Indirect electronic interaction between hydrogen atoms adsorbed on metals

443

Surface Science 159 (1985) 443-465 North-Holland, Amsterdam

INDIRECT ELE~ONIC INTE~C~ON ATOMS ADSORBED ON METALS P. NORDLANDER

4 January

~DROGEN

and S. HOLMSTROM

Institute oj Theoreiical Physics, Chalmers Received

BE~EEN

1985; accepted

University of Technolop,

for publication

25 March

S- 41.2 96 ~othen~urg,

Sweden

1985

A simple theoretical description for the indirect electronic interaction between adsorbates at intermediate distances is presented. The description is based on the effective medium theory and an extension of it, which is derived in the paper. The potential energy for a p(1 X 1) layer of hydrogen atoms on Ni(100) and Pd(100) is calculated. The effects of adsorbate-adsorbate interaction on the equiiib~um distance, chemisorption energy and vibrational frequency of the adsorbates are investigated. The interaction is found to increase the frequency for hydrogen vibrating perpendicular to the surface, in agreement with experimental findings.

1. Introduction The study of adsorbates can provide very useful information for the understanding of technological processes like catalysis and surface reactions. Very detailed information about adsorbates can be obtained today by various types of spectroscopies. In particular, information about the motion of the adsorbate in the chemisorption well can be extracted by electron-energy-loss spectroscopy (EELS). Recent measurements on hydrogen adsorbed on Ni and Pd have revealed spectra with several vibration peaks. At low coverages they show frequency shifts that can be interpreted in terms of clustering [1,2]. Comparisons have been made with theoretical results for singly adsorbed hydrogen, calculated within the effective medium theory [3]. In the harmonic approximation this gives only one mode. The point of this paper is to investigate, how adsorbate-adsorbate interaction effects can affect the vibration spectrum. In addition, consequences on chemisorption energies, equilibrium positions and potential energy surfaces will be considered. The results are on the qualitative and semi-quantitative levels. Considering the great detail of the experimental information, also very simple questions are interesting to get answered from the theory, such as: Do vibration frequencies shift at all? Is the shift positive or negative? What is the order of magnitude of the shift? The effective-medium approach, which is used in this calculation, makes it possible to avoid overwhelmingly large computational effort while still getting physically valuable results. 0039~6028/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

444

P. Nordlander. S. Holmstr?%n / Indrrwt electromc interactron between H atoms

The interactions between adsorbed atoms are known to play an important role in many surface phenomena, such as catalysis, surface diffusion, desorption and phase transitions in adsorbate layers. Conventionally one distinguishes between direct and indirect interactions between adsorbates. The direct interactions are those that could be present also between atoms and molecules in the gas phase, for example covalent bonding [4], Van der Waals interaction [5] and dipole-dipole interaction [6]. The indirect interactions are mediated through the substrate. This category includes for instance the elastic interactions between the displacement fields due to the adsorption of large molecules [7]. There is an electronic analogue to this, which in a simplified picture can be viewed as the interaction between the adsorbate induced electronic distortions of the surface. For adsorbate like hydrogen atoms, where the adsorption induced dipole moment is negligible and the Van der Waals interaction is small due to the low polarizability of the hydrogen atom, the possible interaction mechanisms are the direct and indirect electronic ones. In the present application close packed layers with interadsorbate distances of about 5 a.u. (i.e. 2-3 A) will be considered. These distances are large compared to the range of direct interactions, which is of the order of the bond length of a hydrogen molecule. Hence the indirect electronic interaction dominates. This interaction has been treated in the literature by several authors, using different schemes. A series of model calculations have been performed [8], which focus on solving two-resonant-level Hamiltonians with various types of surface geometries and correlation interactions. Using such models, the changes in the one electron spectrum of the substrate have been evaluated, and the pairwise interaction energy between adsorbates obtained. From these calculations it has been shown that the interaction energy varies in an oscillatory manner along the surface and that at large distances its amplitude decays as r--5 along the surface. The interaction also depends very sensitively upon the adsorbate and substrate geometry of the system considered. In ref. [9] an atomic cluster embedded into jellium has been used to study the interaction. The results agree qualitatively with the picture that has emerged from the model Hamiltonian calculations quoted above. Finally, in ref. [lo] the interaction energy between two hydrogen atoms on a jellium surface has been calculated. Again the interaction energy is found to vary in an oscillatory way as a function of atom-atom distance and to have an amplitude decaying as re5 in the asymptotic region. Calculations of the hydrogen induced charge density give a similar asymptotic behaviour and indicate that the interaction energy in fact closely follows these charge density oscillations [ll]. It would of course be totally impossible to obtain any site dependence from a jellium description of the surface, as this model results in interaction energies depending only on the relative separation of the atoms. Furthermore, jellium models of transition metal surfaces using an ‘; corresponding to the valence

P. Nordlander,

S. Holmstriim

/ Indwect electronic interaction between H atoms

445

electron density fail completely to reproduce the actual charge profile. In the context of adsorbate-adsorbate interaction this defect of the jellium model would be a very severe limitation, since the interactions are mediated through a polarization of the electronic structure of the substrate and therefore depend on the electronic charge densities around the atoms. In this paper a simple method to calculate adsorbate-adsorbate interactions is presented. It extends previous jellium calculations and take into account the proper variation of electronic charge density along the surface. The approach is based on the effective medium theory [12] and explicitly on its recent application to the calculation of potential energy surfaces for a single hydrogen atom outside transition metal surfaces [3]. An extension of the effective medium theory to treat interactions is derived and applied (appendix). In section 2 the influence of adsorbate-adsorbate interactions on the vibration frequencies of hydrogen and deuterium substitutionally disordered in an ordered structure on a surface is investigated. In section 3 and the appendix the effective medium theory is reviewed and extended so that adsorbate-adsorbate interaction effects can be incorporated. In section 4 some qualitative zeroth order thumb rules for how the adsorbate-adsorbate interaction might change the bond-length, the chemisorption energy and the vibration frequencies are derived. In section 5 applications to H/Ni(lOO) and H/Pd(lOO) with comparisons to experiments are presented. Finally, in section 6 the results and the limitations of the approach are discussed. 2. Adsorbate motion The vibrational spectrum from an EELS experiment may then be analyzed in terms of the spectral function S(o) for a dipole-dipole correlation function which is given by the imaginary part of the dipole polarizability atot [13], S(w)

= [l + n(ttw)l

Im a,,,(o),

(2.1) where n( Aw) is the Bose-Einstein distribution function. The polarizability (Y(~~(w)connects the total normal component of the dynamic dipole moment to the homogeneous external field Eext(w) with frequency w, p(w) = %X(Q) EeX’(a). Only the normal displacement u(i) of the atom on site “i ” contributes total normal component of the dynamic dipole moment p, p= Ce*u(i).

