Influence of Electron-Hole Pair Excitation on Dissociative Sticking

Signe Kvist Mengel and Gert D. Billing*Department of Chemistry, H. C. Ørsted Institute, UniVersity of Copenhagen,2100 Copenhagen Ø, Denmark

ReceiVed: June 24, 1997; In Final Form: October 1, 1997X

Using a semiclassical model that treats the phonons and electrons of a metal quantally within a secondquantization formalism and the dynamics of the incoming molecule classically, we have calculated thedissociative sticking and energy accommodation as a function of band structure for the electron-hole pairexcitation. It is shown that the electronic band structure has some influence on the reactivity, such that theinclusion of this aspect makes the 111 surface more reactive than the 100 surfacesin agreement withexperimental findings.

1. Introduction

Recent progress in ab initio density functional calculationson molecule surface interaction1 has made a number of data onthe lowest adiabatic potential energy surface available. The abinitio data give information on the potential barrier andmolecular geometries at a number of selected approach sitesand angles. Usually up to a maximum of a few hundred pointshave been calculated. Often these points are represented by apotential surface in two dimensionsV(r,z) wherer is the bonddistance of the diatomic molecule andz the distance from thecenter of mass to the surface. However, the full dynamicalproblem involves six degrees of freedom for the molecule and3NR - 6 for the solid consisting ofNR atoms. Even for lightmolecules as hydrogen the phonon excitation may also not beneglected a priori.2,3 Previous investigations have shown thatalso electron-hole pair excitation can be important,4 i.e., theassumption that the dynamical process takes place on anadiabatic potential energy surface breaks down. These non-adiabatic processes will give additional friction and enhancethe residence lifetime of the molecule at the surface; i.e.,coupling of the various degrees of freedom is enhanced andeven the phonon-coupling increases.Thus the complete understanding of chemisorption processes

involves the solution of a many-body problem in the phononsand electrons of the system. This also requires a constructionof potential surface functions that include these aspects. Wehave in a number of previous papers formulated a semiclassicaltheory3,5-7 that is capable of including these aspects. The theoryis briefly reviewed below. The present paper improves thedescription of the one-electron wave functions used for describ-ing the electron-hole pair excitation. Here 2D-wave functionsare introduced and a five-band model for the band structure isused. Results are compared with the previous obtained using1D-wave functions and a two-band model and with calculationswhich neglect electron-hole pair excitation altogether. Thesystem chosen for investigation is the hydrogen-copper systemfor which a multidimensional embedded diatomics in molecules(EDIM) potential has recently been constructed3 by fitting theparameters to ab initio density functional data from.8

2. Theory

The semiclassical theory which has been developed in ourprevious publications3-5 treats the phonons and electrons withina quantum description using a second quantization approach.

The motion of the incoming atom/molecule is solved simul-taneously with the excitation processes in a self-consistentmanner. Hence the motion is governed by an effective potential

whereM is the mass andm the reduced mass of the diatom,PX,PY,PZ are the momenta connected to the motion of the centerof mass of the diatom andpx, py, pz the momenta for the internaldegrees of freedom (rotation and vibration). The interactionpotential is expanded around the equilibrium position for themetal atoms. The first term in this expansion isV0. The nextterms (usually only the linear term is included) are representedby Veff

ph - an effective potential

where the expectation value of the phonon coordinatesQk istaken. Vk

(1) is the first derivative of the interaction potentialwith respect to phonon modeQk andΦph(t) the wavefunctionfor the phonons. Thus the phonons are excited duringcollisionsthereby changing the wavefunction and in turn theeffective potential governing the motion. Other excitationprocesses are those due to electron-hole pair excitation. It ispossible3,4 to formulate the response from this excitation in away that gives yet an additional effective potentialVeff

elho

governing the dynamics of the molecule. FinallyEint is theenergy transferred to the metalsit can be divided into energytransfer to the phonons and electrons. The effective potentialwhich arises due to the coupling of the motion of the moleculeand the electrons of the metal is

whereUC is the screened Coulomb potential (see section 5) andΨel(t) the wavefunction for the electrons. We have previouslygiven an approximate solution of the electronic excitationproblem using a second-quantization approach.4 This theoryresulted in an effective potential of the formX Abstract published inAdVance ACS Abstracts,November 15, 1997.

Heff ) 12M

(PX2 + PY

2 + PZ2) + 1

2m(px

2 + py2 + pz

2) +

V0(R, r ) + Veffph + Veff

elho+ Eint (1)

Veffph ) ∑

k

Vk(1) ⟨Φph(t)|Qk|Φph(t)⟩ (2)

Veffelho) ⟨Ψel(t)|UC|Ψel(t)⟩ (3)

10781J. Phys. Chem. B1997,101,10781-10790

S1089-5647(97)02061-0 CCC: $14.00 © 1997 American Chemical Society

where the matrix elementsUij are given by

j0 is a Bessel function,ωji ) (εj - εi)/p, andAii(2) ) ∑j|Aij(1)|2

where

We have introduced the one-electron wave functionsi andthe Fermi-energyεF. The Fermi energy is taken as that of a“free” electron. However, for the 2-D calculations we makethe restriction that the electron is constrained to move in twodimensions. The density of states is determined by how muchspace (∆A) the states occupy in reciprocal space. HavingNelectrons inside the areaA gives∆A ) (2π)2/A. Introducingthe density of statesN/A we get the Fermi vector from theequation

