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Binding of an adatom to a simple metal surface
Surface Science 48 (1975) 187-203 0 North-Holland Publishing Company
BINDING OF AN ADATOM TO A SIMPLE METAL SURFACE?
H.B. HUNTINGTON,
L.A. TURK and W.W. WHITE III*
Deparrment of Physics, Rensselaer Polytechnic institute, Troy, New York 12181, USA
The density functional formalism of Hohenberg and Kohn is used to investigate the energies, charge densities and forces which hold an adatom on the surface of a simple metal. The valence wavefunction of the adatom is fitted to the HermanSkillman solutions at large distance and is simphfied somewhat in the core region. The field of the ion is represented by the Ashcroft pseudopotential. For the metal the jelhum model is used. Detailed calculations are carried out for a sodium adatom on a sodium surface. Simply juxtaposing adatom and surface gives a binding energy of about f eV. This value is approximately twice the surface energy per atom in the close-packed plane. Charge redistributions as determined variationally increase the binding energy by about 10%. The redistribution is primarily a dipole induced on the adatom at close distances, but at somewhat larger distances a prolate quadrupoie aiso appears on the atom. A small amount of charge is also drawn from the metat toward the atom. The equilibrium distance for the adatom turns out to be 1.66 .& from the surface, as compared with 1.52 A, the observed value for one-half the distance between the close-packed planes. Contour plots of the piling-up of electronic charge between the adatom and the metal are presented.
1. Model
In this calculation we have applied the density-functional formalism of Hohenberg and Kohn [ 11 to the problem of an adatom juxtaposed to the surface of a simple metal. SpeciEc application has been made to the case of a sodium adatom on a sodium surface. The model for the atom has been synthetized out of analytic wave functions fitted to the Herman-Skillman tabulations 121 and a potential of the Ashcroft type [3] . For the simple metal (one with a nearly spherical Fermi surface), the jellium model has been used. The density-functional formalism of Hohenberg and Kohn [ 11 has been frequently applied to surface problems and the whole area has been recently reviewed by Lang [4j . The character of electron distribution at the surface of a jcllium metal
t Work supported by the National Aeronautica and Space Administration under Grant No. NGL33-018-003. * Present address: Mission Research Corporation, 735 State Street, Santa Barbara, California 93102, U.S.A.
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H.B. Huntington
et aLlBinding of adatom to simple metal surfhce
has had a long history [5] . Smith [6] was the first to treat it in the density approximation and his results obtained by minimization procedure gave a satisfactory account of the variation of dipole layer and charge fall-off as a function of electron density. Lang [7] and Lang and Kahn [8] considered the problem of the metallic surface energy by the same approximation and obtained values by integrating the variational equation somewhat lower than the observed energies (obtained for liquid metals). The discrepancy was worse at the higher densities, actually giving the wrong sign. By structuring the jellium with the insertion of a lattice of point charges. this disaster was avoided, but the energy values so obtained were still a bit lower than observed. Lang [9] has considered the effect of the adsorbed alkali impurities on the work functions of transition metals by using a jellium model with a step in the positive charge distribution at the surface, whose height was made proportional to the number of adsorbed atoms, and was able to account for the shape of the alteration of metal contact potential with adsorbate coverage. Smith, Ying and Kahn [lo] have used the density approximation to attack the problem of the hydrogen ion adsorbed on a metal of high work function (tungsten) and have developed a detailed picture of the chemisorption process. Ferrante and Smith [l l] have been remarkably successful in using simple metal models to account for the forces of adhesion between clean metal surfaces, without considering any electron redistribution. In the calculation presented in this paper, we have attempted an approach of intermediate accuracy, but with a fairly broad range of applicability. The adatom that we use is reasonably realistic, certainly as detailed as is necessary for the ap‘proximation introduced by the density-functional formalism. The jellium model for the metal is oversimplified, but there seems to be no reason why it can not be somewhat structured at a later time. The positioning of the adatom at distances close to the surface has given a reasonable picture of the atom-metal force. While the binding energy we observe is not soundly based, the value that we get is fortuitously better than one would expect. We have not solved an Euler equation [ 121 to find the correct electron density distribution, but the use of a few variational parameters has served, we feel, to give a good picture of the charge redistribution at the surface. The future of this sort of approach lies less in treatment of very specific adsorbates on highly structured surfaces, which is particularly the realm of catalysis, but more in the consideration of adatoms of simple structure moving over crystalline surfaces. Thus in the future, our primary interest will be in adatom mobility. In the next section we discuss the method in some detail. Section 3 is concerned with the application to a sodium adatom on a sodium surface. Section 4 is the conclusion.
H.B. Huntington et aL/Binding of adatom to simple metal surfhce
189
2. Method The energy functional
3 -7
0II3J 3
?r
for this problem is [ 111
n413 dV-
,,4/3 0.056 jx310079
dV
+j
IVAshQ,+Q
Here atomic units have been used t~~rougllout so e = m = fi = I. The electron charge density is denoted by tz. The first two terms give the electron kinetic energy as measured from the bottom of the conduction band. The n5j3 term is indeed the kinetic energy of the homogeneous Fermi gas. The term in (01~)~ is the first order correction for a Fermi gas whose density fluctuates slowly and resembles in form one that was early introduced by WeizGcker [ 13 ] , except that it is smaller by a factor of l/9. It has been shown by Jones and Young [ 141 that the validity of the WeizGcker expression holds for a situation where small density fluctuations are short in comparison to the electron wavelength. For the situation on the metal surface the charge variation is not particularly short, but the amplitude of the variation is large, especially for high density metals. Nevertheless we feel that the approximation to first order in (Vu)*is adequate in sodium for our purpose. We shall return to this point later. The next two terms are respectively the exchange and correlation energies for the electron distribution. It should be pointed out that we make no distinction between the charge density of the valence electron(s) of the adatom and those of the jellium when calculating these “correlation” terms. The form of the exchange energy is just that average over “k”dependent potentials that would be applied in the case of a ho~nogeneous Fermi gas. The first order correction to this term from density fluctuations has been developed [ 151, but, in view of the uncertainty in the analogous term for the kinetic energy, it seemed more consistent to omit such consideration here. For the correlation energy we have used the standard Wigner expression. It is a relatively smaller term and not quite as sensitive to the electron density. There remain the classical, electrostatic interactions. First the Ashcroft potential, VAsh, interacts with all the charges - the positive jellium charge as well as the composite electronic distribution. It is convenient to regroup these into two parts: (1) n&, , the dipole layer charge composed of the positive jellium charge and the electron distribution of the free-metal surface and (2) n,, the valence charge of the adatom (which simulates the Herman-Skillman density) and the charge redistributions introduced in the minimization process. Next come the self-energy terms. The integral of the dipole layer charge with its own potential gives a constant characteristic of the free surface and quite independent of the atom and its position.