(2.2) to the

(2.3)

The effective charge for the motion is e*. In the presence of a homogeneous external field EeX’ the equation of motion for the displacements reads m(i)

ii(i,

t)=

-m(i)

Q*(i)

+e*Eex’(t),

u(i,

t)-C@‘(i,

k) u(k.

t) (2.4)

446

P. Nordlander, S. Holmstriim / Indirect electronic mteractlon between H atoms

where m(i) is the mass of the atom at site i and Q(i)= i@(i, i)/m(i) its vibrational frequency in the absence of the vibrational interaction @‘(i, j). The disorder in the layer is introduced by the fact that the masses m(i) are distributed randomly for an isotopic mixture. The displacement fields are eliminated in (2.4) by introducing the single site vibrational susceptibilities x(i, w) defined as u(i,

0) = x(i,

w) e*E’“‘(o).

(2.5)

In the absence of vibrational interactions (@’ = 0) the single susceptibilities take a simple form as evident from (2.4)

[( w +

xo(i, Q> = (m(i) It is straightforward ibilities, x(i,

w)=x&,

iO+)* - Q’(i)]

to construct

w)

[

1-l.

k) X(k,

0)

k

From (2.3) it is clear that the total polarizability single site susceptibility X(w), atot

= N(e*J2

(2.6)

from (2.4) a relation

1 -C@‘(i,

site vibrational

between

these suscept-

1.

(2.7)

(Y~,,(o) is given by the average

X(w),

(2.8)

where N is the number of sites in the layer. This description is analogous to the one presented in ref. (141. The best devised analytical method to solve (2.7) for the average polarizability x(w) is the Coherent Potential Approximation (CPA) [15]. On the other hand, CPA is quite elaborate to use in general. For the present purpose such a detailed description of the spectra is not needed. A considerably simpler but still realistic approximation is the Average T-matrix Approximation, denoted by ATA, [15]. The ATA amounts to solving (2.7) by a decoupling of the averages over the product xO(i,

a> x(k,

a>= j&(o)

X(o).

Thus the average susceptibility L,,(w)

= X&)/[l

-

is given by

~,,,,%&41 t

(2.9)

where x,(w)

= W[m.(~2

- %>I

+ Cl/[

m,,(@2 - G>]

3

(2.10)

C, and C, are the relative concentrations of H and D respectively, since we interaction is consider a p(1 X 1) structure, C, + C, = 1. The mean vibrational Gin, = C&(i,

j>.

The spectrum

inferred

(2.11) from (2.9) and (2.10) consists

of two bands

with zero

P. Nordlunder, S. Holmstrijm / Indirect electronic interaction between H atoms

447

width as a function of the composition. This is a deficiency of the ATA since the fluctuations in the local composition will cause a non-zero width. In the variation of split band limit, where +&mu =K L’h - 52&, the compositional the two band frequencies simplifies to ~~=52;+C~&/rn~,

cu=H,D.

(2.12)

This expression represents hydrogen and deuterium-like vibration modes, for cr = H or D respectively. In order to visualize the physical situations involved in this formula, consider first the limit C, equal to unity. The hydrogen like vibration here corresponds to the whole layer vibrating rigidly perpendicular to the surface. This vibrational frequency will be called tzw,. In the dilute limit, where C, is very small, i.e., where a low concentration of hydrogen atoms are embedded in essentially a complete p(1 X 1) structure of deuterium, the hydrogen like vibration in eq. (2.12) corresponds to a single hydrogen atom vibrating, while the surrounding atoms are at rest. This vibration frequency is denoted Aw, and is in general different from ko,. It should be noted that the frequency shifts are only sensitive to the mean vibration interaction & and that no information is obtained about the range of the interaction. Due to the anharmonicity of the potential energy curves, numerical integration has been used to determine the “effective” force constants. The ATA expression has been used in the experimental analysis [1,2]. It gives a good account of the compositional frequency shifts. The use of ATA to model the adsorbate motion for hydrogen and deuterium mixtures has been validated by Persson [17]. The result of the above analysis shows that two different vibration frequencies tto, and Aw, enter the description of a substitutionally isotopically disordered monolayer. These vibrational frequencies differ slightly from that of the single hydrogen adatom, which will be denoted AU,. In the next two sections, the relation between these three vibration frequencies will be discussed using a qualitative model for the adsorbate-adsorbate interactions. 3. Interaction effects from effective medium theory The analysis in the previous section has shown that three mean vibrational frequencies wO, o, and oz can mediate a detailed comparison between theory and experiment. At large, the size of adatom vibrational frequencies is determined by the local properties of a single adsorbate, i.e. adsorption site and nature of the adsorbate-substrate bond. Compared with frequency differences between different adsorption sites and different substrates [3] the frequency shifts caused by the adsorbate-adsorbate interaction are typically small corrections. Against this background, even rather crude estimates of these shifts are useful.