Assuming that each copper atom donates one electron to theelectron gas gives altogether two electrons per unit celld2 andhence the density 2/d2 and a Fermi vectorkF ) 0.982 Å-1. Thusthe Fermi energy 3.545ε (1 ε ) 100 kJ/mol) lies in the lowerband (see Figures 2-4). When performing the summation overstates below the Fermi level we introduce a finite grid inkx, kyspace and use

We have used the same density for the both the 100 and the111 surface, i.e., any difference seen between these two surfacesdoes not come from a difference in the density of states butrather from their “symmetry” in direct as well as momentumspace.The “transition” matrix elementsUij will depend upon time

through the time-dependence of the trajectory and the summationin eq 4 is over statesi with energy less than the Fermi energyand statesj with energy above the Fermi energy. In section 4we discuss how the one-electron wavefunctions are determined.

3. The Potential

To include phonon and electron excitation processes, thepotential must depend on the normal mode coordinates for themetal and the electron coordinates of the electrons. This isacheived using an EDIM (embedded diatomics in molecules)potential9 and by assuming that the excess charge on theincoming moleculesa charge arising from charge transfer fromthe metal to the moleculesinteracts with the electrons in themetal through a screened Coulomb potential. The chargetransfer process is an inherent part of the adiabatic (electronicallyrelaxed) ab initio data for the interaction. However, in orderto simulate the experimental situation we should account forthe fact that the collision process may not take place on theadiabatic surfacesbut rather on a “diabatic” surface whichincludes electronic friction arising from nonadiabatic processes.

The construction of the EDIM potential is based upon acombination of the DIM method for constructing potentials forpolyatomic systems from diatomic fragment potentials and theEAM (embedded atom) method for interaction of an atom withthe electron gas of the metal. The EDIM potential contains anumber of parameters, which are determined by fitting thoseto available ab initio DFT (density functional) calculations. Thescreened Coulomb potential is an often used model for theinteraction of a charged particle with a medium consisting ofelectrons distributed on an “ionic” background. Thus bothmodels are reasonable from a physical point of view and theapproximations quite transparent. Furthermore realistical mo-lecular dynamical calculations can be carried out using thesetwo models.The construction of the EDIM potential has been given in

detail in a previous publication3 and will therefore only bebriefly reviewed here. The lowest adiabatic surface is obtainedby solving a 2× 2 secular equation, i.e.

where the matrix elementsHij are expressed in terms ofCoulomb and exchange integrals

where a and b denote the two atoms and M the metal. TheCoulomb and exchange integrals are represented as sums anddifferences of singlet and triplet curves

For the gas molecule we use Morse and anti-Morse potentialfunctions for the singlet and triplet curves respectively, but forthe atom-metal interaction the embedded atom expressions

are used. The first term is a contribution from an embeddingenergyFa(F), which is a function of the electron density createdby the metal at atom a, and the second term consists of screenedCoulomb pair potentials

whereZa andZR are effective nuclear charges

The effective charge and the embedding function containparameters that has been determined by fitting the energyW11

to available ab initio DFT calculations. The advantage of thisanalytical presentation is that the potential does depend upon

Veffelho)

2Re ∑i,εi<εF

∑j,εj>εF

j0(2xAii(2))Uij

p∫t0t U*ij sin(ωji(t - t′)) dt′ (4)

Uij ) ⟨i |UC|j⟩ (5)

Aij(1) ) 1

p∫t0tUij(t) exp(iωij t′) dt′ (6)

N) 2πkF

2

∆A(7)

∑k

f ∑kx

∑ky

∆kx∆ky

∆A(8)

W11 ) 1/6(H11 + H22 + H12) -1/6xH11

2 + H222 - H11H22 + 2H12(2H12 + H11 + H22) (9)

H11 ) 4(Qab+ QaM + QbM + Jab) - 2(JaM + JbM) (10)

H22 ) 4(Qab+ QaM + QbM + JaM + JbM) - 2Jab (11)

H12) -2(Qab+ QaM + QbM + Jab+ JaM + JbM) (12)

Qab) 1/2(Eabs + Eab

t ) (13)

Jab) 1/2(Eabs - Eab

t ) (14)

EaMs ) Fa(Fa) + ∑

R)1

M

φaR(RaR) (15)

EaMt ) -Fa(Fa) + ∑

R)1

M

φaR(RaR) (16)

φaR(R) ) Za(R) ZR(R)/R (17)

Za(R) ) Z0a(1+ âaRνa) exp(-RaR) (18)

10782 J. Phys. Chem. B, Vol. 101, No. 50, 1997 Mengel and Billing

the position of the metal atoms and hence also upon the normalmode coordinates for the phonons.

