Helium-surface interaction potential: Determination and applications

Surface Science 148 (1984) l-20 North-Holland, Amsterdam

HELIUM-SURFACE AND APPLICATIONS Inder

INTERACTION

DETERMINATION

P. BATRA

IBM Research Laboratory K33/281, Received

POTENTIAL:

4 June 1984; accepted

San Jose, California 95193. USA

for publication

11 June 1984

Extensive interest in the field of elastic helium atom-surface scattering is due to encouraging progress made for surface structure determination. An important ingredient in this development is a reliable atom-surface interaction potential. We briefly review the theoretical activity in this area and present our results for the He-Cu(ll0) interaction potential. From these studies we conclude that the Esbjerg-Nsrskov relation between helium-surface repulsive potential and the surface charge density works well. The addition of the Zaremba-Kohn attractive part to the calculated repulsive part completely specifies the total interaction potential. We also reconcile different values for the Cu(ll0) corrugation obtained with and without using the Esbjerg-Nsrskov relation. For application purposes we find that the Esbjerg-Nsrskov relation with charge densities derived from atomic superposition is more convenient to use. We illustrate this by examining chemisorption of oxygen on Ni(OO1). For this a ~(2x2) structure at low exposures changes to a c(2 X2) at higher exposures. There have been conflicting suggestions about the vertical height for oxygen in the two phases. We compare our calculated corrugations with Rieder’s data and conclude that in both phases oxygen is at a vertical distance of about 0.9 A. Chlorine adsorption on Ag(001) is another system for which He diffraction work of Cardillo et al. has clearly distinguished between two competing structural alternatives. Another application is taken to show how helium diffraction can give site specific information. Here we study the He-H/Pt(lll) system and demonstrate that H chemisorbs on a three-fold hcp site at a vertical distance of about 1 A on Pt(ll1). As a last example, we present some new results for the Cu(llO)-0(2 X 1) system. In particular, we show that He diffraction results are in agreement with the recent findings of EXAFS measurements.

1. Introduction The diffraction of thermal helium atoms from surfaces is proving to be a powerful tool for structural analysis of clean and adsorbate covered surfaces. Diffracted intensities are usually analyzed within the context of a corrugated wall model to obtain the corrugation function. The corrugation function describes an equipotential surface determined by V(x, y) = E,, the normal component of the incident helium atom energy. The bumps in the corrugation function may be related to atomic positions in the surface unit cell. In addition, it is possible to obtain the adsorbate position from an analysis of the magnitude and shape of the corrugation function. A number of good review 0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

2

1.P. Botru / Heltum - surfrrc~ tnteruc~ttonpcm~tul

articles (some of which are included in our bibliography [l-8]) dealing with various aspects of gas-surface interaction are available. Very low coverages [9] of adsorbates can also be detected by analyzing the intensity of the specular beam. In fact, this can be used to detect small amounts of impurities on surfaces. There is. however, a basic problem in obtaining unambiguous, quantitative structural information from He diffraction data. Thermal energy helium atoms have a classical turning point typically 3 A above the centers of the outermost atomic layer of the surface. Thus, the diffraction data sensitively probes only the outer regions of the helium-surface potential. To translate this into structural information, a method of calculating the corrugation function or interaction potential [lo] from an assumed geometry is required. A variety of calculations for scattering potentials have been reported in the literature [lo-361. Even though no completely first principles calculation has yet been presented for a real solid, rather sophisticated model potentials are available to derive at least semi-quantitative information from helium diffraction data. The progress made to date is impressive and the future prospects for helium diffraction as an important analytical tool appear to be promising. In this paper we provide a short survey of the field and present some of our new results. The organization of the paper is as follows. In the following section some recent theoretical attempts for calculating the atom-surface interaction potential are briefly reviewed. In particular, we indicate that the Esbjerg-Norskov [18] relation between helium-surface repulsive potential and the surface charge density is very useful for structural applications. Our ab initio cluster model studies [36] are also described. These calculations offer some additional support in favor of the EsbjerggNarskov [18] relation. We also state how an apparent disagreement between the results of Liebsch et al. [28,29] and Garcia et al. [30,31] for the corrugation of the Cu(ll0) surface can be resolved. In section 3 some current applications of helium diffraction for surface structure determination are presented. These include the chemisorption of oxygen on Ni(OO1). For this system a p(2 x 2) structure at low exposures changes to a c(2 x 2) at higher exposures. There have been conflicting suggestions for the chemisorption distance for the two phases. We compare our calculated corrugations with Rieder’s helium diffraction data [38] and conclude that in both phases oxygen is at a vertical distance of about 0.9 A. Chlorine adsorption on Ag(OO1) is another system for which the He diffraction work of Cardillo et al. [39] has clearly discriminated between two competing structural alternatives. Another application is taken to show how helium diffraction can give site specific information. Here we analyse the He diffraction data of Lee et al. [40] from the H/Pt(lll) system and demonstrate that H chemisorbs on a threefold hcp site at a vertical distance of about 1 A. As a last example we present some new results of our theoretical calculations for helium diffraction

I. P. Butm / Helium -surface

rnterartiun

potential

3

from Cu(llO)-0(2 x 1). We show that the old helium diffraction data [41] can be reinterpreted in the light of our calculations to give the chemisorption site in agreement with the recent EXAFS data [42]. The structural information for O/Ni(OOl), H/Pt(lll) and O/Cu(llO) was obtained by comparing calculated corrugation functions with those deduced from helium diffraction data. Comparison of intensities was not carried out because these depend rather sensitively [23] on the details of the potentials. The level to which the interaction potentials are currently known for some of these systems does not warrant intensity comparisons. Consequently, for semi-quantitative purposes it is sufficient to compare corrugations. One of the important topics which is completely omitted from our discussion is the calculation of diffraction intensities given the form of the interaction potential. There is a lot of sophisticated technology involved in calculating intensities and the subject has been amply discussed in a number of excellent publications 143-511. The reader may consult these representative references for intensity calculations made on important physical systems. We close the paper with some concluding remarks and future prospects.

