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Ab-initio atom cluster models of carbon surfaces
Applications of Surface Science 11/12 (1982) 677—688 North-Holland Publishing Company
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AB-INITIO ATOM CLUSTER MODELS OF CARBON SURFACES William H. FINK Department of Chemistry, University of California, Davis, California 95616, USA Received 8 June 1981; revised manuscript received 28 August 1981
Ab-initio techniques for obtaining a wavefunction which simulates an infinite or semi-infinite solid with a finite cluster of atoms have recently been applied to model the electronic structure of the nalced surfaces of diamond and graphite and of hydrogen atom adsorption on the (100) surface of diamond. The wavefunction for the system is written in the form b=~4~iP~, where ~ is an antisynunetrizer, 4~is frozen and 4i~is determined variationally. Use of a projector to obtain the best possible approximate representation of ~ from a calculation which has been variationally determined in an environment simulating connection to the extended solid, permits the reduction of socalled edge effects in finite clusters. Clusters containing up to eight carbon atoms have been considered in geometries assumed by simple termination of the known bulk structure. While surface reconstruction may be important, it is more difficult to activate in these materials than in metals and an idealized geometry was considered an adequate and in fact necessary first treatment of the problems. The most active H adsorption site on (100) diamond was found to be an overhead site. A bridging site was also modeled. An unusually high value for the heat of H chemisorption was calculated. The surface electronic structure is examined with the aid of population analyses and contour maps. Difference densities were found to be particularly valuable to bring out the detailed changes in structure. A perspective of the approach is presented in view of all the calculations completed to date.
1. Introduction The fundamental idea behind the tecluiiques for electronic structure calculations in condensed phases which we have developed and exploited, is to solve the electronic structure problem for a system that not only has an environment consisting of the site of interest, but also contains an aspect which is selfconsistent with the extended solid. To illustrate the procedure figuratively, consider a two-dimensional problem of a plane covered with triangles such that the electronic structure within each triangle is equivalent to that in each other triangle as depicted schematically in fig. 1 a. The successively distant neighbors of triangle A are lettered B, then C, etc. If the potential within a single triangle of this extended array were known, then a single solution of the electronic structure problem for triangle A using this potential, would solve the problem for the entire plane. Of course, the correct potential is not known; a procedure of systematically improving approximations to it must be used. 0378-5963/82/0000—0000/$02.75 © 1982 North-Holland
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L~WA (a)
V/NV V/NV V
V
(b)
(c)
Fig. I. A schematic representation of the basic approach of the cyclical cluster approach to repeating potential problems; (a) represents the entire two-dimensional plane covered with like unit cells; (b) represents a basic division of a representative cluster into a central unit and its periphery; (c) represents a local aberration within the repeating potential.
In order to rapidly represent the extended problem with a given cluster size, we have developed a cyclical approach which enables us to reflect the solution obtained within a given repeating unit onto like units around the periphery. Again for the plane covered with equivalent triangles consider a cluster of the type depicted in fig. lb where the electronic structure of the peripheral triangles A_1 has been fixed at the solution obtained from a previous calculation. (Initially the solution found for the isolated triangle A for example.) Now the solution for the electronic structure of the central triangle A, is obtained subject to the potential of interaction with the frozen electronic structure of the triangles labeled A,_ and its own internal electronic structure. The solution obtained is then transferred to the peripheral triangles, the peripheral electronic structure frozen in the new form, and a newer solution for the central triangle is obtained. The process of solution, transfer, and fixation is repeated until there is no appreciable change between cycles. The resulting electronic structure of the central triangle A~then represents a good solution of electronic structure for a triangle in the full plane to the extent that only near-neighbor interactions have been included. Of course a hierarchy of clusters may be considered with this cyclical approach as with the simple cluster approach and we have utilized such hierarchies in our applications. With this cyclical approach a better representation of the solution for the full plane is obtained at a given cluster size than will result from a simple cluster approach because the extended nature of the problem is included at the outset. A further benefit of the cyclical approach now becomes apparent as local aberrations in the extended problem can be considered by clusters of the type depicted in fig. Ic. Here the electronic structure of the peripheral triangles is fixed at the solution obtained for the extended problem, and the internal electronic structure of the local aberration Q may be solved for within an environment which includes the peripheral representation of the cyclically consistent extended problem. The local aberration may be an impurity or guest in a host matrix, a point defect in a solid, a local molecular geometry or position change in a crystal, the movement of a proton in a hydrogen bond, or
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the termination of the extended problem by truncation of members of the periphery thus achieving a surface model. Our original motivation in developing this cyclical approach was to address problems of the latter type, specifically adsorption at surfaces or chemisorption with the attendant problems of heterogeneous catalysis. Of course between a general statement of this cyclical approach and its implementation in a reliable procedure for solution of electronic structure problems lies much detailed analysis, computer programming, and implementation. All three of these have been completed within a consistent ab-initio approximation, the single-determinant or Hartree—Fock level of treatment, and important applications to surface phenomena on the carbon allotropes diamond and graphite have resulted. Here we will review the methodology in rough outline, discuss some of the highlights of the results obtained to date, and bring together the generalizations regarding the merits of the procedure as can be discerned.