4. Band Structure

The evaluation of the effective potentialVeffelho requires that

the band structure of the metal is known. This section describesthe two different models of the band structure investigated.4.1. One-Electron Potential. The one-electron wavefunc-

tionsψ and the eigenenergiesψ are the solutions to the time-independent Schro¨dinger equation

whereme is the electron mass. The effective potentialU is thepotential originating from the interaction of the electron andthe positive ions of the lattice and from the interaction of theelectron and the other electrons of the metal.U is assumed tohave the symmetry of the crystal lattice of the metal. Becausethe potentialU is periodic in the lattice it can be written as

where the sum runs over all reciprocal lattice vectorsG.Different models for the bandstructure are obtained by

choosing different forms of the effective potentialU. PreviouslyBilling4 used a potential that restricted the electrons to move inone dimension. The effective potential was assumed to be asimple periodic potential with a periodd1D equal to the nearest-neighbor distance of the atomic nuclei in the copper lattice(Table 1). We have usedd1D ) 2.56 Å. In the following thismodel is referred to as the 1D-model.We have extended the 1D-model so that the electrons instead

of being constrained by the potential to move along a line areallowed to move in a plane. This new model is hereafterreferred to as the 2D-model. As in the 1D-model we have useda simple expression for the one-electron potentialU. But asthe electrons are free to move in a plane it seems natural todemand thatU has the periodicity of the atomic nuclei in thetop layer of the crystal. Because of this requirementU dependsin the 2D-model on the orientation of the crystal.

We have investigated two surfaces: a Cu(100) and a Cu-(111) surface. Those surfaces are illustrated in Figure 1 andthe expressions for the potentials are given in Table 1. Thevectors b1 and b2 in Table 1 are the basis vectors of thereciprocal lattice. If the lattice constant of copper is denotedby d2D and the orthonormal basis vectors of the plane are calledi and j the basis vectors of the lattice are:b1 ) (2π/d2D)i +(2π/d2D)j and b2 ) (2π/d2D)i - (2π/d2D)j for Cu(100) andb1 ) {(2x2π)/d2D}i - {(2x2π)/(x3d2D)}j and b2 ){(4x2π)/(x3d2D)}j for Cu(111), whered2D ) 3.61 Å.4.2. The Approximative Solution. From elementary text-

books on solid state physics it is well-known that the one-electron eigenfunctionsψ and the corresponding eigenenergiesε can be indexed by a wave vectork and a band indexn. Thewave vector can be considered to lie in a unit cell of thereciprocal spacesfor instance in the first Brillouin zone (seeFigure 1) and the band indexn is a natural number. Theeigenfunction can without loss of generality be written as a planewave expansion

where the summation runs over all reciprocal lattice vectorsG.See e.g. ref 10, p 138. The coefficientsck-G,n and theeigenenergiesεk,n are determined by the infinite set of coupledequations

where the free electron energy is defined byεk0 ) (p2k2)/2me.

We now assume that for a given vectork there ism recipro-cal vectorsG1, G2, ..., Gm such that the free energiesεk-G1

0 ,εk-G2

0 , ..., εk-Gm

0 are within the orderV of the Fourier coef-ficients in (20) of each other, but far apart from the otherεk-G0 on the scale ofV

In ref 10, p 155, it is shown that in this case (22) can to leadingorder in V be replaced by the far simpler set ofm coupledequations

The eigenfunction with eigenenergyεk,n is then approximated

TABLE 1: An Overview of the Different One-ElectronPotentials Investigated

1D-model U (x) ) 2U0 cos (2π/d1D)x2D-model, Cu(100) U (r ) ) 2U0 cosr ‚b1 + 2U0 cosr ‚b22D-model, Cu(111) U (r ) ) 2U0 cosr ‚b1 + 2U0 cosr ‚b2 +

2U0 cosr ‚(b1 + b2)

Figure 1. Direct and the reciprocal lattice of the Cu(100) and the Cu(111) surface. The black dots symbolizes the atoms of the lattice for the directand the reciprocal lattices.a1, a2 and b1, b2 are the basis vectors of the direct and reciprocal lattice, respectively. The shaded areas show theBrillouin zones.

(- p2

2me∇2 + U) ψ ) εψ (19)

U(r ) ) ∑G

UG eiG‚r (20)

ψk,n (r ) ) ∑G

ck-G,n ei(k-G)‚r (21)

(εk-G0 - εk,n) ck-G,n + ∑

G′UG′-G ck-G′,n ) 0 (22)

|εk-G0 - εk-Gi

0 |. V, i ) 1, ...,m; G * G1, ...,Gm (23)

(εk-Gj0 - εk,n) ck-Gj,n + ∑

i)1

m

UGi-Gjck-Gi

,n) 0 (24)

Influence of Electron-Hole Excitation J. Phys. Chem. B, Vol. 101, No. 50, 199710783

by the finite sum

We chooseU0 in Table 1 so small that this week coupling limitis fulfilled. Physically this corresponds to the fact that theconducting electrons of copper in many ways behave like freeelectrons.4.2.1. 1D-Model. In accordance with the former investiga-

tion4 we take the interval [0,G1 [whereG1 ) 2π/d1d as the unitcell of the reciprocal space and invoke a two-band model byusing the free-electron energiesεk

0 andεk-G1

0 in (24). Equation24 then determines the two lowest energy bands and thecorresponding normalized eigenvectors. The expressions forthe energy bands are then (εk,1 ) εk

+, εk,2 ) εk-)

The coefficients of the corresponding normalized eigenvectorsare (ck,1 ) ck

+, ck-G1,1 ) ck-G1

+ , ck,2 ) ck-, ck-G1,2 ) ck-G1

- ) with

4.2.2. 2D-Model. We take the first Brillouin zone as theunit cell in the reciprocal space.A good approximation to the bandstructure for the Cu(100)

surface is obtained by invoking a five-band model by using thefree electron energiesεk

0, εk-b10 , εk+b1

0 , εk+b20 , εk+b2

0 in (24). It isexactly those free electron energies which are nearly equal fork in the neighborhood of the center of the Brillouin zone.Equation 24 then reduces to

