The repulsive part of the helium scattering potential at a surface is approximately proportional to the surface electron density. The proportionality coefficient is shown to be a well-defined quantity, which can be related to the electron-helium scattering length. The spread in the values of the proportionality constant suggested in the literature is shown to be due to different definitions of the coefficient or due to inadequate calculational methods. The value calculated using the local density approximation with a self-interaction correction is in very good agreement with the electron-scattering-length measurements.Peer reviewe

Manninen, M.

Nørskov, J. K.

Puska, Martti J.

Umrigar, Cyrus

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This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Manninen, M. & Nørskov, J. K. & Puska, Martti J. & Umrigar, Cyrus
Title: Repulsive interaction of the helium atom with a metal surface
Year: 1984
Version: Final published version
Please cite the original version:
Manninen, M. & Nørskov, J. K. & Puska, Martti J. & Umrigar, Cyrus. 1984. Repulsive
interaction of the helium atom with a metal surface. Physical Review B. Volume 29, Issue
4. 2314-2316. ISSN 1550-235X (electronic). DOI: 10.1103/physrevb.29.2314.
Rights: © 1984 American Physical Society (APS). This is the accepted version of the following article: Manninen, M.
& Nørskov, J. K. & Puska, Martti J. & Umrigar, Cyrus. 1984. Repulsive interaction of the helium atom with a
metal surface. Physical Review B. Volume 29, Issue 4. 2314-2316. ISSN 1550-235X (electronic). DOI:
10.1103/physrevb.29.2314, which has been published in final form at
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.29.2314.
All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, and
duplication or sale of all or part of any of the repository collections is not permitted, except that material may
be duplicated by you for your research use or educational purposes in electronic or print form. You must
obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or
otherwise to anyone who is not an authorised user.
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PHYSICAL REVIEW B VOLUME 29, NUMBER 4
Repulsive interaction of the helium atom with a metal surface
15 FEBRUARY 19S4
M. Manninen
Research Institute for Theoretical Physics, University of Helsinki,
SF-00170 Helsinki 17, Finland
J. K. Nerskov
Nordisk Institut for Teoretisk Atomfysik (NORDITA), Blegdamsvej 17,
DK-2100 Copenhagen g, Denmark
M. J. Puska
Laboratory of Physics, Helsinki University of Technology,
SF-02150 Espoo 15, Finland
Cyrus Urnrigar
Laboratory of Atomic and Solid State Physics, Cornell University,
Ithaca, Net York 14853
(Received 25 October 1983)
The repulsive part of the helium scattering potential at a surface is approximately proportional to the sur-
face electron density. The proportionality coefficient is shown to be a well-defined quantity, which can be
related to the electron-helium scattering length. The spread in the values of the proportionality constant
suggested in the literature is shown to be due to different definitions of the coefficient or due to inade-
quate calculational methods. The value calculated using the local density approximation with a self-
interaction correction is in very good agreement with the electron-scattering-length measurements.
Using the effective-medium theory, ' Esbjerg and
Nrskov showed that the leading repulsive term in the heli-
um scattering potential from any electronic system is pro-
portional to the local unperturbed electron density np( r ) of
the host at the helium site:
VR( r ) =nnp( r )
A more refined treatment" gives
Vs( r ) =n, ffnp( r ) (2)
where n p is the average of n p( r ) over the electrostatic po-
tential C&H, ( r ) of the helium atom, and
A ff=a —J d r@ (Hr) (3)
'
is a new effective proportionality constant. Unfortunately,
the value of n published in the original paper was wrong
due to numerical errors. This together with the occurrence
of two constants o. and a,ff and other approaches for
