J . Phys. Chem. 1985,89, 5576-5586
5576
FEATURE ARTICLE A Theoretical Approach to Heterogeneous Catalysis Using Large Finite Crystals Lionel Salem Laboratoire de Chimie ThPorique (UA 506), Universitt? de Paris-Sud, 91405 Orsay, France (Received: August 6, 1985)
We describe a theoretical approach to heterogeneous catalysis using large finite crystals and an exactly soluble model. We first review, for chemists, some themes which are well-known to physicists but need a “translation” into chemical language: wave vectors, the tight-binding model, and energy bands. Next we describe the finite simple cubic crystal and its analytical wave functions and energies in the Huckel scheme. We also give the analytical Hiickel wave functions for a finite face-centered cubic (FCC) crystal cut along square, (100)-type faces. We then calculate the perturbation interaction energy between H2 and large finite (simple cubic or FCC) crystals of Ni atoms, having up to 13 824 atoms. The interaction energy is shown to be independent of crystal size, whatever the position of attack of the H2 molecule.
1. Introduction Heterogeneous catalysis-the adsorption and dissociation of chemical bonds on metal surfaces-has increasingly struck the fancy of chemists1%* in recent years. The dissociation of the simplest molecules, such as H,, on metals such as nickel or platinum requires no activation barrier. More than 50 years ago Lennard-Jones proposed3 a potential energy diagram to account for the vanishing activation energy (Figure 1). In essence the potential curve 1 for the metal M interacting with the separate atoms A and B has a deep minimum. The curve 2 for the interaction between the metal and the molecule AB is essentially repulsive, with only a small minimum at a large distance from the metal. If the metal-atom cohesive energy is sufficiently large, the curves intersect at K. A molecule may then approach the metal along curve 2 and switch to curve 1 at point K, whence the dissociation without activation energy. Actually a third dimension (the AB distance) is missing in Lennard-Jones’ diagram and curves 1 and 2 are sections of a complicated three-dimensional surface. The Lennard-Jones scheme is only a simplified “after-the-fact” formulation of the effect; it does not provide us with a mechanism for catalysis. The mechanism of catalytic dissociation on metal surfaces remains a mystery. However, the efforts of physicists and chemists alike have helped to lift the veil shrouding the problem. We will mention but only a few of the most striking results. The jellium model, with an electron density for a constant background of positive charge (smeared metal ions), has been used4 to demonstrate the importance of the filling of adsorbate antibonding “affinity levels” as they fall below the Fermi level of the metal. Confirmation that a*-metal mixing predominates has been obtained5 for the activation of H2 and CH4 on Ni (1 11) surfaces. Grimley, however, has warned6 against the deficiencies of the (1) ‘Catalyse par les Metaux”; Imelik, B., Martin, G.-A., Renouprez, A.-J., Eds.: Editions du CNRS: Paris, 1984. (2) ‘Proceedings of the 8th International Congress on Catalysis”; Verlag Chemie: Weinheim, 1984; Vol. I-V. (3) (a) Lennard-Jones, J. E. Trans. Faraday SOC.1932, 28, 233. (b) Robertson, A. J. B. “Catalysis of Gas Reactions by Metals”; Logos Press: London, 1970; p 40. ( c ) Chem. Eng. News, 1980, 34. (4) (a) Lundqvist, B. I.; Gunnarsson, 0.;Hjelmberg, M.; Norskov, J. K. Surf.Sci. 1979, 89, 196. (b) Lundqvist, B. I.; Hellsing, 8.; Holrnstrom, S.; Nordlander, P.; Persson, M.; Norskov, J. K. In?. J . Quantum Chem. 1983, 23, 1083. (5) (a) Saillard, J.-Y.; Hoffmann, R. J . Am. Chem. SOC.1984, 106, 2006. See also: (b) Shustorovich, E.; Baetzold, R. C.; Muetterties, E. L. J . Phys. Chem. 1983, 87, 1100. (c) Shustorovich, E.; Baetzold, R. C. Science 1985, 227, 876. See, however: (d) Ruette, F.; Hernandez, A,; Ludena, E. V. Surf. Sci. 1985, 151, 103.