This system of coupled linear equations is solved numerically.The two lowest energy bands of the solution are shown in Figure2.For the Cu(111) surface it is actually nessecary to use a

seven-band model because bothεk0, εk-b1

0 , εk+b10 , εk-b2

0 , εk+b20 ,

εk-(b1+b2)0 andεk+(b1+b2)

0 are nearly equal whenk is close to thecenter of the first Brillouin zone. Equation 24 then reduces toa set of seven coupled linear equations instead of the five

coupled linear equations given in (29). This set of sevencoupled linear equations is solved numerically. The two lowestenergy bands of the solution are shown in Figure 3. To reducethe computation time we have as well used a five-band modelfor the Cu(111) surface by using (24) with the free electron-energiesεk

0, εk-b10 , εk+b1

0 , εk-b20 , εk+b2

0 . The two lowest energybands of the solution are shown in Figure 4.

5. The Interaction Potential

The last requirement in order to calculate the effectivepotentialVeff

elho is an expression for the matrixelementsUij. Asin the previous work we imagine that the nonadiabatic electronicexcitations is induced by the interaction of the excess chargeon the molecule and the electrons in the metal. We assumethat the interaction can be described by a screened CoulombpotentialUC, i.e.

whereXi is the distance between the electron and the excesschargeeδi(zi,r) on the atom numberi of the molecule,zi is thedistance of the atom numberi from the surface,r is the bondinglength of the molecule, andkc is the screening wavenumber.We consider the screening wavenumber to be a parameter andfor simplicity use the parametera defined by

wherekF is the Fermi vector. As mentioned below the Fermi-Thomas model give a screening proportional to (kF)1/2. We havehowever chosen to parameterize it by the above expression suchthat the magnitude ofa is directly the screening length (in Å).The excess charge is as described in ref 7 determined by an

analytic fit to the ab initio data of Madhaven and Whitten.11

The calculation of the matrixelementsUij ) ⟨ψk,n|UC|ψk′,n′⟩in the expression for the effective potential is faciliated bymaking a Fourier expansion of (30). In the case where theelectrons are restricted to move in one dimension we have:

whereK0 is a Bessel function andx is the position of theelectron. This expression is used in the 1D-model. In the casewhere the electrons are allowed to move in two dimensions (x,y) the result3 is

This expression is then used in the 2D-model. Because boththe interaction potentialUC and the one-electron wave functionsare written as a plane wave expansion it is now possible tocalculate the matrix elements by using the condition:⟨eimx|einx⟩) 2πδn,m, whereδn,m is a Kroneckerδ-function.

ψk,n (r ) ) ∑i)1

m

ck-Gi,nei(k-Gi)‚r (25)

εk+ ) 1

2[εk-G1

0 + εk0 + x(εk-G1

0 - εk0)2 + 4U0] (26)

εk- )

εk0εk-G1

0 - U02

εk+ (27)

ck( ) (

U0

xU02 + (εk

0 - εk()2

(28)

ck-G1( ) ( x1- ck

(2

(εk-b10 - εk,n 0 U0 0 0

0 εk-b20 - εk,n U0 0 0

U0 U0 εk0 - εk,n U0 U0

0 0 U0 εk+b20 - εk,n 0

0 0 U0 0 εk+b10 - εk,n

) ×(ck-b1,n

ck-b2,n

ck,nck+b2,n

ck+b1,n)) (00000) (29)

UC ) ∑i∈{1,2}

-e2δi (zi,r)

Xie-kcXi (30)

kc ) akF (31)

UC ) ∑j∈{1,2}

e2δ(zj,r)

d1Dπ∑

m)-∞

m)∞

eim(x/d1D) K0(zjx( md1D)2

+ kc2)(32)

UC ) ∑j∈{1,2}

e2δ(zj,r)

2π d2D∑

m)-∞

m)∞

∑h)-∞

h)∞

eim(x/d2D)+ih(y/d2D) ×

1

xh2 + m2 + (d2Dkc)2

e-zj{kc2 + (h/d2D)2 + (m/d2D)2}1/2 (33)

10784 J. Phys. Chem. B, Vol. 101, No. 50, 1997 Mengel and Billing

5.1. 1D-model. In the 1D-model we use a two-band modelfor the bandstructure. The expression for the one-electon wavefunction is then

and therefore

whereG1 ) 2π/d1D andm is an integer. Please note that foreach statei ) (k, n) the statesi′ ) (k′, n′) for which Ui,i′ isdifferent from 0 are characterized by only three possible valuesof k′. We therefore say that we have three “selection rules”.5.2. 2D-Model. In the cases where we use a five-band

model for the band structure in the 2D-model (25) reduces to

By using this expression we get

with ⟨ψk,n |eim(x/d2D)+ih(y/d2D)|ψk′,n′⟩ given by

In this case we get 13 “selection rules”. As mentioned in 4.2.2it is most correct to use a seven-band model for the bandstructure of the Cu(111) surface in the 2D-model. In the seven-band model the expression replacing (35) instead of 13 contains19 “selection rules”. This expression is not given here.