2. Current methodology and applications to date In order to develop a working version of the cyclical approach, we need two essential features: (1) An ability to solve the electronic structure problem of a central unit within an environment which contains a fixed electronic structure around its periphery; (2) An ability to transfer the electronic structure of the central unit to each of the peripheral units in the most faithful possible way. Both of these features have a long tradition in quantum chemistry. The first was addressed by Lykos and Parr [1] in their discussion of the a—~r separability problem. These ideas were generalized by McWeeney [2] and McWeeney and Ohno implemented the ideas in a calculation of the water molecule with the IS core of oxygen fixed in the form obtained for the free oxygen atom [3]. A number of more recent formulations have been offered [4—6]including our own matrix formulation for a single-determinant wavefunction [7]. This restriction in the size of the system treated proceeds by writing a wavefunction of the form =
where both ~ and ~ are simply products of molecular orbital functions and l~ is the antisymmetrizer which makes all the electrons in the system equivalent so that the numbering of the electron coordinates has no spatial significance. The question then posed is: what is the best possible solution with a wavefunction of this form where the specific functional form of the molecular orbitals in ~ is not allowed to change? If the molecular orbitals are expanded in a basis set (~}and the variational principle applied optimizing iJ~in the presence of
680 t/’~,
W.H. Fink
/ .4b-initjo atom
cluster models of carbon surfaces
the matrix equation to be solved for a closed shell case becomes:
~YA1=1A1,
(1)
where ~ iq+1
2~-k~+~
2~-k,Ix~),
j1
ctq+l,...,N, f3=q+l,...,n, and (x} is an intermediate orthogonal representation of {~},the first q of which are identical with the orbitals of i/i1 H is the one-electron operator of the hamiltonian which includes kinetic and electron—nuclear attraction terms; .J~ and K. are the Coulomb and exchange operators for the orbitals of ‘J’~J~and K~are the Coulomb and exchange operators for the orbitals of ~ Iterative solution of eq. (1) then leads to the best possible s/ia in the field of itself and of the externally defined 4i~.iJi~may be chosen with a fixed electronic structure to represent the analogy of the peripheral triangles A11 for the cluster examined. By projecting the result for tp11 back onto the space of 4’~[8] a grand iteration becomes possible for the self-definition of a bulk-like electronic structure from a finite cluster. When ~ has been variationally determined in the field of ~ and itself, there will be terms in the molecular orbitals of 4~ which come from atomic orbitals in the peripheral regions of space. These terms must be removed before such an orbital can be transferred to a peripheral unit because the peripheral unit will not have yet another unit further to the outside. We then posed the question of the best possible representation of a set of molecular orbitals under circumstances where some of the atomic orbitals were removed. The answer was a special case of a more general projector technique [8] which we formulated in matrix notation. In this notation the coefficients of the projected molecular orbitals are obtained simply as a product of certain well defined matrices [9]. These projected molecular orbitals may now be positioned on the peripheral units where the coefficients of the atomic orbitals will refer only to orbitals within that unit. All the orbitals on the entire periphery that are to make up ~/i~ are then symmetrically orthogonalized and the resulting orbitals used for ~ The application of these two techniques in a cluster model for the solid has been recently summarized [10]. The techniques have also been successfully applied to problems of molecular structure [11—13]. The allotropes of carbon were chosen as the system for examination in the initial studies because of the variety of crystal structure and because of the intrinsic importance of these materials. Further, ab-initio quantum chemistry is most highly developed for treatment of compounds containing elements up through neon in the periodic table. Some highlights from these studies illustrate resolution of important questions about the viability of our cyclic approach to cluster models.
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Figs. 2—4 show the cluster models used for diamond [14,15] and graphite [16]. The familiar tetrahedral arrangement of five carbon atoms for the diamond lattice is obtained with an atom at the center of the cube shown in fig. 2, and four near-neighbor carbons at each of the corners labeled 1 through 4. Examination of the diamond crystal structure will show that the (100) face of the crystal consists of long parallel rows of zig zag chains, a five carbon fragment of which is shown in fig. 3. The layered structure of graphite planes of hexagonal rings of carbon have two different kinds of carbons in each layer. Those that have a carbon atom above and below them in the adjacent layers, and those that do not. The left hand side of fig. 4 shows the cluster for four and six carbons that have these two different environments for the central carbon. The right hand side depicts a six carbon cluster which begins to extend the cluster’s representation of the in-plane environment and below it an eight carbon cluster which includes both kinds of carbon in a single cluster. The point group of each cluster is in parentheses and serves to distinguish between clusters of the same number of atoms, but different spatial arrangements. Table 1 extracts the pertinent Muiliken population analysis of the cluster models of diamond and graphite that have been examined to date [14—17].The population analysis is a highly reduced presentation of the information in the electronic wavefunction, but represents a consistent definition of the number of electrons which are to be identified with a given atom. For a perfect description of the electronic structure of crystalline carbon, there should, of course be six electrons identified with each atom. In table 1 the first column identifies the cluster and calculation, the next two columns are the numbers of electrons identified in the simple cluster approach with the central and peripheral atoms of the cluster respectively. The last two columns are the numbers of electrons identified in the cyclical approach with the central and peripheral atoms of the cluster. The generalization that emerges from this data is that the cyclical approach portrays a nearly neutral system (six electrons per atom) and that any changes in charge densities seen in passing from a smaller cluster to a larger one with
z
C5~
C0
C5 (Zig-Zag) Fig. 2. The C5 tetrahedral cluster modeling bulk diamond. Fig. 3. The C5 (zig-zag) cluster modeling the (100) face of diamond.
C6
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C® C1’ C4 (D3h)
C6 (D2~)
~C
®C
~C~— C~
C®
~•
C~
C
C
03h1
C
C6 ~
8 (C2~)
Fig. 4. A hierarchy of cluster models for graphite.
the simple approach, are already evident in the smaller cluster with the cyclic approach. Somç of the details of implementation of the cyclical approach all of which have been thoroughly investigated are included in table 1. The first row is of course the result obtained for the tetrahedral cluster. The simple cluster approach puts excessive electron density on the central atom. The next two
Table I Mullikan populations; an exact calculation of a bulk atom in diamond should yield 6.0 electrons Simple cluster
C5 C3 C5 C3 C3
(tetrahedral) (Is, 2s) (zig-zag)2p (Is, 2s, t) (Is, 2s, 2p11)
C4 (D3h)
C6 (D36) C6 (D26) C8 (C2~)~
Cyclical approach
Central
Periph
Central
Periph
6.501 6.319 6.181 6.319 6.319
5.875 5.841 6.104 5.841 5.841
6.014 6.065 5.882 5.440 5.405
5.997 5.967 6.415 6.280 6.298
6.489 6.458 6.297 (6.27)
5.837 5.847 5.852 (5.86)
5.961 5.923 5.956 (5.93)
6.013 6.025 6.022 (6.04)
~ For the C5 cluster there are in fact two different kinds of each type of atom. The average for the two different kin~1sof atoms of each type in the cluster have been entered to simplify the table. The values for each kind were reported in the orginal work [16].