determining the coefficient has resulted in several values for
n and has given an impression that theoretically o. is not
well defined or that the correct value is not known. This
confusion is also increased by the fact that, keeping n as a
free parameter, good agreement with the experimental
results has been achieved for helium diffraction from metal
surfaces. '
The purpose of the present Rapid Communication is to
comment upon the differences between various values of
the proportionality constant. %e point out that n, ff is relat-
ed to the scattering length for slow electrons on atomic heli-
urn. It can thus be determined experimentally. The o.,ff
used in the effective-medium theory is in good agreement
with the experimentally determined value.
The density functional theory' expresses the ground-state
energy as a functional of the tota1 electron density of the
system. In the effective-medium theory one goes a step
further and shows that the embedding energy of the helium
atom in any electronic surrounding can be calculated as a
functional of the unperturbed electron density of the host.
Formally, one can always separate a local term from this
functional and write
VR( r ) =bE(np( r )) +bE I
where bE(np) is the embedding energy of helium in a
homogeneous electron gas of density np, np( r ) is the un-
perturbed surface electron density, and bE„] includes all
nonlocal terms. b,E(np) is calculated by embedding the
helium atom in a homogeneous electron gas with a compen-
sating positive background charge. 9 In the low-density limit
this gives b.E(n) proportional to the density and thereby
Eq. (1). We have calculated bE(n) for low electron densi-
ties using the local density approximation'0 (LDA) for the
exchange and correlation energies with and without a self-
interaction correction" (SIC). The corresponding slopes n
are given in Table I.
In the region outside a metal surface sampled by a low-
energy ( ( 500 meV) helium atom there are no positive
charges. Also, the surface electron density has non-
negligible gradients. In the refined treatment, Eqs. (2) and
(3), the former problem is taken into account by subtracting
from Eq. (1) the electrostatic interaction of the helium atom
with the homogeneous background [Eq. (3)]. This results
in the new proportionality constant n, rr in Eq. (2), which is
also given in Table I." The use of an averaged surface elec-
tron density n p( r ) in Eq. (2) is the lowest-order inclusion
of nonlocality. Outside a metal surface we typically find
n p( r ) = 1.3 np( r ) fairly independent of r for the range
of positions of relevance for low-energy helium scattering.
This means that we can define a third proportionality con-
stant
'Jeff= (n 0( r )/np( r ) ]jeff
2314 1984 The American Physical Society
REPULSIVE INTERACTION OF THE HELIUM ATOM WITH A METAL SURFACE 2315
TABLE I. Proportionality constants for the helium-metal scattering potential.
n (eVap ) A ff {eVap3 ) 0,' ff {eVao3 )
Effective-medium approach
LDA
LDA-SIC
Harris-Liebsch approach
First-order (Ref. 5)
Second-order (Ref. 12)
Scattering length
Calculation (Ref. 13)
Experimental (Ref. 14)
305
329
160
196
203
191, . . . , 203
208
255
1000
—500
so that
Vn( r )=n rrno( r )
o.' ff is also included in Table I. It should be stressed that
(x ff/A rf depends on the kind of system considered and that
6 ff in Table I is only applicable for helium-metal surface in-
teractions over a limited range of distances.
Following Zaremba and Kohn, ' Harris and Liebsch' have
shown that, to first order in the overlap between the surface
and helium states, the repulsive part of the helium-surface
potential can be written
seen how the value cx,ff differs dramatically from the effec-
tive medium value, which is rather independent of the ap-
proximation used for the treatment of exchange and correla-
tion effects. Most of this difference is found again in o.,ff.
Now we want to point out that there is a more direct way
of determining the coefficient n, ff. In the extreme low-
density limit the interaction of electrons with a helium atom