6. The Simulations

In this section we first describe the simulation procedurebriefly. Hereafter we give a survey of how the above definedmodels were tested in the simulations. In the simulations themovement of each atom of the hydrogen molecule was obtainedby numerically integrating the classical equations of motionsfor the diatomic molecule. The simulation was started with themolecule placed with its center of mass (CM) at a distance of7 Å from the copper surface. At this distance the effectivepotential is essentially just the Morse potential of the hydrogenmolecule. The integration of the equations of motion was doneusing a predictor-corrector integrator with variable step lengthand order. The simulation was stopped when one of thefollowing events occurred:1. More than the initial translational part of the kinetic energy

of the molecule was transferred to the phonons. In this casethe molecule was consideredadsorped.2. The distance between the atoms of the molecule was larger

than 2.5 Å. In this case the molecule was considereddissoci-ated.3. The distance, of the center of mass of the molecule, from

the surface was larger than the corresponding initial distance.In this case the molecule was considered to bescattered.In real experiments not all of the initial conditions of the

molecule can be even approximately controlled. Often thoseinitial conditions involve the aiming point, the orientation ofthe molecule, and the binding length. Therefore one choosesthese conditions at random and finally make an average of theresults obtained with different conditions. However the com-putation time when using the 2D-model was so long that thisapproach was not possible because it would require thesimulation of too many trajectories. Instead we first at randomchose 110 sets of those initial valuess“the values of theuncontrolled parameters”. Those 110 sets were then used inall the simulations.

ψk,n ) ck-G1ei(k-G1)x + ck e

ik‚x

U(k,n),(k′,n′) ) ∑j∈{1,2}

2e2δ(zj,r)

d1DK0 (zjx( md1D)

2

+ kc2) ×

{ck,nck′,n′ + ck-G1,nck′-G1,n′ if k′ ) k- (m/d1D)

ck-G1,nck′,n′ if k′ ) k- G1 - (m/d1D)

ck,nck′-G1,n′ if k′ ) k+ G1 - (m/d1D)} (34)

ψk,n ) ck-b1,nei(k-b1)‚r + ck-b2,n

ei(k-b2)‚r + ck,n eik‚r +

ck+b1,nei(k+b1)‚r + ck+b2,n

ei(k+b2)‚r

U(k,n),(k′,n′) )

∑j∈{1,2}

e2δ(zj,r)

2π d2D∑

m)-∞

m)∞

∑h)-∞

h)∞ 1

xh2 + m2 + (d2Dkc)2

×

e-zj{kc2 + (h/d2D)2 + (m/d2D)2}1/2×⟨ψk,n|eim(x/d2D)+ih(y/d2D)|ψk′,n′⟩

Influence of Electron-Hole Excitation J. Phys. Chem. B, Vol. 101, No. 50, 199710785

The rest of the parameters necessary to specify the initialcondition are the size of the center of mass momentum,PCM,the direction of the center of mass momentum given by thepolar angelsθ andφ, the internal state of the molecule specifiedby the quantum numbersJ and V, the screening parameteraand the surface temperatureTs. We have used the values ofthose parameters given in Table 2. Since the same set of randomvariables are used, we in this way ensure that the observeddifference between the results obtained with the three modelsare due to differences in the models and not to differences inthe initial conditions. On the other hand the number oftrajectories is sufficient for some important quantities, calculatedfrom the simulations, to be converged.We now describe which combinations of the 1D-model and

the 2D-model with then-band models of the bandstructure (n∈ {2, 5, 7}) we have used in the simulations.On the Cu(100) surface we used1. The 1D-model together with the two-band model for the

band structure2. The 2D-model together with the five-band model for the

band structure3. A model where the electron-hole excitation is neglected.

This model is in the following called the “-elhomodel”

On the Cu(111) surface we used1. The 1D-model together with the two-band model for the

band structure2. The 2D-model together with the five-band model for the

band structure3. The 2D-model together with the seven-band model for

the band structure4. The -elho modelIt is not relevant to use the two-band model for the

bandstructure together with the 2D-model because the band-structure obtained from the two-band model does not have thesymmetry of the atoms of the top layer of the crystal.The 1D-model is exactly the same for the Cu(100) and Cu-

(111) with respect to the treatment of the electron-holeexcitation. The effective potential of those two surfaces thenonly differ in the adiabatic potential and in the effective potentialoriginating from the coupling to the phonons.In the 2D-model the effective potentials used on the Cu(100)

surface and the Cu(111) surface differ as in the 1D-model inthe adiabatic potential and the effective potential originatingfrom the coupling to the phonons. But the 2D-model also differin the effective potential originating from the electron-holeexcitation in the metal. Even in the case where a five-band

Figure 2. Illustration of the two lowest energy bands of the Cu(100) surface obtained with the five-band model. On the left the 3-D plots of thesurfaces are shown and on the right the corresponding contour plots. The unit of the reciprocal wavevectors is Å-1 and the energy unit isε ) 100kJ/mol.

TABLE 2: An Overview of the Initial Conditions Used in Testing the Models. 1 τ ) 10-14 s

parameter RCM/Å PCM/(amu‚τ/Å) J V a Ts/K θ/deg φ/deg

value 7 1.2, 1.5, 1.7, 2.0 0 0 0.25, 0.5, 1.0 300, 600 0 0, 31.83, 41.41

10786 J. Phys. Chem. B, Vol. 101, No. 50, 1997 Mengel and Billing

model is used on both surfaces this potential is different on thetwo surfaces due to the different basisvectorsb1 andb2.When invoking the 2D-model on the Cu(111) surface it is

most correct to use the seven-band model for the bandstructure,but as this model as mentioned above is numerically quitecomplicated we have used the 2D-model together with the five-band model in many of the simulations. We have chosen toreport results for various values of the screening parameterakF.Application of the Fermi-Thomas model for this parametergives (kF)1/2. Since kF is 1.36 Å-1 for copper this modelcorresponds toa ) 1.3 Å.

