Electron gas-density functional calculations of the repulsive potentials between noble gas atoms and noble metal surfaces

Chemical Physics 321 (2006) 285–292 www.elsevier.com/locate/chemphys

Electron gas-density functional calculations of the repulsive potentials between noble gas atoms and noble metal surfaces Carl Nyeland a b

a,*

, J. Peter Toennies

b

Institute of Chemistry, University of Copenhagen, Universitetsparken 5, DK 2100 Copenhagenø, Denmark Max Planck Institute for Dynamics and Self-Organisation, Bunsenstraße 10, D 37073 Go¨ttingen, Germany Received 30 April 2005; accepted 17 August 2005 Available online 25 October 2005

Abstract An electron gas-density functional approach previously used to calculate the potential between two rare gas atoms has been applied to the calculation of the interaction potential of a rare gas atom with the surface of a noble metal. The method is illustrated by calculating the repulsive potential for helium atoms interacting with copper and silver metal surfaces assuming that the surface is either made up of discrete atoms or is a jellium continuum. The results compare well with published ab initio results. The theory provides estimates of the locations of the jellium edge and of the Nørskov parameter. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Density functional theory calculations; Local density approximation of potentials; Potentials between rare gas atoms and surfaces of noble metals

1. Introduction An improved understanding of the interaction between rare gas atoms and the surfaces of metals has become increasingly important during the last decades due to the advent of new experimental techniques which have helped to elucidate the elementary processes involved [1]. Calculations of intermolecular potentials between small molecules and solid surfaces have through many years been considered to be a complicated matter. The experimental techniques have now reached a very high level of sophistication so that specific model parameters such as the location of the jellium edge in the jellium model of metals are of importance for the interpretation of particle–surface scattering experiments [2]. At the present time an accurate first principles calculation of the potentials even for simple atoms and simple jellium-like metal surfaces is a formidable task [3,4]. In *

Corresponding author. Address: Madvigs alle 16, DK-1829 Frederiksberg C, Denmark. Tel.: +45 35 32 02 17; fax: +45 35 32 03 22. E-mail address: [email protected] (C. Nyeland). 0301-0104/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2005.08.025

previous approximate calculations of molecular interactions with metal surfaces two major limiting models have been considered. In one extreme the potential between an atom and the surface is considered as a pairwise sum of the potentials between the gas atom and all the atoms of the surface lattice [5]. In the other limit the gas atom interacts with the surface modelled by a jellium metal [6–8]. Of course each of these limits have their special relevance for specific mechanisms. The detailed insight provided by effective potentials between a gas particle and the individual atoms of the surface are especially useful for understanding the inelastic interactions involved in energy exchange with the phonons of the surface [9]. The more global potential with the jellium surface is sufficient usually for describing the average interaction and in many cases for describing the elastic scattering of atoms from low index metal surfaces. These two limits are considered in the present investigation of the repulsive potential for a helium atom interacting with copper and silver surfaces. Both calculations use the same electron gas-density functional theory which incorporate all the recently developed corrections for exchange and

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˚ , 1 a.u. not otherwise stated: 1 a.u. length = 0.52917 A energy = 27.211 eV.

correlation, with the exception of the inhomogeneity corrections. The neglect of the latter is justified by several recent studies which indicate that the general non-local density corrections [10,11], in the so-called GGA model, are less important for calculations of intermolecular potentials [12–14]. The technique used is essentially analogous to that used previously for investigating the interaction of two inert gas atoms [15]. This approach is particularly convenient for a comparative study of the above two limiting models. By comparing the results from the two models the calculations can be used to estimate the location of the jellium edge, i.e. the spatial extension of the uniform positive background charge away from the surface. In one of the few previous calculations of the jellium edge position a Hartree–Fock-dispersion model potential was used [16]. The present results obtained for the jellium model also support the relationship between the charge distribution and short range repulsive potential originally suggested by Nørskov et al. [17–19]. In the electron gas-density functional (EGDF) method used here [15] the short range repulsive potential Vrep is given by the following sum of density functionals:

where r is the distance from the center of the atom. This approximation is useful in the intermediate ranges of r considered here [15]. For the parameters B and c results obtained by fitting SCF data [28,29] are listed in Table 1, both for the inert gas atoms and the metal atoms Cu and Ag considered.