0022-3654/85/2089-5576$01.50/0
one-electron model; the singly occupied level of an adsorbed hydrogen atom receives a second electron if the level falls below the metal Fermi level; however the two-electron H- system rises straight back above the Fermi level, whereby it loses the second electron, and falls back down again. Clearly a self-consistent solution is needed with intermediate filling of the level. Cluster models, in which the bulk metal is replaced by a discrete cluster of atoms-of sufficient size so that the metal atoms at the dissociation site have at least all their first nearest-neighbors-can be used to mimic catalytic dissociation by ab initio calc~lations.~ Finally, a model has been proposed* for the dynamics of dissociative diatomic molecular adsorption; it involves potential energy surfaces for the neutral molecule AB, an adsorbed negative ion AB-, and two dissociatively adsorbed atoms (A + B). The negative ion surface corresponds again to those classes of molecules which accept electrons from the metal into vacant antibonding levels. Clearly a coherent explanation of heterogeneous catalysis will require a combination of the physicist’s knowledge of band theory and the chemist’s intuition for orbital interactions. Our purpose is to present an approach which first translates some important solid-state physical properties into chemical language and second uses perturbation theory to calculate the interaction energy between H2 and large finite crystals of nickel atoms.
2. The Wave Vector The standard procedure for describing the electronic properties of periodic crystals is to write out “Bloch sumsn9for the electron at position r
\kk(r) = Ce”.R$g(r - R,) R.
(1)
where the sum is to be extended over the atoms in equivalent positions in all the unit cells of the crystal; each atom at position R, carries an atomic orbital & with quantum numbers symbolized by the subscript g. The atomic orbital coefficients eR*Rnare the values of the plane wave e*.R at the atoms, and the wave vector k indicates the direction of the plane wave and its momentum. The wave vector is generally characterized by its components
0 = ak,,
C#J
= ak,,
x
= ak,
(2)
(6) Grimley, T. B. J . Phys. 1970, 31, Suppl. C1, 85. (7) (a) Avdeev, V. I.; Upton, T. H.; Weinberg, W. H.; Gcddard, W. A. Surf. Sci. 1980, 95, 391. (b) Siegbahn, P. E. M.; Blomberg, M. R. A.; Bauschlicher, C. W. J . Chem. Phys. 1984, 81, 2103. (8) Gadzuk, J. W.; Holloway, S. Chem. Phys. Lett. 1985, 114, 314. (9) Bloch, F. Z. Phys. 1928, 52, 5 5 5 .
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5577
Feature Article
/
Energy
1
M+AB
Distance from
Metal
Figure 2. Basic types of resonance integrals for d and s atomic orbitals.
Figure 1. Interaction of a molecule AB with a metal M according to Lennard-Jones and Robertson.' D is the dissociation energy of molecule
AB.
on a set of orthonormal coordinate axes x , y , zIo. In eq 2 a is the lattice spacing-for simplicity the nearest-neighbor distance in a simple cubic lattice (the nearest-neighbor distance for the FCC lattice is then 21/2aand for the BCC lattice 3Il2a). The angles 0, 4, and x vary from 0 to T. In a finite simple cubic crystal with N atoms per side, each molecular orbital is characterized by three indices j , k , I which define the "wave vector" associated with the stationary molecular orbital:"
e=4 = - Nk?r +l' N + 1' jT
IT
x=- N +
/
1
(3)
I
SS' SS' SS' SS'
SS' SS' SS'
0 CC' CC' CC'
CC'
0 SS' SS' SS'
"'
"'
cc' CC'
cc' CC'
CC'
CC'
here the abbreviation CC'stands for ccf= 4p, COS o COS 4 4px, COS e COS
+
I
FO
(6)
I
+ 4pyzCOS 9 cos (7)
where pXyis the resonance integral between two atomic orbitals adjacent to each other along the ( x - y ) axis, etc. Similarly SS' = -4/3,, sin 0 sin
x
(8)
for ( x z l M s ) , ( x z 1 4 x 2 - y 2 ) , (xzlM3z2 - r 2 ) ,or ( x y l m z ) . A typical matrix element is that linking an s wave to a (3z2 - 9)wave with identical wave vector. Figure 2 shows the different basic types of resonance integrals for d and s functions. For an s orbital surrounded by 12 nearest-neighbor zz orbitals in the directions of the bisectors of the x , y , and z axes (01 1, 110, etc.) eq I gives (slH13z2 - r 2 ) = Psd(-2 cos 0 cos 4
+ cos 0 cos x + cos 4 cos x) (9)
This matrix element is particularly important because it determines the extent of s, d mixing (section 4).
4. Energy Bands (the Chemist's Viewpoint)
C C C C C C
c c c
c c c
50
(4)
(IO) In an orthogonal coordinate system with unit base vectors the components of k will be identical in the direct lattice and the reciprocal lattice. (1 1) Salem, L.; Leforestier, C. Surf.Sci. 1979, 82, 390. (12) Slater, J. C.; Koster, G. F. Phys. Rev. 1954, 94, 1498. (13) For a discussion of this approximation, see ref 7b.