7. Results and Discussion

From the results of each series of 110 simulations we havecalculated the following quantities: The probability of dissocia-tion (Pdis), the probability of adsorption (Pads), the probabilityof scattering (Pscat), the final kinetic (Ekin), rotational (Erot), andvibrational energy (Evib). Also the energy transferred to thephonons (Eint) and to the electrons of the metal (Eelho) arereported.

Below we present some of the results discussing the followingthree questions:1. Do the different models give different values of the just

mentioned quantities for the same initial conditions?2. Do the values of those quantities determined by the

different models have different functional dependences on theinitial conditions?3. On purely theoretical grounds one must expect that the

2D-model is the most realistic. But will a comparison withexperimental results also suggest that it is necessary to use themuch more elaborate 2D-model?7.1. Different Values. The answer to the first question is

yes. The effect is most pronounced for the probability ofdissociation (Pdis) and for the probability of scattering (Pscat)1 - (Pads + Pdis)). The values ofPdis determined with threedifferent values ofa (in Å) are given as a function of the initialkinetic energy (Ei ) PCM

2 /2m wherem is the mass of thehydrogen molecule) in Table 3 for the Cu(100) surface and inTable 4 for the Cu(111) surface. The values of (Pads + Pdis)are in the same way given in Table 5 for the Cu(100) surface

Figure 3. Same as Figure 2 but for the Cu(111) seven-band model.

TABLE 3: The Pdis for Cu(100), TS ) 300 K, θ ) O ) 0, J ) W ) 0. 1 E ) 100 kJ/mol

1D-model2D-Model(five-bands)-elho

model a) 1.00 a) 0.50 a) 0.25 a) 1.00 a) 0.50 a) 0.25

Ei ) 0.36ε 0.00 0.00 0.03 0.19 0.00 0.11 0.30Ei ) 0.56ε 0.00 0.01 0.18 0.48 0.01 0.33 0.60Ei ) 0.72ε 0.01 0.03 0.26 0.57 0.06 0.42 0.72Ei ) 1.00ε 0.11 0.12 0.32 0.62 0.17 0.44 0.74

Influence of Electron-Hole Excitation J. Phys. Chem. B, Vol. 101, No. 50, 199710787

and in Table 6 for the Cu(111) surface. It is seen that the valuesof Pdis obtained with the-elho model are smaller than the resultsobtained with the 1D-model which in turn are smaller than theresults obtained with the 2D-model. The same relation is seenfor (Pads+ Pdis). The absolute difference between the resultsobtained with the three models depends on the initial conditions,but the conclusion that the values obtained by the 2D-modelare the largest and the values obtained by the-elho model arethe smallest holds for all the investigated initial conditions.The mean energy transferred to the phonons (Eint) by the

scattered molecules also shows some dependence on the model.

The results are given in Table 7 for the Cu(100) surface and inTable 8 for the Cu(111) surface. It is seen that in generalEintfor the Cu(111) surface determined with the -elho model wassmaller thanEint determined with the two other models. Theopposite result was observed on the Cu(100) surface. Here thetendency was that the values determined by the -elho modelwere the biggest.In the Tables 10, 11 and 12 the values ofPdis, Pads + Pdis,

andEint respectively are given for the Cu(111) surface in thecase where the angle of incidence was different from 0. Thevalues are given as functions of the so called normal energy

Figure 4. Same as Figure 2 but for the Cu(111) five-band model.

TABLE 4: Pdis for Cu(111), Ts ) 300 K, θ ) O ) 0, J ) W ) 0. 1 E ) 100 kJ/mol

2D-model

1D-model (five-bands)-elhomodel a) 1.00 a) 0.50 a) 0.25 a) 1.00 a) 0.50 a) 0.25

(seven bands)a) 0.5

Ei ) 0.36ε 0.00 0.00 0.00 0.08 0.00 0.23 0.39 0.21Ei ) 0.56ε 0.00 0.00 0.05 0.29 0.03 0.46 0.68 0.41Ei ) 0.72ε 0.00 0.02 0.16 0.37 0.13 0.54 0.78 0.55Ei ) 1.00ε 0.08 0.12 0.24 0.40 0.21 0.63 0.85 0.64

TABLE 5: Pdis + Pads for Cu(100), Ts ) 300 K, θ ) O ) 0, J ) W ) 0. 1 E )100 kJ/mol

1D-model2D-modelfive-bands)-elho

model a) 1.00 a) 0.50 a) 0.25 a) 1.00 a) 0.50 a) 0.25

Ei ) 0.36ε 0.36 0.36 0.40 0.42 0.37 0.34 0.45Ei ) 0.56ε 0.41 0.44 0.65 0.74 0.45 0.67 0.79Ei ) 0.72ε 0.32 0.35 0.57 0.72 0.37 0.66 0.83Ei ) 1.00ε 0.24 0.30 0.50 0.73 0.34 0.61 0.83