H.B. Huntington
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The potential of the dipole layer is a simple rounded step function analytic in one dimension. Its interaction with n,, is next evaluated. The potential from the 12~~can be directly obtained, as will shortly become apparent, and from it the last term in the curly brackets can be evaluated, The final term in the square bracket comes as as correction because the pseudopotential of the Ashcroft potential is coulombic only outside a certain radius, but it is drastically altered inside this radius to exclude the valence electrons from the core region and maintain their orthogonality with the core functions. The positive jellium charge. n,, however, should see only the positive coulomb potential and hence the correction term. Its effect is to provide a very strong repulsion for close distances of adatom approach. The jellium itself is composed of a uniform positive charge density which terminates discontinuously at the geometrical surface of the metal. The negative charge density is made to fall off symmetrically about the surface n = ino e -M for z > 0,
n=no(l
-ie”‘)forz
(2)
where z is measured from the geometrical surface, positive on the vacuum side. Here the curvature parameter a: is chosen to be the same on both sides of the plane of the geometrical surface z = 0. Detailed treatment of the surface distribution through solution of the variational one-dimensional equation in the density approximation [9] has shown that the curvature is actually somewhat smaller on the vacuum side. For the adatom the Ashcroft potential is Za/rforr>R &Jr>
=
0
forr
c
i where the value of R,, the radius at which the ionic potential becomes coulombic, is the only adjustable parameter and Z, is the charge on the adatom ion. Inside R, the zero value for VAshsuggested that the form for the charge density of the valence electron be [sinh (fir)/@] 2. At large distances the Herman-Skilhnan density was fitted by an exponential in r multiplied by a factor (1 - C/rJ2 which gave a maximum at closer distances as required by the ns valence electron density. The two expressions were fitted in slope and magnitude at R, and the composite density duly normalized. There was always concern that the pseudoatom so constructed would not by itself be stable against arbitrary charge deformation in minimizing the energy in the density approximation. For the case of the sodium atom considered in the next section, we required that the atom remain stable under a uniform radial expansion of the charge density, and this condition of stability was used to determine R,. The R, value turned out to be 0.90 A, in gratifying agreement with the value quoted by Ashcroft as appropriate for sodium 0.88 A. One advantage of the analytic expression for the charge density was that it allowed a reduction in the amount of computer storage. In the process of minimizing the overall energy we introduced certain variations in the charge density which for convenience were all of the same type, namely a function of a radial variable times a spherical harmonic. These functions, of course,
191
H.B. Huntington et al/Binding of adatom to simple metal surface
simply displaced charge without creating or destroying any, so that their introduction involved no renormalization. They had the great added advantage that the solution of Poisson’s equation for these charge distributions (as needed for V,r) could be each reduced to two simple quadratures which we could do almost completely analytically, further reducing machine time. (See Appendix A.) We considered a fair number of such charge alterations, but eventually settled on three which we felt were adequate. Two of these were centered at the adatom center and involved the first two cylindrical harmonics, Pl (cos0) and Pz (cos 0). The radial part of the dipolar adaptation, involving PI, was principally r times the original valence density scaled by a variational parameter. Similarly, the radial part of the quadrupolar adaptation was r2 times the same radial function and similarly scaled. The powers of r were chosen to prevent any discontinuity in slope or function at the origin. The third adaptation was centered on the atom-metal axis at a small (variational) distance inside the metal. It was chosen to have the form e,rg exp(-yrs)P1 (COST), where rs = r - 6, and 6 vectors to a position on the atom-metal axis inside the metal. There are then five variational parameters: ed and eq, the amplitudes of the dipolar and quadrupolar terms respectively as centered on the atom and em, y, and 6 for the adaptation inside the metal. The question naturally arises whether these choices were well chosen and were they sufficient for the degree of accuracy sought. Faith and some intuition lead us to believe so. The matter will be discussed again in the concluding section. The actual integrations as performed by computer were carried out over a rectangular mesh in z (along the atom-metal axis) and p (radial distance from the axis). The mesh was 43 by 81 and the shortest distance was about rJ12. For some purposes, this turned out to be a rather coarse mesh. In the interaction term between the Ashcroft potential and the rr,r the results scattered badly as the atom was moved about because of the variation in phasing of the mesh points with the strong discontinuity in the potential at R,. This difficulty was avoided by putting the atom only at the mesh points. Another computational difficulty arose in evaluating the integral of rrdp I/Ash because of the long range nature of the potential. It turned out, however, tl,at this integral can be performed analytically (Appendix B) and gives
s
ndp VAsh d V = 2.rm0Za
1 - eeRclor sinh(Z/o)
for 2 < RC
cash (R&x) eeZ”
forZ>Re
where 2 is the distance of adatom from the geometrical surface. The integration of the final correction terms can also be done analytically. pendix C gives
s
n+(I&lu, - V&,)dV=
In the evaluation
nr~,,(R~-z)~ 0
(4)
Ap-
forZ=GR forZ>Rf.
of the term in (Vn)2 it turned out to be advisable to go to
(5)
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H.B. Huntington
et al/Binding
of adatom to simple rnrtal surfbce
Table 1 Free atom energies n5’3 term (eV) (vn)* term Exchange Correlation Self-valence Valence-ashcroft
0.9766 0.2143 -2.1385 --0.844 1 3.1008 -1.2225
Energy of atom Ionization energy
-5.9134 5.1353
second order in the expansion of the gradient in terms of the finite differences for a really consistent check between analysis and computer results.
3. Application
to sodium atom on sodium metal
By simply placing the atom at various distances from the surface and minimizing the energy by varying the charge distribution, one gets an approximate binding energy curve. The energy of the free atom as evaluated by this procedure is given in table 1. This energy turns out to be -5.91 eV as compared to 5.13 eV for the ionization energy of sodium. This energy is some 15% too low, which is a serious dis‘crepancy as far as the energetics of binding go. We believe that the difficulty lies primarily in using a form of the nearly uniform density approximation in the expansion of the kinetic energy in powers of (17~2)~.Gombas [ 161 has pointed out this problem of obtaining too low values with the application of the nearly uniform density approximation to atoms. In addition the energy functional we are using [eq. (l)] is clearly inappropriate for the separated atom where the 3s valence electron is a unique entity, not subject to statistical treatment. Instead there are included in the energy functional three electron-electron interactions terms: selfvalence, exchange and correlation. Their sum is quite small, 0.12 eV, here as in the metal. When omitted, the discrepancy in ionization energies is further increased. The dashed plot in fig. 1 shows the preliminary binding curve with a binding energy of 0.34 eV. In spite of the difficulty with the value at infinite separation, we believe that this result has some relevance to the adatom problem, mainly because the violent variations of atomic wave function and potential occur in the core region which is very little altered by the approach to the surface. The confidence in this result as a fair approximation to the adatom binding is born out by the following empirical estimate. The formation of two new surfaces by splitting a crystal is primarily that of breaking the interatom bonds that cross the fission surface. It is, however, just these bonds that we allow to form when an atom approachs the surface without charge readjustments. The value for sodrurri surface
H.B. Huntington
et aLlBinding ofadatom
to simple metal surf&e
0.2 ATOM SURFACE BINDING FORCE 0.1 t
t
1
WITHOUT CHARGE DISTORTIONS ATOM SURFACE EI~ING ENERGY WITH CHARGE OISTOF?TlONS
-0.4 -
I
Fig. 1. Binding energy versus atom-metal
distance. Atom surface binding force is in units of
eViA.
