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Inorganica Chimica Acta 360 (2007) 179–189

Metal–ligand bonding in metallocenes: Differentiation betweenspin state, electrostatic and covalent bonding

Marcel Swart

Institucio Catalana de Recerca i Estudis Avancats (ICREA), 08010 Barcelona, Spain

Institut de Quımica Computacional, Universitat de Girona, Campus Montilivi, 17071 Girona, Spain

Received 16 June 2006; received in revised form 26 July 2006; accepted 26 July 2006Available 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 oftheory was recently shown to be the only DFT method able to correctly predict the spin ground state of iron complexes, and similaraccuracy for spin ground states is found here. We considered metallocenes along the first-row transition metals (Sc–Zn) extended withalkaline-earth metals (Mg, Ca) and several second-row transition metals (Ru, Pd, Ag, Cd). Using an energy decomposition analysis, wehave 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. Muchsmaller 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

Predicting chemical bonding within stable organic com-pounds is relatively straightforward, with most of thesemolecules having a closed-shell electronic configuration.This picture changes dramatically when turning to metalcompounds, especially when dealing with (transition) metalatoms having partially filled d-shells [1]. In that case, onehas to consider more than one possible spin state, whichare in many cases close in energy (vide infra) [2,3]. This ishowever not the only concern, as our understanding andthe interpretation of the nature of chemical bonding withineither organic molecules or (transition) metal compounds

0020-1693/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ica.2006.07.073

E-mail address: marcel.swart@icrea.es.

are still under debate, for instance, in the case of the originof the rotational barrier in ethane [4–6]. The most straight-forward and intuitive approach to understanding chemicalbonding is presented by using an energy decompositionanalysis (EDA) [7], which partitions the interaction energyinto physically meaningful components such as Pauli repul-sion, electrostatic interactions and orbital interactions. Assuch it has been applied (among many others) to hydrogenbonding in DNA base pairs [8–10], the origin of the rota-tional barrier in ethane, [4] and hydrogen–hydrogen inter-actions in (non-)planar biphenyl [11,12]. Rayon andFrenking also used EDA [13] to study the nature of chem-ical bonding in transition metal compounds, which enabledthem for instance to differentiate between the d-bondedbis(benzene)chromium and the p-bonded ferrocene. Forunderstanding the differences between accessible spin states

180 M. Swart / Inorganica Chimica Acta 360 (2007) 179–189

in transition metal compounds, it is however not sufficientto understand chemical bonding with EDA, but one shouldalso be able to make accurate predictions about the spinground state of these molecules.

Recently, we have shown that the reliability of densityfunctional (DFT) [14,15] methods for giving a properdescription of relative spin state energies (i.e. spin-statesplittings) depends largely on the functional form of theexchange functional [16]. Standard DFT methods likeBP86 [17,18], BLYP [17,19] and PW91 [20] (that containmainly s2 terms in the formulation of their exchange part,with s being the reduced density gradient) systematicallyfavor low spin states [16]. Hybrid DFT methods likeB3LYP [21,22] that include a portion of Hartree–Fock(HF) exchange systematically favor high spin states andsuffer from spin contamination [16], which result directlyfrom the inclusion of HF exchange. The tendency ofHartree–Fock to favor high spin states can easily be under-stood, most conveniently by looking at a d5 system, i.e. asystem with five d-electrons that can either be all parallel(see Fig. 1, left) to give the high spin (sextet) state, or formtwo electron pairs and one single electron (Fig. 1, right) togive the low spin (doublet) state. For simplification, sup-pose that all five d-orbitals have the same energy level, asindicated in the figure.

One of the characteristics of Hartree–Fock [23] is theabsence of (favorable) electron correlation between unlikespins, leaving only (favorable) electron correlation betweenlike spins through exchange interactions. Now, if we labelthe five electrons as a, b, c, d and e and only look at uniquecombinations, the exchange interactions for the sextet stateon the left in Fig. 1 are a–b, a–c, a–d, a–e, b–c, b–d, b–e,c–d, c–e, d–e, i.e. 10 in total. For the doublet state, thereare only four exchange interactions, i.e., if electrons a–care spin-up and d–e spin-down we find: a–b, a–c, b–c, d–e.Therefore, Hartree–Fock will always favor high(er) spinstates, and as a result, the larger the portion of HFexchange in a hybrid functional, the more the hybrid func-tional will favor high spin states. Almost all hybrid func-tionals suffer from this, including the popular B3LYPfunctional. This has been recognized before by Reiherand co-workers [24,25], which lead them to propose a low-ering of the amount of HF exchange within B3LYP, from20% to 15%. This new functional was called B3LYP*, andalthough it is performing better than the original B3LYPfunctional for spin-state splittings, it is still not as accurateas hoped for (vide infra).

More important than focusing on the shortcomings ofstandard and hybrid functionals is the finding [16] thatrecent and improved pure functionals that also include s4

terms, such as the OPTX [26] exchange functional, havebeen shown to perform better for the spin-state splittings

Fig. 1. Schematic representation of sextet (left) and doublet (right) state ofa d5 system.

of iron complexes. For instance, in a recent study [16] wehave shown that only a small number of DFT functionalsare able to correctly predict the sextet ground state of ahigh-spin iron compound. In that study, we used the crystalstructure for all three spin states, and since the crystalstructure was obtained with for instance the sextet mole-cule in its high spin state, the other spin states (low andintermediate) were disfavored, e.g. they are not in their‘‘natural’’ geometry. In a follow-up paper [27], we per-formed a more thorough check on the validation of densityfunctional methods for spin-state splittings, by letting thestructure of each spin state separately relax towards itsown equilibrium geometry. This resulted in another dra-matic reduction of the list of reliable DFT methods forspin-state splittings, basically leaving OPBE [16,28] as theonly DFT method capable of describing spin-state split-tings of iron complexes correctly.