2. Theoretical dete~ination

of atom-surface

interaction potential

It is a reasonable approximation to think of the metal-atom interaction potential as consisting of two parts. For sufficiently large metal-atom separations, the interaction is attractive due to a correlation term which goes as l/r6 (electric dipole-induced dipole interaction). This interaction when integrated over the half space gives the well known Van der Waals 114,521 form,

The parameters C”, and Z,, have been given by Zaremba and Kohn [14] for a number of metals. At short distances, the net interaction potential is repulsive. Even though the metal electrons feel a strong atomic potential, the energy lost due to He 1s core orthogonalization more than offsets this benefit. In this connection we mention that the work of Kleiman and Landman [12] which approximates the repulsive term simply by an increase in the kinetic energy (which results when metal-atom electrons overlap), gives too high a value for the repulsive term. Zaremba and Kohn ]lSJ (ZK) treated the repulsive and attractive parts separately to ensure proper limiting behaviors. They employed the Hartree-Fock approximation for the repulsive term and showed that to the lowest order in the overlap between the atomic and metallic wave functions, this interaction is given by YR( z,}

= 2X&,.

(lb)

I. P. Barra / Helium

4

surface mterac’tron

potentrul

Here Z, is the position of the helium nucleus and Se, is the change in eigenvalue of the metallic electrons due to the presence of helium atom. The complete interaction potential was then obtained by adding to eq. (lb) the asymptotic Van der Waals interaction. The repulsive term shown in eq. (lb) looks simple because all the complexity is contained in the equation which has to be solved for generating eigenvalues. Consequently, ZK calculated eigenvalues in the density functional approximation and furthermore used a jellium model for the metal. The jellium model gives only a laterally averaged, F,(Z) potential. Van Himbergen and Silbey [16], unlike the ZK method, treated the attractive and repulsive contributions to the interaction energy on an equal footing within the local density functional approach. They followed the electron gas approach used earlier by Gordon and Kim [ll] (GK) who had obtained fairly accurate intermolecular potentials for the rare-gas systems using a superposition of atomic charge densities (obtained from Hartree-Fock approximation). The GK approach was used before by Freeman [13] to calculate the interaction between rare gases and a graphite surface. (We mention that the graphite surface problem has been completely solved by Carlos and Cole [53].) The jellium model was used for the metal. The results of these calculations were further supported by Lang’s self-consistent calculations [22] for Ar-jellium again based on the density functional scheme using the local density approximation for the exchange-correlation potential. He also made the important observation that in the region close to the potential well the inert atom has a pronounced polarization toward the metal similar to that for a covalent bond. He thus argued that the local density approximation should handle this region properly. It should be noted that the correlation contribution (eq. (la)) can not be adequately described in the local density approximation (which gives [22] an exponentially decaying attraction at long range). However, it is not clear at what distance the asymptotic form should really become effective. As we shall see below, there is some evidence which suggests that the asymptotic form is apparently too rapidly varying in the region of interest. The approaches described above require elaborate calculations and can not be readily implemented for real surfaces. An important step in this direction was the work of Esbjerg and Norskov [18] (EN). Using the effective-medium theory [19,20] EN found a linear relationship between the helium-surface repulsive potential and the surface-electron density. The EN relation is, v,(r)

= &P(r).

(2)

These calculations were based on the energy of embedding He in a uniform electron gas. The value of the uniform electron density is taken to be equal to the surface electron density p(r) at the He atom position r. Lang and Norskov [32] have also provided another justification for eq. (2) based on self-consistent helium-jellium model calculations. A slightly refined form of the relation (2) is

I.P. Batra / Helium-surface

interaction potential

obtained [32] by treating the deviations of the electron density ity in the first-order perturbation theory. The result is

5

from homogene-

V(r) = %r,P(r),

(3)

where p is the electrostatic / I P(r)

potential,

A(lr’

- rI> dr’

&(r’)

dr’

+a, averaged

charge density

=

by

(4) /

and in atomic

defined

Rydberg

units

Here za( = 2) is the helium

atomic

number,

cx,rr = (Y- q,,

the integration is over the atomic region only. The value of (Y is - 300 eV bohr3. Unfortunately, in the literature several different values of LU(ranging from 176 to 1000 eV bohr3) have appeared. In a recent paper Manninen et al. [54] have presented a clear discussion of the origin of these different values and have suggested the appropriate choice of 1~. The first application of the EN relation was made by Hamann [21] for He diffraction from GaAs(ll0) and Ni(llO)-H(2 x 1) surfaces. This successful demonstration stimulated a lot of interest both for application and finding further confirmation for the EN relation. Laughlin [23] has suggested a similar relation but with a slightly reduced repulsive term; the reduction is proportional to p’j3. We have verified [36] the EN relation using the self-consistent field (SCF) Hartree-Fock approximation. A cluster model was used in order to investigate the lateral variation of the repulsive potential. In particular we have estimated [36] an average value for (Y- 375 eV bohr’. The calculations were done for the interaction of helium with the Cu(ll0) surface. Two different sites for the approach of He to the surface were considered and appropriate clusters were chosen to represent each site. For the on top site, a Cu,(5, 4) cluster shown as an inset (a) in fig. 1 was used. The notation Cu,(5, 4) indicates that there are 5 atoms in the first layer and 4 in the second layer. The central atom (labelled 1) in the figure is the site which was approached by the He atom. For the bridge site a Cu,,(6, 6) cluster shown as an inset (b) in fig. 1 was used; helium approached the mid point of atoms labelled 1 and 1’ in the figure. The cluster SCF calculations were performed using Gaussian basis sets [36]. For the atoms which are nearest to the incident He, all the electrons were included, but for the Cu atom neighbors of these atoms a pseudopotential

I.P. Bafra / Hehum

6

-

surfaceInteracttonpotenttal

treatment was used. Thus, for the on top site Cu,(5, 4) and Cu,(S, 4)He clusters, the pseudopotential was used for the 8 environmental Cu atoms and only the Cu atom nearest He, 1 in fig. 1, was treated as an all electron atom. Similarly, for the bridge site Cu,,(6, 6) and CU,~(~, 6)He clusters, only the atoms labelled 1 and 1’ in fig. 1, are all electron atoms; the remaining 10 atoms used the one-electron pseudopotential. All other details can be found in ref. [361. Our results for the Cu,He cluster are summarized in table 1. From the calculated repulsive energy and the unperturbed Cu, cluster SCF charge density we obtained a value of the proportionality constant (Yof about 350 eV bohr3 in the Esbjerg-Norskov relation. We have also tabulated the values of E, and the corresponding a’, which corresponds to frozen Cu, and He orbitals. A typical value of (Y, - 500 is considerably larger than (Y which is obtained using the (relaxed) SCF energies. The fact that 01, is larger than (Y is not surprising because one lowers the total energy when the orbitals are allowed to relax to respond to the presence of He and hence the repulsive energy

2

3

4

5

6

7

2 (a.u.) [I 101 Fig. 1. Cu-He repulsive interaction energy for the on top and bridge sites for (A) Cu,(5, 4)He and (B) Cu,,(6, 6)He as a function of the vertical distance, Z. Inset (a) is the cluster Cu,(5, 4) and (b) is the cluster Cu,,(6, 6) for the on top and bridge sites respectively for Cu(l10). Open circles represent Cu atoms in the top layer and shaded circles are Cu atoms in the layer below. The symbol X denotes the site of approach for the helium atom.