10788 J. Phys. Chem. B, Vol. 101, No. 50, 1997 Mengel and Billing

(En) defined byEn ) Ei cos2 θ. It is seen that in this case thesame conclusions concerning the effect of the different modelscan be drawn.We have as well made simulations on the Cu(111) surface

with the surface temperature (Ts) increased from 30 to 600 K.Only the values ofEint were slightly changed. Those values

are given in Table 9. We did not make the correspondingsimulations on the Cu(100) surface because the observedchanges on the Cu(111) surface were quite small.We have only used the 2D-model together with the seven-

band model in the case where the incidence was normal to thesurface and the surface temperature 300 K. The reason for thisis that we in this case only saw minor differences between theresults obtained with the five-band model and the seven-bandmodel. (Compare the results in columns 7 and 9 in Tables 4,6, and 8.)We believe that the general conclusions mentioned above are

independent of the choice of “the values of the uncontrolledparameters” because we obtained the same results when dividingeach series into two equal parts and making the average fromeach part separately. For the other quantities investigated, i.e.,Ekin, Erot, Evib, andEelho, the results obtained were not influencedin quite the same systematic way by the models used. Thismight be due to the fact that the number of simulations (110)in each series is too small.7.2. Different Functional Dependencies.Now the second

question is addressed. We have investigated whether the modelscause the above mentioned quantities to depend in different wayson the following parameters:1. The initial CM momentumPCM2. The surface temperatureTs3. The direction of the initial CM momentum specified by

the polar angelsθ andφ4. The screening factora5. The crystal symmetryThe result was that only the dependence on the crystal

orientation was different for the 3 models. This effect was seenboth in the values ofPdis and in the values of (Pdis + Pads).From the numbers in the Tables 3 and 4 it is clear that wheneither the -elho model or the 1D-model is used the value ofPdis is larger on the Cu(100) surface than on the Cu(111) surfacefor the same values ofEi anda. But when the 2D-model isused the values for the Cu(111) surface are larger than the valuesfor the Cu(100) surface. The results in Tables 5 and 6 show

TABLE 6: Pdis + Pads for Cu(111), Ts ) 300 K, θ ) O ) 0, J ) W ) 0. 1 E ) 100 kJ/mol

2D-model

1D-model (five-bands)-elhomodel a) 1.00 a) 0.50 a) 0.25 a) 1.00 a) 0.50 a) 0.25

(seven-bands)a) 0.5

Ei ) 0.36ε 0.11 0.13 0.29 0.48 0.27 0.59 0.61 0.58Ei ) 0.56e 0.02 0.04 0.19 0.38 0.19 0.68 0.85 0.67Ei ) 0.72ε 0.04 0.07 0.26 0.46 0.25 0.67 0.90 0.67Ei ) 1.00e 0.14 0.16 0.31 0.45 0.29 0.70 0.94 0.70

TABLE 7: Eint for Cu(100), Ts ) 300 K, θ ) O ) 0, J ) W ) 0. 1 E )100 kJ/mol

1D-model2D-model(five-bands)-elho

model a) 1.00 a) 0.50 a) 0.25 a) 1.00 a) 0.50 a) 0.25

Ei ) 0.36ε 0.044 0.047 0.042 0.038 0.044 0.057 0.039Ei ) 0.56ε 0.126 0.128 0.102 0.094 0.136 0.117 0.091Ei ) 0.72e 0.167 0.158 0.141 0.132 0.168 0.143 0.124Ei ) 1.00e 0.215 0.120 0.169 0.156 0.189 0.176 0.173

TABLE 8: Eint for Cu(111), Ts ) 300 K, θ ) O ) 0, J ) W ) 0. 1 E )100 kJ/mol

2D-model

1D-model (five-bands)-elhomodel a) 1.00 a) 0.50 a) 0.25 a) 1.00 a) 0.50 a) 0.25

(seven-bands)a) 0.5

Ei ) 0.36ε 0.066 0.068 0.064 0.051 0.067 0.046 0.048 0.048Ei ) 0.56ε 0.099 0.102 0.108 0.112 0.106 0.114 0.146 0.117Ei ) 0.72ε 0.117 0.120 0.124 0.128 0.123 0.154 0.156 0.179Ei ) 1.00ε 0.150 0.152 0.154 0.177 0.160 0.196 0.164 0.193

TABLE 9: Eint for Cu(111), Ts ) 600 K, θ ) O ) 0, J )W ) 0. 1 E ) 100 kJ/mol

-elhomodel

1D-modela) 0.50

2D-model(five-bands)a) 0.50

Ei ) 0.36ε 0.063 0.064 0.046Ei ) 0.56e 0.098 0.105 0.121Ei ) 0.72ε 0.116 0.122 0.151Ei ) 1.00ε 0.148 0.152 0.193

TABLE 10: Pdis for Cu(111), Ts ) 300 K, Ei ) 1.0E, J )W ) 0. 1 E )100 kJ/mol

-elhomodel

1D-modela) 0.50

2D-model(five-bands)a) 0.50

En ) 0.56ε 0.00 0.09 0.35En ) 0.72ε 0.05 0.22 0.58

TABLE 11: Pdis + Pads for Cu(111), TS ) 300 K, Ei ) 1.0E,J ) W ) 0. 1 E )100kJ/mol

