalent versus electrostatic bonding with respect to the C plane. The OPBE metal–carbon dis- remains rather constant, regardless of the choice of using 5 tances are ca. 0.04 A˚ smaller than either ab initio bench- either neutral or ionic fragments. Ionic fragments have mark studies [56], or experimental values [57] (where been used throughout this paper. available), which is reasonably close. Moreover, the influ- ence of these differences on the energies is expected to be 3. Results and discussion small. Previously, it was assumed that the hydrogens in ferro- We have studied trends in metal–ligand bonding for a cene would be either in the plane (zero angle), or bent series of metal sandwich complexes (metallocenes) that out of the plane and away from the metal atom (negative contain the first-row transition metals (Sc–Zn), extended angle) [56]. Experimental studies using gas-phase electron with alkaline-earth metals (Mg, Ca) and several second- diffraction on ferrocene however showed bent C–H bonds row transition metals (Ru, Pd, Ag, Cd). The geometries with the hydrogens pointing towards the metal, i.e., a posi- of all species were fully optimized (using a convergence cri- tive angle with a value of 5. Later experimental studies terion of 1.0 · 105 a.u.) at the OPBE/TZP [28] level with reduced this number to 3.7, while a recent benchmark the QUILD program [31,46], and characterized by inspect- study using CCSD(T)/TZV2P+f gave a value of 1.03 ing the vibrational frequencies from the analytical Hessian. [56]. With OPBE/TZP we find a value of 1.22 for the angle For the transition metals that are expected to have elec- (see Table 1), which is in good agreement with the bench- tronic states in close proximity, such as the singlet, triplet, mark study. However, it is not true that all hydrogen atoms and quintet state of Fe(II) complexes, the geometry and are pointing towards the metal atom, as can be seen in metal–ligand bonding were studied for each spin state Table 1; out of the 29 metallocenes studied, only seven have separately. the hydrogens pointing towards the metal. The other 22 Although previous studies [51–54] already showed the metallocenes either have almost planar C5H5 rings (5) with eclipsed (D5h) conformation to be lower in energy than an absolute value of the angle <0.2 or have the hydrogens the staggered (D5d) conformation (see Fig. 2), we initially pointing away from the metal (17). M. Swart / Inorganica Chimica Acta 360 (2007) 179–189 183

Table 1 Selected bond distances and anglesa (A˚ and ) of metallocenes in eclipsed conformation Metal R(M–C) R(C–C) R(C–H) \(H–Cp)b Spin Low Interm High Low Interm High Low Interm High Low Interm High Mg 2.343 1.419 1.086 1.18 Ca 2.638 1.415 1.087 2.46 Sc 2.430 1.421 1.087 1.59 Ti 2.283 2.304 1.427 1.424 1.086 1.086 0.53 0.29 V 2.219 2.249 1.424 1.421 1.087 1.086 0.75 0.08 Cr 2.083 2.134 2.323 1.436 1.429 1.420 1.085 1.086 1.086 1.43 0.58 0.31 Mn 2.046 2.201 2.379 1.434 1.426 1.420 1.085 1.086 1.087 1.22 0.11 0.87 Fe 2.007 2.110 2.277 1.431 1.424 1.430 1.087 1.086 1.086 1.22 0.71 0.14 Co 2.070 2.222 1.427 1.422 1.086 1.086 0.70 0.06 Ni 2.148 2.158 1.425 1.424 1.086 1.086 0.20 0.46 Cu 2.247 1.423 1.086 0.29 Zn 2.299 1.424 1.086 0.51 Ru 2.155 2.269 2.452 1.431 1.428 1.422 1.086 1.085 1.086 1.07 0.32 0.79 Pd 2.355 2.356 1.423 1.422 1.086 1.086 0.95 0.91 Ag 2.510 1.421 1.086 1.52 Cd 2.511 1.424 1.086 2.13 a Calculated at OPBE/TZP. b Angle between hydrogens and cyclopentadienyl anion ring, positive value means hydrogen is bent towards metal [56].