I. P. Batra / He/rum - surface interaction poteniiul

7

Table 1 Cu,(5, 4)He interaction energy ‘), Et, at the SCF level as a function of Z. the vertical distance of helium above a copper atom in the on top site; the unperturbed charge density of the Cu, cluster and the resultant a, LY, values are presented: the a, are calculated using frozen orbital repulsive energy E, where orbitals are not allowed to relax (see ref. 1361) Z (a.u.)

EU (mev)

Et (mev)

P

x 10s

LV bohr”)

(eV bohr’)

3.5 4.0 5.0 6.0 7.0

1074.9 654.0 272.2 119.8 52.9

589.7 376.9 171.3 77.7 33.4

2.071 1.256 0.5014 0.2144 0.0970

285 300 342 362 34s

519 521 543 559 546

‘) The reference energy ( = - 1640.992971).

E,=

-1643.854632

“U

H is the sum of E,,

( = -2.861661)

and

Ecu,

decreases. The calculated results for the Cu,,I-Ie cluster are are summarized in table 2. The value of (Yis somewhat larger in this case but we conclude that a value of about 375 eV bohr” is acceptable for both sites. We have also shown in fig. 1 the repulsive interaction energy for the on top (A) and bridge (B) sites as a function of Z. Note that the two straight lines have somewhat different slopes. If the two lines had identical slopes the corrugation would be completely independent of energy which may not be true. For the corrugation function to increase with energy, K~ should be greater than K~. Our calculations give K* = 1.5 A-’ which is in fact greater than KB = 1.4 A’. At a fixed Z, the repulsive interaction energy at the bridge site is lower than at the on top site (due to a reduced charge density): the straight line for the bridge site lies below the on top site line. Note that these lateral variations can not be obtained in the jellium model. Thus our cluster results have shown 1361 that the laterally varying repulsive potential between helium and a metal surface has an exponential fall off. We

Table 2 Cu,,(S, 6)He interaction energy a). E,, at the SCF level as a function of Z, the vertical distance of helium above the midpoint of two copper atoms in the bridge site; the unperturbed charge density of the Cu,, cluster and the resultant a values are also presented (see ref. [36])

z

El

(a-u.)

(mev)

x103

PfV bohr3)

3.5 4.0 5.0 6.0 7.0

317.6 216.9 104.3 49.9 24.1

0.8506 0.5659 0.2741 0.1389 0.0666

373 383 380 359 362

” The reference energy - 3280.431031.

P

I&_ = - 3283.292492

H is the sum of EHe( = - 2.861663)

and EC,,, =

have provided additional support for the linear Esbjerg-Norskov relation and have obtained a value for the proportionality constant (Y from fundamental considerations. The suggested value of 375 eV bohr-? is in good agreement with the value suggested by Manninen et al. [54]. Another significant theoretical development for Heemetal interaction has been due to the work of Harris and Liebsch [26] (HL) and the important extension due to Nordlander [34] and also Nordlander and Harris [35] (NH). They follow the development of ZK [15] and derive eq. (lb) within the density functional scheme. Since they find it to be also valid in the local density scheme they correctly point out that the ZK calculations actually include some intra-atomic and intra-metallic correlation. The approximate correlation term added by ZK then only represents metal-atom correlation. HL then made a series of approximation for actual numerical evaluation. Eq. (lb) was already truncated to first order in overlap between metal and helium. To calculate the shift in eigenvalues they used a perturbation theory expansion using the non-local helium pseudopotential as the expansion parameter. The repulsive potential took a particularly simple form when only the leading term in the pseudopotential was retained and the metal was replaced by jellium. With this set of approximations they obtained:

Here p(~, rN) is the local density of electron states at the position rN of the helium nucleus and g(c) is a universal function of energy. By comparing their calculated results for He-jellium with ZK values, HL find a discrepancy - 15-20% which they explain in terms of neglect of higher order terms in the pseudopotential. This conclusion seems to be validated by the work of Nordlander [34] who included the next order term in the pseudopotential and approached ZK’s results more closely. More recently NH [35] have further extended the ZK [15] theory by removing the assumption of asymptotic behavior of the metal wave functions. They note that because the surface potential reaches its asymptotic value only slowly, deviations of the metal wavefunctions from their asymptotic values are not negligible near and inside the potential well. By using correct numerical wavefunctions and including second order terms in helium pseudopotential, NH found a lowering of - 20-30% in the ground state energies compared with those quoted by ZK. Thus according to these authors, ZK overestimate the repulsive potential and hence underestimate the well depth. However, there are still a number of approximations whose impact is not fully known. The repulsive term is calculated only up to first order in the metal-helium overlap. In addition NH use an important semi-empirical refinement of the metal (jellium) potentials to reproduce the “correct” workfunction of the real surface. They also note that sometimes it is necessary to suppress the divergence for

I. P. Barra / Helium-surface

interuifinn potential

9

2 - Z,, in the asymptotic Van der Waals form. Again a semi-phenomenological procedure is used. The effect of various theoretical approximations and semi-empirical refinements on the calculated values, e.g., the well depth needs to be investigated further. It is important to observe that whereas Norskov et al. [18,32] find the unperturbed surface electron density to be the relevant quantity which determines (the repulsive part of) the atom-surface potential, HL [26] relate it to the local density of states. One can formally write the HL equation as