448

P. Nordlunder,

S. Holmstriim

/ Indirect electronic rnteraction between H atoms

The effects of adsorbate-adsorbate interactions on equilibrium positions, adsorption energies and vibration frequencies are obtained by calculating the difference in embedding energies of a single adsorbate at representative points in the surface region, with and without surrounding adsorbates. The effective medium theory provides a simple scheme for calculating such embedding energies. The starting point for the following analysis is the result from the effective medium theory for single adsorbates. The hypothesis is that, at the distances relevant for the surface structures under study, the major part of the interaction between two adsorbed H atoms is due to the indirect interaction mediated by the conduction electrons of the metallic substrate. With this mechanism, one adatom induces a density change at the location of the other adatom. The latter then registers a change in energy, which can be characterized as the interaction energy. To estimate this energy, the effective medium theory [3] and extentions of it are used in the following. 3.1. Review of the effective medium theory The idea behind the effective medium theory is to replace the true host of an adsorbate by an effective host, which allows a practical theoretical treatment but still accounts well for the physical situation [12,18]. One convenient effective host is the homogeneous electron gas. In the effective medium theory the energy, AE, for embedding an atom at position R into an inhomogeneous host with electron density no(r) is given by [31, AE(R)=AE,h,;“(&)+AE,+AEc+AEhyb,

(3.1)

where Fr,, is an average over n,(r) using essentially the atom induced electrostatic potential A+(r) in a homogeneous electron gas as the sampling function,

The energy AE>~“‘(?i,,) is the embedd’ mg of energy of the atom in a homogeneous electron gas with the electron density Z, plus some minor correction terms defined in ref. [12]. It describes the interaction of the atom with the free electron like electrons of the surface. The energy A E hyh describes the interaction between the atom induced one-electron states and the d-electron states of the substrate. The energy A EC is a core repulsion term and A E, is a small correction term that has to do with the definition of A E hyh. A EC and AE, are of minor significance in chemisorption properties [3]. Eq. (3.1) is derived within the local density approximation for exchange-correlation effects, assuming that outside a region u, containing the adsorbate, the effective

P. Nordlander,

S. Holmstrijm

/ Indirect electronic interaction between H atoms

449

one-electron potential is unaffected by the adsorbate, and that inside this region the effective potential is independent of inhomogeneities in the host

1121. These assumptions are plausible for small adsorbates like hydrogen or oxygen atoms but would be incorrect for large reactive molecules, where the region a would be so extended that additional gradient corrections would enter expression (3.1). The evaluation of eq. (3.1) would also be difficult for atoms like Si and Li, where the polarizability of the embedded atom is large so that the effective potential inside a is sensitive to the inhomogeneities in the host. For higher order corrections to eq. (3.1), see ref. [18]. IJsing eq. (3.1), the potential energy surfaces, AI?(R), for a single hydrogen atom outside a number of transition metal surfaces have been calculated with good agreement with both experimental and other theoretical results [3]. This calculation has shown, that in this case the effective medium term A E$‘“( n) strongly dominates over the hybridization term, and that the latter term can be well described by a Newns-Anderson model Hamiltonian and even, to a good approximation, accounted for by the simple expression AEhyh=2(1-F)II/adj2/(ca-Cd).

(3.3)

This result can also be obtained directly from second order perturbation theory [12]. Vad is the hopping matrix element between the hydrogen induced state and the localized d-orbitals of the substrate. To a first approximation the latter can be described within the atomic sphere approximation [19]. The energy cil is the position of the hydrogen induced resonance for the free-electron like states of the substrate, C, is the centre of the d-band and F is the degree of filling of the d-band. For adsorbed hydrogen, A E,, A E, and A E hyh has been found to vary only weakly with position R and not to influence even quantitatively the shape of the potential energy surfaces around the equilibrium position on close packed surfaces. In the present calculation the interactions between adsorbates through the d-band will be neglected, and the hydrogen atoms in a layer will be assumed to hybridize independently with the substrate. The distances between the adsorbates are large compared to the bond length distance in the hydrogen molecule so any direct hybridization between the adsorbates is not expected. 3.2. Extension

of the effective medium theory

For the calculation of wO, w, and o2 the binding energies for H in three situations are needed: (i) a single H chemisorbed on a clean surface; (ii) a single H as a defect in a full, rigid deuterium layer, and (iii) a full H layer. For (i) the regular effective medium theory applies and the results have been accounted for in ref. [3]. For the situations (ii) and (iii) an extension of effective medium theory is required.

450

P. Nordlander, S. Holmstriim / Indirect electronic interactton between H atoms

In this extension, some care must be taken when treating the total energy of configurations of adsorbates in terms of single adsorbate embedding energies. First of all it is necessary that the distances between the adsorbates are large enough for the regions a not to overlap. This assumption is well justified even for the most dense structure of interest, i.e. pfl x 1)H on Ni(100). Secondly it is important to observe that the region outside a in the case of embedding an adsorbate into a layer contains adsorbates that might be affected by the embedding. Before these questions are discussed, it is necessary to have a model for the electron density of the surface, with and without the adsorbates. This is because in the model all interactions are mediated through the effective medium term AEgm (n), which only depends on the average density at the position of the adsorbate. The density n:,,(r,{ R,}) is the total electronic density with adsorbate i removed and depends on the positions of the other atoms in the adsorbate layer. In principle it could be calculated self-consistently but that would be an extremely difficult task due to the low symmetry of the surface. In order to obtain a reasonable estimate of n:,,(r), the following assumptions have been made: (i) The electron density at the surface is approximated by a linear superposition, n:,,(r,{

R,})

= fi&)

+ &%, Jtfi

RI>,

(3.4)

where An(r, R,) is the density at position r induced by a single hydrogen atom at position Rj. The sum runs over the positions R, of the remaining atoms in the adsorbate layer. The clean-surface density n,(r) may to a good approximation be obtained by overlapping atomic charge densities [20]. (ii) The adatom induced density An(r, R,) can be approximated with the result from a self-consistent calculation for a hydrogen atom at the corresponding position outside a semi-infinite jellium surface which best represents the actua1 surface. Assumption (i) can be justified, since the induced electron densities for r around R, are small compared to the total electron densities. Assumption (ii) means going beyond linear response theory. For close packed p(1 X 1) layers it can be justified by the following argument: The induced electron density is strongly dominated by the contributions from the nearest neighbour atoms. These atoms lie so close that the induced electron density has not started to build up its Friedel oscillations but is still negative, reflecting the fact that the hydrogen atom is electropositive and attracts electrons from its immediate neighbourhood. The results are also quite similar for jellium surfaces corresponding to different bulk densities r, values, where 47~;’ = 3n. This implies that the screening in the neighbourhood of a proton mainly depends on the local electron density.