Apart from the electronic structure, knowledge of themolecular structure forms the basis from which we can pro-ceed with attempting to understand chemistry and molecu-lar biology. By now, molecular structures can be routinelyobtained through experimental techniques, such as X-raydiffraction at crystals or NMR spectroscopy, for moleculesranging in size from a few atoms to biomolecules of severalthousands of atoms. There are however limitations on theapplicability and accuracy of these experimental methods,for example, if crystallization is problematic or if crystalli-zation leads to undesired structural deformations as in thecase of the ‘‘polymerization’’ of manganocene (vide infra).Moreover, reactive intermediates are often of key interestbut too short-lived for experimental characterization. Away out of this problem is provided by quantum chemistry[23] that allows for computing the energy of any geometricconfiguration of a given set of atoms and, thus, also of allstationary points on the energy hypersurface [23]. Animportant contribution to the successes of quantum chem-istry (after the obviously important improvements in theaccuracy of the quantum chemical methods themselves)comes from the ongoing development of still better, i.e.more efficient and numerical accurate techniques for theoptimization of molecular geometries. Essential for per-forming the optimization efficiently is to choose an appro-priate coordinate system, which should both be easy toconstruct and enable the full optimization of any geometricconfiguration of a number of atoms. Baker and co-workers[29] previously showed that delocalized coordinates, whichare easily made, work well for molecules containing onlystrong (i.e. intramolecular) coordinates, but the applicationto weakly bound systems was less successful [30,31].Recently, we adapted [31] the original delocalized setupto be able to treat both strong (intramolecular) and weak(intermolecular) coordinates efficiently and accurately.

2. Experimental

All calculations were performed with the AmsterdamDensity Functional (ADF) program developed by Baer-

M. Swart / Inorganica Chimica Acta 360 (2007) 179–189 181

ends and others [32,33]. Molecular orbitals (MOs) wereexpanded using a large, uncontracted set of Slater-typeorbitals: TZP [34]. The TZP basis is of triple-f quality, aug-mented by one set of polarization functions. Core electrons(e.g. 1s for second period, 1s2s2p for third and fourth per-iod, 1s2s2p3s3p3d for fifth period) were treated by the fro-zen core (FC) approximation [33], which has a negligibleeffect on the accuracy of geometries [35,36] and energies[37–40]. An auxiliary set of s, p, d, f, and g STOs was usedto fit the molecular density and to represent the Coulomband exchange potentials accurately in each SCF cycle. Sca-lar relativistic corrections were included self-consistentlyusing the Zeroth Order Regular Approximation (ZORA)[41].

Energies and gradients were calculated using the localdensity approximation (LDA; Slater exchange and VWN[42] correlation) with non-local corrections due toHandy–Cohen (OPTX exchange) [26] and Perdew–Burke–Ernzerhof (PBEc correlation) [43] added self-con-sistently. This is the OPBE [28] density functional, whichis one of the best DFT functionals for the accuracy ofvibrational frequencies [28], reaction barriers [44], spinstate splittings [16], and geometries [28]; for the accuracyof geometries an estimated unsigned error of 0.009 A incombination with the TZP basis set is observed [28].The restricted and unrestricted formalisms were usedfor closed-shell and open-shell species, respectively. Spincontamination was in all cases but one found to be neg-ligible (the higher lying doublet state of vanadoceneshowed spin contamination, i.e. an expectation value ofS2 of 1.75; a pure spin doublet has an S2 value of 0.75),with typical values of 0.753 for doublet states, or 2.03for triplet states. In a previous study [45], we used spinprojection techniques to correct for the spin contamina-tion and showed that these corrections are negligible,and can safely be ignored, for such a small spin contam-ination. However, for the severely contaminated doubletstate of vanadocene, the spin projection technique [45]was used (both for the energy and the gradients) toobtain the results for the pure spin doublet.

Geometries were optimized using the QUILD (QUan-tum-regions Interconnected by Local Descriptions)[31,46] program, which is designed for QM/QM andQM/MM applications but can also handle QM-only calcu-lations. The QUILD program uses an improved optimiza-tion scheme [31] with adapted delocalized coordinates tohandle weakly bound systems, and serves as a wrapperaround ADF; QUILD handles the geometry optimizationscheme, while ADF only serves to provide the energy andanalytical gradients. Geometries were considered con-verged when the maximum component of the delocalizedgradient was less than 1.0 · 10�5 atomic units. Vibrationalfrequencies were obtained from the analytical Hessian.Enthalpies at 298.15 K and 1 atm (DH298) were calculatedfrom electronic bond energies (DE) and vibrational fre-quencies using standard thermochemistry relations for anideal gas [23], according to Eq. (1):

DH 298 ¼ DEtrans;298 þ DErot;298 þ DEvib;0 þ DðDEvib;0Þ298

þ DðpV Þ ð1Þ

Here, DEtrans,298, DErot,298 and DEvib,0 are the differences be-tween the reactants (i.e. M2+ + 2Cp�, the isolated metal-ionand the cyclopentadienyl anion rings) and product (i.e.MCp2, the metallocene) in translational, rotational andzero point vibrational energy, respectively; D(DEvib,0)298 isthe change in the vibrational energy difference as one goesfrom 0 to 298.15 K. The vibrational energy correctionsare based on our frequency calculations. The molar workterm D(pV) is (Dn)RT; Dn = �2 for three reactants(M2+ + 2Cp�) associating to one product (MCp2). Thermalcorrections for the electronic energy are neglected.