Figure 3 shows the energy bands of nickel14 as a function of the wave vector k (O,O,k,) in the interval 0 d x C T. Instead of the group-theoretical labels proper to solid-state physics, we have introduced chemical labels which define the nature of the atomic orbital (s) corresponding to each band. These chemical labels are obtained" by drawing from our knowledge of symmetry and from the indications given by the Slater-Kaster determinant. First of all the Slater-Koster determinant for a (O,O,k,) wave vector is fully diagonal except for the (z2,s)part in which off-diagonal terms subsist; at the point (O,O,O)even these off-diagonal terms disappear. Hence four of the energy surfaces must correspond respectively to pure xz, xy, y z , and x2 - y 2 waves. Now xz and y z remain degenerate throughout the interval ( 0 , ~ )so that they can be ascribed unambiguously to the pair of degenerate energy curves. Then xy must be the third band which coalesces with the two others at x = 0. The top two levels for x = 0 must then be the degenerate pair ( x 2 - y 2 , 3z2 - r2). One level stays pure ( x 2 - y 2 ) for all values of x. Examination of the Slater-Koster diagonal energy shows that it must be the top slightly ascending level. Finally reasonable values of the resonance integrals (see section 8) show that the lowest diagonal energy for x = 0 and the highest for x = T correspond to the s wave. We can therefore give a label s to the lowest level on the left and predominantly (14) Figure 3 is inspired from: Wakoh, S. J. Phys. SOC.Jpn. 1965, 20, 1894, and from: Moruzzi, V. L.; Janak, J. F.; Williams, A. R. 'Calculated Properties of Metals"; Pergamon Press: New York, 1978; p 92.
5578 The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 En
Salem
Y
/
Ef
“I Figure 3. Calculated energy bands for nickel (average of spin-up and spin-down values)14 along the wave vector k (0, 0, kz). The levels s and z 2 are pure for x = 0 but the corresponding orbitals are hybridized at x = T.
to the highest on the,right. The lowest level on the right should then be predominantly z 2 . We see that the relative position of the z2and s levels is reversed on passing from x = 0 to x = T . Thus, were it not for the ( s , z 2 ) matrix elements, the corresponding energy surfaces would intersect. As things stand, the intersection is avoided. The (s,z2)avoided crossing shows up in the form of a valley on the upper surface and a bump on the lower surface (Figure 3 ) . Figure 4 shows this avoided crossing in more detail, with the intended intersection of the purely diagonal terms shown in dotted lines. Now the matrix element for mixing between s and z2 is given by (9),which reduces here to
(slH13z2- r 2 ) = &(-2
+ 2 cos x)
(10)
It is always positive (& < 0 ) . Hence the lower energy surface corresponds to a wave made of negative s - z2 hybrids while the upper surface corresponds to a wave made of positive s + zz hybrids, which point strongly along the z axis.
5. The Finite Simple Cubic Crystal Let us now consider afinite simple cubic lattice with, say, N atoms along each of the directions x , y , and z . We first assume that each atom carries a single orbital (an s orbital, or a given p-type orbital, or a given d-type orbital). Such a model is exactly soluble within Huckel theory as first shown by Baldock,I5and later independently by othersI5 including ourselves.” Indeed, let us assume, in a first approximation, that each atom interacts only with its six nearest-neighbors, with interactions ox, by, and &, respectively, along the three orthogonal directions. Since the interactions along x , y , and z are independent, the problem is equivalent to that of three independent linear polyenes (15) (a) Baldock, G. R. Proc. Phys. SOC.,London, Sect. A 1953, A66, 2. (b) Hoffmann, T. A. Acta Phys. Hung. 1952,2,97 (but does not give the wave functions). (e) Messmer, R. P. Phys. Reu. 1977, B I S , 1811. Messmer, R. P. In “The Nature of the Surface Chemical Bond”; Rhodin, T. N., Ertl, G., Eds.; North-Holland: Amsterdam, 1979; Chapter 2, pp 65-67. (d) Bilek, 0.; Skala, L. Czech. J. Phys., Secr. B 1978, B28, 1003.
Figure 4. Avoided crossing between s and z2 levels. The purely diagonal terms are shown in dotted lines (drawn approximately).