-elhomodel

1D-modela) 0.50

2D-model(five-bands)a) 0.50

En ) 0.56ε 0.02 0.10 0.41En ) 0.72ε 0.10 0.30 0.65

TABLE 12: Eint for Cu(111), TS ) 300 K, Ei ) 1.0E, J )W ) 0. 1 E ) 100 kJ/mol

-elhomodel

1D-modela) 0.50

2D-model(five-bands)a) 0.50

En ) 0.56ε 0.087 0.092 0.107En ) 0.72ε 0.115 0.131 0.163

Influence of Electron-Hole Excitation J. Phys. Chem. B, Vol. 101, No. 50, 199710789

that the same conclusion holds for (Pdis + Pads) for a ) 0.50anda ) 0.25 at the higher energies.7.3. Experimental Results.Now the third and last question

is discussed. The comparison with experimental results iscomplicated by the fact that in order to reduce computation timethe treatment is simplified in some respects:(1) The motion of the molecule is treated quasi-classically.

This implies that the molecule cannot pass the barrier ofdissociation if its kinetic energy is less than the dissociationbarrier. It also results in that the zero-point vibrational energyis not conserved properly during the simulations. This effectis pronounced because the initial conditionV ) 0 is used in allsimulations. Those two effects will influence the results indifferent ways. The first will tend to reduce the dissociationprobability, while the other will tend to increase it. But it isnot possible to know wether the two effects balance each other.(2) The diabatic coupling between the adiabatic potential

surfaces is neglected and only the lowest adiabatic potentialsurface is used.(3) The motion of each hydrogen molecule is only followed

in a very short real time compared with that of a real experiment.This makes it hard to decide whether the experimentallydetermined dissociation probabilities should be compared toPads,Pdis, or perhaps their sum.Because of these limitations of the model we do not expect

the results to fit the experiments quantitatively. But we find itreasonable to believe that the change in the results caused by achange in the initial conditions will resemble the change of theexperimental results corresponding to a change in the experi-mental conditions. Above we have mentioned thatPdis and toa certain extent (Pdis + Pads) show a different dependence onthe surface structure for the different models investigated. Inref 12 investigations of the sticking probability is reported forboth Cu(100) and Cu(111) as a function ofEkin for Ekin in theinterval [0.2ε,0.4ε]. The result is that for all the experimentalconditions investigated the sticking probability is larger on theCu(111) surface than on the Cu(100) surface. This is inaccordance with the results obtained with the 2D-model bothin the case where the five-band and where the seven-band modelis used for the bandstructure of the Cu(111) surface if thesticking probability is interpreted asPdis. But the results of thesimulations with the -elho model or the 1D-model are in contrastto the results of ref 12 ifPdis is used for the sticking probability.

If the sticking probability is interpreted as (Pdis+ Pads) the sameconclusion holds fora) 0.25 anda) 0.50 for the larger kineticenergies.

8. ConclusionWe have extended an existing model for the coupling of the

hydrogen molecule to the conduction electrons of the coppermetal. The new model includes the symmetry of the crystal inthe bandstructure of the electrons of the metal. Therefore thecorresponding effective potential is different for the Cu(100)and the Cu(111) surface. We have performed simulations withthe extended model to see wether the change does influencethe results. We have only made series of 110 simulations butthis number is big enough for seeing an influence of the modelon the “sticking probability”. We did not expect the results tobe in complete agreement with experimental results because themodel is simplified in some important respects. But we sawthat when using the extended model the functional dependencyof the “sticking probability” on the surface symmetry is inaccordance with experiments. In the case where we used themodels which did not include the symmetry of the crystal inthe band structure the functional dependency of the “stickingprobability” on the surface symmetry was not consistent withthe experimental results. We therefore conclude that theextension of the model is important.

Acknowledgment. This research is supported by the DanishNatural Science Research Council.

References and Notes(1) See e.g.: White, J. A.; Bird, D. M.Chem. Phys. Lett. 1993, 213,

422. White, J.A.; Bird, J. M.; Payne M.; Stich, I.Phys. ReV. Lett. 1994, 93,1404. Hammer, B.; Scheffler, M.; Jacobsen, K. W.; Nørskov, J. K.Phys.ReV. Lett. 1994, 73, 1400. Wiesnekker, G.; Kroes, G. J.; Baerends E. J.;Mowrey, R. C.J. Chem. Phys. 1995, 102, 3873.

(2) Hansen, B. F.; Billing, G. D.Surf. Sci. Lett. 1997, 373, 333.(3) Billing, G. D. J. Phys. Chem. 1995, 99, 15378.(4) Billing, G. D. Chem. Phys. 1987, 116, 269.(5) Billing, G. D. Chem. Phys. 1982, 70, 223.(6) Billing, G. D. Chem. Phys. 1983, 74, 143.(7) Billing, G. D. Comp. Phys. Rep. 1990, 12, 383.(8) White, J. A.; Bird, D. M.Chem. Phys. Lett. 1993, 213, 422.(9) Truong, T. N.; Truhlar, D. G.; Garrett, B. C.J. Chem. Phys. 1989,

93, 8227.(10) Ashcroft, N. W.; Mermin, N. D.Solid State Physics; Samdes

College Publishers, 1976.(11) Madhaven, P.; Whitten, J. L.J. Chem. Phys. 1982, 77, 2673.(12) Angel, G.; Winkler, A.; Rendulic, K. D.Surf. Sci. 1989, 220, 1.

10790 J. Phys. Chem. B, Vol. 101, No. 50, 1997 Mengel and Billing