can be calculated from scattering theory. For X electrons in
a volume V (without the positive background) the embed-
ding of a helium atom increases the total energy of the elec-
trons by
p4F
VR( r ) = „~ deg(e)no(e, r ) (6) ~E 4mb2m V (7)
where g(e) is a universal function of energy only, and
np(e, r ) is the local density-of-states function of the bare
surface. Formally Eq. (6) can be written
where a, is the scattering length. A comparison with Eq.
(1) gives directly
Vs( r ) =g(e)no( r ) 2mb&eff asm (8)
where ~ in principle depends on r . We have made explicit
calculations within the local density approximation, using
the linear augmented plane-wave method for a five-layer
Ni(110) slab, '7 which show that in reality e is basically con-
stant for the range of distance of interest here [no(e, r ) is
dominated by energies close to the Fermi energy]. This has
also been argued by Harris and Liebsch based on jeliium-
surface calculations. The Harris-Liebsch value for n, ff is
thus g (e) and again n, rr will depend on the kind of system
studied. For a low-density homogeneous electron gas the
Harris-Liebsch approach gives n,rr=g(0). The Harris-
Liebsch values for n, ff and n, ff calculated within the
Hartree-Pock approximation are shown in Table I. In the
table we include both the originally published values (first
order in the helium pseudopotential) and the most recent
ones by Nordlander and Harris. ' The latter are calculated
to second order in the helium pseudopotential and the ener-
gy argument in g (e) is replaced by e —up( I' ), where up( r )
is the host one-electron potential at the helium position. "
The Harris-Liebsch approach is not designed for the prob-
lem of helium in a uniform electron gas, and the perturba-
tion theory which involves a polarizability does not converge
well' as the electron gas density and bandwidth go to zero.
%e therefore have not included any second-order value for
A ff in Table I (treated in all orders of the perturbation
theory the Harris-Liebsch approach would lead to the exact
value extracted from the scattering length; see below). It is
This approximation for the electron-helium interaction has
conventionally been used in studying the electron-induced
bubbles in liquid helium (see Ref. 19 and references
therein). The advantage is that the scattering length can be
computed by solving a three-electron problem and a more
elaborate many-body theory can be applied. The o.,ff values
determined from theoretical and experimental scattering
lengths are included in Table I. A comparison shows that
the LDA gives good results and LDA-SIC give excellent
results for the helium interaction with the low-density elec-
tron gas. The results of Table I do, however, indicate some
problems with the Harris-Liebsch value for o.,ff. There are
two possible explanations of the too-large value. It may be
that the expansion in the helium-metal overlap or the per-
turbation theory in the helium pseudopotential is not con-
verged. The other possibility is the difference in the treat-
ment of exchange and correlation effects between the
Hartree-Fock (HF) and LD approximations used in the two
approaches. In the HF approximation of the Harris-Liebsch
approach, ' no correlation effects are included, whereas in
the LD approximation and, in particular, when the self-
interaction correction is included, correlation effects are
treated to a certain approximation. The LDA value of cx,ff
is therefore less repulsive than the Hartree-Pock value. If
the too-large value of a,ff in the Harris-Liebsch-Nordlander
treatment is due to a lack of correlation effects in the HF
treatment, this need not invalidate the approach for calculat-
2316 MANNINEN, NORSKOV, PUSKA, ANI3 UMRIGAR 29
ing the He-surface interaction. All correlation effects are in
principle included in the zeroth order contribution to the
interaction potential in the overlap expansion which is the
van der %aals attraction. In praxis the asymptotic form of
the van der Waals interaction is added to the HF repulsive