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rows are an example of a (limited) hierarchical set of clusters. The C3 cluster is that obtained by removing carbons five and six from the C5 zig-zag cluster of fig. 3. In passing from the C3 to the C5 cluster, the electron density on the central atom in the simple approach decreases, as the influences of cluster terminating effects diminish. The more nearly neutral central atom of the cyclical approach is evident already with the C3 cluster. We expect the effective charge on the central atom to exhibit a damped oscillation about neutral in the cyclic approach, as successive outer layers are added to the ends of the zig-zag cluster in much the same way as calculations of hydrocarbons exhibit charge alternation along the chain. The fourth and fifth rows of table I before the omitted row are illustrative of the detailed examination of alternatives which has been carried out for the cyclical approach. The results in the first three rows are the final best model results where electrons in ls and 2s orbitals have been frozen around the periphery and all other electrons have been variationally optimized. In this way the periphery can function either as an electron source or sink in much the same way that the extended solid does. The results presented in the fourth and fifth rows were obtained when the electrons in the is, 2s, and one 2p orbital were also frozen in the periphery. There are two fundamental problems with this choice of frozen periphery: (1) some orientation must be chosen for the occupied 2p orbital; (2) the periphery may now function only as an electron sink, not as a source. Because of these two deficiencies we have subsequently only frozen spherical orbitals on the periphery and always fewer electrons than the nuclear charge of the atom. The data in the rows below the dashed line are highly condensed results from calculations for clusters modeling the graphite bulk lattice [16], the basal (1000) plane surface [16], and the (1010), (3030) and (1120) prismatic planes [17]. Again in this data emerge the same generalizations as from the diamond models. The cycled approach gives a more nearly uniform electron distribution for a given cluster size and correctly anticipates at this cluster size the change in electron density that will occur in the simple approach as the cluster size is increased. In all the results discussed above, we have avoided questions related to the energy changes with changing internuclear distances, but have focused attention on the static electron density. These questions have been avoided because the size of clusters we have examined result in individual atoms as the central unit and the wavefunction of this unit is orthogonal to the periphery. Thus the cyclic approach in application to these clusters intrinsically is incapable of adequately describing the bonding within the solid. This was evident from our first work on the C5 tetrahedral cluster where the cyclic approach gave a dissociative curve as a function of lattice parameter [14]. Orthogonality between the orbitals of an incoming adsorbate and the surface is not intrinsic to the surface model clusters, however, and we have recently completed work describing atomic hydrogen adsorption on the (100) face of diamond [18] and
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/ Ab-initio atom
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diffusion of adsorbed hydrogen atoms across this face [19]. Unlike silicon, reconstruction of the (100) face proceeds only with difficulty on diamond. Because of the observation of unreconstructed (100) surfaces of diamond and because of the infancy of theoretical studies of chemisorption on diamond, all of our calculations assumed an idealized unreconstructed surface. Thus our present results must be considered a first approximation to the problem. The zig-zag chain of atoms along the (100) face shown in fig. 3 is our basic model, augmented by an incoming hydrogen atom. Both overhead (resting straight on top of C0 of fig. 3) and bridging (along the bisector of the C 1C0C2 angle) sites were considered. Fig. 5 shows a binding energy curve obtained for the overhead site. The extra odd electron contributed by the incoming hydrogen atom can go into an orbital of any symmetry of the point group of the cluster. These symmetries A1, A2, B1, B2 serve as convenient labels to describe the different electronic wavefunctions for the system. In fig. 5 filled circles are A1, open circles are A2 except for the two filled circle points that are explicitly labeled. The energy of the dissociated C5 + H is shown by a triangle and filled circles at the right hand edge of the figure for visual comparison. There is a slight technical difference of the basis set at the dissociative limit hence the two points shown. This difference is unimportant for the main message of the figure which is to conclude that chemisorption is correctly calculated by the model (not shown is that a He atom displays a repulsive nonbonded curve in approaching the cluster) and that hydrogen is strongly bound to the cluster. A summary of the calculated energy parameters associated with H atom —
-187
W ~l88~
R CH Fig. 5. Binding energy of H atom on the overhead site of the (100) surface diamond. The calculated energy as a function of distance from the surface plane is displayed. The asymptotic energy at complete dissociation i~,disp1ayedon the right edge of the figure. Unlabeled filled circles are the A1 electronic state. Open circles are the A2 electronic state.
W.H. Fink / Ab-initlo atom cluster ,nodels of carbon surfaces
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adsorption on the (100) surface of diamond are presented in table 2. These results have been calculated for clusters corresponding with an idealized unreconstructed surface and consequently must be viewed with appreciation that relaxation of the surface will change the values somewhat. There are not good experimental numbers with which to compare these numbers. A critical review of the available data [18] gives a best estimate for the heat of adsorption in the range 80—90 kcal/mol and for the activation energy of adsorption of about 15 kcal/mol. Order of magnitude agreement between the calculated and experimental values has been obtained, but much additional sophistication both in experiment and theory is needed to reach definitive agreement. We have also investigated the three-dimensional quality of the electron distribution in forming the hydrogen—cluster bond. Fig. 6 shows the valence electron density contour map of the C5H model cluster on the left and the naked C5 model cluster on the right. Straight lines connect the nuclear positions. The conversion of the diffuse surface dangling bond to a highly directional hydrogen—cluster bond is clearly evident. As is evident from the preceding discussion, these calculations are most closely related to ab-iitio cluster calculations simulating surface adsorption [20—34].The methods employed in the calculations discussed here offer the sophistication on simple cluster calculations that included in the cluster is a simulation of connection to a more extended solid through the fixed part of the wavefunction t~, thereby reducing edge effects. Other ways of attempting to reduce edge effects in cluster calculations are to employ high spin unrestricted Hartree—Fock wavefunctions for the cluster [26,27] and to surround the cluster with hydrogen atoms [29,32—34].The methods we have employed are somewhat more difficult to implement, but have the advantage of recognizing the extended nature of the problem at the outset. Other cluster approaches have used semi-empirical molecular orbital calculations [35—43]or multiple scattering formalisms [44—48].Both of these groups of approaches have the advantage of being vastly easier computationally than ab-initio approaches and therefore can include larger numbers of atoms in the clusters and can treat the heavier elements of the periodic table. However, the semi-empirical methods suffer from uncertainties associated with their para-
Table 2 Calculated energy parameters characterizing H atom adsorption on (100) diamond Method
Heat of adsorption (kcal/mol)
Activation energy of adsorption (kcal/mol)
Surface diffusion barrier (keal/mol)
C—H wagging frequency (cm~)
SCF Model
147 142
10 10
66 80
1270 1190
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W. H. Fink / A b.initio atom cluster models of carbon surfaces
a
b
~
Fig. 6. Valence electron density of (a) C
5H and (b) C5.
