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Some comments on the damped van der Waals interaction of adatoms on metal and graphite surfaces
Volume 147, number 7
PHYSICS LETTERS A
23 July 1990
Some comments on the damped van der Waals interaction of adatoms on metal and graphite surfaces J.S. Brown Department of Physics, University of Vermont, Burlington, VT 05405, USA Received 24 January 1990; revised manuscript received 16 May 1990; accepted for publication 17 May 1990 Communicated by A.A. Maradudin
A comparison is made ofthe Ruiz—Scoles—Jonsson (RSJ) and Nordlander—Harris (NH) damping functions used in treating the Van der Waals interaction of adsorbed atoms with metal and graphite surfaces. Both damping functions may be written as f= I —P(x)e~with P(x) a polynomial. A general discussion is presented of these two damping functions with reference to their use for adatoms on metal and graphite surfaces and a discussion is given of the need for a more fundamental treatment of the damping function.
In the last few years a number of attempts have been made to develop interaction potentials that adequately describe the interaction of adsorbed atoms with metal substrates and with the graphite (0001) surface [1—41.In dealing with the attractive van der Waals (vdW) potential involved in this interaction it has become fashionable to introduce a damping functionf(x) that modifies the vdW interaction near the surface. The motivation for this, beyond the desire to improve the physical basis and give a better description of the vdW potential, is based on the obvious observation that wavefunction overlap must modify the vdW potential near the surface. Therefore the inclusion of a suitable damping function must modify the bare vdW potential in this region. A difficulty in treating this interaction arises from the form ~‘ of the vdW interaction, which varies as C 3 in the region outside the surface plane, 3/ (z— where z z0) 0 gives the location of the image plane. The fact that this potential diverges as z—~z0 is not, however, physical. it is based, as may be noted in ref. [5], on an asymptotic expansion that breaks down in this region. A popular damping function which has seen considerable use in connection with the treatment of at-
oms adsorbed on metal substrates [7—9] was introduced several years ago by Nordlander and Harris (NH) [10], based on an expression derived by Zaremba and Kohn (ZK), discussed in ref. [5]. Far outside the surface ZK showed that the vdW potential VVdW may be written as VVdW =
—
J
dk
e2k
~~0)
$
~ xA(k, iu)XM(k, iu) 27t (1)
where xA(k, iu) and xM(k, iu) are complex Fourier transforms of the atomic and metallic susceptibilities respectively. NH adopted a long-wavelength expansion of these susceptibilittes in powers of k according to xAk, iu) =~~.(iu) +k~~(iu) +... (2a)
—
~‘
and XM (k,
iu) =~, (iu) + kX’M (iu) +....
(2b)
Introducing a cutoff at /c~in the k-integral of eq. (1) then led NH in a straightforward manner to their damping function, given2by ) e 2x (3) f (x) = 1 — (1 + 2x+ 2x —
See ref. [5] for the introduction of the image plane z 0, and for a more detailed treatment consult ref. [6].
0375-9601/90/s 03.50 © 1990
—
where x = k~ (z z0), with z z0 the distance of the adatom outside the surface from the image plane (see
Elsevier Science Publishers B.V. (North-Holland)
—
—
393
Volume 147, number 7
ref. [5]). becomes
PHYSICS LETTERS A
The vdW interaction potential then
(4) —C3f(x)/(z—z0)3, x=k~(z—zo) It should be stressed that the ZK analysis is based on the model of a neutral adatom interacting with a jelhum half-space of negative charge in a uniform positive background. This model has been used by a number of workers to treat the vdW potential of adatoms on metallic surfaces, using the effective medium approach [11,12] to deal with the repulsive part of the atom—surface interaction, A different vdW potential exists in the literature for treating the interaction of an adatom with the planar (0001) face of graphite. The adatom sees a set of parallel conducting planes in this model, and the vdW potential obtained by the Crowell—Steele [13] procedure leads to a form different from that given in eq. (4). In the Crowell—Steele approach the interaction of an adatom with a given carbon atom in a graphite plane is taken to vary as r~,the atoms in the graphite plane are then spread into a uniform sheet, and the vdW potential is found by integrating over the entire sheet. The resulting vdW potential representing a surface-average potential not displaying the surface corrugations ofa more realistic model, has the form .
