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Physics of Surface, Interface and Cluster Catalysis

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Chapter 1

The reactivity of metals based on delocalizedelectronic states

1.1 IntroductionSome of the theories developed for the chemical reactivity of molecules depend onthe localized orbitals of the reactants. Understanding reactions on metals, however,involves the participation of electronic states that are delocalized. Moreover, theinvestigation of reactions on metal surfaces is the foundation of heterogeneouscatalysis, which has become more relevant in contemporary energy, device, environ-mental and biological materials and applications. Metal surfaces serve as a simpleyet insightful source of the concepts and mechanisms from which heterogeneouscatalysis can be understood. In metal surfaces, electronic states can be directly orindirectly involved in the reaction. This chapter reviews the reactivity modelsand concepts, focusing on their origins, the approximations used, key formulations,their applicability to certain molecules/reactants and their inter-relationships.Explanations for the ‘local’ chemical reactions on a metal surface have also beendeveloped, using some localized orbital analogies in an otherwise delocalized metalsurface, but this chapter mainly focuses on the reactivity models and concepts drawnfrom the latter. The mathematical rigor of the derivation of the formulations will beleft to the reader to explore in the references provided. The chapter is aimed more atdeveloping an understanding of the concepts qualitatively, with mathematicalderivations introduced when appropriate.

1.2 The d-band modelThe reactivity measure based on the center of the d-band or the average energy of thedensity of states (DOS) is commonly described by the equation:

∫∫

ερ

ρ=

E E E

E E

( )d

( )d, (1.1)d

d

d

doi:10.1088/978-0-7503-1164-9ch1 1-1 ª IOP Publishing Ltd 2016

where εd is the d-band center and E and ρd are the energy and d-electron density,respectively. It is now widely used to explain the differences in reactivity betweenmetal surfaces and simple gas molecules [1–7]. The εd has also been employed topredict the reactivity of many metal combinations, such as bimetallic surface alloysand ternary systems [8–16], and is often called the reactivity parameter. Thisparameter actually originated from the description of H2 molecule reactions onseveral metal surfaces: Al(111), Cu(111), Pt(111) and Cu3Pt(111) by Hammer et al[17]. The qualitative description of reactivity differences among these transitionmetals towards H2 and how such differences can be predicted by the εd are discussedin the following subsections.

1.2.1 H2 adsorption on transition metal surfaces

H2 adsorption was first described for Al(111), Cu(111), Pt(111) and Cu3Pt(111)using potential energy (PE) curves [17]. The PE was derived from a fixed atopreaction path. It was noted that activation barriers exist for H2 dissociativeadsorption on Al(111), Cu (of Cu3Pt(111)) and Cu(111), while a barrierlessdissociation was observed for Pt (of Cu3Pt(111)) and Pt(111). The trends in theenergetics of H2 dissociative adsorption did not coincide with the trends in the localDOS (LDOS). Figure 1.1 shows the LDOS for the same clean metal surfaces. It isapparent from figure 1.1 that the LDOS at the Fermi level (LDOSEF) and thenumber of holes in the d-bands (or Nh) cannot account for the observed difference inthe activation barriers. For instance, Pt atoms in pure Pt(111) and Cu3Pt differsignificantly in the LDOSEF, but both can dissociate H2 without a barrier. As acorollary to this, the Al(111), Cu(111), Cu (of Cu3Pt(111)) and Pt (of Cu3Pt(111)) allhave a completely filled d-band, and yet the Pt (of Cu3Pt(111)) reactivity is almostthe same as that of pure Pt(111).

It was argued that the mere use of a narrow point in the d-band of the transitionmetal (e.g. LDOSEF or the Nh) cannot explain the differences observed in the

Figure 1.1. The DOS for (a) Cu(111), (b) (c) Cu3Pt(111) and (d) Pt(111). The d-states projected on a Cu atomin Cu(111) and Cu3Pt(111) are shown in solid lines in (a) and (b), respectively. The d-states projected on a Ptatom in Cu3Pt(111) and Pt(111) are shown in solid lines in (c) and (d), respectively. Dotted curves give theDOS for the corresponding bulk structures. Reproduced with permission from [17]. Copyright 1995 Elsevier.

Physics of Surface, Interface and Cluster Catalysis

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energetics. Thus, a model was derived to qualitatively describe the origin of thedifferences in the reactivity of these metals better. First, it was thought that the H2

molecule at some distance from the surface interacts with the metal’s s and p bands.This would give rise to renormalized states, as shown in figure 1.2.

These states are a consequence of the ‘broadening’ of the molecular states due toweak chemisorption. Such weak chemisorption for H2 is often termed H2/Al(aluminum) or H2/jellium interaction. What happens to the H2 molecular statesafter interaction with metal sp-states can be described as follows. Specifically,the anti-bonding σ⁎( )u and bonding σ( )g states of H2 shift downwards as a result of thebroadening. The σg is now much lower (∼7 eV below the Fermi energy (EF)) and theσ⁎

u is now around the EF. These states are also called resonance states in the similarconcept of chemisorption on metal surfaces developed by Newns [18], which isdiscussed in later sections. Finally, these renormalized states are the ones that splitinto anti-bonding and bonding states due to interaction with the much narrowerd-band of the metal (see figure 1.2(c)). The resulting split states are hybridized statesand the energy levels vary depending on the metal surface catalyst. The strength ofthe chemisorption on the metal surface depends on the relative filling of the anti-bonding states. In figure 1.2(c), the resulting anti-bonding state (a hybridized state) islower than the EF, signifying filling and less binding of the molecule on the surface(as compared to when this anti-bonding state is empty). The renormalized statesresulting from sp-interaction are very important as they differentiate the d-bandmodel from Hoffman’s model for localized states [19]. While the latter considersmetal interaction with molecular levels, the d-band model takes into accountinteraction with renormalized states (or molecular levels already delocalized due tosp-interaction). The differences in the reactivity of the transition metal surfaces to H2

dissociation are therefore described by the interaction of the resonance with thed-band and not directly from the localized states of the molecule in the gas phase.