energy per atom is 0.188 eV and the closeness of this number to one half our binding energy is encouraging. In fig. 2 we show how the variable terms which appear in eq. (1) change with adatonI-metal distance. One observes that the exchange energy with its negative curvature dominates the .5/3 term at large distances. Closer in, the interactions of the dipole Iayer charge with the valence electron-Ashcroft potential for the atom produces a positive curvature near the equilibrium separation. At still closer distances, the core-positive jellium interaction gives a steeply rising repulsion which prevents the atom from entering the metal. The effect of the charge readjustments are relativeiy modest and amount to about 10% at the equilibrium distance. The principal influence is that of thePt term centered on the atom. It is responsible for about 85% of the lowering of the energy, depending somewhat on the atom’s position. Most of the remaining energy lowering is caused by the P2 term; the distortion within the metal was relatively ineffective. The relative variation of the dipole (PI) charge displacements is shown in fig. 3, plotted in units of Debyes. The magnitude of the dipole moment is calculated from the following formula co
p
=
$ed
sr4
n3sWdr,
0
(6)
2
-3
VALENCE DIPOLE
e 5
-1.0
Fig. 2. Various
energy
terms versus adatom-metal
distance.
Where ed is the v3riable coefficient of the dipole term and the Sag is the valence density for sodium. One notices that IpI is small at large distances, increases as the atom approaches the surface and reaches a maximum near the equilibrium position. The direction of the dipole shows that the electron is drawn into the metal: The size of the quadrupole moment Q is calculated from the following formula
Fig. 3. Dipole moment
versus adatom-metal
distance.
H.B. Huntington
et aLlBinding
of adatom
to simple metal surface
Fig. 4. Quadrupole moment versus adatom-metal
195
distance.
where eq is the variable coefficient of the quadrupole term. Fig. 4 shows Q plotted in Debye- versus 2. The quadrupolar moment falls off less rapidly than the dipole at large distances and is prolate. At closer distances, it comes to a maximum and then changes sign near the equilibrium position of the atom. At all distances the atom charge is drawn toward the metal, but at close approach it tends to flatten out a bit against the surface. As for the charge deformation within the metal, it was large only for small Z. The direction was such as to draw charge out to meet the adatom. The center of the distortion came a few tenths of an angstriim inside the metal, about 2.3 a from the center of the adatom. The distortion was moderately localized with y = 1.8 8-l. The largest magnitude of the dipole moment was 0.38 Debye which is an appreciable fraction of the moment at the atom, but oppositely directed. The effect of these variations in the charge density can be seen in the lowering of the binding energy curve to the full line shown in fig. 1. It is only about 10% at the equilibrium distance as remarked before. The adsorption energy is 0.374 eV and comes at an adatom-metal distance of 1.66 8. One half the interplanar spacing for the close-packed (110) planes in sodium is 1.52 A. This agreement may be improved in all probability by structuring the jellium. Ferrante and Smith [ 1 l] found that the lattice-lattice interaction between similar surfaces decreased the intersurface distances. The third curve plotted in fig. 1 shows the force on the adatom perpendicular to the surface. This is an important quantity if one is to treat the kinetics of adatom movement over the surface. It is essential in characterizing the adatom surface bond on a more sophisticated level than a simple sum of two body interactions. In view of the crudeness of the model in describing the free atom, the adsorption
196
H.B. Huntington
et al./Binding of adatom to simple metal surface
energy given by this calculatiorl could well be in serious error. This difficulty is by no means so serious for the adatom close to the surface since the charge varies much less drastically in the region between the atom and the metal when they are near equilibrium. Since the force curve is determined by the changes in this region where the approximations of the method are more nearly fulfilled, its calculation should be relatively more reliable than that of the binding energy. Although the binding energy is increased only slightly by the variation of charge, the distortion themselves are quite evident as can be seen by comparing figs. 5 and 6. In fig. 5 are shown the unmodified charge distributions of adatom and metal along the Z axis at a separation of 2 A, a distance somewhat greater than the equilibrium spacing. In fig. 6 are shown along the same axis the three charge adaptations that have been used tIlroLlgllout. The lowest curve gives the total charge distribution and shows clearly the influence of the modulations in bringing electronic charge up to the surface. The extent to which these modulations fill in and smooth out the region between adatom and metal is illustrated in figs. 7 and 8. In fig. 7 are shown the charge density contours in z-p spaces as plotted by computer for the
-.008
ATOM VALENCE ’ DENSITY
Fig. 5. Unmodulated charge density along the adatom-metal
axis.
H.B. Huntington
et al./Binding of adatom to simple metal surface
197
‘ooO r&e/P] ATOM DIPOLE SURFACE DIPOLE
- ,016 --
-.024-TOTAL ELECTRONIC /DENSITY
Fig. 6. Charge density modulations and total charge density along the adatom-metal axis. adatom simply placed at 2 8, from the surface. One sees the reduced charge at the center of the atom surrounded by a spherical shell of higher density. Between adatom and metal there is a saddle-point falling off steeply on either side. With charge modification, as shown in fig. 8, the saddle-point is gone. The charge density is actually at a maximum in this region and quite a broad one. Undoubtedly the small peaked maxima, also evident here in figs. 6 and 8, are artifacts of the method and would wash out in a more complete variational treatment.
4. Conclusion The principal concerns of this paper are to: (a) Show that our calculations for the juxtaposition of a reasonably carefully constructed atom with a jellium metal of low density, give rise to a binding energy and an equilibrium distance quite consistent with expectation. (b) Explore the charge modifications which minimize the energy of the adatommetal interaction by enhancing the density between the atom and the surface. (c) Analyze charge modifications mainly in terms of cylindrical harmonics about the adatom, and trace the variation of these modifications with atom-surface distance.
H.B. ffuntington et aI./Binding of adatorn to simple metal suv~k~
198
0
I
2
3
4
5
6
VACUUM
-0233
Fig. 7. Unmodulated
charge
contours
in z--p space.
The critical evaluation of our procedures would seem to hinge primariiy on the adequacy of the variational procedure and the validity of the slowly changing density approximation. In regard to the former, there seems little doubt that the energy lowering from charge modulation is indeed small ^u IO%, but the details of the resulting charge distribution may be susceptibIe to visible improvement by more detailed minimization. In any ~ninimi~ation procedure it is almost impossible to say if the final stage in obtaining a specified level of precision has occurred. The forms of charge modulation that have been chosen were to some extent arbitrary and, although the ones we finally settled on seemed quite satisfactory, we can think of others we did not try which might have worked even better. On the whole it seems unlikely that the main results listed above would have been qualitatively altered by the choice of different l~linimization functions. it is even harder to estimate what the shortcomings of the slowly varying expansion for the kinetic energy might be. We do feel that there may be serious difficulties with metals of much higher densities than sodium. Whether these difficulties can be solved by structuring the jellium [S] is an open question. We did try a preliminary effort at putting a sodium atom on an aluminum jellium surface.