potential Eq. (6). Such a procedure gives, for instance, a
reasonable description of the physisorption minimum when
compared to experiment. In the effective-medium ap-
proach, on the other hand, the exchange-correlation energy
of the helium atom in a homogeneous electron gas is al-
ready included in the repulsive part V~ of the potential, and
only the extra exchange-correlation energy due to the inho-
mogeneity of the surface electron density must be added.
Model calculations for jellium surfaces show that this is a
small effect which is important only far from the surface,
where Vg is small. With the inclusion of the extra
exchange-correlation energy the effective-medium approach
provides a good description of the helium-jellium-surface
potential including the physisorption minimum, when com-
pared to a fully self-consistent calculation within the local
density approximation.
Providing the overlap expansion is converged, the choice
of the value for the proportionality constant a,ff in the
repulsive part of the helium-surface potential thus depends
on the choice of the treatment of the exchange and correla-
tion effects. If the asymptotic form of the van der Waals
interaction is included, the corresponding large (HF) value
of n, ff must be used. This description is clearly the most
appropriate far from the surface. Closer in, care must be
taken with the divergence of the asymptotic van der %aals
form. ' If a local density description of the exchange and
correlation is chosen, the small value of n, qf should be used
and only the extra exchange-correlation energy due to the
inhomogeneity of the surface electron density added. This
correction is small. The local density description is expected
to work better the closer the He atom is to the surface.
ACKNOVf LEDGMENTS
Discussions with J. Harris, N. Lang, R. M. Nieminen, and
P. Nordlander are gratefully acknowledged. %e are grateful
to P. Nordlander for providing us with his results before
publication. One of us (C.U.) is grateful to the Office of
Naval Research for financial support.
'J. K. Nerskov and N. D. Lang, Phys. Rev. 8 21, 2131 (1980).
2M. J. Stott and E. Zaremba, Phys. Rev. 8 22, 1564 (1980).
3N. Esbjerg and J. K. Nerskov, Phys. Rev. Lett. 45, 807 (1980).
4N. D. Lang and J. K. Nerskov, Phys. Rev. 8 27, 4612 (1983).
5J. Harris and A. Liebsch, J. Phys. C 15, 2275 (1982).
6R, 8. Laughlin, Phys. Rev. B 25, 2222 (1982).
R. Schinke and A. C. Luntz, Surf. Sci. Lett. 124, L60 (1983).
P. Hohenberg and W. Kohn, Phys. Rev. 136, 8864 (1964);
W. Kohn and L. J. Sham, ibid. 140, A1133 (1965).
M. J. Puska, R. M. Nieminen, and M. Manninen, Phys. Rev. 8 24,
3037 (1981).
' O. Gunnarsson and B. I. Lundqvist, Phys. Rev. 8 13, 4274 (1976).
"J.Perdew and A. Zunger, Phys. Rev. 8 23, S048 (1981).
' P. Nordlander and J. Harris (unpublished).
' N. R. Kestner, J. Jortner, M. H. Cohen, and S. A. Rice, Phys.
Rev. 140, A56 (196S).
' T. F. O' Malley, Phys. Rev. 130, 1020 {1963).
' In principle, the integral in Eq. (3) and in the definition of no is
only over the region close to the He atom where the atomic po-
tential dominates (Ref. 4). In this region the He-induced electron
density in a homogeneous electron gas is approximated very well
by that of the free atom. Furthermore, for this He density, V&
of Eq. (2) is basically independent of the choice of the radius R,
of the near-atom region for 8, )2.S ao, even though e,ff and
na( r )/nc( r ) do depend slightly on 8, (Ref. 4). Here we have
used R, =~.
' E. Zaremba and W. Kohn, Phys, Rev. 8 15, 1769 (1977).
M. Manninen, J. K. Nerskov, and C. Umrigar (unpublished).
' P. Nordlander (private communication).
R. M. Nieminen, I. Valimaa, M, Manninen, and P, Hautoj'arvi,
Phys. Rev. A 21, 1677 (1980),
2 The second-order nonlocal term in the effective-medium theory
involves the polarizability of the helium atom in a homogeneous
electron gas. This term has been calculated for helium outside a
metal surface and it is very small, indicating that the effective-
medium theory is converged with the first-order nonlocal term.