The total interaction energy DE for the heterolytic asso-ciation reaction M2+ + 2Cp� !MCp2 (see below for dis-cussion of heterolytic versus homolytic association)results directly from the Kohn–Sham molecular orbital(KS-MO) model [7] and is made up of two major compo-nents (Eq. (2)):

DE ¼ DEprep þ DEint ð2ÞIn this formula, the preparation energy DEprep is the energyneeded to prepare the ionic fragments and consists of threeterms (Eq. (3)):

DEprep ¼ DEdeform þ DEcyc–cyc þ DEvalexc ð3Þ

The first is the energy needed to deform the separate molec-ular fragments (in this case only for the cyclopentadienylanion (Cp�) rings) from their equilibrium structure to thegeometry that they attain in the overall molecular system(DEdeform), the second (DEcyc–cyc) is the interaction energybetween the two Cp� rings, which results from electrostaticrepulsion between the negatively charged Cp� rings whilemaking one fragment file that contains both Cp� rings.The third term is the valence-excitation energy needed toprepare the metal from its spin-unrestricted (polarized) io-nic state to the spin-restricted (polarized) ionized form(DEvalexc). The valence-excitation energy consists of twoterms: the first (positive) term is the energy difference be-tween the (spherical/non-spherical) spin-polarized metal(2+) cation in its ground state (e.g. the 5D state for Fe2+)and the spin-restricted cationic form used for the metal cat-ion fragment (the fragments need to be spin-restricted). Forthe ground state of the ion, we use the ‘‘average of config-uration’’ approach [47]. The second (negative or zero) termresults from changing the fragment occupations for non-singlet metallocenes to reflect the multiplet character ofthe metal; for singlets, this term is zero. For instance forcopper(II), the spin-restricted cationic fragment is preparedwith 4.5a and 4.5b d-electrons; within the molecule calcula-tion, the occupations of the copper-fragment are changedto make 5a and 4b d-electrons, which will lower the energy.This energy lowering makes up the second part of the va-lence-excitation energy.

The interaction energy DEint is the energy released whenthe prepared fragments (i.e. M2+ + 2Cp�) are brought

eclipsed staggered

Fig. 2. Eclipsed and staggered conformation of metallocenes.

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 ofthe KS-MO model using a Morokuma-type decomposition[48] into electrostatic interaction, Pauli repulsion (orexchange repulsion), and (attractive) orbital interactions(Eq. (4)).

DEint ¼ DV elstat þ DEPauli þ DEorbint ð4ÞThe term DVelstat corresponds to the classical electrostaticinteraction between the unperturbed charge distributionsof the prepared (i.e. deformed) fragments and is usuallyattractive. The Pauli-repulsion, DEPauli, comprises thedestabilizing interactions between occupied orbitals and isresponsible for the steric repulsion. The orbital interactionDEorbint in any MO model, and therefore also in Kohn–Sham theory, accounts for electron-pair bonding, chargetransfer (i.e. donor–acceptor interactions between occupiedorbitals on one fragment with unoccupied orbitals of theother, including the HOMO–LUMO interactions), andpolarization (empty-occupied orbital mixing on one frag-ment due to the presence of another fragment). The orbitalinteraction energy can be decomposed into the contribu-tions from each irreducible representation C of the interact-ing system (Eq. (5)) using the extended transition state(ETS) scheme developed by Ziegler and Rauk [49,50].

DEorbint ¼X

C

DEC ð5Þ

The choice of either neutral (M + 2Cp, i.e. homolytic asso-ciation) or ionic (M2+ + 2Cp�, i.e. heterolytic association)fragments will influence the absolute value of the bondingenergy significantly, and thus also the energy components;however, Rayon and Frenking [13] recently showed thatthe contribution of covalent versus electrostatic bondingremains rather constant, regardless of the choice of usingeither neutral or ionic fragments. Ionic fragments havebeen used throughout this paper.

3. Results and discussion

We have studied trends in metal–ligand bonding for aseries of metal sandwich complexes (metallocenes) thatcontain the first-row transition metals (Sc–Zn), extendedwith alkaline-earth metals (Mg, Ca) and several second-row transition metals (Ru, Pd, Ag, Cd). The geometriesof all species were fully optimized (using a convergence cri-terion of 1.0 · 10�5 a.u.) at the OPBE/TZP [28] level withthe QUILD program [31,46], and characterized by inspect-ing the vibrational frequencies from the analytical Hessian.For the transition metals that are expected to have elec-tronic states in close proximity, such as the singlet, triplet,and quintet state of Fe(II) complexes, the geometry andmetal–ligand bonding were studied for each spin stateseparately.

Although previous studies [51–54] already showed theeclipsed (D5h) conformation to be lower in energy thanthe staggered (D5d) conformation (see Fig. 2), we initially

considered both conformations and found indeed theeclipsed conformation to be the most stable conformation.Because we are interested in the differences in bondingbetween different metals, and inasmuch not the conforma-tion, we will discuss only the results for the eclipsed confor-mations. Likewise, although some compounds may beexpected [55] to exhibit Jahn–Teller distortions to lowerthe symmetry and thus the energy, we explicitly consideredthe molecules only within D5h symmetry, to be able tomake a fair comparison of the bonding in the differentmetallocenes.