V ½q
¼ V kin ½q
þ V coul ½q
þ hexch V exch ½q
þ hcorr V corr ½q
;

2.2. Jellium surfaces

ð1Þ

where Vkin is the contribution to the intermolecular potential from the Thomas–Fermi kinetic energy, Vcoul, the classical electrostatic energy, Vexch, the Dirac exchange energy contribution and Vcorr is the contribution from the local density (LDA) correlation energy. The correction factors hexch and hcorr are introduced to account for self-exchange and self-correlation as discussed in Refs. [20,21] and below. The charge densities q are approximated, as in our previous calculations [15], as the sum of the undisturbed charge distributions of the gas atom and electrons of either the metal atoms or the jellium charges at the metal surface. Further considerations of the redistribution of charges during interaction as considered in Refs. [22–24] were not taken into account. By approximating the charge distributions of the atoms and surface by exponentials the calculations can in large part be carried out analytically. In the density functional calculations considered here only interactions of closed shell systems in which the electron spins are saturated are considered. The present paper is arranged in the following way. In Section 2 the charge distributions used for the free atoms and surfaces are briefly described. In Section 3 the inert gas atom–jellium surface repulsive potential calculated with the electron gas-density functional (EGDF) method are reported. In Section 4 the atom–atom repulsive potentials of the inert gas atom–metal atom systems are calculated following the EGDF method. They are then summed over to obtain the inert gas–surface atom potentials. In the final section the results for the location of the so-called jellium edge and the Nørskov approximation are discussed in the light of these new calculations and compared with earlier ab initio results for the He–Cu and He–Ag potentials. Atomic units are used throughout when

2. Charge distributions 2.1. Atoms Despite the fact that the charge distributions for isolated atoms have been extensively studied in the past [25–27] especially by the use of an asymptotic radial wave function, the Whittaker function [27], a simpler approach is adopted here. Based on SCF results from the literature [28,29] the charge distributions for atoms are rather accurately approximated by exponential functions q ¼ B expðcrÞ;

ð2Þ

The charge distribution at a jellium surface has been calculated by Lang and Kohn [30] using the self-consistent equations of Kohn and Sham [31]. These results which are tabulated in [30] are reproduced for the present regions of interest in Table 2. The distances from the surface z (see Fig. 1) measured from the location of the edge of the positive background are given in reduced units of the Fermi wavelength n = zkF/2p. By interpolating these results the charge densities plotted as continuous lines in Fig. 2 are obtained. By the appropriate values of rS for Cu and Ag, listed in Table 3, the data show nicely an exponential fall-off in the range of n = 0.2–0.6, qj ¼ Bj expðbj zÞ ¼ Bj expðb0j nÞ;

ð3Þ

where b0j ¼ bj 2p=k F is a reduced inverse distance. The best fit values for Bj and bj for ÔCuÕ-jellium and ÔAgÕ-jellium are listed in Table 3.

Table 1 Electron distribution parameters for the free helium gas atom and the free copper and silver atoms Fit range in r(±5%)a Hec Cud Agd

2–5 3–8 3–8

SCF fit Bb

cb

1 .398 0.118 0.177

2.893 1.24 1.24

a Range in r where the deviations between the SCF values and the fit values are less than 5%. b B and c are defined in Eq. (2). c Results for a fit of SCF calculations from [28]. d Results for a fit of SCF calculations from [29].