meterization and the multiple scattering methods with difficulties of the muffin-tin approximation and to a lesser extent with the approximate treatment of exchange. Cluster edge effects are generally handled with these methods by treating very large clusters. Several recent reviews of theoretical treatments of chemisorption from the more delocalized solid state viewpoint are available [49—53].These have generally not been applied to diamond and graphite surfaces, although Ciraci and Batra [54] have applied the extended tight binding method to the (100) diamond surface using slab geometries. The radically different kinds of information which are obtained from the delocalized viewpoint and from the cluster models make them difficult to compare. Additional efforts to bridge this gap between cluster calculations and band methods are to be encouraged. Very roughly qualitative agreement between the orbital composition of the surface states of Ciraci and Batra and the composition of the major contributor to the dangling bond density in the calculations discussed above has been observed [15]. Summarizing the experience with our current methodology to date, it has been highly successful in reducing cluster edge effects. The first portion of the wavefunction simulates connection of the cluster to a more extended solid. In each case where a hierarchy of successively large clusters have been examined, our current methodology successfully develops the quality of electronic structure on the site of interest in a smaller cluster that develops in the simple cluster approach only with a larger cluster. Specifically the simple approach tends to concentrate too much electronic charge on the central units of small clusters leaving the edges deficient in electron density. Our methodology is much more successful in distributing the charge uniformly around the cluster. This feature is evident both in Mulliken population analysis and in the three
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Ab.initio atom cluster models of carbon surfaces
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dimensional quality of the wavefunction as displayed by detailed electronic density contour maps.
References [1] P. Lykos and R.G. Parr, J. Chem. Phys. 24 (1956) 1166. [21 R. McWeeney, Proc. Roy. Soc. (London) A253 (1959) 242. [3] R. McWeeney and K.A. Ohno, Proc. Roy. Soc. (London) A255 (1960) 367. [4] W.J. Taylor, 2nd Canadian Symp. on Quantum Chemistry, Montreal, Canada, 1967, as quoted by E.K. Grimmelman and J.P. Chesick, J. Chem. Phys. 55 (1971) 1690. [5] J. Hinze, Advan. Chem. Phys. 26 (1974) 213. [6] D. Peters, J. Chem. Phys. 67 (1977) 1284. [7] W.H. Fink, J. Chem. Phys. 57 (1972) 1822. [8] N.H.F. Beebe and S. Lunell, Intern. J. Quantum Chem. 6 (1972) 1149. [9] W.H. Fink, Intern. J. Quantum Chem. 7 (1973) 1045. [10] W.H. Fink, A.M. Butkus and J.P. Lopez, Intern. J. Quantum Chem., Symp. Vol. 13 (1979) 331. [II] W.H. Fink, J. Chem. Phys. 60 (1974) 402. [12] W.H. Fink, J. Chem. Phys. 66 (1977) 1968. [13] A.M. Butkus and W.H. Fink, J. Chem. Phys. 71(1979) 5329. [14] W.H. Fink, J. Chem. Phys. 69 (1978) 3325. [15] J.P. Lopez and W.H. Fink, J. Chem. Phys., 75 (1981) 2290. [16] A.M. Butkus and W.H. Fink, J. Chem. Phys. 73 (1980) 2884. [17] A.M. Butkus and W.H. Fink, J. Chem. Phys. 73 (1980) 2893. [18] J.P. Lopez and W.H. Fink, J. Phys. Chem. 85 (1981) 2642. [19] J.P. Lopez and W.H. Fink, J. Chem. Phys., in press. [20] H. Stoll and H. Preuss, Phys. Status Solidi (b) 53 (1972) 519. [21] A.B. Kunz, D.J. McKish and P.W. Dentrach, Solid State Commun. 13 (1973) 35. [22] H. Stoll and H. Preuss, Phys. Status Solidi (b) 64 (1974) 103. [23] C.W. Bauschlicker, D.H. Kislow, CF. Bender and H.F. Schaefer III, J. Chem. Phys. 62 (1975) 4815. [24] R.B. Brewington, C.F. Bender and H.F. Schaefer III, J. Chem. Phys. 64 (1976) 905. [25] C.W. Bauschlicker, C.F. Bender and H.F. Schaefer III, Chem. Phys. 15 (1976) 227. [26] R.F. Marshall, R.J. Blint and A.B. Kunz, Phys. Rev. B13 (1976) 3333. [27] S. Goldstein, L.A. Curtiss and R.N. Euwema, J. Phys. C9 (1976) 4131. [28] J. Wood, Chem. Phys. Letters 50 (1977) 129. [29] L.C. Snyder and Z. Wasserman, Surface Sci. 71(1978) 407. [30] H. Stoll and H. Preuss, Surface Sci. 65 (1977) 229. [31] B.N. Cox and C.W. Bauschlicker, Surface Sci. 115 (1982) 15. [32] A. Redondo, WA. Gooddard III, T.C. McGill and G.T. Surratt, Solid State Commun. 20 (1976) 733. [33] WA. Goodard III, A. Redondo and T.C. McGill, Solid State Commun. 18 (1976) 981. [34] W.A. Goddard III, J.J. Barton, A. Rodondo and T.C. McGill, J. Vacuum Sci. Technol. 15 (1978) 1274. [35] R.P. Messmer, Chem. Phys. Letters 11(1971) 589. [36] GD. Watkins and R.P. Messmer, in: Computational Methods for Large Molecules and Localized States in Solids, Eds. F. Herman and R.K. Nesbet (Plenum, New York, 1973) p. 133. [37] A.B. Anderson, J. Chem. Phys. 63 (1975) 4430. [38] A. Zunger, J. Phys. C7 (1974) 96.
688 [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]
W. H. Fink / Ab~initioatom cluster models of carbon surfaces R.C. Baetzold, Advan. Catalysis 25 (1976) 1. D.J.M. Fassaert and A. van der Avoird, Surface Sci. 55 (1976) 291, 313. TN. Rhodin, CF. Brucker and A.B. Anderson, .1. Phys. Chem. 82 (1978) 894. A.L. Companion, Chem. Phys. 14 (1976) 1. G. Blyholder, J. Chem. Phys. 62 (1975) 3193. K.H. Johnson and R.P. Messmer, Intern. J. Quantum Chem. Symp. 10 (1976) 147. R.P. Messmer and K. Salahub, Chem. Phys. Letters 49 (1977) 59. J.B. Danese and JR. Schrieffer, Intern. J. Quantum Chem. Symp. 10 (1976) 289. A. Rosen, Chem. Phys. Letters 58 (1978) 95. A. Rosen, D.E. Ellis, H. Adachi and F.W. Averill, J. Chem. Phys. 65 (1976) 3629. J.C. Phillips, Surface Sci. 53 (1975) 474. TB. Grimley, in: Chemistry and Physics of Solid Surfaces, Vol. I, Eds. R. Vanselow and S.Y. Tong, Chemical Rubber Company, Cleveland (1977) p. 155. J.A. Appelbaum and DR. Hamann, in: Chemistry and Physics of Solid Surfaces, Vol. I, Eds. R. Vanselow and S.Y. Tong (Chemical Rubber Company, Cleveland, Ohio, 1977) p. 275. AM. Brodsky and M.I. Urbalch, Progr. Surface Sci. 8 (1977) 103. J.P. Muscat and D.M. Newns, Progr. Surface Sci. 9 (1978) 1. S. Ciraci and I.P. Batra, Phys. Rev. B15 (1977) 3254.