VVdW=
~ C(z+nd)~,
where d is the separation of adjacent graphite planes. x is the reduced distance given by x=z/d, and in terms ofthis parameter the vdW potential reduces to ~ (x+n)—4,
(6)
n~O
where B= C/d4. In passing it is noted that this result really corresponds to the G=0 term in a Fourier series expansion of the interaction in the reciprocal space basis of the surface structure. It is also to be noted that this result does not require the C atoms to be uniformly distributed in a given plane parallel to the (0001) face, but emerges from the G=0 term in such a Fourier expansion independent of how the C atoms are distributed. Recently Ruiz, Scotes, and Jonsson (RSJ) [14] used the Crowell—Steele form given by eq. (6) to de394
scribe the interaction of neutral He atoms with the (0001) face ofgraphite. The RSJ treatment included a damping function assumed to be operative only between the He adatom and the first lattice plane (n = 0) of the graphite lattice. The damping function used was one introduced a few years ago by Tang and Toennies [15], which in an ad hoc fashion introduces the effects of wavefunction overlap in modifying the polarization forces involved in treating the adatom—lattice interaction. RSJ did not do a sum over the infinite lattice for two reasons: first, because the damping function rapidly becomes negligible for distances of the order of the second lattice plane from a typical adatom distance; and secondly, because the form of the damping function obtained from the Tang—Toennies treatment of damping is quite involved. The RSJ damping function obtained from the Tang—Toennies atomic wavefunction overlap treatment is cast by RSJ into the form
f= 1+4
ESk(az)
~
,
(7)
k!
k=O
where the functions E~(x)appearing in eq. (7) are the exponential integrals defined in ref. [16] and given by
J
E~(x)=
e
xt -
dt —
I,,
(8)
(5)
n~U
VVdW= —B
23 July 1990
The reductions necessary to arrive at eq. (7) were carried out by RSJ using a method of integration of the interaction over a given graphite plane introduced some years ago by Steele [17]. In passing it is noted that the argument az appearing in eq. (7) is essentially equal to the 2x appearing in the NH damping function of eq. (3), a contention to be supported a bit further on in this note. Eq. (7) may be simplified if we use a set of recurrence relations found in ref. [16] for the exponential integral functions of different orders. The recurrence relation needed here is ~ (x)=e_x_xE~(x) . (9) Use of this recurrence relation allows us to write the functions E 5(x), E4(x), E3(x), and E2(x) all in terms of
Volume 147, number 7
E1(x) =
J
PHYSICS LETTERSA
the second lattice plane anyway. As a rough estimate e
dl I
E 0(x)= Jetdt=e9x
(lOa)
and
1(x)=
J
tdl
=
(1+x)e~/x2.
(lOb)
te-~
Now we can perform the sum given in eq. (7) to obtam the RSJ damping function. When we write it all out one finds that the coefficient of all the terms involving E 1 (x) identically sums to zero, leaving the final result free of any exponential integral functions, and involving only exponentials and powers of x. In an earlier version of this paper a referee kindly pointed that this gratifying result was that foundan byindependent Chow [18], and it isouttherefore confirmation of this point is now available. The RSJ damping function, based on the Tang—Toennies wavefunction overlap approach, then leads to the form fRsJ(x)= 1— (1 +2x+2x2+~x3+~x4 +~x5)e2x.
is noted that for the He atom 1 A outside the first lattice plane, the damping function at the position of the second lattice plane has the value of roughly 0.9996 and is therefore negligible. it
-~‘—.