Figure 1.2. Schematic showing the interaction between (a) molecular levels of H2 and (d) metallic states of Cu(111). (b) Broadening of bonding σ( )g and anti-bonding σ ⁎( )u states due to interaction with the metallic s, pstates. (c) Interaction of the renormalized molecular states with the d-bands. The σ σ ⁎( )g u are shown in solid(dotted) curves and the occupied part is marked with light (dark) gray shading. Reproduced with permissionfrom [17]. Copyright 1995 Elsevier.

Physics of Surface, Interface and Cluster Catalysis

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The energy difference δE( )ts due to the renormalized states coupling with the d-bandsis estimated within the frozen potential approximation:

δε ε ε ε

α= −−

− −−

+σ σ⁎

EV

fV

V2 2(1 ) , (1.2)tsd d

2 22

u g

where the first term is the energy gain due to the hybridization between σ⁎u and the

d-states, the second term describes the degree of filling of the anti-bonding state andthe last term is the repulsion due to the orthogonalization of both σ⁎

u and σg stateswith the metal d-states. In fact, the orthogonalization energy is proportional to SV ,where S is the overlap matrix element. In equation (1.2), S is assumed to beproportional to V and is included in the proportionality constant, α. Also, the σ⁎

ucenter is assumed to be above the EF (i.e. empty). Each of the quantities in equation(1.2) is derived: the εd is the average energy of the d-states, shown in figure 1.1, the fis approximated by the local filling of the surface d-states, the matrix element V iscalculated within the linear muffin-tin orbital atomic sphere approximation(LMTO-ASA) (17 and [42] therein) and the hydrogen εσ⁎

uand εσg

are assumed tobe the same for the metal surfaces. It is deduced that the chemical reactivity of themetal surfaces towards H2 depends largely on the first term of equation (1.2). That

is, since σ⁎u is empty, the first of the two attractive terms −

ε ε−σ⁎2( )V

d

2

u

and

− −ε ε−σ⁎f2(1 )( )V

d

2

u

has a greater effect on the δEts. Next, if the same σ⁎u is used

over different metal surfaces, which happens when the same molecule is used as a‘probe’, then the εd alone can predict the metal surface reactivity well. Thus the

Figure 1.3. Correlation between the reactivity measure δEts and the calculated activation barrier foradsorption (the total energy at (b, Z) = (1.2 Å, 1.5 Å)). α in equation (1.2) is used as a fitting parameter.Reproduced with permission from [17]. Copyright 1995 Elsevier.

Physics of Surface, Interface and Cluster Catalysis

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frequent correlations made between the reactivity differences in metal surfaces andεd in most current theoretical studies on metal surface reactivity.

Now, when the dissociation is complete, in the final state of chemisorption, the εσ⁎u

and εσgconverge to H adsorbate resonance ε( )H below the EF and the change in

chemisorption energy δE( )chem is described as:

δε ε

α∼ − −−

+E fV

V2(1 ) . (1.3)d H

chem

2

chem2

If f and V2 are assumed to be the same for the metal surfaces, then the δEchem shallalso largely depend on the εd . This is the context in which εd has become a reactivityparameter that is often used in physical chemistry studies on chemisorption for theexplanation of activation barriers or dissociated atoms’ binding energies [1–16].

1.2.2 OH adsorption on metal surfaces

What happens when the ‘probing’ molecule is changed? Generally, the same d-bandmodel is applied to describe the differences in the reactivity of metal surfacestowards other molecules such as O2, O and OH. Do we obtain the same trends for allkinds of reactants? Let us first discuss the oxygen cases in comparison to H2. Usingfigure 1.4, Kitchin et al showed the dissociative adsorption energy of H2 and O2

on Pt(111) with subsurface 3d metals [20]. The dissociative energy is defined as− −E E E2 2slabs,ads slab molecule, where Eslabs,ads is the energy of the adsorbed system and

Figure 1.4. Trends in the dissociative adsorption energies for H2 and O2 on Pt(111) slabs containing subsurface3d metals. Reproduced with permission from [20]. Copyright 2004 AIP Publishing LLC.

Physics of Surface, Interface and Cluster Catalysis

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Eslab and Emolecule are the energies of the isolated clean slab and molecule,respectively. This energy can therefore be described by δEchem from equation (1.3).Based on figure 1.4, it was shown that the adsorption energies decrease as the εd

becomes more negative. Furthermore, while the O2 binds with the metal surfacesmore strongly than the H2, a better linear relation was observed for the latter. Still,the differences in the oxygen reactions among transition metal surfaces can bepredicted by the εd .

Next, the adsorption of O atoms on metal surfaces as a function of the εd wasstudied by Xin et al [21]. They used figure 1.5(a) to show that it is possible to achievethe general trend expected from use of the εd (higher εd , larger adsorption energies).

The metal surfaces considered were Pt and Pd, with other transition metals assubsurface layers. The systems are termed Pd or Pt skin alloys. The trend is extractedfrom the numerous adsorption data (as shown in figure 1.5(a)) and depicted using anelliptical-shaped region. The binding energy of O atom increases with less negative(or higher) εd . As also expected, the surface–O distance correlates well with thebinding energies (figure 1.5(c)).

Figure 1.5. Binding energies of (a) O and (b) OH on Pd and Pt skin alloys plotted as a function of the center ofthe d-band projected on surface atoms. (c) Surface–O and (d) surface–OH distances plotted as a function ofbinding energies. The model system and the corresponding d-band centers are shown at the bottom of theplots. Reproduced with permission from [21]. Copyright 2010 AIP Publishing LLC.