P. Nordlander,

S. Holmstriim

/ Indirect electronic interaction between H atoms

451

This property of a jellium surface has recently been pointed out in the regime of linear response to an external charge [21]. There the density response of a jellium surface is well described in terms of that of a homogeneous electron gas with a density equal to the average of the densities at the embedding and the observation points. Furthermore, fully self-consistent calculations of the surface charge density [22], show that the surface electron density varies only slightly along a (110) direction on a clean fcc(100) surface, with a maximum at the bridge position. This fact indicates that the density at a centre position induced by an adsorbed hydrogen atom in an adjacent centre position is reasonably well described by the density response of a jellium surface. The evaluation of An (r, R,) has been made for jellium with rs = 2.5, since the actual electron density profiles perpendicular to the surfaces of Ni(lOO) and Pd(lOO) closely resemble that outside a jellium surface with this r,-value. The results presented in this article do however not depend sensitively upon the particular r, used to model the surface in the calculation of An(r, RI). For instance, calculations performed with An(r, R,) evaluated for jellium with rs = 2.65 give results that agree within 10% with those obtained for r, = 2.5 jellium. The actual calculation of An( r, R,) has been made using the embedding scheme by Gunnarson and Hjelmberg [16]. A straightforward application of the effective medium theory would give that the embedding energy AE(R,,{ R,}) of the hydrogen atom i at the position R, in the presence of the other hydrogen atoms labelled by j with the corresponding locations R, can be written as AE(R,,(R,})=AE~‘“(n,(R,))+AE,(R,)+AE,(R,)+AEhyb(R,) +AE’“‘(R,,{

R,}).

Here the interaction term with the other adsorbates out and is given by AEi”‘(R,,{R,})=ME~“(R,,{R,}).

(3.5) have explicitly

been pulled

(34

The arguments for neglecting the change in the hybridization energy caused by the other adsorbates have been given above. The terms AE, and A E, do not mediate any interaction since they essentially describe the core repulsion between the adsorbate and the substrate atom cores [3]. The term SAE$‘” is the interaction mediated by the effective medium. It contains an electrostatic term SAE’“=

/

Ap(r)

S+(r)

dr,

where Ap(r) is the adsorbate coadsorbate induced electrostatic

(3.7) induced charge density and 6$(r) is the potential. In addition there is an interaction

P. Nordlmder,

452

S. Holmstriim

/ Indirect electronrc rntrructton between H storm

term due to the distortion of the effective medium upon inclusion of the coadsorbates. This term contains kinetic and electronic contributions, as well as effects of exchange and correlation between the electrons, and has the form 6AEh”‘“(R,,{

R,})

= AEh”“(Il’,,,,(R,,{

where A E h”“‘(n( r)) neous electron gas linear in 6Z and is introduced, the sum AE’nt(R,,{

R,})

-AEh”“(Z,(R,)),

R,}))

is the embedd’ mg energy of an with electron density n(r). The typically small. If the averaging of the two terms (3.7) and (3.8)

(3.8)

adsorbate in a homogelast term in 6A EL;;‘“’ is according to eq. (3.2) is can be written

=AE,h,‘,‘“‘(Z;,,,) -AE,_~“‘(G,,).

(3.9)

where ii’,,, is a samples average according to eq. (3.2) of the total electronic charge density around the adsorbate i. In the case of electropositive or electronegative adsorbates at large distances from each other, the dominant contribution in this expression comes from the electrostatic part (3.7). This has been used in a calculation of catalytic promotion and poisoning effects of certain adsorbates [23]. In the present calculation, however. attention is focused on close packed adsorbate layers of hydrogen atoms, where the term (3.8) is dominant due to the charge rearrangements of the surface. In principle, one could proceed and construct the total potential energy for the adsorbed layer by successively building up the structure adsorbate by adsorbate and summing up the corresponding embedding energies. For a configuration where the hydrogen atoms are embedded at points of different substrate density one would find, however, that the total energy would depend upon the order, in which the layer has been built up. This is due to the model for the densities on the surface and to the assumption that the hydrogen atoms already present at the surface can be treated as part of the host and therefore are unaffected by the embedding of a new atom. To circumvent this ambiguity it is assumed that there is an extra contribution to the embedding energy of an atom coming from the fact that when this atom, upon embedding, induces a density change over a region where other atoms are present, the energy of those atoms are allowed to change according to their A EE‘I;I;,terms. This gives an extra contribution to the embedding energy of an atom, which exactly compensates the ambiguity of the total energy and allows the total density of the surface to be estimated by linear superposition. This approximation is further discussed in the appendix. With this approximation the total energy for a layer of hydrogen atoms is well defined and can be written: AE({R,

)> =c

[AE,:;““(

“:,,,( R,,{ R,}))l

+AE\,(R,)

+AE,(R,)

+AEhvh(R,). In

the

next

sections

eq.

(3.10) (3.10)

will

be

used

to evaluate

the

effects

of

P. Nordlander,

S. Holmstriim

/ Indirect electronic interaction between H atoms

adsorbateeadsorbate interactions on the equilibrium frequencies and adsorption energies.

