Can desorption be described by the local density formalism?

Surface Science 141 (1984) L295-L303 North-Holland, Amsterdam

SURFACE

SCIENCE

CAN DESORPTION FORMALISM?

L295

LETTERS BE DESCRIBED

BY THE LOCAL DENSITY

John P. PERDEW Instrlule for Theoretical Physics, Unroersily o/ California, Santa Barbara, California 93106, USA and Department

of Physics,

Tulane University, New Orleans, Louisiana

70118, USA *

and John R. SMITH Institute /or Theoretical Physics, University of California, Santa Barbara, Cali/ornm 93106, USA and Physics Department, General Motors Research Laboratories, Warren, Michrgan 48090 - 9055, USA ** Received

5 December

1983; accepted

for publication

28 February

1984

The local spin density approximation, commonly used in modern chemisorption calculations, sometimes provides a spurious description of desorption. Adsorbates can desorb with fractional charge, leading to qualitatively incorrect binding energy curves and induced dipole moments at large distances, However, the true binding-energy curves exhibit a universal nature which allows them to be simply predicted, as exemplified by atomic and molecular chemisorption of oxygen on Pt(ll1). The exact density functional explanation for the desorption limit is also presented.

The field of chemisorption theory has moved forward rapidly in the last few years, primarily on the strength of numerous successes [1,2] made possible by the combination of the local spin density (LSD) approximation [3,4] and modern computers. This approximation for the exchange-correlation energy in terms of the local (spin polarized) electron density offers a simple and often accurate description of atomic chemisorption. The calculated binding-energy curve predicts the equilibrium distance of the adatom from the substrate metal surface, the energy with which it is bound, and the frequency of its normal vibration. However, as we will show here, the LSD and related approximations can fail both quantitatively and qualitatively when the atom is far outside the * Permanent address. ** Adjunct Professor, 48109, USA.

Physics

Department,

The University

of Michigan,

0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

Ann Arbor,

Michigan

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metal surface. Specifically, in LSD most open-shell atoms desorb from a metal surface as fractionally-charged fragments. This is very different from the real ground-state (T= 0 K) behavior of an atom with ionization potential IP and electron affinity EA, which desorbs from a metal of work function + as a neutral atom when EA < (p < IP as an ion of charge + 1 when IP < +, and as an ion of charge - 1 when + < EA. We will see presently how this real discontinuity is predicted by the exact density functional theory [3,5] but not by the LSD approximation. Chemisorption aside, the binding-energy or “adiabatic” curve serves as the potential for a low-speed atomic or molecular particle scattered [6] or sputtered [7] from a metal surface. We propose the use of a universal form [8] (with measured or LSD-computed input parameters) as an alternative to the flawed LSD binding-energy curve. In LSD the density around the chemisorbed atom will arise partly from localized orbitals (core levels) and partly from resonances [1.2,9] of the extended metallic orbitals. As the atom .is moved further away from the metal, the resonances will eventually narrow down to sharp “atomic levels”, and the density around the atom will approach a limit which can be thought of most simply as if it arose entirely from localized orbitals, some of which might be fractionally occupied. The orbital energies are related to total energies by a theorem of Slater [IO] and Janak [ll]. Let the self-consistent orbital energies of an isolated N-electron system be ordered as c,(N) < c2( N) < The change in the ground-state total energy E(N) due to an infinitesimal change Sf, in the occupation of the highest partly-occupied (ith) orbital is SE(N)=e,(N)Sf,.

(I)

Consider an atom A which, when desorbed from a metal M of work function 9, would in reality have an integer number J of electrons. The LSD energy change of the combined system M + A when SN, electrons are transferred from M to A is

SE,

+ SEM =

[d’(J)++] Sha (6N,O).