The remaining two integrals E0(x) and E1 (x) appearing in eq. (7) are obtained in simplified form via
E
23 July 1990
(11)
A direct comparison with the NH damping function of eq. (3) may now be made. First we note that in eq. (3) the parameter x is given by x = k~ (z z 0). NH chose the cutoff parameter k~to be about 1 au= 1.89 because this rough estimate of the size of theA~ charge cloud ofisana atom. Except for inclusion ofthe image plane position z we maysince note that the relation 2k~z=ctz is essentially0 correct, 2k~=3.78 A-’ and the value of a adopted by RSJ from Tang and Toennies is about this value. Beyond this we cannot go unless a more fundamental treatment of the damping function is developed, a treatment that to our knowledge does not presently exist. The RSJ treatment of He adsorbed on graphite involves only the damping factor included for the first graphite lattice plane. To be sure, this is all that is needed, since fRSJ is essentially 1 at the position of
2x, with A comparison of fRsJ(x) and fNH(x) shows that they are now both of the formf= 1 _P(x)e the polynomialP(x) containing 5, RSJ which presumably arise from theterms formthrough of the xatomic wavefunction overlap assumed in the Tang— Toennies treatment. The NH form, on the other hand, was made by assuming a cutoff in the integral ofeq. (1) which arose from the ZK analysis. It seems very unlikely that these two different approaches can be extended to yield the same P(x) function. A better treatment of the damping function may result from a more fundamental approach to the role of wavefunction overlap in modifying the dispersion forces between the neutral adatom and the surface, but as pointed out earlier we are unaware of the existence of such a treatment at this time. It may be noted that as z—z0 the divergence of3,the is ZK form ofthe i.e. —C3/(z—zo) removed by the vdW effectpotential, ofthe damping function. This same property persists for the Crowell—Steele form of the vdW potential, as given by eq. (6), which approaches x4 behavior as x-40. Again the RSJ damping function removes the divergence. On the other hand the divergences in both cases are not of physical origin, so ittoisthe hard to assess what meaning may be attributed above observations. It is noted in contrast, however, that the NH damping function does not remove the x4 divergence of the
—
Crowell—Steele form of the vdW potential. A simple form that does so is easily obtained by assuming 2x.that By the damping function has the form 1 _P(x)e requiring that this damping function cancel the di4 it is easily shown that the lowest-orvergence ofx der polynomial that will do this contains terms through x3, and all the coefficients are uniquely determined in the Taylor expansion about x=0. The damping function so obtained is given by
f= I— (l+2x+2x2+~x3)e2x. In passing we note that through the cubic term the coefficients in P(x) are those of the RSJ damping function. As stressed earlier no claim is made here ofthe sig395
Volume 147. number 7
PHYSICS LETTERS A
nificance of the cancellation of the divergence of the vdW potential since the limit involved is nonphys-
23 July 1990
References
ical. What we would like to stress here is the need for a more fundamental treatment of damping based on wavefunction overlap, or a more extensive inveStigation of how to extract a better damping function from the ZH integral of eq. (1). What has been
[1] F. Toigo and M.W. Cole, Phys. Rev. B 32 (1985) 6989. [2] J.S. Brown and A.D. Crowell, Surf. Sci. 146 (1984) 61. [3] A. Liebsch, J. Harris and M. Weinert, Surf. Sci. 145 (1984) 207. [4] M. Karimi and G. Vidali, Surf. Sci. 208 (1989) L73.
shown here is that the ad hoc damping function ap-
[5] E. Zaremba and W. Kohn, Phys. Rev. B 13 (1976) 2270.
proach of Tang and Toennies may be cast into standard form, namely 1 _P(x)e2x so that it resembles the damping function of NH. We are indebted to a referee for pointing out that Chow [18] had already arrived at this same conclusion several years ago in his unpublished thesis. Recently Chizmeshya and Zaremba [19] have used the NH damping function in treating the interaction ofrare gas atoms with metal surfaces. They have incorporated the important Tang—Toennies idea that the damping parameter is determined by wavefunction overlap, thus providing a definite but certainly not rigorous prescription for the treatment of an otherwise ill-defined parameter.
[6] B.N.J. Persson and E. Zaremba, Phys. Rev. B 30 (1984) 5669. [7] P. Nordlander, C. Holmberg and J. Harris, Surf. Sci. 175 (1986) L753. [8]P. Nordlander, C. Holmberg and J. Harris, Surf. Sci. 152/ [9] 153 M.G.(1985) Dondi,702. L. Mattera, S. Terreni, F. Tommassini and U. Linke,Phys.Rev.B34(1986) 5897. [10] P. Nordlander and J. Harris, J. Phys. C 17 (1984) 1141. [11] N. Esbjerg and J.K. Norskov, Phys. Rev. Lett. 45(1980) 807. [12] M. Manninen, Norskov, Phys. Rev. B 29J.K. (1984) 2314. M.J. Puska and C. Umrigar,
[13] A.D. Crowell and R.B. Steele Jr., J. Chem. Phys. 34 (1966) 1347. [14] J.C. Ruiz, G. Scoles and H. Jonsson, Chem. Phys. Lett. 129 (1986) 139. [l5]K.T. Tang and J.P. Toennies, J. Chem. Phys. 80 (1984) 3726.
The author wishes to gratefully acknowledge the help of a referee whose illuminating suggestions have contributed substantially to a better version of this paper.
396
[161M. Abramowitz and IA. Stegun, Handbook ofmathematical functions (Dover, New York, 1970). [17] W.A. Steele, Surf. Sci. 36 (1973) 317. [18] O.L. Chow, M.Sc. Thesis, Queen’s University, Kinston, Canada (unpublished, 1988). [19] A. Chizmeshyaand E. Zaremba, Surf. Sci. 220 (1989) 443.