4. Qualitative properties

effects

of adsorbate-adsorbate

configuration,

interactions

453

vibrational

on chemisorption

In this section the general effects of adsorbate-adsorbate interactions on the equilibrium properties will be estimated. The discussion is qualitative with simplifying approximations for the surface electron density and the adsorbate induced densities along the surface as well as the harmonic approximation for the motion of the adsorbates. The purpose of the discussion is to obtain a qualitative framework for the magnitude and the sign of the shifts. 4. I. Shift of equilibrium position The influence of the coadsorbates on the equilibrium position of an adsorbed hydrogen atom can readily be quantified in the present approach. An earlier calculation [3] has shown that for a single hydrogen atom adsorbed on a transition metal surface the effective medium term dominates over the hybridization term. For reactive atoms like hydrogen, the effective medium term AELzrn has a minimum for the density nor,, = 0.01~;’ [12]. This makes the hydrogen atom tend to sit at positions having an average density close to this optimum density. For instance, if the adatom-adatom distance is such that the electron density induced by the other adatoms is negative, then the adsorbates are attracted inwards to a position with higher substrate electron density. A reasonable representation of the surface electron density variation normal to the surface in the vicinity of the chemisorption position is nO( z) = n, ee*‘,

(4.1)

where LYdepends on the lateral position. In the presence of the coadsorbates the equilibrium position for the layer changes from zeq to z& and can be obtained from solving n, e --a’cq + An( z&, z:,) = n, e-“‘cq = nopt, where An(z, z’) adsorbate layer the solution to distance for the

(4.2)

is the electron density induced at point z in a vacancy of the from the coadsorbates located at point z’ above the surface. this equation gives a shift AZ = z& - zeq of the equilibrium adsorbate layer,

AZ z=An&/an,p,.

(4.3)

This shift relates directly to An and might lead to different workfunctions as well as dynamic dipole moments for different adsorbate structures. The size of this shift is in the present situation about 0.3 a.u.

454

P. Nordlander, S. Holmsiri_kn/ Indrrect electronic interactmn between H atoms

4.2. Change inchemisorption

energy

Without the hybridization term there is no energy gain due to adsorbate-adsorbate interaction, the atoms are free to adjust themselves to points having the optimum electron density, nopt. With the hybridization there is a net energy gain due to the shift of equilibrium position, An

(4.4)

AE=AEhYb(~~q)-AEh’.b(ZFy)=~A~hyb(~&-$. opt

This energy gain evidently follows the spatial variations of the electron density induced by the adsorbates and therefore depends on the interadsorbate distances. Due to the approximate description of the hybridization term and the possible interactions through the d-electrons, this result should be regarded as having a more qualitative nature. 4.3. Change in vibrational properties One manifestation of the adsorbate-adsorbate interactions is that the vibration frequencies change with the coverage. When the effective medium term dominates it is possible to derive from eq. (3.10) some simple and useful relations between the change in adsorbate-substrate force constant and the interadsorbate force constants introduced by the adsorbate-adsorbate interactions. In the last part of this section it will be demonstrated in a simple model how the interadsorbate force constants can affect the vibrational properties of isotopic mixtures. The adsorbate-substrate force constant for the perpendicular motion of a single adsorbate against the clean surface is directly related to the slope of the substrate electron density profile. Simple differentiation of (3.10) gives for the force constant $“(i,

i) =

(4.5)

where the fact that the first order derivative vanishes at the equilibrium position has been used. In a harmonic model this force constant corresponds to a vibration frequency of

t?u,,=c* I 1,

(4.6)

;=zcq

where the constant C only depends on properties of the hydrogen atom. In the presence of the coadsorbate the vibration frequencies also depend on the adsorbate induced electron densities along the surface. Consider for instance the motion of the whole p(1 x 1) layer in phase. The effective force

P. Nordlander, S. Holmstrijm / Indirect electronic interaction between H atoms

constant vibration Aw

=

c

1

for this motion can easily be evaluated, frequency takes the form

dncdz)+ZAn(~, d dz

z)

. Z=Z_

and in a harmonic

455

model the

(4.7)

It is seen that the vibration frequency Frw, is shifted compared to Fro,, through the derivative of the term An. This derivative is negative (positive z is defined outward from the surface) for distances corresponding to those between nearest neighbours within p(1 x 1)H on Pd(lOO) and Ni(lOO) and thus gives a higher vibration frequency for the layer vibrating perpendicular to the surface. The force constant for a single hydrogen atom in a layer, +( i, i), can also be evaluated. In a harmonic model the vibration frequency takes the form (4.8) This vibration frequency is found to have a value between ttw, and Ao,. Using the expression for the adsorbate induced density An( z, z’) derived in [21] one obtains in a harmonic model the following relation (ftw, - Aw,)/(

LJ, - B&J*) = 2.

(4.9)

The results of this section are only approximate and of course depend on the harmonic approximation. Nevertheless the results illustrate the qualitative changes one can expect from adsorbate-adsorbate interaction on the equilibrium properties of an adsorbate. In the next section the potential energy curves using eq. (3.10) for a hydrogen atom in the three different situations (1) (2) and (3) on Ni(lOO) and Pd(lOO) will be calculated.

5. Results In this section the results of an application of eq. (3.10) to hydrogen chemisorption on Ni(lOO) and Pd(lOO) will be presented. AE,hym is evaluated using the expression derived in ref. [12]. E:,, is obtained using eq. (3.4). An is calculated within the embedding scheme defined in ref. [16]. AE,, AE, and A E hyh are evaluated according to the prescription in ref. [3]. 5.1. Hydrogen

adsorption

on Ni(lO0)

In fig. la the potential energies for two cases of hydrogen adsorption on Ni( 100) are plotted: (i) a single hydrogen atom adsorbed in the centre position of the surface, (ii) the adsorption energy per hydrogen atom of a p(1 x 1) layer as function of a rigid displacement normal to the surface. A calculation for a c(2 X 2) layer gives almost the same result as that for the single hydrogen atom

v (Zl

v (Z)

f3V

t?V -2.6-

-2.7-

a -2.8

0.5

1.0

1.5

2.0 2.5 z (a.u.1

0.0

0.5

1.0

2.0 1.5 z b.u.1

Fig. 1. {a) Potential energy per hydrogen atom as a function of rigid displacement z perpendicular to the surface for a single hydrogen atom (dashed line) and a ~(1 X 7) layer of hydrogen atoms (full line) outside Ni(lOO). Energies in eV, distance in bohr, measured from the first substrate atom layer. (b) Same as (a) but for Pd(100).