Physics of Surface, Interface and Cluster Catalysis

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However, when the same model systems are used for the OH, the elliptical regionshown in figure 1.5(b) indicates a counterintuitive relationship. The binding energyof the OH decreases as the εd becomes higher. The surface–OH distance also showsan opposite trend. Xin et al explained the interaction between the OH radical andthe metal surfaces as follows. The OH has three kinds of orbitals, σ π⁎4 , 1 and σ3 , asshown in figure 1.6(a). The interaction with the sp-band of the metal obtained usingAl(111) leads to the renormalized molecular states depicted in figure 1.6(b). Theinteraction of the renormalized states with the d-band of the metal surfaces is shownin figure 1.6(c) and (d). In figure 1.6(b), the renormalized states are shifteddownwards with respect to the gas phase molecular levels. The π1 (hybridizedstates) are now below the EF. This populated renormalized anti-bonding stateindicates that a repulsive interaction with the bands of the Pd and Pt skin alloys ishighly likely. Such an interaction was defined in equation (1.3) as α∼SV V 2. Therepulsive term is smaller for surfaces with lower εd (see figure 1.7(a)). This lowerrepulsion is attributed to the surface–OH distance (figure 1.7(b)). For surfaces withlower εd , the surface–OH distance is higher. In this case, the repulsive term issmaller. This higher surface–OH distance is considered to be due to highersp-electron density on the metal surface (electron transfer occurs betweenhetero-metals).

In response to higher electron density, the OH adsorbate, being electron-rich dueto the transfer of electrons from H to O and thus requiring lower optimal density,stabilizes at a greater distance from the surface. Adsorbates with almost-filledvalence shells were expected to exhibit similar adsorption patterns. This is confirmedin figure 1.7(c), which shows the relationship between the binding energies and εd forfluorine (F) and chlorine (Cl). Furthermore, it was assumed that metal surfaces witha completely filled d-band could also interact with simple molecules in a similar

Figure 1.6. (a) DOS of gas phase OH radical in vacuum. (b) DOS projected on the molecular orbitals on atopsite of Al(111). (c) and (d) DOS projected on the molecular orbitals of OH adsorbed on the atom sites of Pdand Pt with 3d subsurface metal. Reproduced with permission from [21]. Copyright 2010 AIP Publishing LLC.

Physics of Surface, Interface and Cluster Catalysis

1-7

fashion. This is also confirmed in figure 1.7(d), which shows the interaction of Agand Au skin alloys with O and OH.

The above findings offer fresh insights into the use of the d-band model. Ingeneral, they indicate which terms in equation (1.3) are important for other types ofadsorbate or metal surfaces. It can therefore be argued that the above study is aspecial case of the d-band model, treating interactions between reactants andoccupied electronic states.

Given the new perspective on the d-band model offered by the understanding ofOH adsorption properties, several points in relation to the derivation of themechanism of the interactions are worth revisiting. (a) The sp-electron density wasused to explain the surface–OH optimal bond distance. However, the sp-electrons,based on the d-band model, are only effective when the molecule is still relatively farfrom the surface. (b) The coupling matrix element was described by the d-statesinteraction with π1 renormalized states, thus the relationship with the surface–OHdistance that was attributed to the sp-states (figure 1.7(b)) cannot be entirelyclarified. We propose other views on the mechanism. Using figure 1.6 (c) and (d),figure 1.7(a) and (b), and Hoffman’s description of four-electron interaction

Figure 1.7. (a) Covalent attraction and Pauli repulsion contributions to the OH binding energy on Pt skinalloys. (b) Surface–OH bond distance and coupling matrix V( )2 plotted as a function of the number ofsp-electrons of the surface atoms. The number of sp-electrons is calculated as the Bader charge minus thed-band filling. (c) and (d) Trends in the binding energy of F, Cl on Pt, Pd skin alloys and of O and OH on Auskin alloys with the d-band center, respectively. Reproduced with permission from [21]. Copyright 2010 AIPPublishing LLC.

Physics of Surface, Interface and Cluster Catalysis

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(or interaction between filled orbitals) [19], we can arrive at the following. For (a),the optimal distance between the molecule and the surface is governed by thed-band. The lower the εd (further from the filled renormalized adsorbate π1 states),the smaller the coupling matrix, πV (figure 1.7(b)), and thus the lower the repulsion(figure 1.7(a)). Next, for (b), when we look at figure 1.6(c), the anti-bonding π1 statesrise above the Fermi level for Pd/Cr, in fact as one goes to the left of the series (figure1.6(d)), these hybridized π1 states rise in energy. In Pd/Cr, such states are no longerfilled. The increase of the binding of OH as one goes to the left could thus be due tothe energy level of the hybridized π1 states. As to how this can be reconciled with thelower εd of the alloys to the left of the series—this may need further study.

1.3 The self-consistent model of chemisorption on surfacesWhere did the formulation of equations (1.2) and (1.3) come from? We recall thatthe most important aspect of the d-band model, which also differentiates it fromprevious local bonding interpretations [22, 23], is the broadening of the molecular orrenormalized states. Consideration of the already broadened states rather than themolecular discrete levels can be traced back to the quantitative description ofbroadening due to sp-band interaction proposed by Lundqvist et al [24]:

∑π δ ϵ ϵΔ = −V ( ), (1.4)k

a ak a k2

where ∆a is the width of the spread of the adatom at state ∣ ⟩a , Vak is the couplingbetween ∣ ⟩a and ∣ ⟩k and ϵa, ϵk are the energy of the substrate state ∣ ⟩k and the atomicresonance energy, respectively. It was assumed that the spread is due to the hopping ofthe electron into the substrate state ∣ ⟩k . This description of the spread of the atomiclevels due to the sp-band of the metal surface was derived from the self-consistentmodel of hydrogen chemisorption developed by Newns [18]. In the following we givesome key formulations and assumptions that lead to equation (1.4).

