DOI: 10.1002/cphc.200600441

d-Band Catalysis in Electrochemistry

Elizabeth Santos*[a, b] and Wolfgang Schmickler[c]

All technologically important electrochemical reactions involvethe breaking or formation of a chemical bond. Well-known ex-amples are the hydrogen oxidation and the oxygen reduction,which form the basis of most fuel cells, and chlorine evolution.The rate of such reactions depends strongly on the electrodematerial, and the search for better catalysts is a never-endingquest for electrochemical engineers. Obviously, it would be ex-tremely useful to have a theory which explains, or even pre-dicts, electrocatalytic effects. However, the existing theories forelectrochemical reactions involving bond rearrangementcannot do this, since they treat the electrode as a metal with awide, structureless band. This is true both for the semiclassicaltheory proposed by Sav'ant,[1] which is essentially an extensionof the Marcus theory,[2] and for the Hamiltonian-based modelpioneered by Koper and Voth.[3] It is well-known, that metalcatalysis generally involves d-bands with energies close to theFermi level, and their interactions with the reactant must playa key role in any useful theory of electrochemical bond-break-ing reactions.In gas-phase catalysis, ab initio calculations have provided

valuable insights into the mechanisms of particular reactions.Recently, these methods have also been applied to electro-chemical reactions by a number of groups.[4] While such calcu-lations may provide valuable details, a full modeling of electro-chemical processes is presently unfeasible, and will remain soat least during the next decade. They cannot account for theinteraction of the reactants with the solvent, in particular notfor solvent fluctuations, and a consistent treatment of the elec-trode potential and the field in the electric double layer is alsobeyond their scope.What is needed is a framework that makes it possible to

treat d-band catalysis within the theory of electrochemical re-actions. This is what we propose here. We consider the basiccase, the reduction of a diatomic molecule at a metal electrodeaccording to the scheme:

A2 þ 2 e� ! 2A� ð1Þ

When this molecule is far from the electrode, its bonding or-bital is filled and its antibonding orbital is empty; during thereaction the antibonding orbital is filled and the bond isbroken. The final states consists of two ions, which strongly in-teract with the polar molecules of the solvent. As is well-known from the theory of Marcus for simple electron transfer,this interaction stabilizes the ions, and the electron transfergenerally involves a reorganization, or fluctuation, of the sol-vent.For a mathematical description, we use the model recently

proposed by Santos et al. ,[5] in which the molecular bond istreated on the H5ckel, or tight binding, level. Thus, the bindingis described by the matrix (H5ckel) element b, which dependsexponentially on the bond distance r between the two atoms:

b ¼ �Aexp�r=l ð2Þ

We take A as positive, as is usually done in H5ckel theory.For the overlap integral we use the Wolfsberg–Helmholtz ap-proximation: S=�ab, where a is a constant. This results in aMorse potential for the isolated molecule, which we character-ize by the H5ckel binding energy De per binding electron.Since we may have more than one electron per orbital, wemust consider spin explicitly, and introduce the Coulomb re-pulsion U between two electrons on the same atomic orbital.This Coulomb interaction is treated in the Hartree–Fock ap-proximation. For details see Supporting Information and theoriginal paper.[5]

When the molecule is in contact with the electrode, it inter-acts with the metal bands. The molecular orbitals are then nolonger sharp, but are characterized by their density of states,which can be calculated within the Anderson–Newnsmodel.[6, 7] A key role is played by the chemisorption functionD(e) (e denotes the electronic energy), which is related to theelectronic density of states 1m(e) of the metal and to the reac-tant–metal interaction.[8]

All previous theories for electrochemical electron transferhave used the so-called wide band approximation, in whichboth the metal density of states 1m(e) and the chemisorptionfunction D(e) are taken as constant.[5, 9,10] The density of statesof the reactant then takes the simple form of a Lorentz-distri-bution; thus, the molecular orbitals just get broadened, andtheir width is determined by the chemisorption function D(e).This approximation is reasonable for reactions at non-catalyticsp metals, which just have a wide, structureless band, but itcannot describe catalytic interactions with narrow d-bands,which are of paramount importance for the breaking of bonds.In particular, it ignores such important effects as a possiblesplitting of the reactant’s orbitals into bonding and antibond-ing states with respect to the d-band.[13] Therefore, we aban-don the wide-band approximation at the expense of somemathematical complexity, and consider a metal with a typicalelectronic structure consisting of a wide sp and a narrow d-band.As is known from the Marcus theory, the reorganization of

the solvent plays a major role in any electron transfer reaction.If the charge transfer affects only classical modes, the state of

[a] Dr. E. SantosZentrum f�r Sonnenenergie und Wasserstoff-ForschungHelmholtzstr. 8, 89081 Ulm (Germany)Fax: (+49)731-9530-666E-mail : elisabeth.santos@uni-ulm.de

[b] Dr. E. SantosFaculdad de Matem3tica, Astronom4a y F4sicaUniversidad Nacional de C8rdoba, C8rdoba (Argentina)

[c] Prof. Dr. W. SchmicklerDepartment of Theoretical Chemistry, University of Ulm89069 Ulm (Germany)

Supporting information for this article is available on the WWW underhttp://www.chemphyschem.org or from the author.