with N atoms. The total wave function is then the product of the corresponding wave functions $jk/
e=-
jT
=
(
(sin r6 sin s4 sin t X ) @ , s r &)3’2
r,s,t
kr
(1 1)
IT
d = -N + l ’ x=1I j,k,l I N (3) N + 1’ N+ 1 where r, s, t label the atom and j , k , 1 characterize the wave vector associated with the molecular orbital $. By the same token the Huckel energy is given by cjjkl = 2px cos 6 + 2py cos 4 + 26, cos x (12) As in the infinite crystal, the off-diagonal matrix elements between orbitals of different wave vector (jkl, j’k’l’) vanish. When several atomic orbital types are present on each atom, both diagonal and off-dia8onal matrix ejements pf H,for instance ( i q k / I m q k l ) ? (!q&/IWflW), and (qw4WqwJ will appear. These matrix elements, for the finite crystal, are identical with the Slater-Koster matrix elements for the infinite crystal with the simple restriction that the wave vector components are given by ( 3 ) instead of (2). In particular there is still no mixing between waves of different wave vectors. Thus replacing the traveling plane waves of an infinite simple cubic crystal by the stationary sine waves of a finite simple cubic crystal has no consequence on the mixing between the waves. 6. The Finite, Square-Cut, FCC C r y ~ t a l ’ ~ ~ ~ ~
We now turn to a finite, face-centered cubic (FCC) crystal, (16) The first (but unknown to us at the time of our work17) determination of the molecular orbitals of finite FCC crystals with (100) surfaces is due to: Bilek, 0.; Kadura, P. Phys. Starus Solidi B 1978,85,225. Bilck and Kadura decompose, like we do,” a simple cubic crystalliteinto two noninteracting FCC sublattices, one with solution (13) and one with solution (13) but changed sign (corresponding to the starred lattice of Figure 5). Bilck and Kadura, however, restrict themselves to the case of one (s) orbital per atom. See also: Bilek, 0.;Skala, L. Czech. J . Phys., Sect. E 1978, B28, 1003 where condition 13b is spelled out. (17) Salem, L.; Leforestier, C. J. Am. Chem. SOC.1985, 107, 2526. Equation 7 of this paper should read N instead of 2N in the normalization factor.
Feature Article
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5579
a
V
x
13
/
/
/n
/
b
L
n
133
333
Figure 5. Numbering of atoms for the finite FCC crystal. The starred atoms belong to the master simple cubic crystal from which the FCC crystal is derived by deleting one atom out of two.
assumed to be cut along square, (001)-type faces. It is convenient to consider this crystal as deriving from a “master” simple cubic crystal in which one atom out of two-the “starred” atoms-has been crossed out (Figure 5). We keep the same principle for atom numbering as in the master cube: the indices r , s, t increase respectively by 1 for each unitary translation along the x, y , and z directions marked by the edges of the cube. The FCC “unstarred” atoms then have the numberings shown in Figure 5. When going from one atom of the FCC crystal to its nearestneighbor, the sum of the atomic indices varies by 0 or 2, instead of 1 in the simple cubic crystal. Let us now first assume that each atom carries a single (s, p, or d) orbital. We then take a central atom surrounded by its 12 nearest-neighbors, as shown in Figure 6a for s atomic orbitals. If we try again a sine solution of form ( l l ) , we find in a straightforward manner elk/
= 40, cos 6 cos $ + 40x2 cos 6 cos
x
+ 4py~cos $ cos x
which is exactly the Slater-Koster diagonal energy (7) for s orbitals in the infinite FCC crystal. The same result holds if r, s, t is a face atom, an edge atom, or a corner atom. Thus sine waves seem to be potential analytical solutions for the MO’s of the finite FCC crystal. However, we cannot choose any arbitrary ensemble of sine waves. The FCC crystal has half as many atoms as the master cubic crystal from which it derives (Figure 5). We must therefore choose combinations of the master-cube functions which vanish on the extra starred atoms. Now it is a remarkable fact that the extra atoms of the master cube and the true atoms of the FCC crystal are like the starred and unstarred atoms of an alternant hydrocarbon.ls To cancel the atomic orbital coefficients on one set of atoms it suffices to remember that the M O s of alternant hydrocarbons occur in pairs, with energies e and -e and with a changed sign on one set of atoms.I8 Hence, in terms of the solution tjJ?/ of the simple cube given in (1 l ) , the exact solution for the finite, square-cut, FCC crystal are 1 tjyk,C = master cube + master c u k 21p(+jk/ $N+l-j,N+l-k,N+l-/)
3 Ij + k
+1I3(N+
1)/2
( l 3,
(13b)
(18) Coulson, C. A.; Rushbrooke, G. S.Proc. Cambridge Philos. SOC. 1940,36, 193. (b) Longuet-Higgins,H. C. J. Chem. Phys. 1950,18,265. (c) Hall, G. G. Proc. R. SOC.1955, 229, 251. (d) Salem, L. “The Molecular Orbital Theory of Conjugated Systems”; W. A. Benjamin: New York, 1966; Section 1-9.
-v
r-l,s-l,t
r +l,s-1,t
Figure 6. Nearest-neighbor atomic orbital interactions in a FCC crystal (a) identical atomic orbitals; (b) different atomic orbitals with matrix element of constant sign in each plane x z , yz, xy; (c) different atomic orbitals with matrix element of variable sign in a plane.