Newns’s self-consistent model of hydrogen chemisorption was restricted to thesimplest ‘neutral’ adsorbate—hydrogen with a 1s orbital denoted by ∣ ⟩a and the self-consistent eigenstates of the unperturbed semi-infinite metal denoted by ∣ ⟩k . The ϵa

and ϵk are the eigenvalues belonging to ∣ ⟩a and ∣ ⟩k when the adatom is infinitelyfar from the surface. After chemisorption, ϵ⟨ ∣ ∣ ⟩ =a H a a, ϵ⟨ ∣ ∣ ⟩ =k H k k and the onlynon-zero matrix elements of the perturbation are given by:

∫ φ φ= ⁎V x H x x x( ) ( ) ( ) d , (1.5)ak k1s HF3

where HHF is the Hartree–Fock (HF) self-consistent Hamiltonian after chemisorp-tion. The Hamiltonian was formulated in a similar way to Anderson’s description ofmagnetic impurities in alloys [25]:

∑ ∑ ∑ϵ ϵ= + + + +σ σ σ

σ σ σ σ σ σ†

−( )H n n V c c h c Un n. . , (1.6)k k

a a k ak a k a a

where σ denotes the spin, the first and second terms denote the set of unperturbedeigenstates, ∣ ⟩a and ∣ ⟩k , the third term gives the hopping terms and the last term

Physics of Surface, Interface and Cluster Catalysis

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describes the Coulomb interaction between electrons of opposite spin in the 1sorbital. Because of the smaller energy spread of the d-type ∣ ⟩k states lying near the EF

for transition metals as compared to the sp-bands, the latter are neglected to the firstapproximation. The difference between Newns’s and Anderson’s formulation of theHamiltonian is the former’s assumption that electron correlation does not play apredominant role. In particular, the solutions are non-magnetic. The FockHamiltonian, which is an unrestricted HF approximation, is derived by replacingthe two-particle interaction σ σ−Un na a by ⟨ ⟩σ σ−U n na a :

∑ ∑ϵ ϵ= + + +σ σ

σσ σ σ σ σ

†( )H n n V c c h c. . , (1.7)k k

a k ak a k

where

ϵ ϵ= +σ σ−U n (1.8)a a

is the effective adatom level spin σ . The one-electron Green’s operator for the σH is:

ϵ ϵ= + − =σ σ − +G is I H s( ) [( ) ] , 0 . (1.9)1

The matrix equation for G is:

ϵ ϵ− =σ σI H G I( ) ( ) . (1.10)

This can be solved in the unperturbed representation to give the Green’s function,Gaa , by [25]:

⎡

⎣⎢⎢

⎤⎦⎥⎥∑ϵ ϵ ϵ

ϵ ϵϵ ϵ ϵ ϵ= − −

− += − − Λ + ∆σ

σ σ

−−G

Vis

i( ) [ ( ) ( )] (1.11)k

aaak

k

21

1

⎡⎣⎢

⎤⎦⎥∑ ∑ϵ

ϵ ϵπ δ ϵ ϵ∆ = −

− += −V

isV( ) Im ( ) (1.12)

k k

ak

kak k

22

∫ϵπ

ϵ ϵϵ ϵ

Λ = ∆ ′ ′− ′−∞

∞P( )

( )d, (1.13)

where the P denotes the Cauchy principal value. The ϵ∆( ) is seen to be the weightedDOS function and ϵΛ( ) is its Hilbert transform. Thus, equation (1.4) can be tracedback to equation (1.12). It is the weighted DOS function for the Green function,

ϵσG ( )aa —the solution to the matrix equation for ϵσG ( ) corresponding to theHamiltonian, σH . How did Newns interpret the ϵ∆( ) and ϵΛ( )? The projectedDOS for the adatom orbital in terms of Fock eigenfunctions σ∣ ⟩m was defined as:

ρ ϵ σ δ ϵ ϵ= Λ∑ −σσm a( ) ( ). (1.14)

maa m

2

The ϵσG ( ) in equation (1.9) may now be expressed in perturbed representation as:

ϵ δ ϵ ϵ= − +σσ′ ′G is( ) /( ), (1.15)mm mm m

Physics of Surface, Interface and Cluster Catalysis

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whence, on transforming into unperturbed representation:

∑ϵ σϵ ϵ

=− +

σ

σG

m ais

( ) . (1.16)m

aam

2

Comparison of equations (1.16) and (1.11) gives the Fock eigenvalues ϵ σm asroots of

∑ϵ ϵϵ ϵ

− −−

=σV

0. (1.17)k

ak

k

2

Referring to equation (1.11), when ∆ = 0, the poles of σGaa may occur outside theband, given by the solutions of

ϵ ϵ ϵ− −Λ =σ ( ) 0. (1.18)

Figure 1.8 shows the formation of the roots of equation (1.18). The intuitiveunderstanding of ϵ∆( ) and ϵΛ( ) was derived using two limiting cases: (1) when thecoupling between the effective level, ϵσ, and the band is weak (i.e. if the separation ofthe ϵσ from any part of the band always greatly exceeds ϵ∆( ), then ϵΛ σ( ) will besmall), there is only one localized state that differs slightly from the atomic orbitalitself; (2) when the adatom–metal coupling is strong (i.e. if ϵ∆( ) over most of the

Figure 1.8. Semielliptical ϵ∆( ) with its Hilbert transform ϵ∧( ). Intersections of ϵ ϵ− σ with ∧ defined localizedstates are shown by lines a, b, c and d. These indicate various cases: a and c—one localized state; b—zerolocalized states; and d—two localized states. Reproduced with permission from [18]. Copyright (1969) by theAmerican Physical Society.

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band greatly exceeds both its width and the separation of ϵσ from its center), theresult must be two localized states widely straddling the band. In other words, inweak coupling conditions a localized state approximating the adatom orbital ∣ ⟩aexists, whereas when strong coupling occurs two localized states approximating thebonding and anti-bonding states of the molecule exist, consisting of ∣ ⟩a and the metalatomic orbitals to which it is directly coupled.