2282 C 2006 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim ChemPhysChem 2006, 7, 2282 – 2285

the solvent can be characterized by a single solvent coordinateq, which has the following meaning: A solvent in the state qwould be in equilibrium with a reactant carrying a formalexcess of q electrons, where q need not be an integer.[10] Amajor effect of the solvent’s interaction with the reactant is ashift of the atomic orbitals by an amount �lq, where l is thesolvent reorganization energy familiar from Marcus theory.[2] Asa result, the reactant’s levels, and hence it density of states,fluctuate with the solvent configuration q.Calculations for real systems would involve a determination

of the metal density of states and the chemisorption functionby quantum-chemical methods. However, much can belearned by using a simple model for a metal containing anarrow d-band, and studying its effect on the bond-breakingreaction. Here, for simplicity, we consider the case in which themolecule lies flat on the surface, so that the two atoms areequivalent. The metal is assumed to contain a wide sp-bandwith a width of 20 eV, centered at the Fermi level, and a d-band with a half-width wd centered at an energy ec. Bothbands are assumed to have a semi-elliptic form.[7] The chemi-sorption function D(e) is then the sum of contributions fromboth bands, which are characterized by a coupling constant D0

for the sp-band, and Dd for the d-band. These coupling con-stants are the amplitude of D(e) at the center of the respectiveband. The main question, which we pursue here, is how the re-action depends on the d-band and its coupling to the reac-tant.It is a great advantage of electrochemical reactions that

their driving force can be varied with the electrode potential :A change Df in the electrode potential shifts the molecularlevels by an amount of e0Df, so that energy shifts of the orderof an electron volt are easily effected. We first consider thecase in which the reaction is at equilibrium.The reactant’s levels fluctuate with the solvent.[14] In the

course of these fluctuations they may get energetically closerto the d-band, or further away, and their density of states(DOS) changes accordingly. A typical situation is depicted inFigure 1, where the d-band has been chosen to lie with its

center below the Fermi-level. It shows three important config-urations; a) when the reactant is in equilibrium, b) when thesystem is at the saddle point, and c) when the products are inequilibrium. In the former case (a), the DOS shows a peak forthe occupied bonding orbital lying well below the Fermi level,and below the d-band as well ; another peak at high energiesindicates the empty anti-bonding orbital. In addition, the mole-cule acquires some density at the position of the d-band. Atthe saddle point (b) the anti-bonding orbital crosses the Fermilevel and becomes occupied. In the case depicted, the interac-tion with the d-band is so strong that it is split into two parts :a bonding and an anti-bonding part with respect to the metal ;for a weak interaction (not shown), it just gets broadened. Inany case the molecular bonding orbital stays well below theFermi-level, but is shifted to slightly higher values. In the finalstate (c) the bond has been broken, there is only one filled or-bital for the two ions well below the Fermi level, and two elec-trons have been transferred. Obviously, these dynamic changesin the density of states have a strong effect on the energy ofactivation; therefore, a normal quantum-chemical calculationperformed for the initial or the final states, which in this exam-ple interact weakly with the d-band, would give little informa-tion about the course of the reaction.Our theory permits the calculation of potential energy surfa-

ces VACHTUNGTRENNUNG(q,r) as a function of the solvent coordinate q and thebond distance r. For explicit calculations, we have set thedecay length in Equation (2) to l=1.3, which amounts to a nor-malization of the bond distance, and let r=0 correspond tothe equilibrium distance in the isolated molecule. An examplefor such a surface, again for equilibrium, is shown in Figure 2.The minimum for the molecule is centered at q=0, r=0; thetwo ions with the bond broken, corresponds to the valley thatappears for large r and is centered near q=2. These two re-gions are separated by an energy barrier with a saddle point.The reaction starts with the system in the well for the mole-cule; a thermal fluctuation makes it pass the barrier near thesaddle point, and then it relaxes into the valley for the twoions. As it passes the barrier, the anti-bonding orbital interacts

strongly with any d-states nearthe Fermi level, and this has apronounced effect on the activa-tion barrier, as we will demon-strate below.Application of an overpoten-

tial h mainly lowers the energyof the final state by an amount2e0h. This induces a correspond-ing lowering of the activationenergy Eact. The change of theactivation energy with the driv-ing force is known as the trans-fer coefficient; for the case dis-cussed here, where two elec-trons are transferred, a=�dEact/dACHTUNGTRENNUNG(2e0h).