0 0 3.2. Spin-state splittings lowest lying orbitals are two E2 orbitals, followed by an A1 00 orbital and two higher lying E1 orbitals. We continue by looking at the spin ground state of the The first d-electron (in scandanocene) occupies one of 0 metallocenes. The relative spin state energies (OPBE/ the E2 orbitals (doublet state), while the second (in titano- 0 TZP) for the metallocenes are given in Table 2, which also cene) occupies the other E2 orbital with parallel spin to contains the metals with only one accessible spin state. The form the triplet state. The third d-electron in vanadocene 0 predicted spin ground state for the metallocenes is in all occupies with parallel spin the close-lying A1 orbital to give cases in perfect agreement with experimental data, thereby the quartet ground state (see Fig. 3, right). contributing to the reliability of the OPBE functional for The additional (fourth) d-electron in chromocene could 00 assessing spin-state splittings in transition metal complexes. in principle occupy one of the higher-lying E1 orbitals to Moreover, the spin ground states are easily comprehensible give an overall high spin quintet state, however a lower by looking at the level diagram of the 3d-orbitals (schemat- energy is obtained when it forms a pair with the first d-elec- ically represented in Fig. 3, left): within D5h symmetry, the tron to give an overall triplet ground state. The same hap- 00 pens with manganocene, which could have occupied an E1 orbital to form the high-spin sextet state, but instead forms Table 2 0 a,b Relative spin state energies (kcal/mol) of metallocenes in eclipsed the second E2 electron pair to give a doublet ground state. conformation Note that at room temperature and in the solid state, Metal d-Electrons Low Intermediate High manganocene is found with a high spin ground state [58]. Mg d0 0.0 (sing) Ca d0 0.0 (sing) ' ' '' Sc d1 0.0 (doub) E2 A1 E1 Ti d2 19.6 (sing) 0.0 (trip) Sc 3 Vd 44.9 (doub) 0.0 (quar) Ti Cr d4 34.8 (sing) 0.0 (trip) 8.0 (quin) 5 '' V Mn d 0.0 (doub) 27.6 (quar) 8.2 (sext) E1 0Fe d6 0.0 (sing) 49.3 (trip) 51.9 (quin) Cr 7 Co d 0.0 (doub) 28.0 (quar) Mn Ni d8 15.3 (sing) 0.0 (trip) 9 Fe Cu d 0.0 (doub) ' 10 A1 Zn d 0.0 (sing) Co Ru d6 0.0 (sing) 76.3 (trip) 114.0 (quin) E ' 2 Ni Pd d8 8.5 (sing) 0.0 (trip) Ag d9 0.0 (doub) Cu 10 Cd d 0.0 (sing) Zn a Calculated at OPBE/TZP. b The terms in parentheses refer to the spin state, e.g. sing for singlet, trip Fig. 3. Schematic representation of 3d-orbital level scheme (left) and for triplet, quin for quintet, doub for doublet, quar for quartet and sext for occupation of the 3d-orbitals in the first-row transition metallocenes sextet. (right). 184 M. Swart / Inorganica Chimica Acta 360 (2007) 179–189

It benefits at elevated temperatures from geometry distor- tions to form a chain-like structure (see Fig. 4), distortions we do not consider in this study, with anti-ferromagnetic coupling between neighboring manganese atoms [58].At lower temperatures, the sandwich structure and the corre- Fig. 4. Chain-like structure for manganocene in the solid state. sponding doublet ground state are retrieved. Manganocene gives a clear example of the problems of hybrid functionals with spin-state splittings (vide supra); ture of manganocene a high spin ground state at ca. Reiher and co-workers [25] used B3LYP* on a small num- 1 kcal/mol below the low spin ground state. Although the * ber of metallocenes and found for the eclipsed D5h struc- performance of B3LYP is better than the original

Table 3 a Decomposition of metal–ligand bonding (kcal/mol) for eclipsed metallocenes in D5h symmetry

MgCp2 CaCp2 ScCp2 TiCp2 VCp2 CrCp2 MnCp2 FeCp2 Spin state singlet singlet doublet triplet quartet triplet doublet singlet