where E in principle depends on r,. In practice < is found [54] to he basically constant for distances of interest and in that case the HL form looks quite like the EN relation. The value of the proportionality constant is, however, - 1000. If one is willing to accept a as parameter, the two theories are rather similar. We favor this alternative because it enables us to extract readily important semi-quantitative structural information. For actual corrugation calculations we use the EN relation with the surface charge density derived from atomic superposition of Hartree-Fock atomic charge densities, much like the GK [II) scheme. Garcia et al. 130,311 determined corrugated repulsive potentials using this scheme and found that it works quite well. A similar conclusion has been reached in several 155,561 other calculations. Liebsch et al. [27-291, on the other hand, from a detailed analysis of He-Cu(ll0) indicated that the relative corrugation derived in this way is about four times larger than the experimentally derived value. Since the experimental data are fitted equally well by potentials constructed by Garcia et al. and Liebsch et al., it might appear that there is a lack of uniqueness in the helium-surface potential as determined from diffraction data. However, Barker et al. [57] have shown that this is definitely not the case. The total potentials of Garcia et al. and of Liebsch et al. are essentially identical. A closer examination by Barker et al. [57] revealed that the apparent disagreement by a factor of 4 arises from two quite different sources. The first is that the corrugation of the rep~~siue part of the empirical potential of Liebsch et al. is smaller than the corresponding corrugation of the repulsive part of the potential of Garcia et al. by a factor of about 2. This arises from the fact that the attractive part of the potential of Liebsch et al. varies about twice as rapidly as the attructiue part of the potential of Garcia et al. in the relevant region. Liebsch et al. used the Van der Waals form given in eq. (la) for the attractive potential while Garcia et al. used the more slowly varying exponential form. The Van der Waals form is asymptotically correct at large distances but varies too rapidly at short distances. It is well known in the study of atomic interactions that it is necessary to apply appropriate short range “damping” factors to the Van der Waals terms if one wishes to represent potentials by simple additive forms [58-603. A similar conclusion has also been reached by

NH [35] for atom-surface interactions. The exponential form used by Garcia et al. evidently mimics the proper behavior at short distances. The remaining disagreement arises [57] from the fact that the corrugations of the repulsive part of the potential calculated from charge densities by Liebsch et al. are /urger by a factor of about 2 than the corresponding results of Garcia et al. This difference is due to a number of approximations used by Liebsch et al., of which the use of an asymptotic expression for a lattice sum and the approximation of the atomic charge density by a single exponential function of distance have the most serious effect. The net effect of all the approximations is that the calculated corrugation of Liebsch et al. is close to twice that of Garcia et al., who avoid all the above approximations, Thus the overall picture is that the e,~per~men~u~ corrugations of the repulsive potential of Liebsch et al. are smaller than those of Garcia et al. by a factor of 2 while their “calculated” corrugations are furger than those of Garcia et al. by a similar factor, leading to a discrepancy of a factor of 4 between calculated and experimental values. We might mention that the corrugation calculated [30,31] for Ni(ll0) using the charge density ideas, which worked well for Cu(l IO). did not fit the data 1371. Experimentally [37], the corrugation along the [OOl] direction was found to be independent of the incident helium atom energy. Calculations, on the other hand, showed [30,31] that the corrugation increases with energy. Experimental results are not easy to rationalize intuitively. There are some important theoretical attempts [33] to understand the energy dependence of corrugatiojl. but the problem is not completely solved yet. In the following illustrations we therefore use the EN relation for calculating corrugation of the repulsive potential, By comparing this corrugation to the experimental corrugation profile, important structural information is obtained. Our calculations are based on superposition of atomic charge densities. Clearly. a more accurate surface charge would obviously be obtained by performing a high precision self-consistent field (SCF) calculation. Such calculations are extremely demanding [21] especially since the charge densities are required near the classical turning points of thermal helium particles. In general an SCF calculation would give lower corrugation values than those obtained using the atomic superposition. However, we make another approximation which at least partly offsets this effect. We only consider the repulsive part of the potential whereas the helium particles sample corrugation in the total potential. The total helium potential is more strongly corrugated [39,573 than the repulsive potential. The difference is proportional 1391 to the slope of the attractive potential in the region of classical turning points of the He in the total potential Thus the two approximations (the lack of self-consistency and repulsive as opposed to total He potential corrugation) tend to compensate each other but the extent of cancellation is uncertain. The net effect is that one gets reasonable values of corrugations using atomic superposition.

I.P. Bmra / Helium-surface

interaction potentral

11

3. Applications and discussion 3.1. Corrugation

analysis for p(2 x 2) and c(2 x 2) phases of O/ Ni(OO1)

Atomic oxygen chemisorbs [61-631 on Ni(OO1) in two different phases depending upon the exposure. At low exposures (1.5 L) p(2 X 2) LEED patterns corresponding to 0.25 monolayer coverage are observed. At higher exposures (20-30 L) the c(2 X 2) structure is obtained. Most of the experimental investigations projected a single vertical height of about 0.9 A for both the phases. Upton and Goddard [64] (UC) using an Ni,,O cluster found two low-lying electronic states: (i) the oxide or closed shell state with all oxygen orbitals doubly occupied and (ii) the radical or open shell state with one u orbital of oxygen singly occupied. They found different values for Z, (the vertical height of the adsorbate) and o (the normal vibration frequency) for the two states. UG assigned the p(2 x 2) coverage to the open shell state. At c(2 x 2) they suggested that the oxygen was in an oxide state and lies much closer to the surface (Z, = 0.26 A). Rahman, Black and Mills [65] used the results obtained by UG and included Ni-Ni (phonon) interactions to successfully account for the 0-Ni phonon spectrum. Their lattice dynamical calculation also supported the short normal distance, Z, = 0.26 A for the c(2 x 2) coverage. The original LEED investigations had given [66] a single value of Z,, = 0.9 A for both the phases. Other quantum chemical cluster model studies [67] have concluded that the oxide state is to be favored for both p(2 X 2) and c(2 x 2) with a single vertical distance (- 0.9 A). A He-diffraction study [38] of the p(2 X 2) and c(2 X 2) phases of oxygen found both phases to be consistent with Gaussian corrugations of height - 0.55 A and the full width at half maximum (W) of - 2.0 A. We calculated [62,63] the corrugation function using the EN relation [18]. The surface charge density was approximated by superposition of self-consistent Hartree-Fock atomic charge densities. The calculated corrugation results for p(2 x 2) and c(2 X 2) phases of oxygen on Ni(OO1) along the [llO] direction fell into two general groups depending on the height of the oxygen atom above the surface. At Z, = 0.9 A the corrugation height {,, ranged from 0.36 to 0.58 A. For Z, = 0.26 A the corresponding range was from 0.14 to 0.27 A. For all values of the calculational parameters it was possible to distinguish between the two heights of oxygen above the Ni surface based on the corrugation height. Our calculations clearly favored a vertical distance of about 0.9 A and not the shorter distance of about 0.26 A. Our calculated corrugation at Z, = 0.9 A is somewhat non-Gaussian and has a width (W - 2.7 A) that is larger than measured experimentally. However, the disagreement with experiment at Z, = 0.26 A was gross and enabled us to rule out that distance.