Equation 13 can also be written tj;c =:
- (-l)‘+”+‘](sin r8 sin s+ sin tx)arsr
which is nonvanishing only on those atoms (numbered in the figure) for which the sum r s t is odd. In (14) N is the number of atoms per edge in the master cubic crystal. If we now allow for several different types of atomic orbitals on each center, the problem becomes more complex. Two cases arise: (1) (Figure 6b) The matrix elements between a central orbital (s, say) and its twelve neighbors (z2, say) have a constant sign in each of the three planes xz, yz, and xy of interaction. T h e n , the corresponding molecular orbitals $&t,t)of form (14)mix only if they correspond to the same wave vector (jkl). The mixings are again identical with the Slater-Koster mixings12 for the infinite FCC crystal with the usual restriction that the wave vector components are given by (3) instead of (2). Sums of cosine products similar to (7) are obtained. The situation for the finite FCC crystal is then analogous to that for the finite simple cubic crystal. (2) (Figure 6c) The matrix elements between a centra1 orbital (s,say) and its nearest-neighbors in a plane (xy, say) change sign
+ +
(ek,,
5580 The Journal of Physical Chemistry, Vol. 89, No. 26, 1985
in the plane from one pair of neighbors to another. Then a complicating factor arises: molecular orbitals corresponding to different wave vectors (jkr) and (j’k’l’) mix together. Let us work out the mixing for the case given in the figure (primed variables refer to j’k‘l’, unprimed variables to j,kJ):
( s l ~ l x ’ y ’= ) -@,Csin
r0 sin s+[sin ( r
IS1
1)0’][sin (s
+ l)#’-
= -4@,, sin 0’ sin 4’6,,.C(sin
+ 1)W
-
Salem
5 3 3 3 -
41
sin ( r -
sin (s - 1)41 sin t x sin tx’ r0 cos rO’)(sin $4 cos 343
3t
---
i
---
2 3 3
3 2 3
3 3 2
rs
(15)
If 0 = 0’ or 4 = 4’ the sums Csin r0 cos r0 = C(sin 2r0) = 0
(16)
r
r
+
vanish because contributions from r and r’ = N 1 - r cancel out (Le., the mixing at one position in the crystal is cancelled out by a mixing of opposite sign at another position in the crystal). Hence the matrix element vanishes if the wave vectors are equal or at least have two identical components ( x = x’ and 0 = 0’ for instance). If on the contrary 0 # 0’ and 4 # $’(while x = x’ as required by the term 6x; in (1 5 ) ) , ( 1 5 ) can be evaluated for large crystals (N m ) by using the transformation
r
221
2 3 2
3 2 2
113
313
3 3 1
321
312
231
--231 113
1 3 1
1 2 3
3 1 2
221
1 2 2
311
’t 01123
1 3 2
213
2 2 2
222
---
-1.
-211
122
2 1 2
1 1 3
131
31 1
--1 1 2
-27
1
_--
-4t
-
-3
212
112
121
1 2 1
211
211
111
111
where j and j‘are integers. Whence
(slHlx’y’)
= -4p,,.
sin 0’sin @’(I - (-I)’+’’) x
SIMPLE
CUBIC
FCC
Figure 7. Orbital energy levels for the 3
X 3 X 3 simple cube and for the associated face-centered cube (energies in units of &). Levels are labeled by indices j , k , 1.
A nonzero matrix element mixes the waves s(jk1) and x‘y’(j‘k‘19 as long as both j & j ‘ and k f k’are odd and 1 = 1 ‘. This result is noteworthy in three respects: The factor -4p, sin 0’sin 4’ linking the MO’s with different wave vectors is strikingly similar to the full matrix element -4p, sin 0 sin 4 found by Slater and Koster (eq 8) for MO’s with identical wave vectors; the mixing between orbitals with different wave vectors in the finite fcc crystal is the price which must be paid for having a lower translational symmetry in the finite fcc crystal than in the finite simple cubic crystal; the requirements for off-diagonal mixing are relatively stringent; they imply that within certain families of orbitals, no (jkllH[l’k’l’) mixing will occur whatsoever. A case in point is any family of orbitals which have indices: jarbitrary kfixed parit) h x e d parity
(19)
5 ) . We assume that each atom carries a simple s orbital. The cube has 27 atoms, the face-centered cube 14 atoms. From eq 5 and 7, the energies in units of &,, the resonance integral for s orbitals, are given by cslmpIe cublc = -2(cOs 0 cos 4 cos x)
+
CFCC
= -4(COS 0
COS
4 + COS 0 COS x
+
+ COS 4 COS X )
(21)
where 0, 4, and x are given in eq 3 with N = 3 (so that cos 0 takes on the values 1/2’12, 0 or -1/2’12). The restriction of eq 14b, 3 a / 4 I0
+ 4 + x I3a/2
applies to the fcc lattice. The corresponding energy patterns are shown in Figure 7 . The most striking feature is that, whereas the levels of the simple cubic lattice are disposed symmetrically about c = 0, in the fcc lattice there is a large density of levels at the top of the band.