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287

Table 2 The reduced electron charge density in front of a jellium surface and the Coulomb potentials / 0 used in the present calculations are tabulated as a function of the reduced distance n for three different values of rs, the radius of the Wigner sphere n

0.200 0.300 0.400 0.500 0.600

rs = 2

rs = 3

rs = 4

qðnÞa q0

/ 0 (n)b

qðnÞa q0

/ 0 (n)b

qðnÞa q0

/ 0 (n)b

0.1194 0.556 (1) 0.251 (1) 0.112 (1) 0.49 (2)

0.666 (1) 0.297 (1) 0.131 (1) 0.57 (2) 0.24 (2)

0.784 (1) 0.286 (1) 0.99 (2) 0.33 (2) 0.11 (2)

0.377 (1) 0.128 (1) 0.42 (2) 0.14 (2) 0.4 (3)

0.533 (1) 0.156 (1) 0.42 (2) 0.10 (2) 0.2 (3)

0.228 (1) 0.59 (2) 0.14 (2) 0.3 (3) 0.1(3)

a 1 q0 ¼ 43 pr3s , where q0 is the bulk electron density. The data for q(n)/q0 were calculated using the Kohn–Sham density functional theory as reported in Table 1 of [30]. b / 0 is defined as /(n)  /(1) in units of EF. The data for /(n) were also taken from Table 1 of [30].

from a best fit of the Lang and Kohn tabulations [30] (Table 2). 3. The density functional energy terms of the jellium model 3.1. The Coulomb energy term In the jellium model the Coulomb potential per unit charge is taken directly from the Kohn–Sham density functional calculation reported in [30] where it is given as /(n) in units of EF, the Fermi energy. Beyond the region of the positive charge the potential falls off with distance very close to exponential, Fig. 1. The solid surface–atom interaction model. ZA is the distance between the atom A and the topmost plane of surface atoms. R is the distance between the atom A and a specific surface atom and zA between the atom A and the jellium edge in the atom–jellium interaction model. zb is the distance between the surface plane and the jellium edge. It is seen that ZA = zb + zA. The coordinates centered on the atom A are given as (r, h, u) or (x, y, z  zA), where x and y are parallel to the jellium surface.

Dondi et al. [32] have also fitted the data of Lang and Kohn using the following expression: h i1=2 2W W þEF qðzÞ ¼ expð2k W zÞ; ð4Þ q0 ½1 þ k F z=9
2 where EF is the Fermi energy of the bulk, W is the work function, kW = (2W)1/2 is an effective work function wavevector and q0 is the bulk density. Calculations based on Eq. (4) are also shown in Fig. 2 as dashed lines for comparison. Another slightly different formula for the electron density has been suggested by Celli [33]: qðzÞ 1=2 ¼ B1 expðð8ðW þ DÞÞ zÞ; q0

ð5Þ

where the energy shift D accounts for the contributions of electrons from below the Fermi energy to the electron spillout. Values for the energy shift D listed in Table 3 were obtained by equating bj from Eq. (3) with 8(W + D)1/2 using the work functions listed in Table 3. The simple exponential form of Eq. (3) was used with parameters obtained

/0 ðzÞ ¼ /0 ðzA Þ expðbc ðz  zA ÞÞ;

ð6Þ

where / 0 (z) = /(z)  /(1) is the Coulomb potential defined as / 0 (1) = 0, and where zA is the location of A with respect to the jellium edge (see Fig. 1). From the best fit of the data in Table 2 the fall-off parameters bc listed in Table 4 are obtained. For the total electrostatic energy between a neutral atom (A) with an electron charge distribution qA(r) and the jellium surface, taking both the charge of the electrons and of the nucleus into account, we have Z V coul ¼ EF ð/0 ðzÞ  /0 ðzA ÞÞqA ðrÞd3 r. ð7Þ Since both distributions in Eq. (7) are exponential the equation can be integrated analytically
 8pBA EF /0 ðzA Þ c3A 2 2 V coul ¼ ððbc  cA Þ  ðbc þ cA Þ Þ  1 4bc c3A ð8Þ and fitted to an exponential with the same fall-off parameter bc V coul ffi C expðbc zA Þ.