3.2. Helium diffraction from Cl/ Ag(001) Another important example is the determination [39] of the structure of adsorbed Cl on the Ag(OO1) surface which forms a c(2 X 2) periodic arrangement much like 0 on Ni(OO1). A LEED [68] analysis for this system was carried out in 1976. Of the four geometrical models considered, two (the on top and bridge bonded Cl atoms) were clearly ruled out. The remaining two, namely a simple overlayer model (SOM) with Cl atoms on the fourfold hollow sites and a mixed layer model (MLM) produced encouraging agreement between theory and experiment. The SOM model consisted of Cl atoms occupying the fourfold symmetrical hollow sites. The MLM model consisted of a mixed Cl-Ag layer, with each of the Cl and Ag atoms on fourfold symmetrical sites with non-coplanar Ag and Cl atoms. However, by varying the angle of incidence the authors [68] did in fact choose the SOM model with a vertical distance of the Cl nucleus above Ag(OO1) in the range 1.57-1.78 A. On the other hand the ultraviolet photoemission spectroscopy data [69] and the subsequent electronic structure calculations [70] of the local density of states favored the MLM structure. Helium diffraction work of Cardillo et al. [39] has clearly resolved this controversy. Using the EN relation with charge density of the Cl-Ag system (calculated self-consistently) Cardillo et al. [39] obtained surface corrugation functions for both SOM and MLM arrangements. The corrugation heights for the two models were strikingly different; - 1 A for the SOM and about an order of magnitude smaller for the MLM geometry (assuming Cl-Ag coplanar). Our own calculations with atomic superposition gave similar results. They [39] noted that the variation of the geometry away from coplanarity may produce some additional corrugation but markedly different from the SOM. The helium diffraction data was found to be consistent [39] with a scattering potential corrugation - 1 A. This firmly established the SOM structure in agreement with the LEED study [68]. 3.3. Corrugation analysis of the H / Pt( 111) system The determination of the geometrical structure of (1 x 1) monolayer of hydrogen on the Pt(ll1) surface has been difficult [71] because of the low sensitivity of low energy electron diffraction (LEED) to hydrogen. Ultraviolet photoemission spectroscopy (UPS) data [72-741 on H/Pt have provided qualitative information about the chemisorption site but the precise location was not obtained. The electron energy loss spectroscopy (EELS) data have been interpreted [75] in terms of a three fold hollow adsorption site with H at a vertical distance of 0.71 A. Lee et al. [40] have studied in detail He atom diffraction from a (1 X 1) adlayer of H on Pt(lll) and obtained the corrugation within the context of a corrugated hard wall model. They report two bumps in the corrugation of

I. P. Batra / Helium-surface

rnteraction potential

13

heights 0.128 and 0.05 A. From the shape of the corrugation function (and the symmetry with respect to incident azimuthal angle) they conclude that H chemisorbs in a threefold site. By assigning the larger bump in the corrugation to H, and the smaller one to Pt, they suggest (by comparison of the He diffraction data to the orientation of the crystal obtained from X-ray diffraction) that the hydrogen atom is at an hcp site (in which there is a Pt atom in the second layer directly below H) and not an fee site. This conclusion is stated tentatively since if the larger bump in the corrugation were due to Pt, not H, the conclusion would be reversed. The atomic arrangement of the (111) surface of Pt covered with a (1 X 1) monolayer of hydrogen in a threefold site is shown in fig. 2. The site shown is a hexagonal closed packed (hcp) site in which there is a Pt atom directly below H in the second layer. The alternate threefold fee site has no second layer Pt atom directly below the H atom. The unit cube edge of Pt is a, = 3.92 A. In our calculations we took the x-axis in the [ilO] direction and the y-axis was chosen parallel to the [112] direction with the origin marked as (0.0) in fig. 2. For the three fold site we calculated corrugation results along the A,A, ([llj] or $ = 0’) and the B,B, ([lOi] or $ = 30”) lines. In table 3 we have given our calculated corrugations as a function of Z, the vertical position of the hydrogen atom relative to the top plane of Pt atoms. The negative Z value corresponds to H below the surface. Complete details about the calculation are given elsewhere [76,77]. It is clear from the table that the calculated results agree with the experimental results for Z = 2 a.u. (- 1 A). The corrugation shows an interesting dependence on Z. At small values of Z (hydrogen below or at the surface), the corrugation is essentially that of the

Pt

0 Ii Fig. 2. Hydrogen

in a (1 X 1) overlayer

structure

on the Pt(ll1)

surface.

I.P. Bafrcr / Heirurn -_surface intercrctiott potenttul

14

clean surface. As the hydrogen monolayer rises above the surface the corrugation decreases, i.e., hydrogen fills in the the areas where the clean surface charge density was unable to reach. As hydrogen rises further above the surface, its charge density sticks out and the corrugation increases. Thus, only when hydrogen is more than about 1.5 a.u. above the surface does it start to become “visible”. At a vertical distance of more than 2.0 a.u. hydrogen sticks out too far and the corrugation increases past the experimental value. At Z = 2.0 au., the maximum corrugation along the [112] (+ = 0’) directly is slightly Lower than the experimental value but along the [lOi] (+ = 30”) direction the calculated value is slightly higher than the experimental value. Thuz Z = 2 a.u. represents a fair compromise for the vertical position of hydrogen above the platinum surface. This implies a Pt-H bond length of about 1.9 A, a value slightly longer than the diatomic Pt-H bond length ( - 1.6 A). Similarly larger than usual diatomic bond lengths have been reported by others ]21,78] for hydrogen chemiso~tion on Ni( 110). It is interesting to note that there is enough room for hydrogen to be accommodated in the plane of Pt atoms without producing any constrained bond length. In plane, hydrogen will produce a bond length of 1.6 A which is quite comparable to diatomic Pt-H bond length. Also based on atomic radii 1791 for Pt (1.35 A) and H (0.3 A), it is not unreasonable to expect H to lie in plane. However, our calculation clearly rules out that site as being in total disagreement with the helium diffraction data. Baro et al. [75] from a careful analysis of their EELS data have concluded that hydrogen adsorbed on Pt(ll1) has a Pt-H bond length of 1.76 A and a hydrogen distance to the surface of 0.71 A. These values are somewhat smaller than our estimates, but considering