3.1. Geometric parameters of metallocenes

We begin by looking at the OPBE/TZP [28] optimizedgeometries of the metallocene molecules. The metal–car-bon, carbon–carbon, carbon–hydrogen distances for allspin states of all metallocenes are reported in Table 1,which also contains the bending angle of the hydrogenswith respect to the C5 plane. The OPBE metal–carbon dis-tances are ca. 0.04 A smaller than either ab initio bench-mark studies [56], or experimental values [57] (whereavailable), which is reasonably close. Moreover, the influ-ence of these differences on the energies is expected to besmall.

Previously, it was assumed that the hydrogens in ferro-cene would be either in the plane (zero angle), or bentout of the plane and away from the metal atom (negativeangle) [56]. Experimental studies using gas-phase electrondiffraction on ferrocene however showed bent C–H bondswith the hydrogens pointing towards the metal, i.e., a posi-tive angle with a value of 5�. Later experimental studiesreduced this number to 3.7�, while a recent benchmarkstudy using CCSD(T)/TZV2P+f gave a value of 1.03�[56]. With OPBE/TZP we find a value of 1.22� for the angle(see Table 1), which is in good agreement with the bench-mark study. However, it is not true that all hydrogen atomsare pointing towards the metal atom, as can be seen inTable 1; out of the 29 metallocenes studied, only seven havethe hydrogens pointing towards the metal. The other 22metallocenes either have almost planar C5H5 rings (5) withan absolute value of the angle <0.2� or have the hydrogenspointing away from the metal (17).

Table 1Selected 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 �1.18Ca 2.638 1.415 1.087 �2.46Sc 2.430 1.421 1.087 �1.59Ti 2.283 2.304 1.427 1.424 1.086 1.086 �0.53 �0.29V 2.219 2.249 1.424 1.421 1.087 1.086 �0.75 0.08Cr 2.083 2.134 2.323 1.436 1.429 1.420 1.085 1.086 1.086 1.43 0.58 �0.31Mn 2.046 2.201 2.379 1.434 1.426 1.420 1.085 1.086 1.087 1.22 0.11 �0.87Fe 2.007 2.110 2.277 1.431 1.424 1.430 1.087 1.086 1.086 1.22 0.71 �0.14Co 2.070 2.222 1.427 1.422 1.086 1.086 0.70 �0.06Ni 2.148 2.158 1.425 1.424 1.086 1.086 0.20 0.46Cu 2.247 1.423 1.086 �0.29Zn 2.299 1.424 1.086 �0.51Ru 2.155 2.269 2.452 1.431 1.428 1.422 1.086 1.085 1.086 �1.07 �0.32 �0.79Pd 2.355 2.356 1.423 1.422 1.086 1.086 �0.95 �0.91Ag 2.510 1.421 1.086 �1.52Cd 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].

M. Swart / Inorganica Chimica Acta 360 (2007) 179–189 183

3.2. Spin-state splittings

We continue by looking at the spin ground state of themetallocenes. The relative spin state energies (OPBE/TZP) for the metallocenes are given in Table 2, which alsocontains the metals with only one accessible spin state. Thepredicted spin ground state for the metallocenes is in allcases in perfect agreement with experimental data, therebycontributing to the reliability of the OPBE functional forassessing spin-state splittings in transition metal complexes.Moreover, the spin ground states are easily comprehensibleby looking at the level diagram of the 3d-orbitals (schemat-ically represented in Fig. 3, left): within D5h symmetry, the

Table 2Relative spin state energiesa,b (kcal/mol) of metallocenes in eclipsedconformation

Metal d-Electrons Low Intermediate High

Mg d0 0.0 (sing)Ca d0 0.0 (sing)Sc d1 0.0 (doub)Ti d2 19.6 (sing) 0.0 (trip)V d3 44.9 (doub) 0.0 (quar)Cr d4 34.8 (sing) 0.0 (trip) 8.0 (quin)Mn d5 0.0 (doub) 27.6 (quar) 8.2 (sext)0Fe d6 0.0 (sing) 49.3 (trip) 51.9 (quin)Co d7 0.0 (doub) 28.0 (quar)Ni d8 15.3 (sing) 0.0 (trip)Cu d9 0.0 (doub)Zn d10 0.0 (sing)Ru d6 0.0 (sing) 76.3 (trip) 114.0 (quin)Pd d8 8.5 (sing) 0.0 (trip)Ag d9 0.0 (doub)Cd d10 0.0 (sing)

a Calculated at OPBE/TZP.b The terms in parentheses refer to the spin state, e.g. sing for singlet, trip

for triplet, quin for quintet, doub for doublet, quar for quartet and sext forsextet.

lowest lying orbitals are two E02 orbitals, followed by an A01orbital and two higher lying E001 orbitals.

The first d-electron (in scandanocene) occupies one ofthe E02 orbitals (doublet state), while the second (in titano-cene) occupies the other E02 orbital with parallel spin toform the triplet state. The third d-electron in vanadoceneoccupies with parallel spin the close-lying A01 orbital to givethe quartet ground state (see Fig. 3, right).