677
AB-INITIO ATOM CLUSTER MODELS OF CARBON SURFACES William H. FINK Department of Chemistry, University of California, Davis, California 95616, USA Received 8 June 1981; revised manuscript received 28 August 1981
Ab-initio techniques for obtaining a wavefunction which simulates an infinite or semi-infinite solid with a finite cluster of atoms have recently been applied to model the electronic structure of the nalced surfaces of diamond and graphite and of hydrogen atom adsorption on the (100) surface of diamond. The wavefunction for the system is written in the form b=~4~iP~, where ~ is an antisynunetrizer, 4~is frozen and 4i~is determined variationally. Use of a projector to obtain the best possible approximate representation of ~ from a calculation which has been variationally determined in an environment simulating connection to the extended solid, permits the reduction of socalled edge effects in finite clusters. Clusters containing up to eight carbon atoms have been considered in geometries assumed by simple termination of the known bulk structure. While surface reconstruction may be important, it is more difficult to activate in these materials than in metals and an idealized geometry was considered an adequate and in fact necessary first treatment of the problems. The most active H adsorption site on (100) diamond was found to be an overhead site. A bridging site was also modeled. An unusually high value for the heat of H chemisorption was calculated. The surface electronic structure is examined with the aid of population analyses and contour maps. Difference densities were found to be particularly valuable to bring out the detailed changes in structure. A perspective of the approach is presented in view of all the calculations completed to date.
1. Introduction The fundamental idea behind the tecluiiques for electronic structure calculations in condensed phases which we have developed and exploited, is to solve the electronic structure problem for a system that not only has an environment consisting of the site of interest, but also contains an aspect which is selfconsistent with the extended solid. To illustrate the procedure figuratively, consider a two-dimensional problem of a plane covered with triangles such that the electronic structure within each triangle is equivalent to that in each other triangle as depicted schematically in fig. 1 a. The successively distant neighbors of triangle A are lettered B, then C, etc. If the potential within a single triangle of this extended array were known, then a single solution of the electronic structure problem for triangle A using this potential, would solve the problem for the entire plane. Of course, the correct potential is not known; a procedure of systematically improving approximations to it must be used. 0378-5963/82/0000—0000/$02.75 © 1982 North-Holland
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L~WA (a)
V/NV V/NV V
V
(b)
(c)
Fig. I. A schematic representation of the basic approach of the cyclical cluster approach to repeating potential problems; (a) represents the entire two-dimensional plane covered with like unit cells; (b) represents a basic division of a representative cluster into a central unit and its periphery; (c) represents a local aberration within the repeating potential.
In order to rapidly represent the extended problem with a given cluster size, we have developed a cyclical approach which enables us to reflect the solution obtained within a given repeating unit onto like units around the periphery. Again for the plane covered with equivalent triangles consider a cluster of the type depicted in fig. lb where the electronic structure of the peripheral triangles A_1 has been fixed at the solution obtained from a previous calculation. (Initially the solution found for the isolated triangle A for example.) Now the solution for the electronic structure of the central triangle A, is obtained subject to the potential of interaction with the frozen electronic structure of the triangles labeled A,_ and its own internal electronic structure. The solution obtained is then transferred to the peripheral triangles, the peripheral electronic structure frozen in the new form, and a newer solution for the central triangle is obtained. The process of solution, transfer, and fixation is repeated until there is no appreciable change between cycles. The resulting electronic structure of the central triangle A~then represents a good solution of electronic structure for a triangle in the full plane to the extent that only near-neighbor interactions have been included. Of course a hierarchy of clusters may be considered with this cyclical approach as with the simple cluster approach and we have utilized such hierarchies in our applications. With this cyclical approach a better representation of the solution for the full plane is obtained at a given cluster size than will result from a simple cluster approach because the extended nature of the problem is included at the outset. A further benefit of the cyclical approach now becomes apparent as local aberrations in the extended problem can be considered by clusters of the type depicted in fig. Ic. Here the electronic structure of the peripheral triangles is fixed at the solution obtained for the extended problem, and the internal electronic structure of the local aberration Q may be solved for within an environment which includes the peripheral representation of the cyclically consistent extended problem. The local aberration may be an impurity or guest in a host matrix, a point defect in a solid, a local molecular geometry or position change in a crystal, the movement of a proton in a hydrogen bond, or
W.H. Fink / Ab-initio atom cluster models of carbon surfaces
679
the termination of the extended problem by truncation of members of the periphery thus achieving a surface model. Our original motivation in developing this cyclical approach was to address problems of the latter type, specifically adsorption at surfaces or chemisorption with the attendant problems of heterogeneous catalysis. Of course between a general statement of this cyclical approach and its implementation in a reliable procedure for solution of electronic structure problems lies much detailed analysis, computer programming, and implementation. All three of these have been completed within a consistent ab-initio approximation, the single-determinant or Hartree—Fock level of treatment, and important applications to surface phenomena on the carbon allotropes diamond and graphite have resulted. Here we will review the methodology in rough outline, discuss some of the highlights of the results obtained to date, and bring together the generalizations regarding the merits of the procedure as can be discerned.