The weak coupling condition is often referred to for the renormalized adsorbatestates in the d-band model [17, 24]. Let us not forget, however, that in the Newnsmodel the metal sp-states are neglected and only the d-states are accounted for.Moreover, the localized states are an approximation to the adatom orbital itself,hence Newns expected that no significant broadening would arise from weakcoupling with metal states. For the mathematical derivation of the final chemisorp-tion energy expression, the reader is referred to [18]. Here it will suffice to trace theorigin of the renormalized states of an adatom in the d-band model, and theassumptions and similarities or differences.

1.4 When the surface electronic properties change: models based onNewns–Anderson

More often than not, the surfaces differ not only in the average energy of the d-band,εd , but also in their magnetic state. Why are the spin-polarized surfaces important?First of all, significant changes in the d-states of transition metals occur because ofspin polarization. Specifically, there are shifts in the energy levels for spin-up andspin-down electrons that cannot be fully accounted for by the εd . Moreover, theincorporation of 3d transition metals into noble metals for cost-effective catalysts isprevalent. This leads to more complicated changes in the reactivity. The chem-isorption on the 3d transition metals alone requires treatment of its magnetic stateand the consequent changes in the d-band. Kasai et al first studied chemisorption ona ferromagnetic 3d transition metal surface (Ni) with the inclusion of exchangecorrelation effects [26]. The alloy analogy of Hubbard’s second approximation[27, 28] was used to calculate the local magnetic moments of the adatom, itsneighbors and a few layers of the ferromagnetic surface. The model Hamiltonianwas described as:

= + + −H H H H , (1.19)A S A S

where,

∑ϵ= +σ

σ ↑ ↓H n U n n , (1.20)A a a a a a

∑ ∑ ∑ ∑∑= + + − −σ σ σ σσ

σ σ σ σ σ σ σ σ σ σ σ+

−+

′ ′+

′′

( )H T n tc c U n n J c c c c n n12

12

,

(1.21)i ij i ij

S i i j i i ij i i j j i j

Physics of Surface, Interface and Cluster Catalysis

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and

∑= +σ

σ σ σ σ−+ +H V c c c c( ), (1.22)A S a a0 0

where HA is the Hamiltonian for an adatom, and ϵa and Ua are the atomic energylevel and the intra-atomic Coulomb interaction, respectively. HS is the Hamiltonianfor the ferromagnetic metal substrate and T , U , J and t are the atomic bindingenergy, the Coulomb interaction for two electrons on the same site, the interatomicexchange interaction and the transfer matrix element between the nearest-neighborlattice sites, respectively. −HA S describes the coupling of the adatom and thesubstrate. The one particle Green’s function, σG E( )ij , satisfies the following self-consistent equations within the alloy analogy of the Hubbard second approximation[26, 27]:

∑δ= +σ σ σF E G E t G E( ) ( ) ( ). (1.23)l

i ij ij il lj

The reader is referred to [26, 29–31] for the detailed formulations and proceduresfor the calculation of equation (1.23). Figure 1.9 gives the local magnetic momentsat the adatom and the other relevant sites.

This study thus showed that upon chemisorption the local magnetic moment atthe adatom site is significantly reduced. Upon chemisorption, the local magneticmoments at the neighboring sites are also reduced. Only when the site is furtheraway does the magnetic moment of the second nearest neighbor on the surface andsubsurface remain the same.

Thus, this study revealed what happens to a ferromagnetic surface afterchemisorption [26]. The study was then revisited to consider what happens to themolecule before adsorption or during the adsorption process. Escaño et al studiedthe reaction pathways and the potential energy of a dissociating molecule (oxygen)

Figure 1.9. The local magnetic moments at an adatom and its neighboring sites in the case where the bulkmagnetic moment is 0.4 and = −V 3.0 eV (marked by solid circles), and those of the corresponding sites in aclean surface (marked with dashed lines). A, 0, 1 and 2 depict the adatom site, the nearest neighbor site, thesecond neighbor site and the second neighbor site in the subsurface, respectively. Reproduced with permissionfrom [26]. Copyright 1983 Elsevier.

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on spin-polarized platinum or ‘ferromagnetic’ Pt using density functional theory(DFT) [32, 33]. This model system was derived by placing a Pt layer on top of aferromagnetic substrate such as Fe(001). Figure 1.10 shows the LDOS of the Pt layer(on Fe(001)) with respect to that of the Pt(001). The εd of the Pt layer (on Fe(001)) isshifted downwards (more negative) by ∼0.43 eV with respect to that of the Pt(001).Of all the d-states, the Pt dzz shows a strong spin polarization around the EF. Sincethe Pt layer is not significantly strained (i.e. there is a very small lattice mismatch of∼0.5%), the dzz strong spin polarization at EF is due to the interaction of the dxy ofthe Fe subsurface (see figure 1.11).

Figure 1.10. The DOS projected on the d-orbitals of the Pt monolayer on Fe(001) is shown in solid lines andthat of the Pt(001) is shown in dotted lines. Reproduced with permission from [32]. Copyright 2008 Elsevier.

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For the reaction of the O2 on Pt/Fe(001), the pathways considered are shown infigure 1.12 and the corresponding potential energy curves (PEC) are shown in figure1.13. It was found that O2 dissociative adsorption prefers the bhb configurationwith practically no barrier. The binding energy of the two adatoms was −1.33 eV.In comparison with Pt(001), the activation barrier for dissociative adsorption was∼0.16 eV, while the binding energy of the two dissociated adatoms was −2.40 eV (seefigure 1.14). It was also found that the tbt configuration rendered a barrierlessadsorption on Pt/Fe(1001), as shown in figure 1.14. However, the final state is a

Figure 1.11. The DOS projected on the d-orbitals of the Fe underlayer is shown in solid lines and that of theFe(001) is shown in dotted lines. Reproduced with permission from [32]. Copyright 2008 Elsevier.