jfixed parity karbitrary lfixed parity
8. Tight-Binding Parameters for Nickel. Comparison with Extended Hiickel Parameters
jfixed parity kRxed parity [arbitrary
Indeed for each family all three products of type (1 - (-l)j+f)(l - (-l)k+k‘)b/[,
(20)
(1 - (-1y+j’)6kk,(l - (-1)“’’) 6,!(1 - (-l)k+k’)(l - (-l)‘+Y) which arise in different matrix elements such as (slHlx’y’), (slHlx’z’), (slHly’z’) vanish. Is suffices to remember that (1 (-l)kfk’) = 0, for instance, if the index k has a fixed parity. 7. Density of States
Let us compare the density of levels for a small ( 3 X 3) simple cubic array of atoms with that for the related FCC lattice (Figure
In 1965 Wakoh c a l ~ u l a t e dthe ’ ~ energy bands of nickel using a self-consistent field method to relate the wave functions to the crystal potential. The average of spin-up and spin-down energies at the points of high symmetry r(k, = 0, ky = 0, k, = 0), A(k, = 0, ky = 0, k, = a/(2a)) X(k, = 0, k, = 0, k, = ./a) provide us with 13 numerical values. To these 13 numerical values correspond 13 analytical formulas which are easily derived from the appropriate Slater-Koster determinant (eq 2 of ref 11). But there are only seven unknown parameters: do and so, the diagonal Coulomb energies of d and s atomic orbitals, and the five resonance integrals Po, a,, ba,&d, and p,, (Figure 2). In order to obtain the (19) Wakoh, S. J . Phys. SOC.Jpn. 1965, 20, 1894.
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5581
Feature Article
TABLE I: Coulomb and Resonance Integrals (in eV) for d and s Atomic Orbitals of Nickel method tight-bindinga extended Hiickelb -0.901 -0.463 +0.637 +0.306 -0.118 -0.154 (-0.3 14) (-0.231) -0.828 -1.26 -2.36 -0.607 -5.158 -9.9 -7.8 -5.994
’
AI
N’i’
b
a
“Tight-binding method (fitted to WakohI9). bExtended Hiickel method.20 The parameter Padj 22, the resonance integral for two adjacent z2 orbitals side-by-side, is given by (6, + 3P6)/4-see Figure 2-and is not an independent parameter. best values of these parameters, we minimized the standard deviation between the Wakoh numbers and the numbers derived from the Slater-Koster equations (using tentative parameter values) by a gradient technique. The results are shown in Table Ia. Here we sound a note of warning. The importance of selfconsistency is an electronic structure calculation pertaining to a solid surface (where catalysis is ”happening”) is much greater than for a bulk structure calculation. This would show up in the form of different tight-binding parameters from those which we have inferred from bulk bands. It is interesting to compare the fitted Coulomb and resonance integrals with those provided by extended Hiickel theory.20 For doand so we refer to the recent paperSaby Saillard and Hoffmann. For the resonance integrals extended Hiickel theory adopts the Wolfberg-Helmholz formula
for orbitals i and j with overlap S , and Coulomb integrals, respectively, Hii and H,j. The results are shown in Table Ib. The discrepancy is important in two respects: the tight-binding Coulomb integrals are significantly higher (by 2-4 eV) than the extended Hiickel ones; the resonance integrals involving Slater orbitals are larger (by a factor of 1-4) than those obtained from the S C F calculation in the solid. This indicates that the Slater orbitals may extend further in space than is realistic.