Thus the LSD energy density unless e?+,(J)>

-$>,fJ”(.U.

will be lowered

by an incorrect

(2) transfer

of electron

(3)

i.e., unless the Fermi level of the metal falls between the lowest unoccupied atomic level and the highest occupied one. Moreover, electron density will continue to be transferred until

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Perdew et al. / Description

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i.e., until the Fermi statistics is satisfied, at which point the atom will generally have a fractional charge. Note that large values of the charge-transfer error will be associated only with large values of the electronegativity difference 9 + ]e ,“+ ,( J) + E;( J)]/2, a quantity which is small (for example) when atom and metal are the same element. If the J-electron atom or ion has its least-bound electron lying in an open magnetic subshell with E$‘+,( J) = ET(J), then clearly eq. (3) cannot be satisfied (except for one very special value of +). For example, Cl (configuration 3p’, IP = 13.0 eV, EA = 3.5 eV) really desorbs from an Al(111) surface ($ = 4.25 eV) as a neutral atom (J = 17) but desorbs in LSD as a fragment of charge q = - 0.45 (N* = 17.45). If the J-electron atom or ion has its least-bound electron lying in a half-filled shell with c;+,(J) > ET(J), there is a “window” (often a narrow one) of possible work functions for which eq. (3) can be satisfied. Outside of this window, the adatom will again desorb in LSD to a charged fragment. For example, a Na atom (configuration 3s’, IP = 5.1 eV, EA = 0.6 eV) really desorbs from an Al(lll) surface ($I = 4.25 eV) as a neutral atom (J = ll), but in LSD it desorbs as a fragment of charge q = +0.25 ( NA = 10.75). Similarly, Na really desorbs from a W(110) surface (4 = 5.25 eV) as an ion of charge + 1 (J = lo), but in LSD it desorbs as a fragment of charge q = +0.45 ( NA = 10.55). Fig. 1 (solid curve) shows how the LSD total energy EA + E, minimizes with a fractional charge on A = Na. This figure was generated from self-consistent LSD calculations for the Na atom, in which first the 3sT and then the 3s J orbital were gradually occupied. It was assumed that LSD would yield the measured work function of the metal, an assumption which is probably correct within about 0.1 eV [12]. How might the fractional-charge error affect the binding-energy curve? Consider how the ionic binding-energy curve of Na on W(110) might be constructed in LSD: The total energy of the MA system is calculated for a range of adatom-substrate separations d (usually small enough to prevent the formation of spin-polarization on the atom), and the minimum with respect to d is located. The whole curve is then shifted upward by subtraction of the LSD total energies for the Na ion of charge + 1 and the W(110) surface of charge - 1, separately. In view of our discussion above, the binding-energy curve so defined will go, not to zero as intended, but to some negative constant, if LSD calculations are carried out to the limit d --, co. For Na on W(llO), this constant is -0.6 eV from fig. 1. This is not a negligible part of the measured ionic binding energy [13,14] of Na on W(llO), 2.4 eV. LSD chemisorption binding-energy curves published to date (e.g., ref. [9]) have not been pushed to large-enough values of d to unveil the error discussed here, perhaps due to the numerical inconvenience of a narrowing resonance. (A second error of the LSD binding-energy curve is its abrupt change of slope at that separation where spin-polarization begins to form on the desorbing atom.)

LZ’)X

J. P. Perdew et al. / Descrrpt~on of desorption I

I

/

SUBSTRATE =AC(H!)

SUBSTRATE = W(110)

I 10.5

10.0

I II.0

11.5

ELECTRON NUMBER ON NO ADATOM Fig. 1. Total energy of a sodium adatom and a metallic substrate when the adatom is far outside the surface of the metal (d - co). as a function of the electron number on the adatom. The substrate is either Al(lll) (upper frame) or W(110) (lower frame). Solid curve: LSD approximation. Dashed curve: exact theory. For clarity, total energies have been zeroed for 11 electrons on the sodium.