H.B. Huntington
199
et al.fBinding of adatom to simple metal surface
p
61 4
3
2 7 “? &
t I
= 0.0311
oa N
0
Fig. 8. Charge
contours
with modulations
in z-p
space.
This resulted in too large an equilibrium distance and too small a surface dipole to account for the decrease of work function found by Porteus [ 171 when he deposited sodium on aluminum. We close by indicating some of the directions which studies along this same line might fruitfully take. One would expect no difficulty in obtaining similar results for metals of lower density - primarily the heavier alkalis. No work on surface mobility can be carried out until the jellium surface can be structured at least crudely. Some cleverness will be needed here if the excessive expense of programming three-dimensional integrals is to be avoided. A still further step in this direction would be going from the close-packed planes to those with atomistic serrations. Such investigations might prove very illuminating for the understanding of surface transport and hence surface reactions.
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H.B. Huntington et al./Binding ofadatom
Appendix A. Evaluation
of self-potentials
The various harmonic charge distortions adatom are of the general form rr,&r, 0) = n&)
to simple metal surface
associated with and centered at the
eaP[ (cos 0) r[,
(A.1)
and inside the jellium metal surface rrrn(rs ,el = cmPI(~0s f? (ys1’ exp t-v6 where E, and em are variational density is n ar = n&)
[l ‘EdPI
J,
tA.2)
parameters. The combined atom and redistribution
(cosO)r
+ EqP2 (cose)2]
(A-3) + fmPl (cosO)r, The atom self-potential sons’s equation
exp(-yr&). associated with one of the distortions
is found from Pois-
02V,(r, 0) = - 4afaPI(COS8),In3s(‘). Eq. (A.4) can be rewritten,
(A.4)
assuming the following separation of variables,
V&C 0) = f,P[ (cos 0) tir),
(A.5)
as 1(2+ 1)
--p(r)
= -
which becomes upon substituting r/f”(F) t 2(1+ I)+’ Multiplying
f(r)
47W$(‘)‘~’
(A.6)
O(r) = r’f(r) = - 4an&)/.
(A.7)
eq. (A.7) by r1+2, we obtain
which by two quadratures
gives at the field point t
(A.9) Thus the atom self-potential
for any 1 is t
I VJ4‘) = 4ne,P1 (cos 0) &
c 0
&I [
(Yy(l+l)
.f
“&)
F
dr’ + j
AZ&‘)
ti:,
1 (k.10)
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H.3. ~~~ti~gton et al./Sinding of adatom to simple metal surface
from which Vat), V&, V,,, the undistorted, dipole and quadrupole self-potential, are obtained. An analogous procedure holds for the surface adaptations. We obtain for a surface dipole distortion with x z -yZ (A.1 1) The combined
self-potential
is given by (A.12)
v,r = v,O + %d + %q + ‘rnd.
Appendix B. Ascroft-dipole The interaction charge distribution
layer interaction
between the Ashcroft potential is given by
[eq. (3)] and the dipole layer
2
I Ash-de = f ndp adV r
forr>Re.
tB.1)
It follows I Ash-de = 2rrza 7 “dp(ll+Z)ja pI dXzpdp -cm
dz
where
z’ =z-2,
(B.2)
and n dp is 2 > 0
-eazla,
n,p(z) = in0
ez/”
3
Z-CO
with nO the density in the bulk metal. The integration over p gives [pz + (.z’)~]~‘~ - [pf f (z’)~] l”. For large p2 the term at the upper limit can be expanded to give p2 + 1 (z’)~/P~, neither part of which contributes, since (a) the constant p2 can be taken outside the integral and Jn dz vanishes for the dipole layer, and (b) the second part goes to zero as p2 goes to infinity. At the lower limit,
iz'l >R c, p1= 0 and (pi + (z’)*]~‘~ = \.a’[, (B.4) IZ’IGRc,
p1 =
For the integration letZ>R,,then I Ash-dp = 7rZ,no
[R:- (z')~]"~ and [p: + (z’)~]~‘~ = Rc. over z it is necessary to consider two cases separately.
$% -co
lz’\ &’ .+ e-Zb
7’ -z
e-z’,a
iz’\ &’
First
H.B. Huntington
202
+ e-z/a
et aL/Binding of adatom to simple metal surface
RC
s
e-i'/& Rc &‘t
Evaluation
Rc
of these elementary
IAsh-dp
.
(B.5)
integrals leads to
21r.Za11*~a~ exp (-R&)
=
1
,-‘I@ f eda /z’I dz’
-R,
u3.6)
sinh(Z/a).
Similarly for Z
= nZano
Ash-dp
eZya Iz’( &’ + eZILY s
er’lcu R, dz’
-R,
-cc
(B.7) and this gives I Ash-dp
=
2nZ,noo2
[cash (Rc/cu)e-“@
-I]
.
(B.8)
However, the energy which we calculated in the energy minimization is the energy difference for bringing the atom in from infinity. A term 27rZ,noo2 must be added throughout. For Z > R,, EAsh-dp
=
27rZar~~o1~cosh(RJo)
=
2nZ,n0~2
exp (-Z/a),
(B.9
and forZ
[ 1 - exp (-RJa)
Appendix C. Positive jellium-adatom
sinh (Z/o)]
(B.lO)
core interaction
The repulsive interaction, E,, , between the overlap of positive ions of the jellium n,, and the adatom core represented by a coulombic potential must be added in the energy equation (1) as a correction term. The region of integration over the overlap volume, due to the symmetry of the Ashcroft potential, is a spherical cap. We thus obtain for EC,, E
cm
=
s
‘A&Jdv
n+(v&,-
msy
= 2Tn+p z
which is proportional
r sin 0 d0 dr = rrr~+(R,--Z)~,
0
to the square of the jellium-core
overlap distance.
(C.1)
H.B. Huntington
et aLlBinding of adatom to simple metal surface
203
References [ 1 ] P. Hohenberg
and W. Kohn,
Phys. Rev. 136 (1964)
B 864.