Table 3 Maximum corrugation height, lo (measured with [llf] (+ = 0”) and along the direction [lOi] ($I = for M in the threefold site as a function of Z, the &(I 11) surface; results are shown for the normal ref. 1771)

-0.5 0.0 0.5 1.0 1.5 1.8 2.0 2.2 Experiment

[40]

respect to the value at A,), along the direction 30”) (measured with respect to the value at B, ) vertical height of the hydrogen layer above the energy, E, = 43 meV and o = 350 eV bohr’ (see

0.057 0.055 0.052 0.045 0.075 0.104 0.125 0.144

0.025 0.002 0.011 0.012 0.051 0.079 0.098 0.117

0.128

0.096

I. P. Batra / HeIrurn - surface interaction potential

15

the mutual uncertainties involved, the agreement is satisfactory. Both analyses imply that hydrogen chemisorbs in a threefold site at a sizeable distance above the surface plane of Pt atoms. For (1 X 1) H monolayer in a top configuration we tried values of Z ranging from 1.6 to 2.0 A. For Z = 1.6 A along the C,C, direction we found a maximum corrugation value of 0.18 A. However, Lee et al. [40] find that a 0.13 A corrugation amplitude is required to fit the intensity data. Consequently this site can be safely ruled out. This site was also ruled out by Lee et al. [40] because it did not satisfy the observed symmetry of the diffraction results with respect to azimuthal angle of incidence. For the bridge site, Lee et al. [40] attempted a large number of parameters but none fitted the intensity data. Since no corrugation parameters are available we are unable to give any consideration to this site. We therefore conclude that the hydrogen overlayer chemisorbs in a threefold site at a vertical distance of about 1 A. We have shown by comparing calculated corrugations for H(l X 1) on Pt(ll1) with those deduced by 0fitting He diffraction data that H chemisorbs at a vertical distance of about 1 A above the Pt surface in a threefold site. We are unable to distinguish the two threefold sites (fee and hcp) on the basis of the corruption function alone. However, we show that the larger hill in the corrugation function determined by Lee et al. is due to H atoms (rather than Pt atoms) and this, taken together with the results of Lee et al. [40] on the orientation of the He diffraction pattern with respect to X-ray diffraction, shows that H adsorbs in the hcp site. 3.4. Structure

of 0 / Cu(l IO) using helium diffruction

It is well known that the adsorption of oxygen on Cu(ll0) gives rise to a (2 x 1) LEED pattern [80] at low exposures (< 100 L) at room temperature. A number of models have been proposed [41,42,81-831 for this super structure of oxygen on Cu(ll0). A recent angular dependent surface extended X-ray-absorption fine structure (EXAFS) study [42] has given the nearest- and secondnearest-neighbor distances to be 1.84 and 2.00 A respectively. The reader may also consult this paper for references to some of the early work on O/Cu(llO). From these distances the authors conclude that the oxygen atom is located slightly (- 0.35 A) above the the surface on the long bridge site along the [OOl] direction. They conclusively rule out the sub-surface site [81]. Unfortunately the EXAFS work is not sensitive to the third nearest neighbors and therefore is unable to conclude whether the observed (2 X 1) LEED pattern is due to 0 or by a missing row of Cu atoms. Helium diffraction experiments [41] performed on the Cu(llO)-0(2 x 1) have indicated a nearly one-dimensional corrugation function with a corrugation amplitude of - 0.7 A. However, this corrugation function was used to imply that the oxygen atom occupied a sub-surface site - 0.7 A below the top

Cu layer along the [OOl] direction. The calculations we have performed show that the above corrugation function is in fact consistent with the site deduced from the EXAFS analysis provided we introduce the missing row model. Oxygen in a (2 X I) configuration on Cu(ll0) is shown in fig. 3. We take the x-axis in the [Ii01 direction and the y-axis in the [OOl] direction. The ideal two-dimensional surface (1 x 1) unit cell dimensions are ~,/fi and a,, where a, = 3.615 A. The presence of oxygen atoms in every other ideal (1 x 1) unit cell along the [liO] direction doubles the periodicity along the x-direction which is then responsible for the (2 X 1) super structure. Oxygen atoms occupy twofold long bridge sites in the [OOlJ direction. The missing row model which is obtained by removing each second [OOl] row adjacent to the oxygen containing row does not alter the (2 X 1) periodicity. As before, For calculating surface charge density we used superposition of atomic charge densities arising from the two-dimensional periodic system. Three layers of copper were included along the ]llO] direction. For oxygen we assumed an O- charge distribution. We used the ionic basis set consisting of three exponential functions for 2s and five functions for 2p corresponding [84] to the 2P state. All copper layers were treated as neutral (infinite reservoir of electrons) and each Cu atom was described by quadruple zeta functions [84] ( 2S). We performed calculations for various positions of oxygen below, in and above the Cu(l10) surface in the long bridge positions along the .y-axis. In fig. 4 the calculated corrugation function {(x, y) for the (2 X 1) phase of oxygen on Cu(ll0) is shown. The results are for 100 meV (normal component of incident helium energy) and (Y= 350 eV bohr”. Electrostatic potential averaging of the target charge density was performed. Each curve is labelled according to the height (Z) of oxygen atom above the surface. Z = 0 corresponds to oxygen being coplanar with the surface Cu layer atoms. It is clear

Fig. 3. Oxygen in a (2 x 1) overlayer structure on the Cu(ll0) surface

I. P. Botra / Helium - surfuce interaction potential

11

that for Z - 0.2 A the corrugation is nearly one-dimensional with a corruga- 0.8 A. If oxygen atom is moved above or below this position tion amplitude even by 0.2 A, discernible deviation from one-dimensionality occurs. A corruas can be clearly seen from gation - 0.08 A is produced along the y-direction the figure. For negative value of 2 = -0.2 A (oxygen below the surface) corrugation profile is two-dimensional; the amplitude being - 0.13 A along the y-direction. This is not consistent with helium-diffraction data [41] and hence the sub-surface site can be ruled out. A site - 0.2 A above the surface Cu layer is favored by our calculation and is in reasonable agreement with the EXAFS data [42]. Our calculated corrugation function for Z = 0.2 A can be adequately represented by the following one-dimensional cosine Fourier series: (9)

1.0

(0.4)

0.8

(-0.2) 0.6 G 3 $ *n 0.4 Cu( 1 10)-0(2X 1)

0.2

0.0

Fig. 4. Calculated corrugation function for Cu(llO)-0(2

x 1).