English

Repulsive interaction of the helium atom with a metal surface

Aaltodoc Publication Archive

This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Manninen, M. & Nørskov, J. K. & Puska, Martti J. & Umrigar, CyrusTitle: Repulsive interaction of the helium atom with a metal surfaceYear: 1984Version: Final published versionPlease cite the original version:Manninen, M. & Nørskov, J. K. & Puska, Martti J. & Umrigar, Cyrus. 1984. Repulsiveinteraction of the helium atom with a metal surface. Physical Review B. Volume 29, Issue4. 2314-2316. ISSN 1550-235X (electronic). DOI: 10.1103/physrevb.29.2314.Rights: © 1984 American Physical Society (APS). This is the accepted version of the following article: Manninen, M.& Nørskov, J. K. & Puska, Martti J. & Umrigar, Cyrus. 1984. Repulsive interaction of the helium atom with ametal surface. Physical Review B. Volume 29, Issue 4. 2314-2316. ISSN 1550-235X (electronic). DOI:10.1103/physrevb.29.2314, which has been published in final form athttp://journals.aps.org/prb/abstract/10.1103/PhysRevB.29.2314.All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)PHYSICAL REVIEW B VOLUME 29, NUMBER 4Repulsive interaction of the helium atom with a metal surface15 FEBRUARY 19S4M. ManninenResearch Institute for Theoretical Physics, University of Helsinki,SF-00170 Helsinki 17, FinlandJ. K. NerskovNordisk Institut for Teoretisk Atomfysik (NORDITA), Blegdamsvej 17,DK-2100 Copenhagen g, DenmarkM. J. PuskaLaboratory of Physics, Helsinki University of Technology,SF-02150 Espoo 15, FinlandCyrus UrnrigarLaboratory of Atomic and Solid State Physics, Cornell University,Ithaca, Net York 14853(Received 25 October 1983)The repulsive part of the helium scattering potential at a surface is approximately proportional to the sur-face electron density. The proportionality coefficient is shown to be a well-defined quantity, which can berelated to the electron-helium scattering length. The spread in the values of the proportionality constantsuggested in the literature is shown to be due to different definitions of the coefficient or due to inade-quate calculational methods. The value calculated using the local density approximation with a self-interaction correction is in very good agreement with the electron-scattering-length measurements.Using the effective-medium theory, ' Esbjerg andNrskov showed that the leading repulsive term in the heli-um scattering potential from any electronic system is pro-portional to the local unperturbed electron density np( r ) ofthe host at the helium site:VR( r ) =nnp( r )A more refined treatment" givesVs( r ) =n, ffnp( r ) (2)where n p is the average of n p( r ) over the electrostatic po-tential C&H, ( r ) of the helium atom, andA ff=a —J d r@ (Hr) (3)'is a new effective proportionality constant. Unfortunately,the value of n published in the original paper was wrongdue to numerical errors. This together with the occurrenceof two constants o. and a,ff and other approaches fordetermining the coefficient has resulted in several values forn and has given an impression that theoretically o. is notwell defined or that the correct value is not known. Thisconfusion is also increased by the fact that, keeping n as afree parameter, good agreement with the experimentalresults has been achieved for helium diffraction from metalsurfaces. 'The purpose of the present Rapid Communication is tocomment upon the differences between various values ofthe proportionality constant. %e point out that n, ff is relat-ed to the scattering length for slow electrons on atomic heli-urn. It can thus be determined experimentally. The o.,ffused in the effective-medium theory is in good agreementwith the experimentally determined value.The density functional theory' expresses the ground-stateenergy as a functional of the tota1 electron density of thesystem. In the effective-medium theory one goes a stepfurther and shows that the embedding energy of the heliumatom in any electronic surrounding can be calculated as afunctional of the unperturbed electron density of the host.Formally, one can always separate a local term from thisfunctional and writeVR( r ) =bE(np( r )) +bE Iwhere bE(np) is the embedding energy of helium in ahomogeneous electron gas of density np, np( r ) is the un-perturbed surface electron density, and bE„] includes allnonlocal terms. b,E(np) is calculated by embedding thehelium atom in a homogeneous electron gas with a compen-sating positive background charge. 