DEprep 78.3 67.6 76.0 88.5 78.3 182.8 301.4 240.6 DEdeform 0.3 0.3 0.5 0.6 0.4 1.3 2.3 1.6 DEcyc–cyc 78.0 67.3 74.4 79.9 82.7 90.7 99.1 103.4 DEvalexc 0.0 0.0 1.1 7.9 4.8 90.8 200.0 135.6 DEint 641.0 557.6 627.9 689.7 718.7 800.4 884.8 907.9 DEPauli 54.9 81.1 139.5 177.5 202.3 271.7 321.6 345.7 DEelstat 500.5 482.3 516.2 537.5 562.1 583.2 599.3 619.1 DEorbint 195.4 156.4 251.3 329.8 358.8 488.9 607.0 634.5 % Orbintb 28.1 24.5 32.7 38.0 39.0 45.6 50.3 50.6 0 A1 38.2 23.1 32.1 39.7 43.4 47.5 52.1 53.2 0 E1 57.5 27.0 36.4 44.1 48.5 56.6 63.8 67.6 0 E2 12.7 9.5 28.5 34.4 29.9 65.3 78.0 68.3 00 A2 27.5 12.9 17.0 20.7 22.1 26.1 30.3 31.1 00 E1 48.0 75.3 126.1 177.7 201.3 277.4 365.1 396.2 00 E2 11.5 8.5 11.1 13.2 13.6 15.9 17.8 18.1 BSSE 2.0 1.6 2.2 2.2 2.7 2.0 1.1 2.4 DE 560.7 488.4 549.7 599.1 637.7 615.6 582.4 665.0 DZPE 6.5 5.9 6.5 6.4 7.5 8.4 7.6 9.0

DDH298 1.5 1.3 1.5 1.6 1.8 2.0 2.1 2.4 DH 555.7 483.9 544.7 594.3 632.0 609.3 576.8 658.4 Exp.c 606 572 635 ± 6 Theory 655 ± 15

CoCp2 NiCp2 CuCp2 ZnCp2 RuCp2 PdCp2 AgCp2 CdCp2 Spin state doublet triplet doublet singlet singlet triplet doublet singlet

DEprep 166.2 83.0 82.8 80.9 185.9 71.7 71.2 72.4 DEdeform 1.1 0.7 0.6 0.7 1.7 0.5 0.5 0.8 DEcyc–cyc 96.2 88.6 83.0 80.3 89.8 77.5 71.4 71.6 DEvalexc 68.9 6.3 0.8 0.0 94.5 6.3 0.6 0.0 DEint 850.9 780.7 757.1 695.5 891.5 755.2 719.1 617.1 DEPauli 304.5 189.8 147.6 108.0 411.4 199.1 127.2 111.3 DEelstat 602.9 576.7 551.5 542.5 674.8 576.7 525.7 524.4 DEorbint 552.5 393.8 353.2 261.0 628.0 377.6 320.6 204.0 % Orbintb 47.8 40.6 39.0 32.5 48.2 39.6 37.9 28.0 0 A1 56.7 57.9 60.6 74.3 57.7 48.1 47.3 63.2 0 E1 68.2 68.7 65.9 79.4 56.3 49.7 44.7 59.6 0 E2 48.7 28.9 21.8 16.1 67.3 19.6 13.5 12.3 00 A2 31.8 32.7 31.8 37.3 25.1 24.4 22.8 29.2 00 E1 331.0 191.6 160.8 41.9 404.2 224.2 183.4 29.9 00 E2 16.1 14.0 12.2 12.1 17.4 11.6 8.9 9.7 BSSE 2.4 2.5 2.3 1.9 2.5 2.6 1.9 1.4 DE 682.4 695.3 672.0 612.7 703.1 680.9 646.0 543.3 DZPE 8.3 6.9 6.4 5.5 8.7 6.2 5.8 4.7