An incorrect fractional charge q on the desorbed dramatically in the induced dipole moment p=

-

/

d3rzAn(r),

atom will be reflected

most

(5)

where An(r) is the change in density due to introduction of the adatom, and z is the component of r along the direction normal to the surface, with the origin of r at the nucleus of the atom. For q f 0, the induced dipole moment p will tend to 2qd as the distance d of the atom from the surface tends to infinity. A precursor of this behavior is discernible in fig. 9 of ref. [9], which displays p versus d from local density calculations for Na, Si and Cl on an cY= 2 jellium surface (- aluminum). Further evidence for fractional desorption in the local density approximation can be found in fig. 6 of ref. [9], which shows how the Si 3p and Cl 3p resonances narrow and approach the Fermi level as d increases. Einstein, Hertz and Schrieffer [15] have interpreted this behavior as evidence that the local

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density orbital energies cannot be interpreted as ionization energies. Here we are making a different point: These resonances are approaching the Fermi level, i.e., eq. (4) is becoming realized as d --* co, via the retention of spurious fractional charge on the atom. Fractional dissociation can also be obtained in the separation of two gas-phase open-shell atoms [10,16,17]. The problem is not limited to LSD, but extends to related approximations. For example, the Langreth-Mehl [18] generalized gradient expansion is markedly superior to LSD in its description of the energy and density of an atom or a metal surface. However, even in its spin-polarized version which is not yet available, the Langreth-Mehl scheme will be unable to prevent fractional dissociation. Ying, Smith, Kohn and Kahn [19,14] have studied chemisorption via the linear response of the metallic electrons to a monovalent positive ion described by a pseudopotential. In this approach the adatom always desorbs to an ion of charge + 1, which is the correct ground state for Na, K, Rb and Cs on W(llO), but an excited state for H and Li on the same substrate. There is of course an exact density functional theory [3,5] which in principle predicts the exact ground-state density and energy of any electronic system. The way in which this exact theory prevents fractional desorption is shown in fig. 1 (dashed curve). Recall the basic result [16,17] of the ground-level density functional theory of isolated open systems [3,5] with fluctuating electron number: Regarded as a function of the continuous variable N (time-averaged electron number), the energy E(N) is a linkage of straight line segments with possible slope discontinuities at integer values of N. For N between the integers j and j + 1, the ground-level of the N-electron system is a statistical mixture of the j and j + 1 electron ground states. Thus for Na desorbed from Al(lll), EA + E, minimizes with a cusp at NA = 11; for Na desorbed from W(llO), EA + E, minimizes with a cusp at NA = 10. These cusps develop in the limit d + cc from more rounded features. (The desorbed adatom is an “open system”, free to exchange electrons with the substrate “reservoir”. The LSD approximation satisfies an exact sum-rule [17] on the exchange-correlation hole on/__ when the electron number in the open system does not fluctuate, i.e., only for integer electron number. Thus in fig. 1 the LSD and exact curves, forced to agree at N = 11, also agree better at N = 10 than at most non-integer values of N.) There is also an exact Kohn-Sham self-consistent one-electron theory [3] in which eq. (1) remains true [ll]. To see how this theory avoids fractional desorption, consider the case of neutral desorption (EA < 9 < IP). From eq. (I), and from the exact behavior of E(N) described in the preceding paragraphs, the exact orbital energies are [16] e;(N)= c;+,(N)=

-IP -EA

(J-l
(6)

(J
(7)

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If the atom has an open subshell [c,“,, (N)=cT(N)], then c;(N) must jump discontinuously from - IP to - EA as N increases through the integer J. At N = J, c,“(N) can take my value between - IP and - EA; in particular the resonance which tends to e,“(J) can approach the Fermi level - $J as d --* co with no accumulation of fractional charge on the atom. This is achieved by the formation around the desorbed atom of a region of non-zero, “constant” exchange-correlation potential [17]. Obviously, approximations will be hardpressed to reproduce the discontinuous behavior and infinite range of the exact exchange-correlation potential. Despite the significant advances provided by the combination of the LSD approximation and modern computers, one cannot always describe desorption in this manner. Configuration-interaction (CI) calculations should describe desorption correctly, but are prohibitively costly for solids. Methods like the induced-covalent-bond scheme [20], which are computationally less demanding than a full CI calculation, can also yield the correct desorption limit. The discovery of universality [8] provides a simple and accurate way to obtain the binding energy curve for desorption of atoms. Evidence has been provided [8] that the total binding-energy E(a) as a function of the distance a between an adatom and a metal surface can, in many cases, be written as E(u)=AEE*(a*),

(8)

where

u* = (a - u,)/l,

(9)

AE and a, are the equilibrium binding energy (desorption energy) and separation respectively, and I is a scaling length. E*( a*) is a universal function which is valid not only for many cases of chemisorption, but also for bimetallic adhesion and cohesion [8]. The scaling length I can be determined from AE and the second derivative of the binding energy with respect to displacement at equilibrium:

I=

AE

d2E(u)/du21,,

l/2 1 .