9 In the low-density limitthis gives b.E(n) proportional to the density and therebyEq. (1). We have calculated bE(n) for low electron densi-ties using the local density approximation'0 (LDA) for theexchange and correlation energies with and without a self-interaction correction" (SIC). The corresponding slopes nare given in Table I.In the region outside a metal surface sampled by a low-energy ( ( 500 meV) helium atom there are no positivecharges. Also, the surface electron density has non-negligible gradients. In the refined treatment, Eqs. (2) and(3), the former problem is taken into account by subtractingfrom Eq. (1) the electrostatic interaction of the helium atomwith the homogeneous background [Eq. (3)]. This resultsin the new proportionality constant n, rr in Eq. (2), which isalso given in Table I." The use of an averaged surface elec-tron density n p( r ) in Eq. (2) is the lowest-order inclusionof nonlocality. Outside a metal surface we typically findn p( r ) = 1.3 np( r ) fairly independent of r for the rangeof positions of relevance for low-energy helium scattering.This means that we can define a third proportionality con-stant'Jeff= (n 0( r )/np( r ) ]jeff2314 1984 The American Physical SocietyREPULSIVE INTERACTION OF THE HELIUM ATOM WITH A METAL SURFACE 2315TABLE I. Proportionality constants for the helium-metal scattering potential.n (eVap ) A ff {eVap3 ) 0,' ff {eVao3 )Effective-medium approachLDALDA-SICHarris-Liebsch approachFirst-order (Ref. 5)Second-order (Ref. 12)Scattering lengthCalculation (Ref. 13)Experimental (Ref. 14)305329160196203191, . . . , 2032082551000—500so thatVn( r )=n rrno( r )o.' ff is also included in Table I. It should be stressed that(x ff/A rf depends on the kind of system considered and that6 ff in Table I is only applicable for helium-metal surface in-teractions over a limited range of distances.Following Zaremba and Kohn, ' Harris and Liebsch' haveshown that, to first order in the overlap between the surfaceand helium states, the repulsive part of the helium-surfacepotential can be writtenseen how the value cx,ff differs dramatically from the effec-tive medium value, which is rather independent of the ap-proximation used for the treatment of exchange and correla-tion effects. Most of this difference is found again in o.,ff.Now we want to point out that there is a more direct wayof determining the coefficient n, ff. In the extreme low-density limit the interaction of electrons with a helium atomcan be calculated from scattering theory. For X electrons ina volume V (without the positive background) the embed-ding of a helium atom increases the total energy of the elec-trons byp4FVR( r ) = „~ deg(e)no(e, r ) (6) ~E 4mb2m V (7)where g(e) is a universal function of energy only, andnp(e, r ) is the local density-of-states function of the baresurface. Formally Eq. (6) can be writtenwhere a, is the scattering length. A comparison with Eq.(1) gives directlyVs( r ) =g(e)no( r ) 2mb&eff asm (8)where ~ in principle depends on r . We have made explicitcalculations within the local density approximation, usingthe linear augmented plane-wave method for a five-layerNi(110) slab, '7 which show that in reality e is basically con-stant for the range of distance of interest here [no(e, r ) isdominated by energies close to the Fermi energy]. This hasalso been argued by Harris and Liebsch based on jeliium-surface calculations. The Harris-Liebsch value for n, ff isthus g (e) and again n, rr will depend on the kind of systemstudied. For a low-density homogeneous electron gas theHarris-Liebsch approach gives n,rr=g(0). The Harris-Liebsch values for n, ff and n, ff calculated within theHartree-Pock approximation are shown in Table I. In thetable we include both the originally published values (firstorder in the helium pseudopotential) and the most recentones by Nordlander and Harris. ' The latter are calculatedto second order in the helium pseudopotential and the ener-gy argument in g (e) is replaced by e —up( I' ), where up( r )is the host one-electron potential at the helium position. "The Harris-Liebsch approach is not designed for the prob-lem of helium in a uniform electron gas, and the perturba-tion theory which involves a polarizability does not convergewell' as the electron gas density and bandwidth go to zero.%e therefore have not included any second-order value forA ff in Table I (treated in all orders of the perturbationtheory the Harris-Liebsch approach would lead to the exactvalue extracted from the scattering length; see below). It isThis approximation for the electron-helium interaction hasconventionally been used in studying the electron-inducedbubbles in liquid helium (see Ref. 19 and referencestherein). The advantage is that the scattering length can becomputed by solving a three-electron problem and a moreelaborate many-body theory can be applied. The o.,ff valuesdetermined from theoretical and experimental scatteringlengths are included in Table I. A comparison shows thatthe LDA gives good results and LDA-SIC give excellentresults for the helium interaction with the low-density elec-tron gas. The results of Table I do, however, indicate someproblems with the Harris-Liebsch value for o.,ff. There aretwo possible explanations of the too-large value. It may bethat the expansion in the helium-metal overlap or the per-turbation theory in the helium pseudopotential is not con-verged. The other possibility is the difference in the treat-ment of exchange and correlation effects between theHartree-Fock (HF) and LD approximations used in the twoapproaches. In the HF approximation