PHYSICS LETTERS A
23 July 1990
Some comments on the damped van der Waals interaction of adatoms on metal and graphite surfaces J.S. Brown Department of Physics, University of Vermont, Burlington, VT 05405, USA Received 24 January 1990; revised manuscript received 16 May 1990; accepted for publication 17 May 1990 Communicated by A.A. Maradudin
A comparison is made ofthe Ruiz—Scoles—Jonsson (RSJ) and Nordlander—Harris (NH) damping functions used in treating the Van der Waals interaction of adsorbed atoms with metal and graphite surfaces. Both damping functions may be written as f= I —P(x)e~with P(x) a polynomial. A general discussion is presented of these two damping functions with reference to their use for adatoms on metal and graphite surfaces and a discussion is given of the need for a more fundamental treatment of the damping function.
In the last few years a number of attempts have been made to develop interaction potentials that adequately describe the interaction of adsorbed atoms with metal substrates and with the graphite (0001) surface [1—41.In dealing with the attractive van der Waals (vdW) potential involved in this interaction it has become fashionable to introduce a damping functionf(x) that modifies the vdW interaction near the surface. The motivation for this, beyond the desire to improve the physical basis and give a better description of the vdW potential, is based on the obvious observation that wavefunction overlap must modify the vdW potential near the surface. Therefore the inclusion of a suitable damping function must modify the bare vdW potential in this region. A difficulty in treating this interaction arises from the form ~‘ of the vdW interaction, which varies as C 3 in the region outside the surface plane, 3/ (z— where z z0) 0 gives the location of the image plane. The fact that this potential diverges as z—~z0 is not, however, physical. it is based, as may be noted in ref. [5], on an asymptotic expansion that breaks down in this region. A popular damping function which has seen considerable use in connection with the treatment of at-
oms adsorbed on metal substrates [7—9] was introduced several years ago by Nordlander and Harris (NH) [10], based on an expression derived by Zaremba and Kohn (ZK), discussed in ref. [5]. Far outside the surface ZK showed that the vdW potential VVdW may be written as VVdW =
—
J
dk
e2k
~~0)
$
~ xA(k, iu)XM(k, iu) 27t (1)
where xA(k, iu) and xM(k, iu) are complex Fourier transforms of the atomic and metallic susceptibilities respectively. NH adopted a long-wavelength expansion of these susceptibilittes in powers of k according to xAk, iu) =~~.(iu) +k~~(iu) +... (2a)
—
~‘
and XM (k,
iu) =~, (iu) + kX’M (iu) +....
(2b)
Introducing a cutoff at /c~in the k-integral of eq. (1) then led NH in a straightforward manner to their damping function, given2by ) e 2x (3) f (x) = 1 — (1 + 2x+ 2x —
See ref. [5] for the introduction of the image plane z 0, and for a more detailed treatment consult ref. [6].
0375-9601/90/s 03.50 © 1990
—
where x = k~ (z z0), with z z0 the distance of the adatom outside the surface from the image plane (see
Elsevier Science Publishers B.V. (North-Holland)
—
—
393
Volume 147, number 7
ref. [5]). becomes
PHYSICS LETTERS A
The vdW interaction potential then
(4) —C3f(x)/(z—z0)3, x=k~(z—zo) It should be stressed that the ZK analysis is based on the model of a neutral adatom interacting with a jelhum half-space of negative charge in a uniform positive background. This model has been used by a number of workers to treat the vdW potential of adatoms on metallic surfaces, using the effective medium approach [11,12] to deal with the repulsive part of the atom—surface interaction, A different vdW potential exists in the literature for treating the interaction of an adatom with the planar (0001) face of graphite. The adatom sees a set of parallel conducting planes in this model, and the vdW potential obtained by the Crowell—Steele [13] procedure leads to a form different from that given in eq. (4). In the Crowell—Steele approach the interaction of an adatom with a given carbon atom in a graphite plane is taken to vary as r~,the atoms in the graphite plane are then spread into a uniform sheet, and the vdW potential is found by integrating over the entire sheet. The resulting vdW potential representing a surface-average potential not displaying the surface corrugations ofa more realistic model, has the form .