DDH298 2.2 1.7 1.3 1.0 2.3 1.4 1.0 0.7 DH 676.2 700.5 666.9 608.2 696.7 676.2 641.3 539.3 Exp.c 652 a 2+ Computed at OPBE/TZP for the reaction M + 2Cp ! MCp2, given here for the spin ground state of the metallocenes. b Contribution of orbital interactions to total attraction. c Experimental values should probably be corrected: they are obtained by taking the average of sequentially disrupting the metallocenes (see text). M. Swart / Inorganica Chimica Acta 360 (2007) 179–189 185

B3LYP, which predicts the high spin state to be even lower between CCSD(T) and experiment, however it should be by ca. 7 kcal/mol [25], neither of these functionals can be kept in mind that the difference of 30 kcal/mol was ob- trusted to provide the low spin ground state of this served for homolytic dissociation, not for heterolytic disso- molecule. ciation as in the theoretical studies. Therefore, we are 0 Within ferrocene also the A1 orbital is doubly occupied unable to draw any conclusions from comparing our other to give the overall singlet ground state. Continuing along computed enthalpy values with experimental data at the 00 the first-row transition metals, the higher lying E1 orbitals moment. It should be noted that our computed values for have to be occupied leading to, respectively, a doublet vanadocene, manganocene, ferrocene and nickelocene are (cobaltocene), triplet (nickelocene), doublet (cuprocene) all similar to the results from an earlier DFT study [61] and singlet (zincocene). For the second-row transition met- at the BP86/TZP level. als, the spin ground state does not change compared to the These enthalpies however do not reflect the stability of first-row metals in the same group, i.e. for ruthenocene we the metallocene compounds, as it is well-known that nick- find similar to ferrocene a singlet ground state, and for pal- elocene is easily oxidized, vanadocene readily reacts with ladocene a triplet ground state, similar to nickelocene. carbonmonoxide, and titanocene is even found in an unu- sual fulvalene-hydride structure. Neither of these effects, nor others such as protonation or oxygenation, are consid- 3.3. Metal–ligand bonding ered in this paper. Moreover, the difference in enthalpy is mainly determined by two terms: the valence-excitation Finally, we come to the analysis of the metal–ligand energy and the interaction energy. The former is part of bonding in the metallocenes. We analyze the metal–ligand the preparation energy, and does not contribute to the bonding in the metal sandwich complexes in terms of the metal–ligand bonding, but does influence the enthalpy heterolytic association reaction (Eq. (6)): trend. For instance, the valence-excitation energy first 2þ increases when going down the first-row transition metals M þ 2Cp ! MCp2 ð6Þ (see Table 3), from 1 kcal/mol for Sc to 8 for Ti and 5 The results of the energy decomposition analysis for the for V to 91 for Cr and 200 kcal/mol for Mn; then it spin ground states of the sixteen metallocene molecules decreases again to 135 kcal/mol for Fe, 69 for Co, 6 for are given in Table 3. The enthalpy of association for all Ni, and 1 for Cu. However, the focus of this study is these compounds is found in the range from 483 (calcio- on trends in metal–ligand bonding, which is determined cene) to 701 kcal/mol (nickelocene). We can compare entirely by the interaction energy. several of our computed values with either experiment or The interaction between the metal cation and the cyclo- high-level ab initio calculations. The experimental value pentadienyl anion ligands is in all cases very favorable, for the bond disruption enthalpy of ferrocene is 636 kcal/ although this may have been biased by having chosen mol [59] (i.e. an enthalpy of association of 636 kcal/ charged fragments. Rayo´n and Frenking [13] studied both mol), which is ca. 20 kcal/mol lower than the theoretical charged and neutral interacting ligands, and found a much best estimates of 657 (CASPT2) and 654 (CCSD(T)) smaller interaction energy with neutral fragments kcal/mol [53,60]; with CCSD(T) a value of 653 was found (274 kcal/mol) than with charged fragments (894 kcal/ for the staggered D5d conformation [60], which should be mol). However, the choice of fragments affected not only corrected with the energy difference of 1.15 kcal/mol [56] the electrostatic component of the interaction energy but between the staggered and eclipsed conformation. Our also the orbital interactions, and by almost the same factor. computed value of 658 kcal/mol is in excellent agreement Therefore, the relative contribution for each of the two with both these benchmark values. components remained rather constant. This changes when The discrepancy between theory and experiment is we focus on the trend for the metallocenes considered in unsatisfactory, especially since the theory benchmarks are this study. Both the absolute value as well as the relative of high quality. The origin of the discrepancy might be contribution of covalent bonding, i.e. the orbital interac- found in the experimental data, as they refer to the average tions, first increase if we go along the row of transition met- enthalpy of sequentially disrupting the M–Cp bonds [59]. als (see Table 3 and Fig. 5) until a maximum is reached at For example, for the homolytic dissociation of ferrocene, ferrocene. the experimental enthalpy is obtained by taking the average Continuing along the row, the absolute value of the value of the enthalpy for reactions (7a) and (7b). orbital interactions decreases and because the electrostatic interaction remains rather constant (because of the charged FeCp2 ! FeCp þ Cp ð7aÞ fragments), also the contribution of covalent bonding FeCp ! Fe þ Cp ð7bÞ decreases. For the few metallocenes we studied from the The removal of the first cyclopentadiene ligand was found second-row transition metals, a similar pattern emerges to be more endothermic than the second one by 30 kcal/ with strong covalent bonding in ruthenocene (the second- mol [59], which led to a value for the homolytic dissocia- row analog of ferrocene), which decreases if we continue tion of 79 kcal/mol instead of 95. This difference of along the row. Both the absolute value and contribution 16 kcal/mol is almost exactly equal to the difference of covalent bonding decrease if we go down the periodic 186 M. Swart / Inorganica Chimica Acta 360 (2007) 179–189