2. Current methodology and applications to date In order to develop a working version of the cyclical approach, we need two essential features: (1) An ability to solve the electronic structure problem of a central unit within an environment which contains a fixed electronic structure around its periphery; (2) An ability to transfer the electronic structure of the central unit to each of the peripheral units in the most faithful possible way. Both of these features have a long tradition in quantum chemistry. The first was addressed by Lykos and Parr [1] in their discussion of the a—~r separability problem. These ideas were generalized by McWeeney [2] and McWeeney and Ohno implemented the ideas in a calculation of the water molecule with the IS core of oxygen fixed in the form obtained for the free oxygen atom [3]. A number of more recent formulations have been offered [4—6]including our own matrix formulation for a single-determinant wavefunction [7]. This restriction in the size of the system treated proceeds by writing a wavefunction of the form =
where both ~ and ~ are simply products of molecular orbital functions and l~ is the antisymmetrizer which makes all the electrons in the system equivalent so that the numbering of the electron coordinates has no spatial significance. The question then posed is: what is the best possible solution with a wavefunction of this form where the specific functional form of the molecular orbitals in ~ is not allowed to change? If the molecular orbitals are expanded in a basis set (~}and the variational principle applied optimizing iJ~in the presence of
680 t/’~,
W.H. Fink
/ .4b-initjo atom
cluster models of carbon surfaces
the matrix equation to be solved for a closed shell case becomes:
~YA1=1A1,
(1)
where ~ iq+1
2~-k~+~
2~-k,Ix~),
j1
ctq+l,...,N, f3=q+l,...,n, and (x} is an intermediate orthogonal representation of {~},the first q of which are identical with the orbitals of i/i1 H is the one-electron operator of the hamiltonian which includes kinetic and electron—nuclear attraction terms; .J~ and K. are the Coulomb and exchange operators for the orbitals of ‘J’~J~and K~are the Coulomb and exchange operators for the orbitals of ~ Iterative solution of eq. (1) then leads to the best possible s/ia in the field of itself and of the externally defined 4i~.iJi~may be chosen with a fixed electronic structure to represent the analogy of the peripheral triangles A11 for the cluster examined. By projecting the result for tp11 back onto the space of 4’~[8] a grand iteration becomes possible for the self-definition of a bulk-like electronic structure from a finite cluster. When ~ has been variationally determined in the field of ~ and itself, there will be terms in the molecular orbitals of 4~ which come from atomic orbitals in the peripheral regions of space. These terms must be removed before such an orbital can be transferred to a peripheral unit because the peripheral unit will not have yet another unit further to the outside. We then posed the question of the best possible representation of a set of molecular orbitals under circumstances where some of the atomic orbitals were removed. The answer was a special case of a more general projector technique [8] which we formulated in matrix notation. In this notation the coefficients of the projected molecular orbitals are obtained simply as a product of certain well defined matrices [9]. These projected molecular orbitals may now be positioned on the peripheral units where the coefficients of the atomic orbitals will refer only to orbitals within that unit. All the orbitals on the entire periphery that are to make up ~/i~ are then symmetrically orthogonalized and the resulting orbitals used for ~ The application of these two techniques in a cluster model for the solid has been recently summarized [10]. The techniques have also been successfully applied to problems of molecular structure [11—13]. The allotropes of carbon were chosen as the system for examination in the initial studies because of the variety of crystal structure and because of the intrinsic importance of these materials. Further, ab-initio quantum chemistry is most highly developed for treatment of compounds containing elements up through neon in the periodic table. Some highlights from these studies illustrate resolution of important questions about the viability of our cyclic approach to cluster models.
W.H. Fink / Ab-initio atom cluster models of carbon surfaces
681
Figs. 2—4 show the cluster models used for diamond [14,15] and graphite [16]. The familiar tetrahedral arrangement of five carbon atoms for the diamond lattice is obtained with an atom at the center of the cube shown in fig. 2, and four near-neighbor carbons at each of the corners labeled 1 through 4. Examination of the diamond crystal structure will show that the (100) face of the crystal consists of long parallel rows of zig zag chains, a five carbon fragment of which is shown in fig. 3. The layered structure of graphite planes of hexagonal rings of carbon have two different kinds of carbons in each layer. Those that have a carbon atom above and below them in the adjacent layers, and those that do not. The left hand side of fig. 4 shows the cluster for four and six carbons that have these two different environments for the central carbon. The right hand side depicts a six carbon cluster which begins to extend the cluster’s representation of the in-plane environment and below it an eight carbon cluster which includes both kinds of carbon in a single cluster. The point group of each cluster is in parentheses and serves to distinguish between clusters of the same number of atoms, but different spatial arrangements. Table 1 extracts the pertinent Muiliken population analysis of the cluster models of diamond and graphite that have been examined to date [14—17].The population analysis is a highly reduced presentation of the information in the electronic wavefunction, but represents a consistent definition of the number of electrons which are to be identified with a given atom. For a perfect description of the electronic structure of crystalline carbon, there should, of course be six electrons identified with each atom. In table 1 the first column identifies the cluster and calculation, the next two columns are the numbers of electrons identified in the simple cluster approach with the central and peripheral atoms of the cluster respectively. The last two columns are the numbers of electrons identified in the cyclical approach with the central and peripheral atoms of the cluster. The generalization that emerges from this data is that the cyclical approach portrays a nearly neutral system (six electrons per atom) and that any changes in charge densities seen in passing from a smaller cluster to a larger one with
z
C5~
C0
C5 (Zig-Zag) Fig. 2. The C5 tetrahedral cluster modeling bulk diamond. Fig. 3. The C5 (zig-zag) cluster modeling the (100) face of diamond.
C6
W.H. Fink / Ab-initio atom cluster models of carbon surfaces
682
C® C1’ C4 (D3h)
C6 (D2~)
~C
®C
~C~— C~
C®
~•
C~
C
C
03h1
C
C6 ~
8 (C2~)
Fig. 4. A hierarchy of cluster models for graphite.
the simple approach, are already evident in the smaller cluster with the cyclic approach. Somç of the details of implementation of the cyclical approach all of which have been thoroughly investigated are included in table 1. The first row is of course the result obtained for the tetrahedral cluster. The simple cluster approach puts excessive electron density on the central atom. The next two
Table I Mullikan populations; an exact calculation of a bulk atom in diamond should yield 6.0 electrons Simple cluster
C5 C3 C5 C3 C3
(tetrahedral) (Is, 2s) (zig-zag)2p (Is, 2s, t) (Is, 2s, 2p11)
C4 (D3h)
C6 (D36) C6 (D26) C8 (C2~)~
Cyclical approach
Central
Periph
Central
Periph
6.501 6.319 6.181 6.319 6.319
5.875 5.841 6.104 5.841 5.841
6.014 6.065 5.882 5.440 5.405
5.997 5.967 6.415 6.280 6.298
6.489 6.458 6.297 (6.27)
5.837 5.847 5.852 (5.86)
5.961 5.923 5.956 (5.93)
6.013 6.025 6.022 (6.04)
~ For the C5 cluster there are in fact two different kinds of each type of atom. The average for the two different kin~1sof atoms of each type in the cluster have been entered to simplify the table. The values for each kind were reported in the orginal work [16].