where u = u,,d2 . The values of the corrugation coefficients (in A) are: n, = 0.75, e2 = - 0.12, D3 = 0.03. We find an average softness parameter, K = 1.75 + 0.25 A-’ for the repulsive potential. The information provided here should be useful for future refinements of the vertical height of oxygen which may he deduced from more rigorous diffracted intensity calculations. 4. Concluding remarks We have shown by some recent examples how atom beam diffraction from surfaces is attaining a prominent status in surface structure determination. This is largely due to important contributions of experimentalists. The standards for interaction potentials have been set by the pioneering work of Zaremba, Kohn and Lang on interaction of inert atoms with jellium. Esbjerg, Norskov and Land have provided us with a simple working hypothesis. Hamann was the first to demonstrate its usefulness for surface applications and a number of other applications are in the literature now. Among others, the work of Lauglin, Harris, Liebsch and Nordlander on incorporating detailed electronic structure into the interaction potential seems very encouraging. A complete first principles determination of the helium surface interaction potential continues to be the goal [SS]. However, at the present time we have to be content with the semi-quantitative structural inf~~rmation given the uncertainty in the theoretical interaction potential. Acknowledgements I acknowledge useful discussions with Dr. J.A. Barker, Dr. P.S. Bagus, and Dr. D.J. Auerbach. I am grateful to Dr. K.H. Rieder and Dr. T. Engei who taught me about O/Ni(llO) and I was able to transfer that knowledge to O/Cu(llO). I am indebted to Dr. J. Weare for useful comments on the manuscript; to Dr. M. Cardillo for looking over the section on Cl/Ag(OOl). I am also thankful to Professor V. Celli for inviting me to participate in the US-Italy Conference on Gas-Surface Interaction and Physisorption in Virginia (June 4-7, 1984) where the contents of this manuscript were presented as an invited lecture. References [I] T. Engel and K.H. Rieder. in: Structural

Studies of Surfaces with Atomic and Molecular Beam Scattering, Springer Tracts in Modern Physics. Vol. 91 (Springer, Berlin. 1982) p. 55. [2] M.J. Cardillo. Ann. Rev. Phys. Chem. 32 (1980) 331; J.A. Barker and D.J. Auerbach. Surface Sci. Rept., in press. [3] H. Hoinkes. Rev. Mod. Phys. 52 (1980) 933. [4] G. Armand and J. Lapujoulade, in: Proc. 11th Rarefied Gas Dynamics Symp.. Ed. R. Campargue (Commisariat B I’Energie Atomique. Paris, 1979) p. 1329.

I. P. Burra / Helium

surface interaction

potentid

19

[51 H. Wilsch, in: Topics in Surface Chemistry, Eds. E. Kay and P.S. Bagus (Plenum. New York, 1978) p. 135. 161 M.W. Cole and D.R. Frankl. Surface Sci. 70 (1978) 585. Dynamics of Gas-Surface Scattering (Academic Press, [71 F.O. Goodman and H.Y. Wachmann. New York, 1976). PI J.P. Toennies, Appl. Phys. 3 (1974) 91. I91 B. Poelsema, R.L. Palmer, S.T. de Zwart and G. Comsa. Surface Sci. 126 (1983) 641: B. Poelsema, G. Mechtersheimer and G. Comsa. Surface Sci. 111 (1981) 519. PO1 W.A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon. Oxford, 1974). 1111 R.G. Gordon and Y.S. Kim, J. Chem. Phys. 56 (1972) 3122; Y.S. Kim and R.G. Gordon, J. Chem. Phys. 60 (1974) 1842. [12] G.G. Kleiman and U. Landman, Phys. Rev. B8 (1973) 8484. (131 D.L. Freeman, J. Chem. Phys. 62 (1975) 941. [14] E. Zaremba and W. Kohn, Phys. Rev. 813 (1976) 2270. [15] E. Zaremba and W. Kohn, Phys. Rev. B15 (1977) 1769. [16] J.E. van Himbergen and R. Silbey, Solid State Commun. 23 (1977) 623. [17] V.I. Gerasimenko, Soviet Phys-Solid State 19 (1977) 1677. [18] N. EsbJerg and J.K. Nerskov, Phys. Rev. Letters 45 (1980) 807. [I91 J.K. Norskov and N.D. Lang, Phys. Rev. 821 (1980) 2131. M.J. Stott and E. Zaremba, Phys. Rev. B22 (1980) 1564. PI 1211 D.R. Hamann, Phys. Rev. Letters 46 (1981) 1227. [221 N.D. Lang, Phys. Rev. Letters 46 (1981) 824. ~231 R.B. Laughlin, Phys. Rev. B25 (1982) 2222. Surface Sci. 119 (1982) L292. v41 J. Pereau and J. Lapujoulade, ~251 G. Armand and J.R. Manson. Surface Sci. 119 (1982) L299. WI J. Harris and A. Liebsch. J. Phys. C (Solid State Phys.) 15 (1982) 2275. v71 J. Harris and A. Liebsch, Phys. Rev. Letters 49 (1982) 341. WI A. Liebsch, J. Harris, B. Salanon and J. LapuJouIade, Surface Sci. 123 (1982) 338. 1291 A. Liebsch and J. Harris, Surface Sci. 123 (1982) 355. 30 (1983) [301 N. Garcia, J.A. Barker and I.P. Batra, J. Electron Spectrosc. Related Phenomena 137; see also N. Garcia, J.A. Barker and K.H. Rieder, Solid State Commun. 45 (1983) 567. [311 N. Garcia, J.A. Barker and I.P. Batra, Solid State Commun. 47 (1983) 485. 1321 N.D. Lang and J.K. Nsrskov, Phys. Rev. B27 (1983) 4612. [331 A. Liebsch and J. Harris, Surface Sci. 111 (1981) L721; H.O. Beckmann. J.L. Whitten and I.P. Batra, J. Vacuum Sci. Technol. A2 (1984) 1042; J.F. Annett and R. Haydock. Phys. Rev. Letters 53 (1984) 838. Surface Sci. 126 (1983) 675. 1341 P. Nordlander, and J. Harris, Surface Sci., to be published. 1351 P. Nordlander I361 1.P. Batra, P.S. Bagus and J.A. Barker, Phys. Rev., in press. [371 K.H. Rieder and T. Engel, Phys. Rev. Letters 43 (1979) 373; K.H. Rieder, Surface Sci. 117 (1982) 12. [38] K.H. Rieder, Phys. Rev. B27 (1983) 6978. [39] M.J. Cardillo, G.E. Becker, D.R. Hamann. J.A. Serri. L. Whitman and L.F. Mattheiss, Phys. Rev. B28 (1983) 494. 1401 J. Lee, J.P. Cowan and L. Wharton, Surface Sci. 130 (1983) 1. [41] J. Lapujoulade, Y. Le Cruer, M. Lefort, Y. Lejay and E. Maurel, Phys. Rev. B22 (1980) 5740; J. Lapujoulade, Y. Le Cruer, M. Lefort, Y. Lejay and E. Maurel, Surface Sci. 118 (1982) 103. 142) U. Dobler, K. Baberscke, J. Hasse and A. Puschmann, Phys. Rev. Letters 52 (1984) 1437. [43] U. Garibaldi, A.C. Levi, R. Spadacini and G.E. Tommei, Surface Sci. 48 (1975) 649. [44] R.I. Masel, R.P. Merrill and W.H. Miller, J. Chem. Phys. 65 (1976) 2690.