121 F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall, Englewood Cliffs, N.J., 1963). [3] N.W. Ashcroft, Phys. Letters 23 (1966) 48. [4] N.D. Lang, in: Solid State Physics, Vol. 28, Eds. H. Ehrenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1973) p. 225. [S] C. Herring, in: Metal Interfaces (Am. Sot. Metals, Cleveland, Ohoi, 1952) p. 6; H.B. Huntington, Phys. Rev. 81 (1951) 1035; R. Stratton, Phil. Mag. [7] 44 (1953) 1236; H.J. Juretschke, in: The Surface Chemistry of Metals and Semiconductors, Ed. H.C. Gatos (Wiley, New York, 1960) p. 40. [6] J.R. Smith, Phys. Rev. 181 (1969) 522. [7] N.D. Lang, Solid State Commun. 7 (1969) 1047. [S] N.D. Lang and W. Kohn, Phys. Rev. B 1 (1970) 4555. [9] N.D. Lang, Phys. Rev. B4 (1971) 4234. [lo] J.R. Smith, SC. Ying and W. Kohn, Phys. Rev. Letters 30 (1973) 610. [ll] J. Ferrante and J.R. Smith, Surface Sci. 38 (1973) 77. [12] Ref. 4, p. 231. (131 C.F. von Weizdcker, Z. Physik 96 (1935) 43. [14] W. Jonesand W.H. Young, J. Phys. C4 (1971) 1322. [ 151 F. Herman, J.P. Van Dyke and LB. Ortenberger, Phys. Rev. Letters 22 (1969) 807. [16] P. Gombas, Phys. Letters 28 A (1969) 585. 1171 J.P. Porteus, Surface Sci. 41 (1974) 515.
BINDING OF AN ADATOM TO A SIMPLE METAL SURFACE?
H.B. HUNTINGTON,
L.A. TURK and W.W. WHITE III*
Deparrment of Physics, Rensselaer Polytechnic institute, Troy, New York 12181, USA
The density functional formalism of Hohenberg and Kohn is used to investigate the energies, charge densities and forces which hold an adatom on the surface of a simple metal. The valence wavefunction of the adatom is fitted to the HermanSkillman solutions at large distance and is simphfied somewhat in the core region. The field of the ion is represented by the Ashcroft pseudopotential. For the metal the jelhum model is used. Detailed calculations are carried out for a sodium adatom on a sodium surface. Simply juxtaposing adatom and surface gives a binding energy of about f eV. This value is approximately twice the surface energy per atom in the close-packed plane. Charge redistributions as determined variationally increase the binding energy by about 10%. The redistribution is primarily a dipole induced on the adatom at close distances, but at somewhat larger distances a prolate quadrupoie aiso appears on the atom. A small amount of charge is also drawn from the metat toward the atom. The equilibrium distance for the adatom turns out to be 1.66 .& from the surface, as compared with 1.52 A, the observed value for one-half the distance between the close-packed planes. Contour plots of the piling-up of electronic charge between the adatom and the metal are presented.
1. Model
In this calculation we have applied the density-functional formalism of Hohenberg and Kohn [ 11 to the problem of an adatom juxtaposed to the surface of a simple metal. SpeciEc application has been made to the case of a sodium adatom on a sodium surface. The model for the atom has been synthetized out of analytic wave functions fitted to the Herman-Skillman tabulations 121 and a potential of the Ashcroft type [3] . For the simple metal (one with a nearly spherical Fermi surface), the jellium model has been used. The density-functional formalism of Hohenberg and Kohn [ 11 has been frequently applied to surface problems and the whole area has been recently reviewed by Lang [4j . The character of electron distribution at the surface of a jcllium metal
t Work supported by the National Aeronautica and Space Administration under Grant No. NGL33-018-003. * Present address: Mission Research Corporation, 735 State Street, Santa Barbara, California 93102, U.S.A.
188
H.B. Huntington
et aLlBinding of adatom to simple metal surfhce
has had a long history [5] . Smith [6] was the first to treat it in the density approximation and his results obtained by minimization procedure gave a satisfactory account of the variation of dipole layer and charge fall-off as a function of electron density. Lang [7] and Lang and Kahn [8] considered the problem of the metallic surface energy by the same approximation and obtained values by integrating the variational equation somewhat lower than the observed energies (obtained for liquid metals). The discrepancy was worse at the higher densities, actually giving the wrong sign. By structuring the jellium with the insertion of a lattice of point charges. this disaster was avoided, but the energy values so obtained were still a bit lower than observed. Lang [9] has considered the effect of the adsorbed alkali impurities on the work functions of transition metals by using a jellium model with a step in the positive charge distribution at the surface, whose height was made proportional to the number of adsorbed atoms, and was able to account for the shape of the alteration of metal contact potential with adsorbate coverage. Smith, Ying and Kahn [lo] have used the density approximation to attack the problem of the hydrogen ion adsorbed on a metal of high work function (tungsten) and have developed a detailed picture of the chemisorption process. Ferrante and Smith [l l] have been remarkably successful in using simple metal models to account for the forces of adhesion between clean metal surfaces, without considering any electron redistribution. In the calculation presented in this paper, we have attempted an approach of intermediate accuracy, but with a fairly broad range of applicability. The adatom that we use is reasonably realistic, certainly as detailed as is necessary for the ap‘proximation introduced by the density-functional formalism. The jellium model for the metal is oversimplified, but there seems to be no reason why it can not be somewhat structured at a later time. The positioning of the adatom at distances close to the surface has given a reasonable picture of the atom-metal force. While the binding energy we observe is not soundly based, the value that we get is fortuitously better than one would expect. We have not solved an Euler equation [ 121 to find the correct electron density distribution, but the use of a few variational parameters has served, we feel, to give a good picture of the charge redistribution at the surface. The future of this sort of approach lies less in treatment of very specific adsorbates on highly structured surfaces, which is particularly the realm of catalysis, but more in the consideration of adatoms of simple structure moving over crystalline surfaces. Thus in the future, our primary interest will be in adatom mobility. In the next section we discuss the method in some detail. Section 3 is concerned with the application to a sodium adatom on a sodium surface. Section 4 is the conclusion.
H.B. Huntington et aL/Binding of adatom to simple metal surfhce
189
2. Method The energy functional
3 -7
0II3J 3
?r
for this problem is [ 111
n413 dV-
,,4/3 0.056 jx310079
dV
+j
IVAshQ,+Q
Here atomic units have been used t~~rougllout so e = m = fi = I. The electron charge density is denoted by tz. The first two terms give the electron kinetic energy as measured from the bottom of the conduction band. The n5j3 term is indeed the kinetic energy of the homogeneous Fermi gas. The term in (01~)~ is the first order correction for a Fermi gas whose density fluctuates slowly and resembles in form one that was early introduced by WeizGcker [ 13 ] , except that it is smaller by a factor of l/9. It has been shown by Jones and Young [ 141 that the validity of the WeizGcker expression holds for a situation where small density fluctuations are short in comparison to the electron wavelength. For the situation on the metal surface the charge variation is not particularly short, but the amplitude of the variation is large, especially for high density metals. Nevertheless we feel that the approximation to first order in (Vu)*is adequate in sodium for our purpose. We shall return to this point later. The next two terms are respectively the exchange and correlation energies for the electron distribution. It should be pointed out that we make no distinction between the charge density of the valence electron(s) of the adatom and those of the jellium when calculating these “correlation” terms. The form of the exchange energy is just that average over “k”dependent potentials that would be applied in the case of a ho~nogeneous Fermi gas. The first order correction to this term from density fluctuations has been developed [ 151, but, in view of the uncertainty in the analogous term for the kinetic energy, it seemed more consistent to omit such consideration here. For the correlation energy we have used the standard Wigner expression. It is a relatively smaller term and not quite as sensitive to the electron density. There remain the classical, electrostatic interactions. First the Ashcroft potential, VAsh, interacts with all the charges - the positive jellium charge as well as the composite electronic distribution. It is convenient to regroup these into two parts: (1) n&, , the dipole layer charge composed of the positive jellium charge and the electron distribution of the free-metal surface and (2) n,, the valence charge of the adatom (which simulates the Herman-Skillman density) and the charge redistributions introduced in the minimization process. Next come the self-energy terms. The integral of the dipole layer charge with its own potential gives a constant characteristic of the free surface and quite independent of the atom and its position.