W. H. Fink / A b-initio atom cluster models of carbon surfaces
683
rows are an example of a (limited) hierarchical set of clusters. The C3 cluster is that obtained by removing carbons five and six from the C5 zig-zag cluster of fig. 3. In passing from the C3 to the C5 cluster, the electron density on the central atom in the simple approach decreases, as the influences of cluster terminating effects diminish. The more nearly neutral central atom of the cyclical approach is evident already with the C3 cluster. We expect the effective charge on the central atom to exhibit a damped oscillation about neutral in the cyclic approach, as successive outer layers are added to the ends of the zig-zag cluster in much the same way as calculations of hydrocarbons exhibit charge alternation along the chain. The fourth and fifth rows of table I before the omitted row are illustrative of the detailed examination of alternatives which has been carried out for the cyclical approach. The results in the first three rows are the final best model results where electrons in ls and 2s orbitals have been frozen around the periphery and all other electrons have been variationally optimized. In this way the periphery can function either as an electron source or sink in much the same way that the extended solid does. The results presented in the fourth and fifth rows were obtained when the electrons in the is, 2s, and one 2p orbital were also frozen in the periphery. There are two fundamental problems with this choice of frozen periphery: (1) some orientation must be chosen for the occupied 2p orbital; (2) the periphery may now function only as an electron sink, not as a source. Because of these two deficiencies we have subsequently only frozen spherical orbitals on the periphery and always fewer electrons than the nuclear charge of the atom. The data in the rows below the dashed line are highly condensed results from calculations for clusters modeling the graphite bulk lattice [16], the basal (1000) plane surface [16], and the (1010), (3030) and (1120) prismatic planes [17]. Again in this data emerge the same generalizations as from the diamond models. The cycled approach gives a more nearly uniform electron distribution for a given cluster size and correctly anticipates at this cluster size the change in electron density that will occur in the simple approach as the cluster size is increased. In all the results discussed above, we have avoided questions related to the energy changes with changing internuclear distances, but have focused attention on the static electron density. These questions have been avoided because the size of clusters we have examined result in individual atoms as the central unit and the wavefunction of this unit is orthogonal to the periphery. Thus the cyclic approach in application to these clusters intrinsically is incapable of adequately describing the bonding within the solid. This was evident from our first work on the C5 tetrahedral cluster where the cyclic approach gave a dissociative curve as a function of lattice parameter [14]. Orthogonality between the orbitals of an incoming adsorbate and the surface is not intrinsic to the surface model clusters, however, and we have recently completed work describing atomic hydrogen adsorption on the (100) face of diamond [18] and
684
W. H. Fink
/ Ab-initio atom
cluster models of carbon surfaces
diffusion of adsorbed hydrogen atoms across this face [19]. Unlike silicon, reconstruction of the (100) face proceeds only with difficulty on diamond. Because of the observation of unreconstructed (100) surfaces of diamond and because of the infancy of theoretical studies of chemisorption on diamond, all of our calculations assumed an idealized unreconstructed surface. Thus our present results must be considered a first approximation to the problem. The zig-zag chain of atoms along the (100) face shown in fig. 3 is our basic model, augmented by an incoming hydrogen atom. Both overhead (resting straight on top of C0 of fig. 3) and bridging (along the bisector of the C 1C0C2 angle) sites were considered. Fig. 5 shows a binding energy curve obtained for the overhead site. The extra odd electron contributed by the incoming hydrogen atom can go into an orbital of any symmetry of the point group of the cluster. These symmetries A1, A2, B1, B2 serve as convenient labels to describe the different electronic wavefunctions for the system. In fig. 5 filled circles are A1, open circles are A2 except for the two filled circle points that are explicitly labeled. The energy of the dissociated C5 + H is shown by a triangle and filled circles at the right hand edge of the figure for visual comparison. There is a slight technical difference of the basis set at the dissociative limit hence the two points shown. This difference is unimportant for the main message of the figure which is to conclude that chemisorption is correctly calculated by the model (not shown is that a He atom displays a repulsive nonbonded curve in approaching the cluster) and that hydrogen is strongly bound to the cluster. A summary of the calculated energy parameters associated with H atom —
-187
W ~l88~
R CH Fig. 5. Binding energy of H atom on the overhead site of the (100) surface diamond. The calculated energy as a function of distance from the surface plane is displayed. The asymptotic energy at complete dissociation i~,disp1ayedon the right edge of the figure. Unlabeled filled circles are the A1 electronic state. Open circles are the A2 electronic state.
W.H. Fink / Ab-initlo atom cluster ,nodels of carbon surfaces
685
adsorption on the (100) surface of diamond are presented in table 2. These results have been calculated for clusters corresponding with an idealized unreconstructed surface and consequently must be viewed with appreciation that relaxation of the surface will change the values somewhat. There are not good experimental numbers with which to compare these numbers. A critical review of the available data [18] gives a best estimate for the heat of adsorption in the range 80—90 kcal/mol and for the activation energy of adsorption of about 15 kcal/mol. Order of magnitude agreement between the calculated and experimental values has been obtained, but much additional sophistication both in experiment and theory is needed to reach definitive agreement. We have also investigated the three-dimensional quality of the electron distribution in forming the hydrogen—cluster bond. Fig. 6 shows the valence electron density contour map of the C5H model cluster on the left and the naked C5 model cluster on the right. Straight lines connect the nuclear positions. The conversion of the diffuse surface dangling bond to a highly directional hydrogen—cluster bond is clearly evident. As is evident from the preceding discussion, these calculations are most closely related to ab-iitio cluster calculations simulating surface adsorption [20—34].The methods employed in the calculations discussed here offer the sophistication on simple cluster calculations that included in the cluster is a simulation of connection to a more extended solid through the fixed part of the wavefunction t~, thereby reducing edge effects. Other ways of attempting to reduce edge effects in cluster calculations are to employ high spin unrestricted Hartree—Fock wavefunctions for the cluster [26,27] and to surround the cluster with hydrogen atoms [29,32—34].The methods we have employed are somewhat more difficult to implement, but have the advantage of recognizing the extended nature of the problem at the outset. Other cluster approaches have used semi-empirical molecular orbital calculations [35—43]or multiple scattering formalisms [44—48].Both of these groups of approaches have the advantage of being vastly easier computationally than ab-initio approaches and therefore can include larger numbers of atoms in the clusters and can treat the heavier elements of the periodic table. However, the semi-empirical methods suffer from uncertainties associated with their para-
Table 2 Calculated energy parameters characterizing H atom adsorption on (100) diamond Method
Heat of adsorption (kcal/mol)
Activation energy of adsorption (kcal/mol)
Surface diffusion barrier (keal/mol)
C—H wagging frequency (cm~)
SCF Model
147 142
10 10
66 80
1270 1190
686
W. H. Fink / A b.initio atom cluster models of carbon surfaces
a
b
~
Fig. 6. Valence electron density of (a) C
5H and (b) C5.