VVdW=
~ C(z+nd)~,
where d is the separation of adjacent graphite planes. x is the reduced distance given by x=z/d, and in terms ofthis parameter the vdW potential reduces to ~ (x+n)—4,
(6)
n~O
where B= C/d4. In passing it is noted that this result really corresponds to the G=0 term in a Fourier series expansion of the interaction in the reciprocal space basis of the surface structure. It is also to be noted that this result does not require the C atoms to be uniformly distributed in a given plane parallel to the (0001) face, but emerges from the G=0 term in such a Fourier expansion independent of how the C atoms are distributed. Recently Ruiz, Scotes, and Jonsson (RSJ) [14] used the Crowell—Steele form given by eq. (6) to de394
scribe the interaction of neutral He atoms with the (0001) face ofgraphite. The RSJ treatment included a damping function assumed to be operative only between the He adatom and the first lattice plane (n = 0) of the graphite lattice. The damping function used was one introduced a few years ago by Tang and Toennies [15], which in an ad hoc fashion introduces the effects of wavefunction overlap in modifying the polarization forces involved in treating the adatom—lattice interaction. RSJ did not do a sum over the infinite lattice for two reasons: first, because the damping function rapidly becomes negligible for distances of the order of the second lattice plane from a typical adatom distance; and secondly, because the form of the damping function obtained from the Tang—Toennies treatment of damping is quite involved. The RSJ damping function obtained from the Tang—Toennies atomic wavefunction overlap treatment is cast by RSJ into the form
f= 1+4
ESk(az)
~
,
(7)
k!
k=O
where the functions E~(x)appearing in eq. (7) are the exponential integrals defined in ref. [16] and given by
J
E~(x)=
e
xt -
dt —
I,,
(8)
(5)
n~U
VVdW= —B
23 July 1990
The reductions necessary to arrive at eq. (7) were carried out by RSJ using a method of integration of the interaction over a given graphite plane introduced some years ago by Steele [17]. In passing it is noted that the argument az appearing in eq. (7) is essentially equal to the 2x appearing in the NH damping function of eq. (3), a contention to be supported a bit further on in this note. Eq. (7) may be simplified if we use a set of recurrence relations found in ref. [16] for the exponential integral functions of different orders. The recurrence relation needed here is ~ (x)=e_x_xE~(x) . (9) Use of this recurrence relation allows us to write the functions E 5(x), E4(x), E3(x), and E2(x) all in terms of
Volume 147, number 7
E1(x) =
J
PHYSICS LETTERSA
the second lattice plane anyway. As a rough estimate e
dl I
E 0(x)= Jetdt=e9x
(lOa)
and
1(x)=
J
tdl
=
(1+x)e~/x2.
(lOb)
te-~
Now we can perform the sum given in eq. (7) to obtam the RSJ damping function. When we write it all out one finds that the coefficient of all the terms involving E 1 (x) identically sums to zero, leaving the final result free of any exponential integral functions, and involving only exponentials and powers of x. In an earlier version of this paper a referee kindly pointed that this gratifying result was that foundan byindependent Chow [18], and it isouttherefore confirmation of this point is now available. The RSJ damping function, based on the Tang—Toennies wavefunction overlap approach, then leads to the form fRsJ(x)= 1— (1 +2x+2x2+~x3+~x4 +~x5)e2x.
is noted that for the He atom 1 A outside the first lattice plane, the damping function at the position of the second lattice plane has the value of roughly 0.9996 and is therefore negligible. it
-~‘—.