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molecular O2. The comparison of the bhb and tbt configurations in terms ofenergetics is also depicted in figure 1.14.

Incorporating Pt/Co into the systems gives the trend on the binding energy withthe εd shown in figure 1.15. The adsorption energies of the dissociated adatoms at thefinal state follow the trend predicted by the d-band model and in agreement withother chemisorption trend studies on O2 [20, 21]. However, the trend for theactivation barriers is counterintuitive [34]. This is explained as follows: because ofthe very localized dzz state at EF caused by spin polarization, for instance in Pt/Fe(001),

Figure 1.12. Reaction pathways for O2 dissociative adsorption on Pt/Fe(001) and Pt(001). Parallel pathways aredescribed in terms of the directions of both ends of the parallel O2 axis (towards hollow h, bridge b, or top t) andthe O2 center-of-mass (over h, b, or t). Vertical pathways are shown as solid circles and are labeled based on the siteof the two oxygen atoms. Reproduced with permission from [33]. Copyright 2009 American Chemical Society.

Figure 1.13. PEC for the dissociative adsorption of O2 on Pt/Fe(001) in different reaction pathways. The zeropotential energy is the energy of the isolated molecule and the Pt/Fe(001) surface (or slab). Vertical pathwaysare unstable, hence they are not included in this plot. Reproduced with permission from [33]. Copyright 2009American Chemical Society.

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Figure 1.14. PEC for the dissociative adsorption of O2 on Pt (001) in two reaction pathways (in accordancewith the minimum energy paths for Pt/Fe(001): tbt and bhb. The zero potential energy is the energy of theisolated molecule and the Pt (001) surface (or slab). Reproduced with permission from [33]. Copyright 2009American Chemical Society.

Figure 1.15. Relationship of the binding energies of O2 dissociative adsorption on pure Pt and Pt/M (M = Feand Co) with εd . Reproduced with permission from [34]. Copyright 2011 Institute of Physics Publishing.

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a strong coupling with the molecular states at a distance far from the surface can beexpected. Figure 1.16 shows the molecular states at a distance of ∼2.80 Å for Pt(001),Pt/Co(001) and Pt/Fe(001) systems.

In accordance with the Newns model for weak coupling, the anti-bonding states ofthe O2 molecule deviate slightly from that of the gas phase due to the interaction withthe pure Pt(001) d-states. This can be seen clearly in figure 1.16(a). However, themolecular states are already significantly broadened for Pt/Co(001) (figure 1.16(b))and there is already splitting of the states, which indicates strong coupling, for Pt/Fe(001) (figure 1.16(c)). Due to such splitting, the occupation of the anti-bonding states,which governs stability at the transition state, is prevalent on the spin-polarized Pt.Figure 1.17 explains the interaction in a simple schematic model. This schematicmodel shows that the creation of a localized state near EF in a metal surface via

Figure 1.16. The DOS projected on the bonding π( ) and anti-bonding states π⁎( ) of O2 in the gas phase (solidlines) and at the transition state (bold lines) on (a) Pt(001), (b) Pt/Co(001) and (c) Pt/Fe(001). The Pt dzz state isshown by dotted lines. Reproduced with permission from [34]. Copyright 2011 Institute of Physics Publishing.

Figure 1.17. Schematic diagram of the π⁎–dzz interaction at the transition state for O2 reaction on Pt/Fe(001).Based on the DOS in figure 1.16, interaction with anti-bonding states parallel to the surface is also evident,however, only the anti-bonding states perpendicular to the surface are shown here.

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methods such as spin polarization can lead to strong molecular states–metal surfacecoupling at a distance far from the surface, which changes the energetics trend at thetransition state. At the final state, however, the molecule already interacts with theentire d-band, paving the way for the effect of the rest of the d-orbitals and thus εd

correlates well with the oxygen atom adsorption energy. Finally, it has also beenconfirmed that the oxygen adatoms reduce the local magnetic moment of Pt sitesin Pt/Fe [32], in accordance with the study of Kasai et al on chemisorption onferromagnetic surfaces [26]. Deviations from the d-band can indeed occur whenspecial properties of the metal d-states arise. It is also worth noting that the abovestudies are in accordance with the Newns model of chemisorption on metal surfaces.

1.5 The LDOSEF modelAnother earlier parameter for the reactivity of metal surfaces is the LDOSEF. It hasbeen studied and considered since the 1980s. Feibelman et al first introduced theLDOSEF concept using sulfur (S) on Rh(001) [35]. It was found that the LDOSEF ofRh(001) changes significantly at low S coverage. Since the LDOSEF is unscreened, italso changes in the regions where S is not adsorbed, and it is therefore considered tobe a ‘responsive’ property of metal surfaces towards reactants. Figure 1.18 depictsthe changes in the LDOSEF at different sites on the S-covered surface with respect tothe clean Rh(001).

Figure 1.18. Muffin-tin LDOS for clean two-layer Rh(001) and for the three inequivalent Rh sites of theS-covered film. Rh numbers 3, 2 and 1 depict the LDOS for sites with no S neighbor, same-side S neighbors but noS directly below on the other side of the film and same-side S neighbors with S directly below on the other side ofthe film, respectively. Reproduced with permission from [35]. Copyright 1984 by the American Physical Society.

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Moreover, it was found that the presence of S creates electronic perturbationthat is not screened, leading to changes in the properties of CO co-adsorption(i.e. reduced binding energy). Here the early ‘poisoning effect’ case arises, whichdoes not come from charge transfer effects. Figures 1.19 and 1.20 show the plots for

Figure 1.19. Valence charge densities (in atomic units) for a two-layer Rh(001) film (a) with and (b) without aS(3 × 1) adlayer. The density changes by a factor of =2.154 ( 10 )1/3 from one contour to the next. Forcomparison, in the region between the contours of charge density ∼ −10 3 a.u., a hatched region is transcribedfrom the S/Rh onto that for the clean Rh. The geometry of the plot is indicated in the inset. Reproduced withpermission from [35]. Copyright 1984 by the American Physical Society.