The additional (fourth) d-electron in chromocene couldin principle occupy one of the higher-lying E001 orbitals togive an overall high spin quintet state, however a lowerenergy is obtained when it forms a pair with the first d-elec-tron to give an overall triplet ground state. The same hap-pens with manganocene, which could have occupied an E001orbital to form the high-spin sextet state, but instead formsthe second E02 electron pair to give a doublet ground state.Note that at room temperature and in the solid state,manganocene is found with a high spin ground state [58].

E1''

A1'

E2'

Sc

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

Zn

E1''A1

'E2'

Fig. 3. Schematic representation of 3d-orbital level scheme (left) andoccupation of the 3d-orbitals in the first-row transition metallocenes(right).

Fig. 4. Chain-like structure for manganocene in the solid state.

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), distortionswe do not consider in this study, with anti-ferromagneticcoupling between neighboring manganese atoms [58]. Atlower temperatures, the sandwich structure and the corre-sponding doublet ground state are retrieved.

Manganocene gives a clear example of the problems ofhybrid functionals with spin-state splittings (vide supra);Reiher and co-workers [25] used B3LYP* on a small num-ber of metallocenes and found for the eclipsed D5h struc-

Table 3Decomposition of metal–ligand bondinga (kcal/mol) for eclipsed metallocenes

MgCp2 CaCp2 ScCp2 TiCp2

Spin state singlet singlet doublet tripletDEprep 78.3 67.6 76.0 88.DEdeform 0.3 0.3 0.5 0.

DEcyc–cyc 78.0 67.3 74.4 79.

DEvalexc 0.0 0.0 1.1 7.

DEint �641.0 �557.6 �627.9 �689.DEPauli 54.9 81.1 139.5 177.DEelstat �500.5 �482.3 �516.2 �537.DEorbint �195.4 �156.4 �251.3 �329.% Orbintb 28.1 24.5 32.7 38.

A01 �38.2 �23.1 �32.1 �39.

E01 �57.5 �27.0 �36.4 �44.

E02 �12.7 �9.5 �28.5 �34.

A002 �27.5 �12.9 �17.0 �20.

E001 �48.0 �75.3 �126.1 �177.

E002 �11.5 �8.5 �11.1 �13.

BSSE 2.0 1.6 2.2 2.DE �560.7 �488.4 �549.7 �599.DZPE 6.5 5.9 6.5 6.

DDH298 �1.5 �1.3 �1.5 �1.

DH �555.7 �483.9 �544.7 �594.Exp.c

Theory

CoCp2 NiCp2 CuCp2 ZnCp

Spin state doublet triplet doublet singleDEprep 166.2 83.0 82.8 80.DEdeform 1.1 0.7 0.6 0.

DEcyc–cyc 96.2 88.6 83.0 80.

DEvalexc 68.9 �6.3 �0.8 0.

DEint �850.9 �780.7 �757.1 �695.DEPauli 304.5 189.8 147.6 108.DEelstat �602.9 �576.7 �551.5 �542.DEorbint �552.5 �393.8 �353.2 �261.% Orbintb 47.8 40.6 39.0 32.

A01 �56.7 �57.9 �60.6 �74.

E01 �68.2 �68.7 �65.9 �79.

E02 �48.7 �28.9 �21.8 �16.

A002 �31.8 �32.7 �31.8 �37.

E001 �331.0 �191.6 �160.8 �41.

E002 �16.1 �14.0 �12.2 �12.

BSSE 2.4 2.5 2.3 1.DE �682.4 �695.3 �672.0 �612.DZPE 8.3 6.9 6.4 5.

DDH298 �2.2 �1.7 �1.3 �1.

DH �676.2 �700.5 �666.9 �608.Exp.c �652

a Computed at OPBE/TZP for the reaction M2+ + 2Cp� !MCp2, given heb Contribution of orbital interactions to total attraction.c Experimental values should probably be corrected: they are obtained by ta

ture of manganocene a high spin ground state at ca.1 kcal/mol below the low spin ground state. Although theperformance of B3LYP* is better than the original

in D5h symmetry

VCp2 CrCp2 MnCp2 FeCp2

quartet triplet doublet singlet5 78.3 182.8 301.4 240.66 0.4 1.3 2.3 1.6

9 82.7 90.7 99.1 103.4

9 �4.8 90.8 200.0 135.6

7 �718.7 �800.4 �884.8 �907.95 202.3 271.7 321.6 345.75 �562.1 �583.2 �599.3 �619.18 �358.8 �488.9 �607.0 �634.50 39.0 45.6 50.3 50.6

7 �43.4 �47.5 �52.1 �53.2

1 �48.5 �56.6 �63.8 �67.6

4 �29.9 �65.3 �78.0 �68.3

7 �22.1 �26.1 �30.3 �31.1

7 �201.3 �277.4 �365.1 �396.2

2 �13.6 �15.9 �17.8 �18.1

2 2.7 2.0 1.1 2.41 �637.7 �615.6 �582.4 �665.04 7.5 8.4 7.6 9.0

6 �1.8 �2.0 �2.1 �2.4

3 �632.0 �609.3 �576.8 �658.4�606 �572 �635 ± 6

�655 ± 15

2 RuCp2 PdCp2 AgCp2 CdCp2

t singlet triplet doublet singlet9 185.9 71.7 71.2 72.47 1.7 0.5 0.5 0.8

3 89.8 77.5 71.4 71.6

0 94.5 �6.3 �0.6 0.0

5 �891.5 �755.2 �719.1 �617.10 411.4 199.1 127.2 111.35 �674.8 �576.7 �525.7 �524.40 �628.0 �377.6 �320.6 �204.05 48.2 39.6 37.9 28.0