Fig. 5. Components of interaction energy (kcal/mol) for metal–ligand bonding in metallocenes.

˚ table, i.e. covalent bonding decreases from 195 to ScCp2 to FeCp2, the ML distance decreases from 2.43 A 156 kcal/mol when comparing magnesiocene and calcio- (Sc) to 2.30 (Ti), 2.25 (V), 2.13 (Cr), 2.05 (Mn) to 2.01 cene, from 635 kcal/mol for ferrocene to 628 for ruthe- (Fe); continuing towards the end of the row leads to an nocene, from 394 kcal/mol for nickelocene to 378 for increase of the distance again, from 2.07 A˚ (Co), 2.16 palladocene, from 353 kcal/mol for cuprocene to 321 (Ni), 2.25 (Cu) to 2.30 (Zn). for argentocene, and from 261 kcal/mol for zincocene Rayo´n and Frenking [13] made the distinction between to 204 for cadmocene. d-bonded CrBz2 and p-bonded FeCp2, and therefore it is Although covalent bonding, i.e. orbital interactions, is interesting to see if a similar difference in bonding appears the component of the interaction energy that changes most for chromocene versus ferrocene. Similar to their results, significantly for the different metals, also the other two we find ferrocene to be p-bonded, i.e. the largest orbital 00 components do vary along the set of metallocenes. For interactions are seen in the E1 irrep (396 kcal/mol, see instance, Pauli repulsion is relatively speaking small for Table 3), which is analogous to E1g in D5d symmetry; the 0 alkali-earth metallocenes (55–81 kcal/mol), but then E2 irrep, the analog of the E2g irrep from D5d symmetry, increases rapidly when the d-orbitals start to be occupied shows much smaller orbital interactions (68 kcal/mol). 00 in the transition metal series. Like the orbital interactions, The same is true for chromocene with values of 277ðE1Þ 0 it has a maximum for ferrocene (346 kcal/mol) and then and 65 ðE2Þ kcal/mol. Almost all other metallocenes are decreases again towards the end of the row, to reach found to be p-bonded with large orbital interactions for 00 108 kcal/mol for zincocene. The variation shown by Pauli the E1 irrep. However, for metallocenes with either fully repulsion along the first-row transition metals (ca. occupied or fully unoccupied d-shells such as zincocene 0 260 kcal/mol) is roughly half the variation in orbital inter- or magnesiocene, the orbital interactions of the A1 and 0 actions (ca. 480 kcal/mol). Electrostatic interactions, E1 irreps are almost as large as or larger than those in 00 although considerable in size (482–619 kcal/mol), vary the E1 irrep. the least among the three components (ca. 135 kcal/mol). Not surprisingly, the electrostatic interactions also have 3.4. Spin-state dependent metal–ligand bonding the maximum value for ferrocene, similar to the other two components. This suggests a synergy between the com- For the metallocenes with more than one accessible spin ponents: more favorable covalent bonding leads to shorter state, it is interesting to look at the differences in metal– metal–ligand distances, resulting in an increase of Pauli ligand bonding. Given in Table 4 is the decomposition of repulsion that is overcome by the joint increase in electro- the interaction energy for all spin states. The difference in static and covalent bonding. The metal–ligand (ML) dis- interaction energy between the several spin states is for tances from Table 1 confirm this trend, i.e. going from some metallocenes small (titanocene and vanadocene), with M. Swart / Inorganica Chimica Acta 360 (2007) 179–189 187