case but with a slightly more repulsive V(z) towards vacuum. The reason for the ~(2 X 2) structure being relatively unaffected by adsorbate-adsorbate interaction is the larger separations between the adatoms in this case. From the figure the curves appear similar. The total energies differ with about 25 meV. The potential minimas are located at about 1.2-1.5 a.u. outside the surface atom layer. The equilibrium position for the p(1 x 1) layer curve is displaced inwards some~~hat reflecting the fact that the adsorbate induced electron densities are negative and that the adsorbates therefore find the optimum density II.,,, (see section 4) closer to the surface. The effects of hydrogen-hydrogen interactions can most easily be seen far from the surface, where the potential V(z) for the p(1 x 1) structure is steeper towards the vacuum side than the corresponding curves for a single hydrogen atom. This is a pure screening effect. A hydrogen atom embedded into the low density region of the surface attracts charge from its neighbourhood. Thereby the background charge density at the positions of the coadsorbates is reduced. This effect increases AE,h,Fm and gives rise to a higher total energy for hydrogen atoms far from the surface compared to the single hydrogen atom case. For hydrogen atoms placed in the region of the surface with slightly higher density charge is still removed, but the more effective screening in this case makes the effect less pronounced. In the high density region the removal of charge lowers the embedding energy of a hydrogen atom resulting in an energy gain. However, at densities higher than a certain value the induced charge densities at the adsorption sites become positive, and a weak repulsion results.

P. Nordlander,

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457

The stiffening of the potential curve towards the vacuum side increases the frequency for perpendicular vibrations. The vibration frequency for the calculated p(1 x 1) curves corresponds to 93 meV, while for the single hydrogen atom we find 76 meV. Ref. [24] describes a full quantum mechanical calculation of the motion of a single hydrogen atom around the centre position. It takes into account the detailed variation of the three-dimensional potential energy surface V(r). The hydrogen atom is there found to exhibit several vibration modes with parallel and perpendicular motions involved. The first excited band corresponds to a purely parallel motion, however, and shows a small dispersion. The second band corresponds to a motion pe’vendicular to the surface, also with some dispersion. The totally symmetric vibration mode at the r point has the frequency 86 meV, which agrees fairly well with the estimate of 76 meV. The reason for a higher vibration frequency in the three-dimensional calculation is that the parallel motion samples the potential energy over a region around the centre position, which has the lowest perpendicular vibration frequency of all lateral positions of the surface. The experimental value obtained in the low coverage region is 67 meV [2]. The difference in vibration frequencies between the single hydrogen atom and that for the p(1 X 1) structure implies a higher zero point energy for the

5 (a.u.1

Fig. 2. Interaction energies for a single hydrogen atom in a p(1 x 1) layer of hydrogen atoms on Ni(100) and Pd(100) as a function of displacement 1 in the (110) direction. The geometry of the problem is shown in the insert of the figure. The large circles represent the surface lattice atoms of the metal. The dashed large circles are the second lattice atom layer. The filled circles represent the hydrogen atoms. The hydrogen atoms are chemisorbed in the C position and 5 is a distance measured in the (110) direction. For Ni (interadsorbate distances 4.70 a.u.) there is a large repulsion which tends to confine the H atoms towards the centre position. In the case of Pd, where the lattice constant is larger (interadsorbate distance 5.20 au.), the corresponding interaction energies are smaller and therefore the localization weaker. For c(2 X 2) structures the interaction energies are very small due to the large interatomic distances.

458

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/ Indirect electronic mteraction between H atoms

Table 1 Different vibration frequencies AuO, hw, and hw, defined in section 2 are shown for hydrogen outside Ni(lOO) and Pd(100); for comparison results Pd(lOO) [I], are shown within brackets; units in meV

A% f2w1 h%

from

the experiments

Ni(100)

Pd(lOO)

76 (67) 93 (78) 84 (69)

56 63 (64) 59 (60)

on Ni(lOO) [2], and

p(1 X 1) structure. In addition, due to lateral interactions the hydrogen atoms in this structure are more strongly confined to the centre positions. Fig. 2 shows the hydrogen-hydrogen interaction for a single hydrogen atom moving in the (110) direction within the p(1 x 1) layer as a function of distance [ from its equilibrium position. It implies that the hydrogen atoms have higher parallel zero point energy in the p(1 X 1) structure than singly adsorbed. The total increase in zero point vibration energy balances the lower potential energy for the p(1 X 1) structure apparent in fig. la. Thus predictions about the stability of different structures are uncertain, in particular in view of the approximations for the hybridization term. The three different vibration frequencies tZwO, Aw, and tto, defined in section 2 are obtained by numerical integration of the potential. The results are collected in table 1 and compared with the experimental results found in ref. [12]. The agreement between theory and experiment is within 10% and the shifts of the vibration frequencies have the right sign. 5.2. Hydrogen adsorption on Pd(100)

In fig. 1 b, the calculated potential energy curves for hydrogen adsorption on Pd(lOO) are plotted. Due to the larger lattice constant of Pd the electron density is typically lower and the potential energy wells are centred closes to the surface than in the case of Ni(lOO). There is no steep repulsive potential wall for the hydrogen atom moving inwards, and the potential V(z) has a much flatter shape. As in the case of Ni, a calculation for the c(2 x 2) hydrogen structure shows a potential very similar to that of the single hydrogen atom but with a slightly more repulsive V(z) towards vacuum. The larger lattice constant of Pd also make the effects of adatom-adatom interaction smaller than on Ni, albeit qualitatively very similar. From fig. 2 it can be seen that the lateral hydrogen-hydrogen interactions in the p(1 x 1) layer are also very small. It can be seen that the hydrogen interaction tends to somewhat displace the hydrogen atoms on Pd(lOO) away from the centre position. This effect is balanced by the forces from the surface and only a full quantum calculation could reveal the most probable position.