[11] Marcus theory pre-dicts a=1/2 near equilibrium, avalue which corresponds to per-

Figure 1. Density of states at three different points: a) equilibrium for the molecule, b) saddle point and c) afterbond breaking. Parameters : D0=0.1 eV, Dd=2.0 eV, wd=1 eV, ec=�0.5 eV, l=0.75 eV, De=2 eV. The inset at theright shows the position of the bands.

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fect symmetry between the forward and backward reaction. Inour model, the presence of the d-bands breaks this symmetry,and typical values lie in the range of 0.4�a�0.6. For a suffi-ciently high overpotential the energy barrier will disappear,and the molecule will dissociate upon adsorption. The sameeffect can also be observed, if the intramolecular bond isweak, and the interaction with the d-band strong.A d-band near the Fermi level will affect the energy of acti-

vation. A natural question is : For a given interaction strengthDd, what is the optimum position of the d-band? One mayexpect the optimum to be a band centered at the Fermi-level,where it strongly affects the anti-bonding orbital as the systemcrosses the saddle point. Figure 3 shows that this is indeed the

case. In our example a suitably placed d-band can lower theactivation energy from a high value, where the reaction wouldpractically not occur, to a lower range where it proceeds withease. However, it is not just the density of states right at theFermi level which affects the rate. As can be seen by the crossin Figure 3, a broader band centered below the Fermi level canbe just as effective as a narrower band centered at the Fermilevel. Thus, the detailed distribution of the metal density ofstates near the Fermi level must be taken into account. For agiven position of the d-band, the activation energy decreasesstrongly as the coupling strength Dd is made larger, as shownin Figure 3.Our model calculations give some justification for a recent

suggestion, that the weighted d-band center of an electrodecan be used to characterize its catalytic activity.[12] This hasbeen based on the observation, that the d-band center oftencorrelates with the adsorption energies of a particular reac-tant.[13] But adsorption seldom determines the rate of an elec-trochemical reaction, in particular it does not do so for theoxygen or chlorine evolution. Therefore, up to now, correla-tions of the d-band center with the rates of bond-breaking re-actions were quite unfounded. Our investigation of the effectof the d-band on the activation energy, as demonstrated inFigure 3, now provides some substance to this notion.However, our model has more to offer than a qualitative un-

derstanding: it opens a path for the investigation of real sys-tems. For this purpose, the key parameters of our theory mustbe calculated by modern computing methods. The electronicstructure of a metal, in particular the density of the d-band,can be obtained from standard ab initio methods. The cou-pling constants between the metal and the reactant are not soeasy to calculate directly; however, the local density of states

(DOS) of the reactant can be cal-culated with ease at the equilib-rium position, and from this thecoupling constants can be ob-tained by fitting. For a realisticcalculation, one would also re-place the H5ckel description ofthe molecular binding by the re-sults of quantum-chemical meth-ods. The quantity that is mostdifficult to obtain with accuracyis the energy of reorganizationof the solvent. In principle, it canbe obtained from molecular dy-namics simulations of the classi-cal or of the quantum type;[15–17]

but these methods have a great-er uncertainty, of the order of�0.1 eV, than ab initio methods.Thus, realistic calculations for thecatalysis of electrochemical reac-tions, with the ultimate aim ofmaking predictions, are nowpossible.

Figure 2. Potential energy surface for equilibrium. Parameters: D0=0.1 eV,Dd=2.0 eV, wd=1 eV, ec=�0.5 eV, l=0.5 eV, De=2 eV.

Figure 3. The energy of activation for various interactions with the d-band; a) as a function of the position of thed-band center ; b) as a function of the coupling strength. Parameters: D0=0.1 eV, wd=1 eV, l=0.5 eV, De=2 eV;in (a) Dd=2.0 eV; the cross indicates the value for wd=2 eV. In (b) ec=�0.5 eV. The lines at the top indicate theenergy of activation in the absence of d-band catalysis.

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Acknowledgements

Financial support of the European Union under contract 505906,project NENA, is gratefully acknowledged. E.S. thanks CONICETfor continued support. This project was also partially fundedthrough a cooperation program between SECYT and DAAD.

Keywords: bond breaking · metal d-bands · electrocatalysis ·electron transfer · potential energy surfaces

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Received: July 20, 2006Published online on September 27, 2006

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