Table 4 a b Decomposition of metal–ligand bonding (kcal/mol) for eclipsed metallocenes in D5h symmetry

TiCp2 TiCp2 VCp2 VCp2 CrCp2 CrCp2 CrCp2 singlet triplet doublet quartet singlet triplet quintet

DEPauli 205.1 177.5 245.8 202.3 304.1 271.7 183.0 DEelstat 541.7 537.5 569.5 562.1 580.7 583.2 547.7 DEorbint 367.4 329.8 415.6 358.8 586.1 488.9 324.4 0 A1 40.1 39.7 57.5 43.4 407.9 47.5 42.5 0 E1 45.1 44.1 50.6 48.5 60.0 56.6 49.2 0 E2 61.6 34.4 58.8 29.9 230.1 65.3 20.1 00 A2 21.0 20.7 22.6 22.1 27.3 26.1 23.0 00 E1 186.0 177.7 211.8 201.3 303.1 277.5 177.6 00 E2 13.6 13.2 14.3 13.6 17.9 15.9 12.0 DEint 704.0 689.7 739.3 718.7 862.7 800.4 689.1

MnCp2 MnCp2 MnCp2 FeCp2 FeCp2 FeCp2 CoCp2 CoCp2 doublet quartet sextet singlet triplet quintet doublet quartet

DEPauli 321.6 243.7 121.9 345.7 250.5 154.2 304.5 161.4 DEelstat 599.3 570.0 536.2 619.1 574.5 552.4 602.9 558.5 DEorbint 607.0 430.8 239.2 634.5 496.0 302.9 552.5 355.8 0 A1 52.1 48.1 46.0 53.2 232.0 51.0 56.7 55.2 0 E1 63.8 57.8 54.0 67.6 62.8 60.4 68.2 64.0 0 E2 78.0 43.6 15.0 68.3 117.7 27.0 48.7 28.4 00 A2 30.3 27.3 25.5 31.1 29.4 29.1 31.8 31.9 00 E1 365.1 239.9 87.4 396.2 274.5 122.8 331.0 163.0 00 E2 17.8 14.1 11.2 18.1 15.0 12.7 16.1 13.3 DEint 884.8 757.1 653.5 907.9 820.0 701.1 850.9 752.9

NiCp2 NiCp2 RuCp2 RuCp2 RuCp2 PdCp2 PdCp2 singlet triplet singlet triplet quintet singlet triplet

DEPauli 244.5 189.8 411.4 344.6 175.2 255.1 199.1 DEelstat 579.3 576.7 674.8 608.8 551.8 577.1 576.7 DEorbint 475.1 393.8 628.0 529.1 292.7 456.0 377.6 0 A1 57.8 57.9 57.7 57.8 51.9 49.2 48.1 0 E1 67.1 68.7 56.3 51.8 45.8 48.3 49.7 0 E2 32.4 28.9 67.3 50.4 23.4 22.7 19.6 00 A2 31.8 32.7 25.1 24.9 22.3 23.5 24.4 00 E1 271.9 191.6 404.2 329.6 138.5 301.0 224.2 00 E2 14.1 14.0 17.4 14.5 10.7 11.3 11.6 DEint 809.9 780.7 891.5 793.3 669.2 778.0 755.2

a 2+ Computed at OPBE/TZP for the reaction M + 2Cp ! MCp2, given here for different spin states of several metallocenes.