(10)

Thus al1 one needs is a computed or measured vibrational frequency and desorption energy to obtain the full binding energy curve. One could presumably use the LSD approximation to obtain reasonably accurate values of both of these quantities. While LSD fails in the asymptotic limit, it is expected to be reasonably accurate around the equilibrium position. For the metal plus separated atom, care must be taken to constrain the electron number on the atom to its correct integer value. The desorption energy AE was correctly computed in this fashion in ref. [9].

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J.P. Perdew et al. / Descriptron of desorption

All that remains be well represented E*(a)

=f*(u*)

wheref*(u*) E*(u*)

= -(l

is to specify E*(a*). by the form

It has been found [8] that ,!?*(a*)

e-O*, is a slowly varying

can

(II) function.

+ a*) ecu*.

The simplest

form is (12)

As an example of E(u), we combine eqs. (8)-(10) and (12) for the case of oxygen atoms adsorbed on Pt( 111). The measured atomic vibrational frequency and desorption energy were taken from the data of refs. [21-231 as described in ref. [24]. The resultant E(u) is plotted in fig. 2 (labeled 0 + 0). Universality has not yet been applied to molecular adsorption. There are unfortunately no first-principles calculations for binding-energy curves of molecules on crystalline solids. Thus one cannot assess the validity of universality in molecular adsorption to the same degree that one can for atomic adsorption. Further, the molecular binding energy curve tends to lose a precise meaning as the molecular and atomic curves approach each other. The measured 0, stretch frequency [21,22] on the surface, 870 cm-‘, is considerably less than that measured for gaseous 0, (1550 cm-‘). One could conclude that

a

-6.0

SEPARATION (nm) Fig. 2. Binding energy of two oxygen atoms (0 + 0) or an oxygen molecule (0,) as a function of distance from a Pt(ll1) surface (g = 5.7 eV), predicted by the universal curve with empirical input. (Although no LSD calculations exist for this system, the LSD error is expected to be rather small: Each 0 atom desorbs with spurious fractional charge q = -0.15, and the LSD 0+ 0 binding-energy curve tends in the infinite-separation limit to -0.3 eV.)

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even in its equilibrium state the bond between the two 0 atoms is weakened upon chemisorption. As the molecular curve approaches the atomic curve, the bond is weakened further and, in the adiabatic limit, the molecule will dissociate when the atomic energy is lower at a given adsorbate position. In addition, the adsorbates can move parallel to the surface as well as perpendicular (so that the binding-energy relation becomes a potential surface and not just a curve). Further, the results are coverage dependent. This is clearly a very complicated situation. It is perhaps instructive, however, to plot the one-dimensional binding-energy curve for 0, chemisorbed on Pt(lll) which results only from a knowledge [21-22,241 of a Pt-0, vibrational frequency at the minimum, an 0, desorption energy, and the universal binding energy relation eqs. (8)-(10) and (12). Here we ignore the previously-mentioned dissociation effects as well as the three-dimensional nature of the potential surface. We also need to estimate the position of the minimum of the 0, curve relative to the 0 + 0 (atomic) curve. By simply adding the metallic radius of Pt to the covalent radius of oxygen, we estimate the atomic oxygen to be 0.126 nm above the plane through the nuclei of the Pt atoms (the oxygen is assumed to be in the three-fold symmetric site). For O,, we refer by analogy to a Pt(0,) complex where the distance between the line through the 0 nuclei and the Pt nucleus was given [25] to be 0.186 nm. If we assume [21-233 that the 0, bridges a Pt atom and lies flat on the surface, then the 0, minimum is 0.186 - 0.126 = 0.06 nm further out than the 0 minimum. The resultant molecular curve is also plotted in fig. 2. It is quite different from the atomic curve, as expected. This is the first time to our knowledge that such binding-energy curves have been determined for either atomic or molecular chemisorption on a transition metal. While the binding-energy curves can be determined very simply, one should bear in mind the cautionary remarks given above when interpreting them. We gratefully acknowledge discussions with D.C. Langreth, G.B. Fisher, D.L. Mills, and J.R. Schrieffer, and support by the National Science Foundation, Grant Nos. DMR 80-16117 and PHY 77-27084.