ð9Þ

The best fit values of C are listed in Table 4. 3.2. The kinetic energy term and the exchange energy term In the present calculations the charge distribution q in Eq. (1) is approximated by the sum of the charge distributions of the separated systems [15],

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Fig. 2. Comparison of the reduced electronic charges q/q0 as a function of the reduced distance n = zkF/2p, with respect to the edge of the positive background based on parameters in Tables 2 and 3 (full line) with reduced charges obtained from Eq. (4) for EF and W from Table 3 (broken line).

Table 3 Jellium and jellium surface charge distribution parameters for Cu and Ag

Cu Ag

rs[au]a

EF [eV]a

Bjb[a.u.]

bj[a.u.]b

W [eV]c

D [eV]d

2.67 3.02

7.00 5.49

0.125(1) 0.709(2)

1.255 1.121

3.64 3.50

1.71 0.76

for which the Thomas–Fermi kinetic energy contribution to the potential Eq. (11), can be written as V kin ¼ C kin 2p½qj ðzA Þ


5=3

Að5=3; bj ; cA ; cÞ.

ð14Þ

Here

a

Data obtained from [34]. b Best fit values of the jellium density data reported in [30] and listed in Table 2 and plotted in Fig. 2 of the present article. c The jellium results for W were taken from [35]. d Calculated from a best fit to the data in Table 2 using Eq. (5) together with the values of W in the preceding column. Table 4 Best fit exponential potential parameters for the Coulomb energy in front of a jellium surface

He–Cu He–Ag a

Ca

bca

0.960 (1) 0.384 (1)

1.26 1.12

For definitions see Eq. (9).

qeff ¼ qj þ qA ;

ð10Þ

to obtain an effective charge distribution of the composite system. In the local density approach (LDA) the kinetic energy contribution is, according to Thomas–Fermi [15,20], given by Z h i 5=3 5=3 5=3 V kin ¼ C kin qeff  qj  qA d3 r; ð11Þ where C kin ¼ 103 ð3p2 Þ2=3 . The Dirac exchange energy contribution to the potential is given by [15,20] Z h i 4=3 4=3 4=3 V exch ¼ C exch qeff  qj  qA d3 r; ð12Þ  1=3 where C exch ¼  34 p3 . In the coordinates of Fig. 1 the charge distribution of Eq. (10) is conveniently expressed as qeff ¼ qj ðzA Þebj r cos h þ BA ecA r

ð13Þ

c ¼ BA =qj ðzA Þ

ð15Þ

and Aðp;bj ;cA ;cÞ Z Z  b rt  p ¼ ðe j þ cecA r Þ  epbj rt  cp epcA r r2 dtdr;

ð16Þ

where t = cosh, p = 5/3, and the integral over t extends from 1 6 t 6 1. With A(p, bj, cA, c) defined in Eqs. (16) and (12) leads to the following result for the Dirac exchange energy contribution to the potential 4=3

V exch ¼ C exch 2p½qj ðzA Þ


Að4=3; bj ; cA ; cÞ.

ð17Þ

As defined in Eq. (1) the exchange energy is weighted with a prefactor hexch to account for the self-exchange effect [21]. In the following the exchange energy prefactor for He–metal surface interactions was obtained by use of a combining rule proposed by Ihm and Cole [36]. A value of hexch = 0.36 was obtained. For the correlation energy prefactor hcorr = 0.50 was used, as justified in our earlier publication [15]. 3.3. The correlation energy term In the LDA approach the correlation contribution to the potential energy is given by Z V corr ¼ ½qeff e0LDA ðqeff Þ  qj e0LDA ðqj Þ  qA e0LDA ðqA Þ
d3 r;

ð18Þ

which was calculated numerically using the LDA function of [20], based on results from [37,38].

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289

3.4. The atom–jellium repulsive potentials The total short range repulsive potential denoted as V jA rep is obtained from Eq. (1) with the prefactors hexch and hcorr given in Section 3.2. V jA rep ¼ V kin þ V coul þ 0:36V exch þ 0:50V corr .