Figure 1.20. Fermi-level DOS for two-layer Rh(001) films (a) with and (b) without a S(3 × 1) adlayer. As infigure 1.19, a hatched region is transcribed from the S/Rh onto that for the clean Rh, for comparison.Reproduced with permission from [35]. Copyright 1984 by the American Physical Society.

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contours of charge density (or full charge density) and contours of constant Fermi-energy LDOS (or charge density associated with all the states that lie within ±2 eVof the EF, which we shall call charge density) for S-covered Rh(001), respectively.

The figures indicate that the S adsorption induces greater changes in the chargedensity within the small energy window around the EF than in the full chargedensity, even at distances further than the neighboring sites. The authors extendedthe study to other ‘surface modifiers’, such as P, Cl and Li and noted the varyingamount of reduction of the LDOSEF on the changes in the charge density ofthe surface atoms, and on the different kinds of effects on CO adsorption (i.e.poisoning or promoting effects) [36]. The reduction in the LDOSEF over distancesgreater than the nearest Rh neighbors was greatest in Cl, followed by S and then P. Itwas found that such reductions are substantial for small coverage, making theLDOSEF effective for small coverage of surface modifiers. The opposite was notedfor Li, where the LDOSEF increased.

The full charge density and the charge density obtained from states at the smallenergy window around EF also vary in Li-covered Rh(001). The effect of thesesurface modifiers on CO adsorption is that the barrier to adsorption is higher on Cl,S and P, and it is lower on Li. This was attributed to the trajectory of CO adsorption.It was considered that as the molecule approaches the metal surface, the potentialbetween the CO and the surface is lowered and the metal electronic distribution‘bulges’ outward, lowering the total energy of the system. The magnitude of thebulge depends on the static polarizability, which in turn is strongly affected by theLDOSEF.

Finally, the LDOSEF also was used as an analogy for ‘local’ bond formation inmetal surfaces [37]. In general, the chemical reactivity of a molecule is characterizedby the local softness rs( ) [38]:

⎡⎣⎢

⎤⎦⎥

ρμ

= ∂∂ υ

rr

s( )( )

, (1.24)r( )

where ρ r( ) is the electron density, μ is the chemical potential and υ r( ) is the nuclearpotential acting on the electrons at point r. The ρ r( ) and μ are given exactly in theKohn–Sham formulation of DFT [39], where the ρ r( ) is related to μ by

∫ρ =μ

r rg E E( ) ( , )d , (1.25)

where rg E( , ) is the local Kohn–Sham DOS. Using equations (1.24) and (1.25), thelocal softness is:

⎡⎣⎢

⎤⎦⎥∫μ

μ= + ∂

∂

μ

υ

r rr

s gg E

E( ) ( , )( , )

d . (1.26)r( )

Yang and Parr [40] first found the relationship of the rs( ) and μ rg( , ), based onDFT. The equation (1.26) is solved by

∫ μ= ′ ′ ′−rs r K r r g r( ) d ( , ) ( , ), (1.27)1

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where ′r rK ( , ) is the transpose of the response function κ ′r r( , ). Details of thesolution can be found in [41]. The κ ′r r( , ) asymptotically approaches the Hartreeapproximation to the static Kohn–Sham dielectric function, ϵ ′r r( , )H . The electronicsoftness, therefore, is regarded as a response to the LDOSEF. Nuclear reactivity hasalso been introduced, and it is considered to be more fundamental to the under-standing of chemical reactivity [41]. The softness of the nucleus a, σa is the derivativeof the force on the nucleus with respect to μ. It has been shown that the σa is linearlyrelated to the μ rg( , ) via the electron–nuclear Coulomb force screened by κ ′r r( , ),such that nuclear softness is also considered to be a response to the LDOSEF [41].Whether electronic or nuclear, the LDOSEF or μ rg( , ) in the (1.26) formulation aboveis considered to be crucial.

1.6 Cluster reactivity and the link to surface modelsMetal clusters are widely used as catalysts in industry. Because of the varying sizesand the presence of vertices and edges, understanding their reactivity is largelydependent on the local nature of the adsorption sites. However, there are caseswhere surface reactivity models are still applicable to clusters. In this section, theconnection with the surface reactivity model is discussed and other electronicmechanisms are also introduced.

1.6.1 Strain in very large clusters

Strain has a direct correlation with the d-band model via the extent of the overlap ofthe d-orbitals of the metals [42]. In simple terms, the greater the overlap, the morebroadened the d-states become, and the εd shifts accordingly. Using figure 1.21, Liet al showed that interplanar compression of Pt clusters approaches the bulk limitfor particles larger than 10 nm [43]. It is apparent, therefore, that for very largeclusters surface reactivity models are still applicable.

1.6.2 Beyond quantum size effects in very small clusters

Small clusters have more complicated structures, so using a surface model such asthe d-band one is quite a simplification. Quantum size effects emerge as a viableexplanation because of the discreteness of the energy spectrum of small clusters.However, additional electronic phenomena can exist at such a size regime. Kleis et alstudied Au clusters in the 13-atom to 1415-atom (∼3.7 nm) size range and found acritical size of 561 atoms (or 2.7 nm) (see figure 1.22) [44]. Above this cluster size, theadsorption properties tend to the surface limit, while below this cluster size, markedchanges in the adsorption energies can be noted.