C
Figure 8. Approaches of H2 to the surface of a finite lattice of nickel atoms. We have considered three different possible approaches of an H2 molecule to the surface (Figure 8): (a) parallel, on top of a Ni-Ni bond, (b) parallel, on top of a Ni atom, and (c) perpendicular, on top of a Ni atom. Let us first consider approach a. The interactions between H2 and the y z and xy orbitals of nickel vanish by symmetry. We neglect the interactions between H2 and the xz - y 2 orbitals, which have little amplitude in the direction of the substrate. We denote respectively by flxr,h, f122,h, Bs,h the resonance integrals for interaction between a hydrogen 1s orbital and xz, z2,or s on the adjacent nickel atom. Then the second-order interaction between nH2*and the filled orbitals of nickel is the sum
wI
2
- 2-2Shh
full
c
jk/
(cIxz+ CZXZ)2@xz,hZ
-
Ea.-El - C2z2)bpz2,h+
full [ ( C l z 2 2 E 2 - 2Shh J k l (primed term, with E , )
+
(Cis
- c~)cp~,h]z
+
E,, - E, (double-primed term, with E6) (24)
In (24) clxzrefers to the coefficient of molecular orbital $ , k y on the first nickel atom, cIz2to the coefficient of molecular orbital $Jjkr2on the same atom, etc. For given j , k , 1, eq 11 applies to all types of molecular orbitals so that clxz E c1z2 E
E
(25)
C1
The total perturbation energy can then be expressed as the sum of four terms
9. Interaction between H2and Large Finite Simple Cubic Nickel Crystalsz1 The energy levels of the finite cubic crystal are given by the solutions of (4); the first three levels correspond to pure eigenfunctions x z , y z , and x y while the other three are obtained by diagonalizing a 3 X 3 determinant between x 2 - y2, 3z2 - 9 and S:
E,
-
$1
xz
w,=
(23)
w4 = E4 E,
E6
-
-
-+
q4= a(x2 - y 2 ) + b(z2)+ c(s)
$,
$6
= ar(x2- y 2 ) + b’(z2) + c’(s) = U”(XZ - y 2 ) -k brr(z2)+ C r r ( S )
The diagonalization program arranges the solutions in the order of increasing index (E4 < E5 < E 6 ) . All energies are functions of j , k , and I via 8, @, and x. (20) Hoffmann, R. J . Chem. Phys. 1963,39, 1397.
-1 - Shh
Jk/
Similar expressions exist for the perturbation energy of interactions between the filled uH2molecular orbital of the adsorbate and the empty metal orbitals. They will not be given here, since throughout our calculations this energy was found to be an order of magnitude smaller than that for filled metal empty adsorbate orbital interactions. However we give the results for the two other approaches, in the same approximation. For approach b
-
5582
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985
TABLE II: Numerical Interaction Energies (Filled Metal-Empty UH,*) for Approach a, in eV-’ (Simple Cubic Lattice) I. Approach to a Central Bond 8 16 24 N no. of atoms 512 4096 13824 3072 24576 82944 no. of levels bond label 4.4,1-5,4,1 8,8,1-9,8,1 12,12,1-13,I2,l coeff of 0.2194 0.2143 -[Pxz,h2/(1 - Shh)l 0.2152 0.1467 0.1524 -[Pz2.h2/(1 - Shh)] 0.1375 0.1103 0.1131 0.1034 -[Ps.h2/(1 - shh)] 0.0522 0.0483 -[br2,h’br,h/(1 - Shh)] 0.0549 no. of surface states 8 56 208 11. Approach to an Edge Bond 8 16 I,l,l-2,1,1 1,1,1-2,1,1
N
bond label coeff of -[Pxz,h2/(I - Shh)] -[Pz2,h2/(1 - shh)] -[Ps.h2/(l - shh)] -[Pr2,h’Ps.h/(l - Shh)l no. of surface states
0,2179 0.1389 0,1143 0.0409 0
0.2201 0.1468 0.1191 0.04 17 0
w, = w3 = w4
0.2186 0.1616 0.1287 0.0298 0
Y)
Ys,h
2
1-Shh
co2( jkl
(26c)
C’* +E,. - E4 E,* - E5 C2
E,* - E6
Before proceeding further we must ensure that it is legitimate to use second-order perturbation theory. This appears to be indeed the case, the empty and full levels of H2 lying respectively at 4.25 and -17.57 eV while the extrema of the metal bands are well aproximated by the highest and lowest possible energies of $,: so - 6p,, = -2.4 eV SO
+ 6ps, = -9.6
(27)
eV
In practice the uppermost s levels are pushed up slightly by mixing with z2 and x2 - y 2 , but inspection shows that no metal level appears above so - 8p,, = -1.14 eV
I. Approach to a Central N 8 no. of atoms 512 no. of levels 3072 atom label 4,4,1 coeff of -[@xz,h2/(l- Shh)l“ 0.4203 -[Y2rzh2/(1 - Shh)lb 0.0807 -[?>,h’/(I - Shh)lb 0.0633 -[Yi2,h‘Ys,h/(l - Shh)lb 0.0190 no. of surface states 8
Atom 16 4096 24576 83,1
24 13824 82944 12,12,1
0.4258 0.0816 0.0621 0.0157 56
0.4204 0.0825 0.0625 0.0133 208
11. Amroach _ . to a Corner 8 no. of atoms 512 no. of levels 3012 atom label 1,1,1 coeff of 0.4234 0.0818
Atom 16 4096 24576 1,1,1
24 13824 82944 1,1,1
0.4261 0.0858 0.0731 0.0041 56
0.4252 0.0882 0.0745 0.0002 208
no. of surface states
where co is the coefficient of orbital qjk1 on the nickel atom under attack. For approach c (the resonance integrals are denoted by