3 �57.7 �48.1 �47.3 �63.2

4 �56.3 �49.7 �44.7 �59.6

1 �67.3 �19.6 �13.5 �12.3

3 �25.1 �24.4 �22.8 �29.2

9 �404.2 �224.2 �183.4 �29.9

1 �17.4 �11.6 �8.9 �9.7

9 2.5 2.6 1.9 1.47 �703.1 �680.9 �646.0 �543.35 8.7 6.2 5.8 4.7

0 �2.3 �1.4 �1.0 �0.7

2 �696.7 �676.2 �641.3 �539.3

re for the spin ground state of the metallocenes.

king 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 lowerby ca. 7 kcal/mol [25], neither of these functionals can betrusted to provide the low spin ground state of thismolecule.

Within ferrocene also the A01 orbital is doubly occupiedto give the overall singlet ground state. Continuing alongthe first-row transition metals, the higher lying E001 orbitalshave to be occupied leading to, respectively, a doublet(cobaltocene), triplet (nickelocene), doublet (cuprocene)and singlet (zincocene). For the second-row transition met-als, the spin ground state does not change compared to thefirst-row metals in the same group, i.e. for ruthenocene wefind similar to ferrocene a singlet ground state, and for pal-ladocene a triplet ground state, similar to nickelocene.

3.3. Metal–ligand bonding

Finally, we come to the analysis of the metal–ligandbonding in the metallocenes. We analyze the metal–ligandbonding in the metal sandwich complexes in terms of theheterolytic association reaction (Eq. (6)):

M2þ þ 2Cp� !MCp2 ð6ÞThe results of the energy decomposition analysis for thespin ground states of the sixteen metallocene moleculesare given in Table 3. The enthalpy of association for allthese compounds is found in the range from �483 (calcio-cene) to �701 kcal/mol (nickelocene). We can compareseveral of our computed values with either experiment orhigh-level ab initio calculations. The experimental valuefor the bond disruption enthalpy of ferrocene is 636 kcal/mol [59] (i.e. an enthalpy of association of �636 kcal/mol), which is ca. 20 kcal/mol lower than the theoreticalbest estimates of 657 (CASPT2) and 654 (CCSD(T))kcal/mol [53,60]; with CCSD(T) a value of 653 was foundfor the staggered D5d conformation [60], which should becorrected with the energy difference of 1.15 kcal/mol [56]between the staggered and eclipsed conformation. Ourcomputed value of 658 kcal/mol is in excellent agreementwith both these benchmark values.

The discrepancy between theory and experiment isunsatisfactory, especially since the theory benchmarks areof high quality. The origin of the discrepancy might befound in the experimental data, as they refer to the averageenthalpy of sequentially disrupting the M–Cp bonds [59].For example, for the homolytic dissociation of ferrocene,the experimental enthalpy is obtained by taking the averagevalue of the enthalpy for reactions (7a) and (7b).

FeCp2 ! FeCpþ Cp ð7aÞFeCp! Feþ Cp ð7bÞ

The removal of the first cyclopentadiene ligand was foundto be more endothermic than the second one by 30 kcal/mol [59], which led to a value for the homolytic dissocia-tion of 79 kcal/mol instead of 95. This difference of16 kcal/mol is almost exactly equal to the difference

between CCSD(T) and experiment, however it should bekept in mind that the difference of 30 kcal/mol was ob-served for homolytic dissociation, not for heterolytic disso-ciation as in the theoretical studies. Therefore, we areunable to draw any conclusions from comparing our othercomputed enthalpy values with experimental data at themoment. It should be noted that our computed values forvanadocene, manganocene, ferrocene and nickelocene areall similar to the results from an earlier DFT study [61]at the BP86/TZP level.

These enthalpies however do not reflect the stability ofthe metallocene compounds, as it is well-known that nick-elocene is easily oxidized, vanadocene readily reacts withcarbonmonoxide, 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-ered in this paper. Moreover, the difference in enthalpy ismainly determined by two terms: the valence-excitationenergy and the interaction energy. The former is part ofthe preparation energy, and does not contribute to themetal–ligand bonding, but does influence the enthalpytrend. For instance, the valence-excitation energy firstincreases when going down the first-row transition metals(see Table 3), from 1 kcal/mol for Sc to 8 for Ti and �5for V to 91 for Cr and 200 kcal/mol for Mn; then itdecreases again to 135 kcal/mol for Fe, 69 for Co, �6 forNi, and �1 for Cu. However, the focus of this study ison trends in metal–ligand bonding, which is determinedentirely by the interaction energy.

The interaction between the metal cation and the cyclo-pentadienyl anion ligands is in all cases very favorable,although this may have been biased by having chosencharged fragments. Rayon and Frenking [13] studied bothcharged and neutral interacting ligands, and found a muchsmaller interaction energy with neutral fragments(�274 kcal/mol) than with charged fragments (�894 kcal/mol). However, the choice of fragments affected not onlythe electrostatic component of the interaction energy butalso the orbital interactions, and by almost the same factor.Therefore, the relative contribution for each of the twocomponents remained rather constant. This changes whenwe focus on the trend for the metallocenes considered inthis study. Both the absolute value as well as the relativecontribution of covalent bonding, i.e. the orbital interac-tions, first increase if we go along the row of transition met-als (see Table 3 and Fig. 5) until a maximum is reached atferrocene.