References [l] For a review, see J.R. Smith, Ed., Theory of Chemisorption (Springer, Berlin, 1980). [2] Applications of the LSD approximation are also reviewed by N.D. Lang, in: Theory of the Inhomogeneous Electron Gas, Eds. S. Lundqvist and N.H. March (Plenum, 1983). [3] W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) A1133. [4] U. von Barth and L. Hedin, J. Phys. C5 (1972) 1629. [5] N.D. Mermin, Phys. Rev. 137 (1965) A1441. [6] J.C. Tully and M.J. Cardillo, Science 223 (1984) 445.

J. P. Perdew et (11./ Descriptron oj desorption

L303

[7] N.D. Lang, Phys. Rev. B27 (1983) 2019. [8] J. Ferrante, J.R. Smith and J.H. Rose. Phys. Rev. Letters 50 (1983) 1385; Phys. Rev. B28 (1983) 1835; F. Guinea, J.H. Rose, J.R. Smith and J. Ferrante. Appl. Phys. Letters 44 (1984) 53. [91 N.D. Lang and A.R. Williams, Phys. Rev. B18 (1978) 616. Field for Molecules and Solids (McGraw-Hill, New York, [lOI J.C. Slater, The Self-Consistent 1974). [ill J.F. Janek, Phys. Rev. 818 (1978) 7165. [121 J.P. Perdew, D.C. Langreth and V. Sahni, Phys. Rev. Letters 38 (1977) 1030. [I31 C.J. Todd and T.N. Rhodin, Surface Sci. 42 (1974) 109. P41 L.M. Kahn and S.C. Ying, Surface Sci. 59 (1976) 333. [I51 T.L. Einstein, J.A. Hertz and J.R. Schrieffer, in ref. [l]. 1161 J.P. Perdew, R.G. Parr, M. Levy and J.L. Balduz, Jr., Phys. Rev. Letters 49 (1982) 1691. 1171 J.P. Perdew and M. Levy, in: Many-Body Phenomena at Surfaces, Eds. D.C. Langreth and H. Suhl (Academic Press, 1984); J.P. Perdew, in: Density Functional Methods in Physics, Eds. R.M. Dreizler and J. da Providencia (Plenum, to be published). [I81 D.C. Langreth and M.J. Mehl. Phys. Rev. 828 (1983) 1809. 1191 S.C. Ying, J.R. Smith and W. Kohn, Phys. Rev. Bll (1975) 1483. WI R.H. Paulson and J.R. Schrieffer, Surface Sci. 48 (1975) 329. 1211 J.L. Gland, B.A. Sexton and G.B. Fisher, Surface Sci. 95 (1980) 587. P21 H. Steininger, S. Lehwald, and H. Ibach, Surface Sci. 123 (1982) 1. ~231 C.T. Campbell, G. Ertl, H. Kuipers and J. Segner, Surface Sci. 107 (1981) 220. ]241 The authors of ref. 1211 reported a value of 490 cm-’ for the Pt-0 stretch and 390 cm-’ for the Pt-0, stretch. The authors of ref. [22] listed a value of 470 cm-’ for the Pt-0 stretch and 380 cm-’ for the Pt-0, stretch. We chose to use the values of 470 and 390 cm-’ respectively, although the plot in fig. 2 is not sensitive to the choice. The molecular and atomic desorption energies were shown to be coverage dependent by Gland et al. [21] and by the authors of ref. [23]. The atomic desorption energy we chose, 3.67 eV, agrees approximately with the results of refs. [21] and [23] over a range of coverage from 0.3 to 1.0 monolayers. The molecular desorption energy, 0.38 eV, is also representative of the data [21] for 0.6 to 0.8 monolayers. 1251 J.G. Norman, Jr., Inorgan. Chem. 16 (1977) 1328.