H.B. Huntington
190
et al/Binding
of adatom to simple metal surface
The potential of the dipole layer is a simple rounded step function analytic in one dimension. Its interaction with n,, is next evaluated. The potential from the 12~~can be directly obtained, as will shortly become apparent, and from it the last term in the curly brackets can be evaluated, The final term in the square bracket comes as as correction because the pseudopotential of the Ashcroft potential is coulombic only outside a certain radius, but it is drastically altered inside this radius to exclude the valence electrons from the core region and maintain their orthogonality with the core functions. The positive jellium charge. n,, however, should see only the positive coulomb potential and hence the correction term. Its effect is to provide a very strong repulsion for close distances of adatom approach. The jellium itself is composed of a uniform positive charge density which terminates discontinuously at the geometrical surface of the metal. The negative charge density is made to fall off symmetrically about the surface n = ino e -M for z > 0,
n=no(l
-ie”‘)forz
(2)
where z is measured from the geometrical surface, positive on the vacuum side. Here the curvature parameter a: is chosen to be the same on both sides of the plane of the geometrical surface z = 0. Detailed treatment of the surface distribution through solution of the variational one-dimensional equation in the density approximation [9] has shown that the curvature is actually somewhat smaller on the vacuum side. For the adatom the Ashcroft potential is Za/rforr>R &Jr>
=
0
forr
c
i where the value of R,, the radius at which the ionic potential becomes coulombic, is the only adjustable parameter and Z, is the charge on the adatom ion. Inside R, the zero value for VAshsuggested that the form for the charge density of the valence electron be [sinh (fir)/@] 2. At large distances the Herman-Skilhnan density was fitted by an exponential in r multiplied by a factor (1 - C/rJ2 which gave a maximum at closer distances as required by the ns valence electron density. The two expressions were fitted in slope and magnitude at R, and the composite density duly normalized. There was always concern that the pseudoatom so constructed would not by itself be stable against arbitrary charge deformation in minimizing the energy in the density approximation. For the case of the sodium atom considered in the next section, we required that the atom remain stable under a uniform radial expansion of the charge density, and this condition of stability was used to determine R,. The R, value turned out to be 0.90 A, in gratifying agreement with the value quoted by Ashcroft as appropriate for sodium 0.88 A. One advantage of the analytic expression for the charge density was that it allowed a reduction in the amount of computer storage. In the process of minimizing the overall energy we introduced certain variations in the charge density which for convenience were all of the same type, namely a function of a radial variable times a spherical harmonic. These functions, of course,
191
H.B. Huntington et al/Binding of adatom to simple metal surface
simply displaced charge without creating or destroying any, so that their introduction involved no renormalization. They had the great added advantage that the solution of Poisson’s equation for these charge distributions (as needed for V,r) could be each reduced to two simple quadratures which we could do almost completely analytically, further reducing machine time. (See Appendix A.) We considered a fair number of such charge alterations, but eventually settled on three which we felt were adequate. Two of these were centered at the adatom center and involved the first two cylindrical harmonics, Pl (cos0) and Pz (cos 0). The radial part of the dipolar adaptation, involving PI, was principally r times the original valence density scaled by a variational parameter. Similarly, the radial part of the quadrupolar adaptation was r2 times the same radial function and similarly scaled. The powers of r were chosen to prevent any discontinuity in slope or function at the origin. The third adaptation was centered on the atom-metal axis at a small (variational) distance inside the metal. It was chosen to have the form e,rg exp(-yrs)P1 (COST), where rs = r - 6, and 6 vectors to a position on the atom-metal axis inside the metal. There are then five variational parameters: ed and eq, the amplitudes of the dipolar and quadrupolar terms respectively as centered on the atom and em, y, and 6 for the adaptation inside the metal. The question naturally arises whether these choices were well chosen and were they sufficient for the degree of accuracy sought. Faith and some intuition lead us to believe so. The matter will be discussed again in the concluding section. The actual integrations as performed by computer were carried out over a rectangular mesh in z (along the atom-metal axis) and p (radial distance from the axis). The mesh was 43 by 81 and the shortest distance was about rJ12. For some purposes, this turned out to be a rather coarse mesh. In the interaction term between the Ashcroft potential and the rr,r the results scattered badly as the atom was moved about because of the variation in phasing of the mesh points with the strong discontinuity in the potential at R,. This difficulty was avoided by putting the atom only at the mesh points. Another computational difficulty arose in evaluating the integral of rrdp I/Ash because of the long range nature of the potential. It turned out, however, tl,at this integral can be performed analytically (Appendix B) and gives
s
ndp VAsh d V = 2.rm0Za
1 - eeRclor sinh(Z/o)
for 2 < RC
cash (R&x) eeZ”
forZ>Re
where 2 is the distance of adatom from the geometrical surface. The integration of the final correction terms can also be done analytically. pendix C gives
s
n+(I&lu, - V&,)dV=
In the evaluation
nr~,,(R~-z)~ 0
(4)
Ap-
forZ=GR forZ>Rf.
of the term in (Vn)2 it turned out to be advisable to go to
(5)
192
H.B. Huntington
et al/Binding
of adatom to simple rnrtal surfbce
Table 1 Free atom energies n5’3 term (eV) (vn)* term Exchange Correlation Self-valence Valence-ashcroft
0.9766 0.2143 -2.1385 --0.844 1 3.1008 -1.2225
Energy of atom Ionization energy
-5.9134 5.1353
second order in the expansion of the gradient in terms of the finite differences for a really consistent check between analysis and computer results.