459

P. Nordlander, S. Holmstrijm / Indirect electronic interaction between H atoms

The potential energy curves for H on Pd are clearly more anharmonic than for Ni. The reason for this is the open structure of the Pd lattice which gives a very soft repulsive wall for diffusion into the bulk. As for Ni the hydrogen atom should show quantum band effects due to the mixing of parallel and perpendicular motions and the tunneling of hydrogen atoms along the surface. In spite of the increased lattice constant, those effects are not expected to be larger on Pd since the equilibrium position for the hydrogen atoms are about 0.5 a.u. closer to the surface and the barriers that localize the hydrogen atom are very high due to the surface lattice atoms. This property can be seen in fig. 3, where the potential energy contours for a single hydrogen atom in a cut across the centre position on Ni(lOO) and Pd(lOO) surfaces are plotted. The repulsion from the surrounding substrate ions makes the potential well parallel to the surface rather narrow and the first excited vibration state perpendicular to the surface can be reasonably well approximated from the V(z) curves in fig. lb. In table 1 the results for the vibrational frequencies Fro,, Ao, and Aw, from the present theoretical analysis are presented together with the experimental results from ref. [I]. The calculated vibration frequency Fzw, should be regarded with some caution. The corresponding potential energy curve is clearly anharmonic and effects of parallel motions might alter the frequency. The experimental value for a c(2 X 2) structure is 63 meV [l], and is therefore in disagreement with our finding of a relatively unaffected c(2 x 2) structure with a vibration frequency close to that for the single hydrogen. The calculated frequency shift for the isotope experiment have the right sign and magnitude however.

7.0

;;

k,a,, , ,~, , ,A i:ik;&, ,m,,

00

10

2.0

3.0

4.0

5.0

6.0

0.0

1.0

2.0

3.0

/

4.0

5.0

6.0

J 7.0

Fig. 3. (a) Contours of constant potential energy for a single hydrogen atom outside a Ni(lOO) surface. The cut is perpendicular to the surface along the (110) direction. The contours are shown in steps of 0.10 eV with the highest values indicated in the figure. The energy value at the chemisorption point is also shown. (b) Same as (a) but for Pd(lOO).

460

P. Nordlander,

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/ Indirect electronic mteraction between H atoms

6. Discussion and conclusions The results of the applications of the proposed model show that adsorbate-adsorbate interaction can affect chemisorption properties like the chemisorption energy, the equilibrium position and the vibration properties of the chemisorbed layer. The calculated vibration frequencies agree well with the experimental ones [1,2]. The predicted shifts of perpendicular vibration frequency have the right sign and size. The agreement between theory and experiment is surprisingly good in view of the complexity of the problem and the simplicity of the approach. This agreement indicates that the theory incorporates the essential physical effects. The present calculation considers interactions between adsorbates located close to each other on the scale of Friedel oscillations but sufficiently far from each other for the effects of covalent bonding between the adsorbates to be negligible. For these distances it is argued that the effective medium term dominates over the hybridization changes. The interaction between the adsorbates is mediated by the density distortions a chemisorbed adsorbate causes on the surface. For such small distances as considered in the present calculation, the induced electron densities are mainly determined by the local electron densities around the adsorbate. For large interadsorbate distances the proposed model still applies but the adsorbed induced densities would have to be calculated with a more realistic model than that of a jellium surface. In the case of a c(2 X 2) structure, for instance, the surface charge density between two neighbouring adsorbates varies very much, since a substrate atom is located half way between the adsorbates. This atom creates a very high electron density barrier between the adsorbates and affects the polarization properties of the surface. For this case the use of a simple jellium model to describe the density response of an atom along the surface is not adequate. For very short interadsorbate distances the regions u around each adsorbate overlap, and the effective medium could not without caution be taken as a homogeneous electron gas. A straightforward application of eq. (3.10) without the hybridization terms gives for two hydrogen atoms in vacuum close to each other, a total energy of 4.8 eV and a bondlength of 1.9 a.u. These values lie fairly close to the values for a hydrogen molecule, 4.6 eV and 1.4 a.u., respectively [26]. This agreement indicates that the present theory can be extrapolated down to fairly small interadsorbate distances. In these cases the interaction is of direct nature, since there are no induced electron densities in the absence of a host. In the present application the interaction effects mediated by the hybridization term are neglected. This approximation is strictly only valid for metals with no d-electrons or with a large filling factor F for d-electron states, i.e. metals like Ni, Pd and noble metals, where the hybridization term, eq. (3.3) is

P. Nordlander,

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/ Indirect electronic interaction berween H atoms

461

small. With a better description of hybridization effects, like that in ref. [9], also metals where the hybridization effects are expected to be large could be treated. In this calculation only hydrogen adsorbates are treated. The extension to other adsorbates is in principle straightforward. In the case of oxygen, other theoretical calculations [27] have shown that adsorbate-adsorbate interaction effects can alter the vibration properties in such a manner that different structures have different perpendicular vibration frequencies. The major conclusions of this paper are that adsorbate-adsorbate interaction can change the equilibrium properties of chemisorbed hydrogen with small but significant amounts, and that the effective medium theory can provide a reasonable estimate of such changes. It should however be noted that the qualitative aspects of the results would derive from any theory of chemisorption, where the dominant contribution to the embedding energy comes from a term that only depends on the local electron density around the embedding point. As long as there exists an optimum density for which such a embedding function has a minimum the results of section 4 do apply. The conceptual picture of an indirect electronic interaction mediated by the induced electron density, and a corresponding change in the chemisorption bond, applies beyond the effective medium theory.

Acknowledgements The authors are very much indebted to S. Andersson, S. Holloway, B. Lundqvist, J.K. Norskov and M. Persson for many valuable comments and suggestions. This work has been supported in part from the Natural Science Research Council and the Swedish Board for Technical Development,

Appendix In this appendix it is shown that within conventional effective medium theory [3,12], and within the linear superposition approximation for the total electron density, the total energy for a configuration of atoms embedded in a host depends upon the order in which they are embedded. A suitable extension of the effective medium theory to circumvent this problem is introduced. For illustrative purposes only two atoms and a host are considered. The atoms are denoted 1 and 2. The substrate densities at their corresponding positions are denoted ni and ns respectively. The atom induced changes of the density are denoted An,, for the change in electron density at position 1 induced by atom 2 and An,, for the density induced by atom 1 at the position of atom 2. When atom 1 is embedded, the effective medium theory states that