w3=--
TABLE 111: Numerical Interaction Energies (Filled Metal-Empty uH,*) for Approaches b and c, in eV-’ (Simple Cubic Lattice)
N
24 l,l,l-2,l.l
=0
w, = 0
Salem
(28)
Hence the entire band of metal levels is well separated from the two adsorbate levels.22 Our procedure is fairly simple. The cubic lattice has N atoms per side. There are therefore 6N3 levels, which should all be contained in the limits (so + 6&, so - 8@,). We divide the 114&I overall bandwidth into N intervals, and for security add one interval at each end. We now choose an interval and run through all possible wave vectors (j = 1, ..., N,k = 1, ..., N,I = 1, ...,N). For each wave vector the program calculates the six energy levels: three of them (EI-E3, (eq 23)) are obtained directly from the Slater-Koster diagonal terms while the other three (E4-&) require (21) For previous calculations see: (a) Einstein, T. L.; Schrieffer, J. R. Phys. Reu. E . 1973, 7, 3629. (b) Van Doorn, W.; Koutecky, J. Int. J . Quanrum Chem.1977, 12, Suppl. 2, 13. (c) Salem, L.; Elliott, R. J . Mol. Struct. (THEOCHEM)1983, 93,15. (22) The same holds true for the finite FCC crystal, even though the lower bound of the metal band is now so + 12& (-13.2 eV) and the upper bound no greater than so - 8@,, again (diagonal energy 50 - 4&)
Approach b.
0.0714 0.0048 8
Approach c.
diagonalization of a 3 X 3 matrix (see (4)). It is then straightforward to see whether any (or all) of the six levels are included in the interval under consideration. Whenever a level is found to belong to the interval, its contribution to the interaction energy is calculated. For instance a +rtype level will contribute to all three terms W,, W3,and W4 (eq 26a). On the other hand, q2and q3-type levels give no contribution whatsoever to the interaction energy. Of course the total number of levels in each interval is counted, and a count is also made of the total number of levels included since the beginning. When the latter reaches 5N3, the Fermi level is reached since there are now 10 times as many electrons as there are atoms. The interactions of type 26a (filled metal level empty adsorbate level) are then switched off while the opposite interactions (empty metal level filled adsorbate level) are switched on. For N = 24, the Fermi level is level number 69 120, lying in the interval (-4.32 eV, level number 64 209; -3.97 eV, level number 70 274). We therefore estimate by interpolation the Fermi energy to be close to -4.05 eV. This agrees well, not unexpectedly, with Wakoh’s value of -3.77 eV and with the experimental valuesz3of -5.22 and -4.75 eV for the work functions of clean (100) nickel surfaces. Table I1 shows the results for approach a-whether on a central bond of the metallic surface or on a corner bond-while Table I11 shows the results for approaches b and c. The numbers (in units of eV-I) are the coefficients of the respective factors &,,h2/( 1 - &,), etc. which multiply the perturbation sums in (26a), (26b), and (26c). Except for the coefficients of the cross terms (4th line of each subtable), which show a slight variation with N but are very small, the energies are remarkably constant as a function of N, with the values for 13 824 atoms hardly differing (1-10%) from the values for 512 atoms. Hence there is no apparent effect of crystallite size. We have checked further this insensitivity of the energies to N by varying the depth of the finite cubic lattices. Table IV shows how the interaction energies vary with crystal depth for approach a on a 16 X 16 X N,-atom crystal. The minute oscillations can be ascribed to imperfections of the rather than to any significant influence of crystal size. We can also calculate the number of surface states, which are arbitrarily defined by
-
-
(23) (a) Farnsworth, H. E.; Madden, H. H. J . Appl. Phys. 1964,32, 1933. (b) Gerlach, R. L.; Rhodin, T. N. In “Structure and Chemistry of Solid Surfaces”; Somorjai, G., Ed.; Wiley: New York, 1969; p 55-1. (24) In particdar the orbitals are spanned by first running entirely through I , then through k,then throughj. The labeljF, kF,IF of the Fermi level (and the nature of the levels just below and above it) depends on this order.
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5583
Feature Article
+ 1)3/2(1c1),Ic21) > 0.90 (approach a ) ( N + 1)3/21co(> 0.90 (approaches b, c)
(N
(29)
The existence of surface. states for attack on a central bond but not for attack on an edge bond (Table 11) is easily explained by using (1 1) for the orbital amplitudes. If we choose the families of orbitals j = 1, 3, 5,
...