3. Application
to sodium atom on sodium metal
By simply placing the atom at various distances from the surface and minimizing the energy by varying the charge distribution, one gets an approximate binding energy curve. The energy of the free atom as evaluated by this procedure is given in table 1. This energy turns out to be -5.91 eV as compared to 5.13 eV for the ionization energy of sodium. This energy is some 15% too low, which is a serious dis‘crepancy as far as the energetics of binding go. We believe that the difficulty lies primarily in using a form of the nearly uniform density approximation in the expansion of the kinetic energy in powers of (17~2)~.Gombas [ 161 has pointed out this problem of obtaining too low values with the application of the nearly uniform density approximation to atoms. In addition the energy functional we are using [eq. (l)] is clearly inappropriate for the separated atom where the 3s valence electron is a unique entity, not subject to statistical treatment. Instead there are included in the energy functional three electron-electron interactions terms: selfvalence, exchange and correlation. Their sum is quite small, 0.12 eV, here as in the metal. When omitted, the discrepancy in ionization energies is further increased. The dashed plot in fig. 1 shows the preliminary binding curve with a binding energy of 0.34 eV. In spite of the difficulty with the value at infinite separation, we believe that this result has some relevance to the adatom problem, mainly because the violent variations of atomic wave function and potential occur in the core region which is very little altered by the approach to the surface. The confidence in this result as a fair approximation to the adatom binding is born out by the following empirical estimate. The formation of two new surfaces by splitting a crystal is primarily that of breaking the interatom bonds that cross the fission surface. It is, however, just these bonds that we allow to form when an atom approachs the surface without charge readjustments. The value for sodrurri surface
H.B. Huntington
et aLlBinding ofadatom
to simple metal surf&e
0.2 ATOM SURFACE BINDING FORCE 0.1 t
t
1
WITHOUT CHARGE DISTORTIONS ATOM SURFACE EI~ING ENERGY WITH CHARGE OISTOF?TlONS
-0.4 -
I
Fig. 1. Binding energy versus atom-metal
distance. Atom surface binding force is in units of
eViA.
energy per atom is 0.188 eV and the closeness of this number to one half our binding energy is encouraging. In fig. 2 we show how the variable terms which appear in eq. (1) change with adatonI-metal distance. One observes that the exchange energy with its negative curvature dominates the .5/3 term at large distances. Closer in, the interactions of the dipole Iayer charge with the valence electron-Ashcroft potential for the atom produces a positive curvature near the equilibrium separation. At still closer distances, the core-positive jellium interaction gives a steeply rising repulsion which prevents the atom from entering the metal. The effect of the charge readjustments are relativeiy modest and amount to about 10% at the equilibrium distance. The principal influence is that of thePt term centered on the atom. It is responsible for about 85% of the lowering of the energy, depending somewhat on the atom’s position. Most of the remaining energy lowering is caused by the P2 term; the distortion within the metal was relatively ineffective. The relative variation of the dipole (PI) charge displacements is shown in fig. 3, plotted in units of Debyes. The magnitude of the dipole moment is calculated from the following formula co
p
=
$ed
sr4
n3sWdr,
0
(6)
2
-3
VALENCE DIPOLE
e 5
-1.0
Fig. 2. Various
energy
terms versus adatom-metal
distance.
Where ed is the v3riable coefficient of the dipole term and the Sag is the valence density for sodium. One notices that IpI is small at large distances, increases as the atom approaches the surface and reaches a maximum near the equilibrium position. The direction of the dipole shows that the electron is drawn into the metal: The size of the quadrupole moment Q is calculated from the following formula
Fig. 3. Dipole moment
versus adatom-metal
distance.
H.B. Huntington
et aLlBinding
of adatom
to simple metal surface
Fig. 4. Quadrupole moment versus adatom-metal
195
distance.
where eq is the variable coefficient of the quadrupole term. Fig. 4 shows Q plotted in Debye- versus 2. The quadrupolar moment falls off less rapidly than the dipole at large distances and is prolate. At closer distances, it comes to a maximum and then changes sign near the equilibrium position of the atom. At all distances the atom charge is drawn toward the metal, but at close approach it tends to flatten out a bit against the surface. As for the charge deformation within the metal, it was large only for small Z. The direction was such as to draw charge out to meet the adatom. The center of the distortion came a few tenths of an angstriim inside the metal, about 2.3 a from the center of the adatom. The distortion was moderately localized with y = 1.8 8-l. The largest magnitude of the dipole moment was 0.38 Debye which is an appreciable fraction of the moment at the atom, but oppositely directed. The effect of these variations in the charge density can be seen in the lowering of the binding energy curve to the full line shown in fig. 1. It is only about 10% at the equilibrium distance as remarked before. The adsorption energy is 0.374 eV and comes at an adatom-metal distance of 1.66 8. One half the interplanar spacing for the close-packed (110) planes in sodium is 1.52 A. This agreement may be improved in all probability by structuring the jellium. Ferrante and Smith [ 1 l] found that the lattice-lattice interaction between similar surfaces decreased the intersurface distances. The third curve plotted in fig. 1 shows the force on the adatom perpendicular to the surface. This is an important quantity if one is to treat the kinetics of adatom movement over the surface. It is essential in characterizing the adatom surface bond on a more sophisticated level than a simple sum of two body interactions. In view of the crudeness of the model in describing the free atom, the adsorption
196
H.B. Huntington
et al./Binding of adatom to simple metal surface
energy given by this calculatiorl could well be in serious error. This difficulty is by no means so serious for the adatom close to the surface since the charge varies much less drastically in the region between the atom and the metal when they are near equilibrium. Since the force curve is determined by the changes in this region where the approximations of the method are more nearly fulfilled, its calculation should be relatively more reliable than that of the binding energy. Although the binding energy is increased only slightly by the variation of charge, the distortion themselves are quite evident as can be seen by comparing figs. 5 and 6. In fig. 5 are shown the unmodified charge distributions of adatom and metal along the Z axis at a separation of 2 A, a distance somewhat greater than the equilibrium spacing. In fig. 6 are shown along the same axis the three charge adaptations that have been used tIlroLlgllout. The lowest curve gives the total charge distribution and shows clearly the influence of the modulations in bringing electronic charge up to the surface. The extent to which these modulations fill in and smooth out the region between adatom and metal is illustrated in figs. 7 and 8. In fig. 7 are shown the charge density contours in z-p spaces as plotted by computer for the
-.008
ATOM VALENCE ’ DENSITY
Fig. 5. Unmodulated charge density along the adatom-metal
axis.
H.B. Huntington
et al./Binding of adatom to simple metal surface
197
‘ooO r&e/P] ATOM DIPOLE SURFACE DIPOLE
- ,016 --
-.024-TOTAL ELECTRONIC /DENSITY
Fig. 6. Charge density modulations and total charge density along the adatom-metal axis. adatom simply placed at 2 8, from the surface. One sees the reduced charge at the center of the atom surrounded by a spherical shell of higher density. Between adatom and metal there is a saddle-point falling off steeply on either side. With charge modification, as shown in fig. 8, the saddle-point is gone. The charge density is actually at a maximum in this region and quite a broad one. Undoubtedly the small peaked maxima, also evident here in figs. 6 and 8, are artifacts of the method and would wash out in a more complete variational treatment.
4. Conclusion The principal concerns of this paper are to: (a) Show that our calculations for the juxtaposition of a reasonably carefully constructed atom with a jellium metal of low density, give rise to a binding energy and an equilibrium distance quite consistent with expectation. (b) Explore the charge modifications which minimize the energy of the adatommetal interaction by enhancing the density between the atom and the surface. (c) Analyze charge modifications mainly in terms of cylindrical harmonics about the adatom, and trace the variation of these modifications with atom-surface distance.