Continuing along the row, the absolute value of theorbital interactions decreases and because the electrostaticinteraction remains rather constant (because of the chargedfragments), also the contribution of covalent bondingdecreases. For the few metallocenes we studied from thesecond-row transition metals, a similar pattern emergeswith strong covalent bonding in ruthenocene (the second-row analog of ferrocene), which decreases if we continuealong the row. Both the absolute value and contributionof covalent bonding decrease if we go down the periodic

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

186 M. Swart / Inorganica Chimica Acta 360 (2007) 179–189

table, i.e. covalent bonding decreases from �195 to�156 kcal/mol when comparing magnesiocene and calcio-cene, from �635 kcal/mol for ferrocene to �628 for ruthe-nocene, from �394 kcal/mol for nickelocene to �378 forpalladocene, from �353 kcal/mol for cuprocene to �321for argentocene, and from �261 kcal/mol for zincoceneto �204 for cadmocene.

Although covalent bonding, i.e. orbital interactions, isthe component of the interaction energy that changes mostsignificantly for the different metals, also the other twocomponents do vary along the set of metallocenes. Forinstance, Pauli repulsion is relatively speaking small foralkali-earth metallocenes (55–81 kcal/mol), but thenincreases rapidly when the d-orbitals start to be occupiedin the transition metal series. Like the orbital interactions,it has a maximum for ferrocene (346 kcal/mol) and thendecreases again towards the end of the row, to reach108 kcal/mol for zincocene. The variation shown by Paulirepulsion along the first-row transition metals (ca.260 kcal/mol) is roughly half the variation in orbital inter-actions (ca. 480 kcal/mol). Electrostatic interactions,although considerable in size (482–619 kcal/mol), varythe least among the three components (ca. 135 kcal/mol).Not surprisingly, the electrostatic interactions also havethe maximum value for ferrocene, similar to the othertwo components. This suggests a synergy between the com-ponents: more favorable covalent bonding leads to shortermetal–ligand distances, resulting in an increase of Paulirepulsion that is overcome by the joint increase in electro-static and covalent bonding. The metal–ligand (ML) dis-tances from Table 1 confirm this trend, i.e. going from

ScCp2 to FeCp2, the ML distance decreases from 2.43 A(Sc) to 2.30 (Ti), 2.25 (V), 2.13 (Cr), 2.05 (Mn) to 2.01(Fe); continuing towards the end of the row leads to anincrease of the distance again, from 2.07 A (Co), 2.16(Ni), 2.25 (Cu) to 2.30 (Zn).

Rayon and Frenking [13] made the distinction betweend-bonded CrBz2 and p-bonded FeCp2, and therefore it isinteresting to see if a similar difference in bonding appearsfor chromocene versus ferrocene. Similar to their results,we find ferrocene to be p-bonded, i.e. the largest orbitalinteractions are seen in the E001 irrep (�396 kcal/mol, seeTable 3), which is analogous to E1g in D5d symmetry; theE02 irrep, the analog of the E2g irrep from D5d symmetry,shows much smaller orbital interactions (�68 kcal/mol).The same is true for chromocene with values of �277ðE001Þand �65 ðE02Þ kcal/mol. Almost all other metallocenes arefound to be p-bonded with large orbital interactions forthe E001 irrep. However, for metallocenes with either fullyoccupied or fully unoccupied d-shells such as zincoceneor magnesiocene, the orbital interactions of the A01 andE01 irreps are almost as large as or larger than those inthe E001 irrep.

3.4. Spin-state dependent metal–ligand bonding

For the metallocenes with more than one accessible spinstate, it is interesting to look at the differences in metal–ligand bonding. Given in Table 4 is the decomposition ofthe interaction energy for all spin states. The difference ininteraction energy between the several spin states is forsome metallocenes small (titanocene and vanadocene), with

Table 4Decomposition of metal–ligand bondinga (kcal/mol) for eclipsed metallocenes in D5h symmetryb

TiCp2 TiCp2 VCp2 VCp2 CrCp2 CrCp2 CrCp2

singlet triplet doublet quartet singlet triplet quintetDEPauli 205.1 177.5 245.8 202.3 304.1 271.7 183.0DEelstat �541.7 �537.5 �569.5 �562.1 �580.7 �583.2 �547.7DEorbint �367.4 �329.8 �415.6 �358.8 �586.1 �488.9 �324.4A01 �40.1 �39.7 �57.5 �43.4 �407.9 �47.5 �42.5

E01 �45.1 �44.1 �50.6 �48.5 �60.0 �56.6 �49.2

E02 �61.6 �34.4 �58.8 �29.9 230.1 �65.3 �20.1

A002 �21.0 �20.7 �22.6 �22.1 �27.3 �26.1 �23.0

E001 �186.0 �177.7 �211.8 �201.3 �303.1 �277.5 �177.6

E002 �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 quartetDEPauli 321.6 243.7 121.9 345.7 250.5 154.2 304.5 161.4DEelstat �599.3 �570.0 �536.2 �619.1 �574.5 �552.4 �602.9 �558.5DEorbint �607.0 �430.8 �239.2 �634.5 �496.0 �302.9 �552.5 �355.8A01 �52.1 �48.1 �46.0 �53.2 �232.0 �51.0 �56.7 �55.2