20

1.P. Bum

/ Heirurn - surfuce lnteractton potentutl

[45] F. Toigo, A. Marvin, V. Celli and N.R. Hill. Phys. Rev. B15 (1977) 5618. [46] G. Wolken, J. Chem. Phys. 58 (1973) 3047. [47] N. Garcia, J. Chem. Phys. 67 (1977) 897: N. Garcia and N. Cabrera, Phys. Rev. B18 (1978) 576. [48] K.L. Wolfe and J.H. Weare, Phys. Rev. Letters 41 (1978) 1663. [49] V. Celli, N. Garcia, and J. Hutchinson, Surface Sci. 87 (1979) 112. [SO] N. Garcia, V. Celli and F.O. Goodman, Phys. Rev. B19 (1979) 634. (511 G. Armand and J.R. Manson, Phys. Rev. B25 (1982) 6195; J.M. Hutson and C. Schwartz, J. Chem. Phys. 79 (1983) 5179: also, see T. Maniv and M.H. Cohen. Phys. Rev. Letters 53 (1984) 78. [52] E.M. Lifshitz, Zh. Eksperim. Teor. Fiz. 29 (1955) 94 [Soviet Phys-JETP 2 (1956) 73951. [53] W.E. Carlos and M.W. Cole, Phys. Rev. Letters 43 (1979) 697. [54] M. Manninen, J.K. Norskov, M.J. Puska and C. Umrigar. Phys. Rev. B29 (1984) 2314. [55] D. Haneman and R. Haydock, J. Vacuum Sci. Technol. 21 (1983) 330. [56] J.K. Norskov, M. Manninen and C. Umrigar, Surface Sci. 119 (1982) L393; M. Manninen, J.K. Norskov and C. Umrigar, J. Phys. F12 (1982) L7. [57] J.A. Barker, N. Garcia. I.P. Batra and M. Baumberger. Surface Sci. 141 (1984) L317. [58] K.T. Tang and J.P. Toennies, J. Chem. Phys. 66 (1977) 1496; 68 (1978) 5501; 80 (1984) 3726. [59] K.C. Ng, W.J. Meath and A.R. Allnatt, Mol. Phys. 37 (1979) 237; Chem. Phys. 32 (1978) 175. [60] G. Stoles, Ann. Rev. Phys. Chem. 31 (1980) 81. [61] J. Stohr, R. Jaeger and T. Kendelewicz. Phys. Rev. Letters 49 (1982) 142. [62] J.A. Barker and I.P. Batra, Phys. Rev. B27 (1983) 3138. [63] I.P. Batra and J.A. Barker, Phys. Rev. B29 (1984) 5286. [64] T.H. Upton and W.A. Goddard, Phys. Rev. Letters 46 (1981) 1635. [65] T.S. Rahman. J.E. Black and D.L. Mills, Phys. Rev. Letters 46 (1981) 1469; Phys. Rev. B25 (1982) 883. [66] J.E. Demuth. P.M. Marcus and D.W. Jepsen. Phys. Rev. Letters 31 (1973) 540; M.A. Van Hove and S.Y. Tong, J. Vacuum Sci. Technol. 12 (1975) 230; P.M. Marcus, J.E. Demuth and D.W. Jepsen. Surface Sci. 53 (1975) 501. [67] C.W. Bauschlicher. S.P. Walsh, P.S. Bagus and C.R. Brundle. Phys. Rev. Letters 50 (1983) 864. [68] E. Zanazzi, F. Jona, D.W. Jepsen and P.M. Marcus, Phys. Rev. B14 (1976) 432. [69] S.P. Weeks and J.E. Rowe, J. Vacuum Sci. Technol. 16 (1979) 470; S.P. Weeks and J.E. Rowe, Solid State Commun. 27 (1978) 885. [70] H.S. Greenside and D.R. Hamann, Phys. Rev. 823 (1981) 4879. [71] K. Christmann. G. Ertl and T. Pignet, Surface Sci. 54 (1976) 365. [72] J.E. Demuth, Surface Sci. 65 (1977) 369. [73] D.M. Collins and W.E. Spicer, Surface Sci. 69 (1977) 114. [74] W. Eberhard, F. Greuter and E.W. Plummer. Phys. Rev. Letters 46 (1981) 1085. [75] A.M. Bare, H. Ibach and H.D. Bruchmann, Surface Sci. 88 (1979) 384. 1761 I.P. Batra, Surface Sci. 137 (1983) L97. [77] 1.P. Batra, J.A. Barker and D.J. Auerbach, J. Vacuum Sci. Technol. A2 (1984) 943. [78] K. Christmann, R.J. Behm, G. Ertl, M.A. Van Hove and W.H. Weinberg. J. Chem. Phys. 70 (1979) 4168. [79] L. Pauling. The Nature of the Chemical Bond (Cornell University Press. Ithaca, NY, 1960). [SO] G. Ertl, Surface Sci. 6 (1972) 208. [Sl] A.G.J. de Wit, R.P.N. Bronckers and J.M. Fluit, Surface Sci. 82 (1979) 177. [82] R.P.N. Bronckers and A.G.J. de Wit, Surface Sci. 112 (1981) 111. [83] A. Spitzer and H. Luth, Surface Sci. 118 (1982) 121. 1841 E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14 (1974) 177. [85] W. Kohn, Presentation at the US-Italy Conference on Gas-Surface Interaction and Physisorption. Virginia, 1984.