H.B. ffuntington et aI./Binding of adatorn to simple metal suv~k~
198
0
I
2
3
4
5
6
VACUUM
-0233
Fig. 7. Unmodulated
charge
contours
in z--p space.
The critical evaluation of our procedures would seem to hinge primariiy on the adequacy of the variational procedure and the validity of the slowly changing density approximation. In regard to the former, there seems little doubt that the energy lowering from charge modulation is indeed small ^u IO%, but the details of the resulting charge distribution may be susceptibIe to visible improvement by more detailed minimization. In any ~ninimi~ation procedure it is almost impossible to say if the final stage in obtaining a specified level of precision has occurred. The forms of charge modulation that have been chosen were to some extent arbitrary and, although the ones we finally settled on seemed quite satisfactory, we can think of others we did not try which might have worked even better. On the whole it seems unlikely that the main results listed above would have been qualitatively altered by the choice of different l~linimization functions. it is even harder to estimate what the shortcomings of the slowly varying expansion for the kinetic energy might be. We do feel that there may be serious difficulties with metals of much higher densities than sodium. Whether these difficulties can be solved by structuring the jellium [S] is an open question. We did try a preliminary effort at putting a sodium atom on an aluminum jellium surface.
H.B. Huntington
199
et al.fBinding of adatom to simple metal surface
p
61 4
3
2 7 “? &
t I
= 0.0311
oa N
0
Fig. 8. Charge
contours
with modulations
in z-p
space.
This resulted in too large an equilibrium distance and too small a surface dipole to account for the decrease of work function found by Porteus [ 171 when he deposited sodium on aluminum. We close by indicating some of the directions which studies along this same line might fruitfully take. One would expect no difficulty in obtaining similar results for metals of lower density - primarily the heavier alkalis. No work on surface mobility can be carried out until the jellium surface can be structured at least crudely. Some cleverness will be needed here if the excessive expense of programming three-dimensional integrals is to be avoided. A still further step in this direction would be going from the close-packed planes to those with atomistic serrations. Such investigations might prove very illuminating for the understanding of surface transport and hence surface reactions.
200
H.B. Huntington et al./Binding ofadatom
Appendix A. Evaluation
of self-potentials
The various harmonic charge distortions adatom are of the general form rr,&r, 0) = n&)
to simple metal surface
associated with and centered at the
eaP[ (cos 0) r[,
(A.1)
and inside the jellium metal surface rrrn(rs ,el = cmPI(~0s f? (ys1’ exp t-v6 where E, and em are variational density is n ar = n&)
[l ‘EdPI
J,
tA.2)
parameters. The combined atom and redistribution
(cosO)r
+ EqP2 (cose)2]
(A-3) + fmPl (cosO)r, The atom self-potential sons’s equation
exp(-yr&). associated with one of the distortions
is found from Pois-
02V,(r, 0) = - 4afaPI(COS8),In3s(‘). Eq. (A.4) can be rewritten,
(A.4)
assuming the following separation of variables,
V&C 0) = f,P[ (cos 0) tir),
(A.5)
as 1(2+ 1)
--p(r)
= -
which becomes upon substituting r/f”(F) t 2(1+ I)+’ Multiplying
f(r)
47W$(‘)‘~’
(A.6)
O(r) = r’f(r) = - 4an&)/.
(A.7)
eq. (A.7) by r1+2, we obtain
which by two quadratures
gives at the field point t
(A.9) Thus the atom self-potential
for any 1 is t
I VJ4‘) = 4ne,P1 (cos 0) &
c 0
&I [
(Yy(l+l)
.f
“&)
F
dr’ + j
AZ&‘)
ti:,
1 (k.10)
201
H.3. ~~~ti~gton et al./Sinding of adatom to simple metal surface
from which Vat), V&, V,,, the undistorted, dipole and quadrupole self-potential, are obtained. An analogous procedure holds for the surface adaptations. We obtain for a surface dipole distortion with x z -yZ (A.1 1) The combined
self-potential
is given by (A.12)
v,r = v,O + %d + %q + ‘rnd.
Appendix B. Ascroft-dipole The interaction charge distribution
layer interaction
between the Ashcroft potential is given by
[eq. (3)] and the dipole layer
2
I Ash-de = f ndp adV r
forr>Re.
tB.1)
It follows I Ash-de = 2rrza 7 “dp(ll+Z)ja pI dXzpdp -cm
dz
where
z’ =z-2,
(B.2)
and n dp is 2 > 0
-eazla,
n,p(z) = in0
ez/”
3
Z-CO
with nO the density in the bulk metal. The integration over p gives [pz + (.z’)~]~‘~ - [pf f (z’)~] l”. For large p2 the term at the upper limit can be expanded to give p2 + 1 (z’)~/P~, neither part of which contributes, since (a) the constant p2 can be taken outside the integral and Jn dz vanishes for the dipole layer, and (b) the second part goes to zero as p2 goes to infinity. At the lower limit,
iz'l >R c, p1= 0 and (pi + (z’)*]~‘~ = \.a’[, (B.4) IZ’IGRc,
p1 =
For the integration letZ>R,,then I Ash-dp = 7rZ,no
[R:- (z')~]"~ and [p: + (z’)~]~‘~ = Rc. over z it is necessary to consider two cases separately.
$% -co
lz’\ &’ .+ e-Zb
7’ -z
e-z’,a
iz’\ &’
First
H.B. Huntington
202
+ e-z/a
et aL/Binding of adatom to simple metal surface
RC
s
e-i'/& Rc &‘t
Evaluation
Rc
of these elementary
IAsh-dp
.
(B.5)
integrals leads to
21r.Za11*~a~ exp (-R&)
=
1
,-‘I@ f eda /z’I dz’
-R,
u3.6)
sinh(Z/a).
Similarly for Z
= nZano
Ash-dp
eZya Iz’( &’ + eZILY s
er’lcu R, dz’
-R,
-cc
(B.7) and this gives I Ash-dp
=
2nZ,noo2
[cash (Rc/cu)e-“@
-I]
.
(B.8)
However, the energy which we calculated in the energy minimization is the energy difference for bringing the atom in from infinity. A term 27rZ,noo2 must be added throughout. For Z > R,, EAsh-dp
=
27rZar~~o1~cosh(RJo)
=
2nZ,n0~2
exp (-Z/a),
(B.9
and forZ
[ 1 - exp (-RJa)
Appendix C. Positive jellium-adatom
sinh (Z/o)]
(B.lO)
core interaction
The repulsive interaction, E,, , between the overlap of positive ions of the jellium n,, and the adatom core represented by a coulombic potential must be added in the energy equation (1) as a correction term. The region of integration over the overlap volume, due to the symmetry of the Ashcroft potential, is a spherical cap. We thus obtain for EC,, E
cm
=
s
‘A&Jdv
n+(v&,-
msy
= 2Tn+p z
which is proportional
r sin 0 d0 dr = rrr~+(R,--Z)~,
0
to the square of the jellium-core
overlap distance.
(C.1)
H.B. Huntington
et aLlBinding of adatom to simple metal surface
203
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