meterization and the multiple scattering methods with difficulties of the muffin-tin approximation and to a lesser extent with the approximate treatment of exchange. Cluster edge effects are generally handled with these methods by treating very large clusters. Several recent reviews of theoretical treatments of chemisorption from the more delocalized solid state viewpoint are available [49—53].These have generally not been applied to diamond and graphite surfaces, although Ciraci and Batra [54] have applied the extended tight binding method to the (100) diamond surface using slab geometries. The radically different kinds of information which are obtained from the delocalized viewpoint and from the cluster models make them difficult to compare. Additional efforts to bridge this gap between cluster calculations and band methods are to be encouraged. Very roughly qualitative agreement between the orbital composition of the surface states of Ciraci and Batra and the composition of the major contributor to the dangling bond density in the calculations discussed above has been observed [15]. Summarizing the experience with our current methodology to date, it has been highly successful in reducing cluster edge effects. The first portion of the wavefunction simulates connection of the cluster to a more extended solid. In each case where a hierarchy of successively large clusters have been examined, our current methodology successfully develops the quality of electronic structure on the site of interest in a smaller cluster that develops in the simple cluster approach only with a larger cluster. Specifically the simple approach tends to concentrate too much electronic charge on the central units of small clusters leaving the edges deficient in electron density. Our methodology is much more successful in distributing the charge uniformly around the cluster. This feature is evident both in Mulliken population analysis and in the three
W.H. Fink
/
Ab.initio atom cluster models of carbon surfaces
687
dimensional quality of the wavefunction as displayed by detailed electronic density contour maps.
References [1] P. Lykos and R.G. Parr, J. Chem. Phys. 24 (1956) 1166. [21 R. McWeeney, Proc. Roy. Soc. (London) A253 (1959) 242. [3] R. McWeeney and K.A. Ohno, Proc. Roy. Soc. (London) A255 (1960) 367. [4] W.J. Taylor, 2nd Canadian Symp. on Quantum Chemistry, Montreal, Canada, 1967, as quoted by E.K. Grimmelman and J.P. Chesick, J. Chem. Phys. 55 (1971) 1690. [5] J. Hinze, Advan. Chem. Phys. 26 (1974) 213. [6] D. Peters, J. Chem. Phys. 67 (1977) 1284. [7] W.H. Fink, J. Chem. Phys. 57 (1972) 1822. [8] N.H.F. Beebe and S. Lunell, Intern. J. Quantum Chem. 6 (1972) 1149. [9] W.H. Fink, Intern. J. Quantum Chem. 7 (1973) 1045. [10] W.H. Fink, A.M. Butkus and J.P. Lopez, Intern. J. Quantum Chem., Symp. Vol. 13 (1979) 331. [II] W.H. Fink, J. Chem. Phys. 60 (1974) 402. [12] W.H. Fink, J. Chem. Phys. 66 (1977) 1968. [13] A.M. Butkus and W.H. Fink, J. Chem. Phys. 71(1979) 5329. [14] W.H. Fink, J. Chem. Phys. 69 (1978) 3325. [15] J.P. Lopez and W.H. Fink, J. Chem. Phys., 75 (1981) 2290. [16] A.M. Butkus and W.H. Fink, J. Chem. Phys. 73 (1980) 2884. [17] A.M. Butkus and W.H. Fink, J. Chem. Phys. 73 (1980) 2893. [18] J.P. Lopez and W.H. Fink, J. Phys. Chem. 85 (1981) 2642. [19] J.P. Lopez and W.H. Fink, J. Chem. Phys., in press. [20] H. Stoll and H. Preuss, Phys. Status Solidi (b) 53 (1972) 519. [21] A.B. Kunz, D.J. McKish and P.W. Dentrach, Solid State Commun. 13 (1973) 35. [22] H. Stoll and H. Preuss, Phys. Status Solidi (b) 64 (1974) 103. [23] C.W. Bauschlicker, D.H. Kislow, CF. Bender and H.F. Schaefer III, J. Chem. Phys. 62 (1975) 4815. [24] R.B. Brewington, C.F. Bender and H.F. Schaefer III, J. Chem. Phys. 64 (1976) 905. [25] C.W. Bauschlicker, C.F. Bender and H.F. Schaefer III, Chem. Phys. 15 (1976) 227. [26] R.F. Marshall, R.J. Blint and A.B. Kunz, Phys. Rev. B13 (1976) 3333. [27] S. Goldstein, L.A. Curtiss and R.N. Euwema, J. Phys. C9 (1976) 4131. [28] J. Wood, Chem. Phys. Letters 50 (1977) 129. [29] L.C. Snyder and Z. Wasserman, Surface Sci. 71(1978) 407. [30] H. Stoll and H. Preuss, Surface Sci. 65 (1977) 229. [31] B.N. Cox and C.W. Bauschlicker, Surface Sci. 115 (1982) 15. [32] A. Redondo, WA. Gooddard III, T.C. McGill and G.T. Surratt, Solid State Commun. 20 (1976) 733. [33] WA. Goodard III, A. Redondo and T.C. McGill, Solid State Commun. 18 (1976) 981. [34] W.A. Goddard III, J.J. Barton, A. Rodondo and T.C. McGill, J. Vacuum Sci. Technol. 15 (1978) 1274. [35] R.P. Messmer, Chem. Phys. Letters 11(1971) 589. [36] GD. Watkins and R.P. Messmer, in: Computational Methods for Large Molecules and Localized States in Solids, Eds. F. Herman and R.K. Nesbet (Plenum, New York, 1973) p. 133. [37] A.B. Anderson, J. Chem. Phys. 63 (1975) 4430. [38] A. Zunger, J. Phys. C7 (1974) 96.
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