ð19Þ

The results for the four electron gas terms for the systems He atom–ÔCuÕ jellium and He atom–ÔAgÕ jellium and V jA rep for a few selected values of zA are listed in Table 5. They are also plotted in Figs. 3 and 4 for comparison with earlier ab initio results for the same He atom–jellium surface systems. From Figs. 3 and 4 it is seen that the potentials can be accurately approximated by a simple exponential distance dependence aja zA V jA . rep  U jA e

ð20Þ

The corresponding best fit parameters for UjA and ajA are listed in Table 7. 4. The density functional energy terms for the atom–atom model. The estimation of the jellium edge In the jellium model of metals the positive charge ends abruptly at a fixed distance from the top layer of atoms called the jellium edge. This distance has been shown to be accurately approximated by half the layer spacing of the crystal (Fig. 1) [7]. For a real surface the jellium edge is expected to deviate from this prediction. Since the jellium potential derived here is defined with respect to the edge of the positive background, whereas the pairwise addition model is defined with respect to the top layer of atoms a comparison of the two potentials can be used to estimate an effective jellium edge zb. The procedure used for the atom–atom potential was exactly the same as in the earlier work [15]. Results for the potential between a single metal atom and an inert gas atom V MA rep ðRÞ following Eq. (1) are listed in Table 6 and also plotted in Figs. 3 and 4 where the potential energy is

Fig. 3. Comparison of the jellium total short range potential for He–Cu from the present calculations (full line) with previous He atom–jellium potentials (dotted line) [8], (broken line) [6]. A comparison of the corresponding atom–atom potentials (full line) with previous SCF calculations (O) [39] is also shown.

on the right ordinate. Due to the uncertain corrections for self-correlation [20] the results for V MA rep ðRÞ are reliable only for R < 6 a.u. In these atom–atom calculations the same correction prefactors hcorr = 0.50 and hexch = 0.36 were used as in the atom–jellium model calculations. Again the resulting potentials have to a good approximation a simple exponential form aMA R V MA . rep ðRÞ  U MA e

ð21Þ

The best fit parameters UMA and aMA are listed in Table 7. Celli [40] has shown that an exponential potential can be summed in a straightforward way over a single monolayer

Table 5 Calculated terms for the repulsive He atom–jellium metal surface potential, Eq. (19), for a Cu and an Ag jellium metala,b Vcoul

Vkin

Vexch

Vcorr

c V jA rep

He–ÔCuÕ (rs = 2.67) 2 0.1014(2) 3 0.2893(3) 4 0.8256(4) 5 0.2356(4) 6 0.6723(5)

0.7724(2) 0.2191(2) 0.6215(3) 0.1763(3) 0.5000(4)

0.2873(1) 0.9318(2) 0.2960(2) 0.9224(3) 0.2841(3)

0.1562(1) 0.6207(2) 0.2397(2) 0.9034(3) 0.3370(3)

0.2344(2) 0.1114(2) 0.5166(3) 0.2336(3) 0.1024(3)

0 .1421(1) 0.4335(2) 0.1217(2) 0.3041(3) 0.6158(4)

He–ÔAgÕ (rs = 3.02) 2 0.7550(3) 3 0.2463(3) 4 0.8035(4) 5 0.2622(4) 6 0.8554(5)

0.4088(2) 0.1334(2) 0.4352(3) 0.1420(3) 0.4633(4)

0.2029(1) 0.7309(2) 0.2559(2) 0.9040(3) 0.3121(3)

0.1180(1) 0.5065(2) 0.2215(2) 0.8715(3) 0.3518(3)

0.1875(2) 0.9507(3) 0.4697(3) 0.2264(3) 0.1064(3)

0.1102(1) 0.3676(2) 0.1092(2) 0.3351(3) 0.8592(4)

zA

a b c

qj(zA)

The data for densities and for the Coulomb term are from the Tables 1,4. The theory used is discussed in Section 3. V jA rep is defined in Eq. (19).