The remaining two integrals E0(x) and E1 (x) appearing in eq. (7) are obtained in simplified form via
E
23 July 1990
(11)
A direct comparison with the NH damping function of eq. (3) may now be made. First we note that in eq. (3) the parameter x is given by x = k~ (z z 0). NH chose the cutoff parameter k~to be about 1 au= 1.89 because this rough estimate of the size of theA~ charge cloud ofisana atom. Except for inclusion ofthe image plane position z we maysince note that the relation 2k~z=ctz is essentially0 correct, 2k~=3.78 A-’ and the value of a adopted by RSJ from Tang and Toennies is about this value. Beyond this we cannot go unless a more fundamental treatment of the damping function is developed, a treatment that to our knowledge does not presently exist. The RSJ treatment of He adsorbed on graphite involves only the damping factor included for the first graphite lattice plane. To be sure, this is all that is needed, since fRSJ is essentially 1 at the position of
2x, with A comparison of fRsJ(x) and fNH(x) shows that they are now both of the formf= 1 _P(x)e the polynomialP(x) containing 5, RSJ which presumably arise from theterms formthrough of the xatomic wavefunction overlap assumed in the Tang— Toennies treatment. The NH form, on the other hand, was made by assuming a cutoff in the integral ofeq. (1) which arose from the ZK analysis. It seems very unlikely that these two different approaches can be extended to yield the same P(x) function. A better treatment of the damping function may result from a more fundamental approach to the role of wavefunction overlap in modifying the dispersion forces between the neutral adatom and the surface, but as pointed out earlier we are unaware of the existence of such a treatment at this time. It may be noted that as z—z0 the divergence of3,the is ZK form ofthe i.e. —C3/(z—zo) removed by the vdW effectpotential, ofthe damping function. This same property persists for the Crowell—Steele form of the vdW potential, as given by eq. (6), which approaches x4 behavior as x-40. Again the RSJ damping function removes the divergence. On the other hand the divergences in both cases are not of physical origin, so ittoisthe hard to assess what meaning may be attributed above observations. It is noted in contrast, however, that the NH damping function does not remove the x4 divergence of the
—
Crowell—Steele form of the vdW potential. A simple form that does so is easily obtained by assuming 2x.that By the damping function has the form 1 _P(x)e requiring that this damping function cancel the di4 it is easily shown that the lowest-orvergence ofx der polynomial that will do this contains terms through x3, and all the coefficients are uniquely determined in the Taylor expansion about x=0. The damping function so obtained is given by
f= I— (l+2x+2x2+~x3)e2x. In passing we note that through the cubic term the coefficients in P(x) are those of the RSJ damping function. As stressed earlier no claim is made here ofthe sig395
Volume 147. number 7
PHYSICS LETTERS A
nificance of the cancellation of the divergence of the vdW potential since the limit involved is nonphys-
23 July 1990
References
ical. What we would like to stress here is the need for a more fundamental treatment of damping based on wavefunction overlap, or a more extensive inveStigation of how to extract a better damping function from the ZH integral of eq. (1). What has been
[1] F. Toigo and M.W. Cole, Phys. Rev. B 32 (1985) 6989. [2] J.S. Brown and A.D. Crowell, Surf. Sci. 146 (1984) 61. [3] A. Liebsch, J. Harris and M. Weinert, Surf. Sci. 145 (1984) 207. [4] M. Karimi and G. Vidali, Surf. Sci. 208 (1989) L73.
shown here is that the ad hoc damping function ap-
[5] E. Zaremba and W. Kohn, Phys. Rev. B 13 (1976) 2270.
proach of Tang and Toennies may be cast into standard form, namely 1 _P(x)e2x so that it resembles the damping function of NH. We are indebted to a referee for pointing out that Chow [18] had already arrived at this same conclusion several years ago in his unpublished thesis. Recently Chizmeshya and Zaremba [19] have used the NH damping function in treating the interaction ofrare gas atoms with metal surfaces. They have incorporated the important Tang—Toennies idea that the damping parameter is determined by wavefunction overlap, thus providing a definite but certainly not rigorous prescription for the treatment of an otherwise ill-defined parameter.
[6] B.N.J. Persson and E. Zaremba, Phys. Rev. B 30 (1984) 5669. [7] P. Nordlander, C. Holmberg and J. Harris, Surf. Sci. 175 (1986) L753. [8]P. Nordlander, C. Holmberg and J. Harris, Surf. Sci. 152/ [9] 153 M.G.(1985) Dondi,702. L. Mattera, S. Terreni, F. Tommassini and U. Linke,Phys.Rev.B34(1986) 5897. [10] P. Nordlander and J. Harris, J. Phys. C 17 (1984) 1141. [11] N. Esbjerg and J.K. Norskov, Phys. Rev. Lett. 45(1980) 807. [12] M. Manninen, Norskov, Phys. Rev. B 29J.K. (1984) 2314. M.J. Puska and C. Umrigar,
[13] A.D. Crowell and R.B. Steele Jr., J. Chem. Phys. 34 (1966) 1347. [14] J.C. Ruiz, G. Scoles and H. Jonsson, Chem. Phys. Lett. 129 (1986) 139. [l5]K.T. Tang and J.P. Toennies, J. Chem. Phys. 80 (1984) 3726.
The author wishes to gratefully acknowledge the help of a referee whose illuminating suggestions have contributed substantially to a better version of this paper.
396
[161M. Abramowitz and IA. Stegun, Handbook ofmathematical functions (Dover, New York, 1970). [17] W.A. Steele, Surf. Sci. 36 (1973) 317. [18] O.L. Chow, M.Sc. Thesis, Queen’s University, Kinston, Canada (unpublished, 1988). [19] A. Chizmeshyaand E. Zaremba, Surf. Sci. 220 (1989) 443.