462

P. Nordlander, S. Holmstrtim / Indirect electronic interuction between H atoms

the embedding energy is AE,$,,(iii) + AEhyb. The A Et:,‘, term is local in the sense that it only depends on the substrate properties within the region u, around the atom. The region b, outside a, does not affect the AE,“:,,, term since the average of the substrate electron density is to be taken over a,. The AEhyh term represents the interaction between the atom and region b, of the substrate. In the case of a single adsorbate on a transition metal surface, this term represents the hybridization interaction between the atom and the d-band of the substrate. In the case, where there are several atoms on the substrate, there is also an interaction with the coadsorbates. This interaction is crucial and the conventional effective medium theory must be extended to take this into account. To see this, effective medium theory will be applied to the above situation. When atom 1 is embedded the electron density of the system is changed. When atom 2 is embedded at its position the average electron density around it is n; + An,, and therefore the embedding energy is A”&,( Fi; + An,,). If now, as conventional effective medium theory suggests, the effects on atom 1 which is already present on the surface, can be neglected, when embedding atom 2, the total energy for the configuration is AE,,,,=AE;;~~~((n~)+AE~f,f,(n;+An,,). On the other hand, if the total energy is constructed first and the atom 1 the total energy is AE,,,,=AE,“f,f,(n’,)+AE,‘~~“,(n;+An,,).

(A.1) by embedding

the atom 2

(A.2)

This expression is different from expression (A.l). This discrepancy comes from the assumption of treating the already embedded atom as a part of the host. To resolve this dilemma the effect of the second atom on the first one must be included. The simplest way to do this is to assume that the change of electron density on the surface, when an atom is added, is allowed to change the energy of the already embedded atom through the change in its AE$i” terms. By doing this the energy becomes well defined (not dependent upon the order atoms were embedded). The total energy can then be obtained through successive embedding. When the first atom 1 is embedded, the energy gain is therefore A EEl,f,,( iii). Th’1s contr.ibution comes from the region a, surrounding atom 1. The region h, does not contain any other atom and the contribution from this region can be obtained from eq. (3.3). The embedding of atom 2 involves an energy A Eiflfm( ii;, ) coming from region uZ. The region h, contains a,, and there is an extra energy contribution coming from the change in surface density in region u,, when atom 2 is embedded. With the proposed prescription this energy is mediated by the effective medium term and therefore equal to AE,=AE~~~~((n;+An,,)-AE,‘f,f,(TiP).

(A.3)

P. Nordlander, S. Holmstriim / Indirect electronic interaction between H atoms

463

This extra contribution to the embedding energy of an atom makes the total energy well defined (not dependent upon the order in which the two atoms were embedded). The total energy for embedded atoms 1 and 2 is thus AE’“‘=AE$!,,(ti;

+ATi,,)+AE;;~~((Ti+An,,).

(A.4)

This result is obtained under the assumption that the densities on the surface can be obtained with linear superposition. An alternate way to understand (A.4) is provided by the density functional theory, which states that the total energy of an electronic system is a functional of the total electron density n(r) [25]. E’“‘=E[n(r)].

(A.5)

For illustrative purposes energy can be written E[ n(r)]

=/drn(r)

the functional

F[n(r)]

is introduced

so that the total

F[n(r)].

(‘4.6)

In the so called local density approximation F[n(r)] is a function of the local density n(r), but so far no approximations have been made. Using (A.6) the embedding energy for an atom into a host with the unperturbed electron density n,(r) is simply

AE=

/ dr{[n,,(r)+n.(r)]

~[n,,+n,]-n,(r)

~[n,]},

(A.7)

where n,(r) is the density induced by the embedded atom. An expression with this structure applies within the effective medium theory in situations when the dominant contribution to the chemisorption energy can be assumed to come from a region a around the atom. The AE,$,,, term in eq. (3.1) only depends on the average density within a local region a around the atom. An effective medium functional F,[n] that only depends on the host properties within a region a around the hydrogen can be introduced and the integral (A.7) can then be written

AE=jdr{[no(r)+n.(r)]~_[no+n~l-n,(r) a

F,[n,l)

= AE;;;“,( no). In the case of two adsorbates energy is A.E’2=jdr{[n,(r)+n,(r)+n2(r)]

(A.8) 1 and 2 on a substrate

F[n,+n,+n,]-n,(r)

the total

embedding

~[n,]}. (A.9)

464

P. Nordlander,

S. Holmstriim

Provided that the regions overlap, the corresponding

AE,” = / dr {b,(r) 01

/ Indirect electronic interaction

between H atoms

a, and a, corresponding to atoms 1 and 2 do not effective medium expression can be written

+n,(r) +n2b91 Eh,, + n, f n2l -no(r) Ehol}

+/ dr {[no(r) +n,(r) +n,(~-)l F,[n, + nI + n21-no(r) ~_[n,l} u2

=/dr{[n,(r)+n,(r)+n,(r)l F,[n,,+n, +n21

l’l,,(r) + n,(r)1

+/‘ dr {[nil a,

F,[~o+ ~21>

+n2(r)1 F,[n, + n21-n,(r)

&[n,l}

If the density on the surface can be obtained by linear superposition, the integrals can be identified with single atom embedding energies. One obtains AE~‘=AE~&(ii;+Aii,,)+/dr

{[n,(r)+n,(r)]

F,[n,+n,]

Ul -no(r)

F,[n,l}

+AE;~;(n;+An,,)

-%W

+/

dr {[no(r)

aI

+n,(r)l

<[no

+

n,l (A.ll)

F,bol).

The remaining integrals vanish, since the integrand corresponds to embedding outside the region of integration and the interaction has been assumed to be confined to regions around the atoms. The total energy for two atoms is then (A.12)

AE’“‘=AE,‘~~(f,(il~+An,,)+AE~~~(n;+An2,). The above result several atoms.

can easily

be generalized

to eq. (3.10)

in situations

with

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465

References [l] [2] [3] [4] [5] [6] [7] [8]

[9] [lo] [11] [12] [13] [14] [15] [16] [17] [lS] [19] [20] [21] [22] [23] [24] [2S] [26] [2’7]

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