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C. Nyeland, J.P. Toennies / Chemical Physics 321 (2006) 285–292 Table 7 MA Coefficients for the energy V jA rep and V rep for the intermolecular potentials. The parameters a and U were obtained from fits of the data in Figs. 3 and 4 zA

ajA

UjA

He–Cu 3 4 5

1.187 1.270 1.387

0.153 0.196 0.312

He–Ag 3 4 5

1.098 1.214 1.181

0.099 0.140 0.123

R

aMA

UMA

He–Cu 4 5 6

1.086 1.161 1.234

0.764 1.031 1.486

He–Ag 4 5 6

1.072 1.141 1.223

1.062 1.402 2.092

Fig. 4. Comparison of the total short range potentials for He–Ag. The notation is the same as in Fig. 3.

to give the following analytical expression for the laterally averaged surface rare gas atom potential [40,5] V SA rep ðZ A Þ ¼

2p ð1 þ aMA Z A ÞU MA eaMA Z A ; a2MA Ac

ð22Þ

where ZA is the distance between the inert gas atom and a plane passing through the top layer of atoms (Fig. 1) and Ac is the surface area per surface atom. Since aMAd  4, where d is the distance between layers the contribution from deeper lying layers contributes only about 4 per cent and can be neglected. The following values for the lattice ˚ ) for Cu(0 0 1) constant [41], were used: 6.818 a.u. (3.608 A ˚ and 7.706 a.u. (4.078 A) for Ag(0 0 1),which for (0 0 1)

structures are twice the layer spacing. These lattice constants were also used to calculate the following surface ˚ 2) for areas per surface atom: Ac = 23.2 a.u. (6.509 A 2 ˚ Cu(0 0 1) and Ac = 29.7 a.u. (8.315 A ) for Ag(0 0 1). To estimate the size of zb the potentials obtained from the jellium model are compared with those obtained from the gas atom–laterally averaged surface atom potentials. jA V SA rep ðZ A Þ  V rep ðzA Þ.

ð23Þ

Since ZA = zA + zb (Fig. 1) this equation provides an implicit relationship for determining zb. Since as found in Section 3 the jellium model potentials are also close to exponential, Eq. (20), one gets from Eqs. (20)–(23) without further approximations that

Table 6 Calculated terms for the repulsive He atom–metal atom potential (Eq. (1)) for a Cu atom and an Ag atoma,b R

Vcoul

Vkin

Vexch

Vcorr

c V MA rep

He–Cu 2 3 4 5 6

0.2979(1) 0.1316(1) 0.4190(2) 0.1260(2) 0.3682(3)

0.1386(0) 0.5297(1) 0.1868(1) 0.6276(2) 0.2046(2)

0.5335(1) 0.2457(1) 0.1045(1) 0.4224(2) 0.1650(2)

0.5808(2) 0.3178(2) 0.1613(2) 0.7783(3) 0.3587(3)

0.8670(1) 0.2938(1) 0.9921(2) 0.3106(2) 0.9045(3)

He–Ag 2 3 4 5 6

0.4469(1) 0.1974(1) 0.6285(2) 0.1890(2) 0.5523(3)

0.1983(0) 0.7638(1) 0.2706(1) 0.9155(2) 0.2990(2)

0.7149(1) 0.3323(1) 0.1426(1) 0.5815(2) 0.2291(2)

0.7290(2) 0.4037(2) 0.2072(2) 0.1009(2) 0.4722(3)

0.1242(0) 0.4266(1) 0.1461(1) 0.4668(2) 0.1377(2)

a b c

The data used for densities are taken from Table 1. The theory used is discussed in Section 4, and in [15]. V MA rep as defined from Eq. (1).