of the Harris-Liebschapproach, ' no correlation effects are included, whereas inthe LD approximation and, in particular, when the self-interaction correction is included, correlation effects aretreated to a certain approximation. The LDA value of cx,ffis therefore less repulsive than the Hartree-Pock value. Ifthe too-large value of a,ff in the Harris-Liebsch-Nordlandertreatment is due to a lack of correlation effects in the HFtreatment, this need not invalidate the approach for calculat-2316 MANNINEN, NORSKOV, PUSKA, ANI3 UMRIGAR 29ing the He-surface interaction. All correlation effects are inprinciple included in the zeroth order contribution to theinteraction potential in the overlap expansion which is thevan der %aals attraction. In praxis the asymptotic form ofthe van der Waals interaction is added to the HF repulsivepotential Eq. (6). Such a procedure gives, for instance, areasonable description of the physisorption minimum whencompared to experiment. In the effective-medium ap-proach, on the other hand, the exchange-correlation energyof the helium atom in a homogeneous electron gas is al-ready included in the repulsive part V~ of the potential, andonly the extra exchange-correlation energy due to the inho-mogeneity of the surface electron density must be added.Model calculations for jellium surfaces show that this is asmall effect which is important only far from the surface,where Vg is small. With the inclusion of the extraexchange-correlation energy the effective-medium approachprovides a good description of the helium-jellium-surfacepotential including the physisorption minimum, when com-pared to a fully self-consistent calculation within the localdensity approximation.Providing the overlap expansion is converged, the choiceof the value for the proportionality constant a,ff in therepulsive part of the helium-surface potential thus dependson the choice of the treatment of the exchange and correla-tion effects. If the asymptotic form of the van der Waalsinteraction is included, the corresponding large (HF) valueof n, ff must be used. This description is clearly the mostappropriate far from the surface. Closer in, care must betaken with the divergence of the asymptotic van der %aalsform. ' If a local density description of the exchange andcorrelation is chosen, the small value of n, qf should be usedand only the extra exchange-correlation energy due to theinhomogeneity of the surface electron density added. Thiscorrection is small. The local density description is expectedto work better the closer the He atom is to the surface.ACKNOVf LEDGMENTSDiscussions with J. Harris, N. Lang, R. M. Nieminen, andP. Nordlander are gratefully acknowledged. %e are gratefulto P. Nordlander for providing us with his results beforepublication. One of us (C.U.) is grateful to the Office ofNaval Research for financial support.'J. K. Nerskov and N. D. Lang, Phys. Rev. 8 21, 2131 (1980).2M. J. Stott and E. Zaremba, Phys. Rev. 8 22, 1564 (1980).3N. Esbjerg and J. K. Nerskov, Phys. Rev. Lett. 45, 807 (1980).4N. D. Lang and J. K. Nerskov, Phys. Rev. 8 27, 4612 (1983).5J. Harris and A. Liebsch, J. Phys. C 15, 2275 (1982).6R, 8. Laughlin, Phys. Rev. B 25, 2222 (1982).R. Schinke and A. C. Luntz, Surf. Sci. Lett. 124, L60 (1983).P. Hohenberg and W. Kohn, Phys. Rev. 136, 8864 (1964);W. Kohn and L. J. Sham, ibid. 140, A1133 (1965).M. J. Puska, R. M. Nieminen, and M. Manninen, Phys. Rev. 8 24,3037 (1981).' O. Gunnarsson and B. I. Lundqvist, Phys. Rev. 8 13, 4274 (1976)."J.Perdew and A. Zunger, Phys. Rev. 8 23, S048 (1981).' P. Nordlander and J. Harris (unpublished).' N. R. Kestner, J. Jortner, M. H. Cohen, and S. A. Rice, Phys.Rev. 140, A56 (196S).' T. F. O' Malley, Phys. Rev. 130, 1020 {1963).' In principle, the integral in Eq. (3) and in the definition of no isonly over the region close to the He atom where the atomic po-tential dominates (Ref. 4). In this region the He-induced electrondensity in a homogeneous electron gas is approximated very wellby that of the free atom. Furthermore, for this He density, V&of Eq. (2) is basically independent of the choice of the radius R,of the near-atom region for 8, )2.S ao, even though e,ff andna( r )/nc( r ) do depend slightly on 8, (Ref. 4). Here we haveused R, =~.' E. Zaremba and W. Kohn, Phys, Rev. 8 15, 1769 (1977).M. Manninen, J. K. Nerskov, and C. Umrigar (unpublished).' P. Nordlander (private communication).R. M. Nieminen, I. Valimaa, M, Manninen, and P, Hautoj'arvi,Phys. Rev. A 21, 1677 (1980),2 The second-order nonlocal term in the effective-medium theoryinvolves the polarizability of the helium atom in a homogeneouselectron gas. This term has been calculated for helium outside ametal surface and it is very small, indicating that the effective-medium theory is converged with the first-order nonlocal term.

Manninen, Matti

Puska, M. J.

Physical Review B

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Repulsive interaction of the helium atom with a metal surface

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