Apart from quantum size effects, it was found that the charge density distributionupon oxygen adsorption only resembles that of the (111) surface when the Au clusteris 561-atom size (see figure 1.23), while at smaller sizes, the charge densitydistribution is very different. For the 309-atom cluster, the electron density alongthe edges is already involved in the reaction. For the much smaller ones, the electrondensity response extends all the way to the bottom. Such a ‘screening cloud’ affectsthe nearest and the next nearest Au atoms, which in turn affects the adsorption.

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Figure 1.21. (a) The continuum model for the interplanar compression (d/deq) of Pt clusters with size. Thetriangles are the calculated values and the open circles are the experimental values. (b) The interplanar surfacestrain. The blue line is for cuboctahedra and the red is for Wulff-shaped clusters. Reproduced with permissionfrom [43]. Copyright 2015 American Chemical Society.

Figure 1.22. Size dependence of CO and O adsorption energies on inside facets (left) and on edges (right). Thehorizontal lines represent adsorption on (111) and (211) slabs. Reproduced with permission from [44].Copyright 2011 by Springer Science + Business Media.

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Pt clusters show the same behavior as Au clusters in terms of the presence ofscreening clouds for small ones. However, in the former, the critical size is muchsmaller (147 atoms or ∼1.6 nm) [45].

1.6.3 Clusters at finite temperature

It is often expected that metal clusters, especially small ones, can exhibit largechanges in structure due to temperature effects. Escaño et al conducted an ab initiomolecular dynamics simulated annealing of 1–2 nm Os metal clusters and obtainedthe changes in structure with respect to size, as shown in figure 1.24 [46]. It wasfound that at 300 K (see table 1.1) the average distance between atoms on the 0001facet does not change considerably with size. This implies the validity of thetheoretical predictions based on DFT calculations at 0 K for close-packed facets.However, for open facets (e.g. 1010) a different scenario can be observed: the effectof temperature is only insignificant when the size of the cluster is 2 nm. In addition,another kind of strain is found in metal clusters because of temperature effects,

Figure 1.23. Charge redistribution upon oxygen adsorption plotted at an isosurface value of 0.001 e Å−3. Redindicates accumulated electron density and blue indicates depleted electron density. Reproduced withpermission from [44]. Copyright 2011 by Springer Science + Business Media.

Table 1.1. Geometry of Os–NP at 300 K. aave, bave and ∠cave correspond to the Os–Os distances and angle, asshown in figure 1.24. The corresponding distances on Os–NP at 0 K are given in parenthesis. Reproduced withpermission from [46]. Copyright 2014 The Royal Society of Chemistry.

Os–NP aave bave ∠cave

Os–57 2.60 (2.64) 2.94 (2.66) 168.5° (180.0°)Os–89 2.68 (2.67) 2.81 (2.68) 168.8° (180.0°)Os–214 2.71 (2.69) 2.71 (2.69) 172.4° (180.0°)

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the buckling of edge atoms, or <cave in figure 1.24. The effect of temperature on thebuckling of edge atoms is found to be minimal for the 2 nm Os cluster.

1.7 SummaryExcept for the earlier LDOSEF, models for the reactivity of metal surfaces are basedon the self-consistent model of chemisorption developed by Newns. For instance, thed-band model, can trace its origins to the Hamiltonian describing the chemisorptionof hydrogen on metal surfaces. The d-band model, however, describes the broad-ening of molecular states because of interaction with metal sp-states, which isotherwise neglected in the Newns model (interaction with the d-states is considered).New insights into the d-band model have been derived, especially for cases where thereactant properties are different from H2’s. It has been found that opposite trends forthe center of the d-band can be obtained when the reactant’s relevant anti-bondingstates are filled, or when the metal surface d-band is filled. The mechanisms for suchnew findings have been discussed and revisited. Some variations to the d-band modelhave been incorporated to better capture reactivity on metal surfaces. For instance,the effect of the shape of the DOS and the bandwidth have been included to bettercharacterize the reaction of specific molecules, such as C, N and O and theirhydrogenated species on metal surfaces [8, 47]. It can be stated that the d-bandmodel is powerful in terms of describing variations in the chemisorption energy ofthe same molecule over a wide ensemble of different metal surfaces. Care has to betaken, however, when representing deeper mechanisms for specific molecule–surfaceinteractions.

When the metal surfaces are changed—especially when this involves morecomplicated shifts in the d-band, such as spin polarization—more detailed andfundamental concepts of reactivity are derived. Based on the Newns–Andersonmodel, changes in the magnetic surface upon chemisorption are observed, such asthe lowering of the local magnetic moment on sites other than the chemisorption

Figure 1.24. Final structures of Os metal clusters at (a) ∼1.0 nm, (b) ∼1.5 nm and (c) ∼2.0 nm. The aave

denotes the distance between one edge atom and one vertex atom at the 0001 facet, bave gives the distancebetween the atoms on a more open facet, and 1010 and <cave correspond to the buckling angle of the edgeatoms at the 1100 facet. Reproduced with permission from [46]. Copyright 2014 The Royal Society ofChemistry.

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site. It has also been found that because of spin polarization, the metal surface canrender two different energetics at the transition and the final states of dissociation.Overall, the bonding of a molecule on a metal surface is governed by many factorsand the choice of the suitable reactivity concept/model is in fact largely dependent onthe reactants (adsorbate) and the metal electronic properties. Metal surfacereactivity models can extend to metal clusters when they are sufficiently large,even at finite temperature. For small clusters, electronic mechanisms such asscreening effects are observed in addition to quantum size effects.

1.8 Future prospectsThe reactions of molecules on metal surfaces are not as simple as they seem.Reaction per se is a complex process. The complexity can be due to the size of themolecules and their composition, the sites on the surface, the composition of thesurface and the presence of defects. Making improvements to the current modelsthat can capture more realistic scenarios such as these can be challenging. Forinstance, using surface reactivity models to describe the supported cluster’s reactivitymay be unsuitable or limited, hence the development of new reactivity models in thisdirection is worthwhile.

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