E01 �63.8 �57.8 �54.0 �67.6 �62.8 �60.4 �68.2 �64.0

E02 �78.0 �43.6 �15.0 �68.3 117.7 �27.0 �48.7 �28.4

A002 �30.3 �27.3 �25.5 �31.1 �29.4 �29.1 �31.8 �31.9

E001 �365.1 �239.9 �87.4 �396.2 �274.5 �122.8 �331.0 �163.0

E002 �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 tripletDEPauli 244.5 189.8 411.4 344.6 175.2 255.1 199.1DEelstat �579.3 �576.7 �674.8 �608.8 �551.8 �577.1 �576.7DEorbint �475.1 �393.8 �628.0 �529.1 �292.7 �456.0 �377.6A01 �57.8 �57.9 �57.7 �57.8 �51.9 �49.2 �48.1

E01 �67.1 �68.7 �56.3 �51.8 �45.8 �48.3 �49.7

E02 �32.4 �28.9 �67.3 �50.4 �23.4 �22.7 �19.6

A002 �31.8 �32.7 �25.1 �24.9 �22.3 �23.5 �24.4

E001 �271.9 �191.6 �404.2 �329.6 �138.5 �301.0 �224.2

E002 �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 Computed at OPBE/TZP for the reaction M2+ + 2Cp� !MCp2, given here for different spin states of several metallocenes.

M. Swart / Inorganica Chimica Acta 360 (2007) 179–189 187

a difference in interaction energy of only 14–21 kcal/mol.However, the differences in the components are larger.For instance for vanadocene, the Pauli repulsion increasesby 43 kcal/mol in going from the quartet ground state tothe doublet state that is being counteracted by a smallincrease in DEelstat (7 kcal/mol) and a substantial increaseof DEorbint (57 kcal/mol). Similarly for titanocene, the elec-trostatic component of the triplet ground state and singletstate is almost equal (4 kcal/mol), but an increase in Paulirepulsion (27 kcal/mol) and DEorbint (37 kcal/mol) lead toan increase in interaction energy of 14 kcal/mol.

For ferrocene, the singlet ground state has the largestinteraction energy (�908 kcal/mol) with the triplet andquintet states significantly lower, at respectively �820and �701 kcal/mol. The difference in interaction energyresults from significant reductions in both DEPauli (�95and �190 kcal/mol, respectively, for the triplet and quintetstate) and DEorbint (�139 and �331 kcal/mol, respectively,for triplet and quintet). The largest change in DEorbint is

seen for the E001 irrep, i.e. the unoccupied d-orbitals in sin-glet ferrocene in which they serve as acceptor orbitals forM L donor interactions. In triplet and quintet ferrocene,these orbitals are partially occupied which results in thembeing less suited as acceptor orbitals. Finally, for mangano-cene we see a similar trend as for ferrocene, with the largestinteraction energy for the low spin state (�885 kcal/mol),followed by the intermediate and high spin state, at �757and �654 kcal/mol, respectively. Also in the case of mang-anocene is the change in interaction energy mainly result-ing from significant reductions in both DEPauli and DEorbint.

The spin ground state of metallocenes can thus beunderstood in terms of two opposing effects: metal–ligandbonding that favors low spin states over higher spin statesbecause of the better suitability of metal acceptor d-orbitalsin the former, and Hund’s rule of maximum multiplicity,which states that for atoms a larger total spin makes theatom more stable. Therefore, from the point of view ofthe 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-ter acceptor orbitals. The compromise between these twoeffects is delicate, and needs a careful and accurate methodsuch as OPBE to be able to correctly predict spin groundstates of transition metal compounds.

4. Conclusions

We have studied metal–ligand bonding in metallocenesusing density functional theory at the OPBE/TZP level.This level of theory was previously shown to be the onlyDFT method capable of correctly predicting the spinground state of iron compounds, and we find similar goodperformance in the current study. We considered metalloc-enes along the first-row transition metals (Sc–Zn), extendedwith alkaline-earth metals (Mg, Ca) and several second-row transition metals (Ru, Pd, Ag, Cd). Although we ini-tially considered both the staggered D5d and eclipsed D5h

conformation of the metallocenes, we report here onlythe results for the more stable eclipsed conformations. Toenable a fair comparison, we use D5h symmetry for all spinstates of all compounds even though some compounds maybe expected to break symmetry to lower the energy.

The metal–ligand bonding was analyzed in terms of anenergy decomposition analysis to produce physically mean-ingful terms such as Pauli repulsion, electrostatic interac-tions and orbital interactions. For the analysis, theinteraction was obtained between charged fragments, i.e.a metal2+ cation and a bicyclopentadienyl anion2� frag-ment. The OPBE/TZP enthalpy of heterolytic associationfor ferrocene (�658 kcal/mol) as obtained from this analy-sis is in excellent agreement with benchmark CCSD(T) andCASPT2 calculations.

Covalent bonding, i.e. orbital interactions, was shown tobe the determining factor for understanding metal–ligandbonding in metallocenes and is found in the range from�155 to �635 kcal/mol. Smaller roles are played by Paulirepulsion (from 55 to 345 kcal/mol) and electrostatic bond-ing, although the latter is in all cases strong (from �480 to�620 kcal/mol). The covalent bonding, and thus themetal–ligand bonding, is larger for low spin states thanfor higher spin states, due to better suitability of acceptord-orbitals of the metal in the low spin state. Therefore, spinground states of transition metal complexes can be seen asthe result of a delicate interplay between metal–ligandbonding and Hund’s rule of maximum multiplicity.

Acknowledgement

MS thanks F.M. Bickelhaupt for helpful discussionsregarding details of the energy decomposition analysis.

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