C. Nyeland, J.P. Toennies / Chemical Physics 321 (2006) 285–292

zb ¼ ðajA Þ

1

 ln

   2pU MA aMA ð1 þ a Z Þ þ 1  ZA. MA A ajA a2MA U jA Ac ð24Þ

The jellium edge location zb can now be obtained from jA exponential fits to results for V MA rep and V rep in Figs. 3 and 4. As a consequence of the slight deviation from exponential behaviour for the two repulsive potentials, one has to be careful when using potential parameters for both potentials in the same calculation, as for instance in calculations of zb from Eq. (24). From Eqs. (21) and (22) it is seen that the major part of V SA rep ðZ A Þ is from the atom–atom interactions where R is close to ZA, which means that the assumpMA tion in Eq. (23) also implies V jA rep ðzA Þ  V rep ðRÞ. To account for the fact that the potential parameters in Eqs. (20) and (21) depend slightly on the actual values of zA or R, respectively they were chosen according to the region under consideration. The results for zb are tabulated for two values of ZA in Table 8. 5. Discussion and comparisons with other computational results There are only a few explicit results in the literature for comparison with the present atom–jellium repulsive potentials. The results of Nordlander and Harris [6], which are based on a modification of the original Zaremba and Kohn method [7], and the recent results of Chizmeshya and Zaremba [8], based on a scattering theory, are compared with the present results in Figs. 3 and 4. The absolute potentials agree to better than 20% for zA 6 5 a.u. and the slopes are identical. From the SCF calculation by Batra et al. [39] for He–Cu the five points (Fig. 3, upper part) all lie about 30% higher but the slopes are nearly identical. This sort of agreement can be considered quite satisfactory. In 1980 Esbjerg and Nørskov [17] showed that the repulsive potential is approximately proportional to the local electron density in front of the surface. In Table 9 the ratios R ¼ V jA rep =qj ðzA Þ for He–ÔCuÕ and He–ÔAgÕ are listed for a range of distances zA from the surface. Since R depends somewhat on zA—it tends to decrease only slightly with increasing zA—an average ratio was calculated. It is gratifying that the average value of R = 13.2 a.u. ± 15% agrees nicely with the recommended value of 12.1 a.u. or 329 eV [18]. One of the prime objectives of the present study was to gain some insight into the displacement of the jellium edge from the top layer of surface atoms (Fig. 1). The values of Table 8 Results obtained for the jellium edge distance zb

291

Table 9 jA The repulsive potential V jA rep and the ratio R ¼ V rep =qj ðzÞ for the atom– jellium interaction model zA

a V jA rep

R ¼ V jA rep =qj ðzA Þ

He–ÔCuÕ 2 3 4 5 6 Average

0.1421(1) 0.4335(2) 0.1217(2) 0.3041(3) 0.6158(4)

14.0 15.0 14.7 12.9 9.2 13.2 ± 2.1

He–ÔAgÕ 2 3 4 5 6 Average

0.1102(1) 0.3676(2) 0.1092(2) 0.3351(3) 0.8592(4)

14.6 14.9 13.6 12.8 10.0 13.2 ± 1.8

a

V jA rep , LDA results of Table 5.

between 1.98 and 2.08 obtained for He–ÔCuÕ and between 2.30 and 2.32 for He–ÔAgÕ( Table 8) are somewhat larger than values obtained from the most often used relationship zb = 0.5d which predicts zb = 1.704 for Cu(0 0 1) and zb = 1.927 for Ag(0 0 1). The only experimental value is zb = 1.20 ± 0.38, which was obtained from an elegant interference experiment involving a clean and a CO coated Cu(0 0 1) surface [2]. There are only two theoretical values for He–Cu. Lenarcic-Poljanec et al. [16] obtained a value of zb = 1.697 from SCF-HF calculations while Chizmeshya estimated zb  2 [42]. 6. Conclusion The method of the electron gas-density functionals for estimations of intermolecular potentials was extended to repulsive interactions of inert gas atoms with surfaces of noble metals. This easily used method appears to give results very close to results obtained by the more complicated Hartree–Fock and Kohn–Sham methods at least for the He atom-metal surface systems considered. Finally a contribution to the present knowledge of the location of the jellium edges was presented. Acknowledgement This project was supported in part by the Danish Natural Science Research Council. References

ZA

zb

He–Cu 5 6

1.98 2.08

He–Ag 5 6

2.30 2.32

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