00 a difference in interaction energy of only 14–21 kcal/mol. seen for the E1 irrep, i.e. the unoccupied d-orbitals in sin- However, the differences in the components are larger. glet ferrocene in which they serve as acceptor orbitals for For instance for vanadocene, the Pauli repulsion increases M L donor interactions. In triplet and quintet ferrocene, by 43 kcal/mol in going from the quartet ground state to these orbitals are partially occupied which results in them the doublet state that is being counteracted by a small being less suited as acceptor orbitals. Finally, for mangano- increase in DEelstat (7 kcal/mol) and a substantial increase cene we see a similar trend as for ferrocene, with the largest of DEorbint (57 kcal/mol). Similarly for titanocene, the elec- interaction energy for the low spin state (885 kcal/mol), trostatic component of the triplet ground state and singlet followed by the intermediate and high spin state, at 757 state is almost equal (4 kcal/mol), but an increase in Pauli and 654 kcal/mol, respectively. Also in the case of mang- repulsion (27 kcal/mol) and DEorbint (37 kcal/mol) lead to anocene is the change in interaction energy mainly result- an increase in interaction energy of 14 kcal/mol. ing from significant reductions in both DEPauli and DEorbint. For ferrocene, the singlet ground state has the largest The spin ground state of metallocenes can thus be interaction energy (908 kcal/mol) with the triplet and understood in terms of two opposing effects: metal–ligand quintet states significantly lower, at respectively 820 bonding that favors low spin states over higher spin states and 701 kcal/mol. The difference in interaction energy because of the better suitability of metal acceptor d-orbitals results from significant reductions in both DEPauli (95 in the former, and Hund’s rule of maximum multiplicity, and 190 kcal/mol, respectively, for the triplet and quintet which states that for atoms a larger total spin makes the state) and DEorbint (139 and 331 kcal/mol, respectively, atom more stable. Therefore, from the point of view of for triplet and quintet). The largest change in DEorbint is the metal, it wants to make the spin larger to satisfy Hund’s 188 M. Swart / Inorganica Chimica Acta 360 (2007) 179–189 rule, and at the same time make it smaller for creating bet- [4] F.M. Bickelhaupt, E.J. Baerends, Angew Chem. 115 (2003) 4315. ter acceptor orbitals. The compromise between these two [5] V. Pophristic, L. Goodman, Nature 411 (2001) 565. effects is delicate, and needs a careful and accurate method [6] F. Weinhold, Nature 411 (2001) 539. [7] F.M. Bickelhaupt, E.J. Baerends, Reviews in Computational Chem- such as OPBE to be able to correctly predict spin ground istry, 15, Wiley-VCH, New York, 2000, pp. 1–86. states of transition metal compounds. [8] C. Fonseca Guerra, F.M. Bickelhaupt, J.G. Snijders, E.J. Baerends, Chem.-Eur. J. 5 (1999) 3581. 4. Conclusions [9] C. Fonseca Guerra, F.M. Bickelhaupt, J.G. Snijders, E.J. Baerends, J. Am. Chem. Soc. 122 (2000) 4117. We have studied metal–ligand bonding in metallocenes [10] M. Swart, C. Fonseca Guerra, F.M. Bickelhaupt, J. Am. Chem. Soc. 126 (2004) 16718. using density functional theory at the OPBE/TZP level. [11] J. Poater, M. Sola`, F.M. Bickelhaupt, Chem.-Eur. J. 12 (2006) 2889. This level of theory was previously shown to be the only [12] J. Poater, M. Sola`, F.M. Bickelhaupt, Chem.-Eur. J. 12 (2006) 2902. DFT method capable of correctly predicting the spin [13] V.M. Rayo´n, G. Frenking, Organometallics 22 (2003) 3304. ground state of iron compounds, and we find similar good [14] R.G. Parr, W. 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