2850
J. Phys. Chem. 1082,86,2850-2853
Ab-Initio Model Cluster Calculations of Hydrogen Atom Dlffuslve Motion Across the (100) Face of Diamond Jesus P. Lopezt and William H. Fink" Department of Chemistry, University of California, Davis, Californie 95616 (Receivd: January 15, 1982)
Ab-initio atom cluster calculations are reported modeling hydrogen atom diffusive motion across the (100) face of diamond. Previous work had identified overhead and bridging sites for chemisorption by examining energy as a function of hydrogen motion perpendicular to the (100)face of diamond for several models of atomic clusters. This work extends the calculations to hydrogen motion parallel with the (100) face. Edge effects of the cluster calculations are minimized by using a portion of the wave function to simulate connection of the cluster to a more extended solid. Extraordinarily high barriers to diffusion were obtained; the bridging site is the saddle point for diffusion between overhead si-. The energy profiie for diffusion of an excited state of the chemisorbed system was also calculated and displayed a minimum energy about one quarter of the distance between the overhead and bridging sites. Parabolic fita to the energy profile about the ground-stateminimum give a predicted vibrational energy of about 1200 cm-l for the C-H wag of the chemisorbed system.
In a recent paper' the chemisorption process of H atoms on the idealized (100) surface of diamond was investigated by ab-initio model cluster calculations. We present here extension of that work to an examination of the energy profile for diffusive motion of the H atom across the idealized (100) face. The approach of a hydrogen atom to model clusters in a direction simulating the perpendicular to the (100) face showed chemisorption on both overhead (single coordination) and bridging (bicoordinate) sites.' The calculations showed the most favorable of these to be the overhead site, and it becomes of immediate interest to determine whether these sites are both within the same potential energy well or whether there is a maximum between them. Therefore we present here calculated results examining lateral movement of the hydrogen atom between these two sites, which is equivalent to diffusion across the face. Figure 1 depicts the diamond (100) surface viewed perpendicularly. Overhead sites for chemisorbed H atoms would be directly above each of the heavy filled circles and lines lettered A, B, and C would constitute alternative diffusion paths between overhead sites. Path A provides the only route between the overhead and bridging sites. The entire length of path A has either a surface layer or second layer atom beneath it whereas path B alternates between surface and third layer atoms and path C does not even incorporate third layer atoms. Thus paths B and C would appear to be higher energy diffusion routes because of the lack of carbon atoms to sustain the chemisorption bond along the way. For the above reasons we have examined only path A as a model for diffusion across the (100) face. While this choice appears to be correct, the possibility of paths of lower energy than A may exist. The general problem of surface diffusion is discussed by Boer2 and by Tsong and C ~ w a n . Experimental ~ observation of hydrogen surface diffusion is difficult and there are no reports of diffusion on diamond. However, Gomer et have reported an activation energy of 16 kcal/mol for the diffusion of hydrogen on tungsten at very low coverages. Method of Calculation The same basic ab-initio model approach to calculations of electronic structure has been employed here as was used a 1 4 p 6
'Surface Analytic Research, Inc., 465A Fairchild Drive, Suite 128, Mountain View, CA 94043. 0022-3654/82/2086-2850$0 1.2510
in previous applications to bulk diamond and its naked to bulk graphite and its surfaces,9'l0 and to chemisorption of hydrogen on diamond (lOO).l All calculations reported here are ab-initio with the Gaussian lobe basis set of Whittenll for carbon, and a five-term Gaussian expansion for hydrogen12with an exponential scaling factor of d2. In addition to solving the Hartree-Fock-Roothaan SCF equations within this basis set for the clusters used, a model calculation is employed which attempts to represent connection of the cluster with a more extended solid. The clusters employed in the study of this problem are of general formula C,H. Since n is a small number the model calculation intends to minimize the edge effeds due to such small clusters. For this, advantage is taken of the calculations performed on naked where two ideas widely developed in the literature were used (a) the possibility of defining a highly localized representation of electronic structure, and (b) the transferability of the highly localized representation from one molecular environment to another of similar characteristics. Two steps are involved in the definition of the highly localized representation. Firstly, the one-determinant wave function of the naked cluster is written as an antisymmetrized product of two portions
II/ = 4 & is kept fiied during the calculation, and consequently all variational flexibility of the wave function is contained in #n. Under this prescription, the variational equations lead to a modified Hartree-F~k-Roothaad~problem, but of reduced dimension. A matrix formulation for this (1) Lopez, J. P.; Fink, W. H. J. Phys. Chem. 1981, 85, 2642. (2) de Boer, J. H. 'Molecular Processes on Solid Surfaces"; Druglis, E.; Gretz, R. D.; Jaffee, R. I. Ed.; McGraw-Hill: New York, 1969; p 3. (3) Tsong, T. T.; Cowan, P. L. Cn't. Rev. Solid State Mater. Sci. 1978, 7, 289. (4) Gomer, R. h o c . Znt. Congr. Surf. Act., 2nd 1957, 236. (5) Gpmer, R.; Wortman, R.; Lundy, R. J.Chem. Phys. 1957,26,1147. (6) Fink, W. H. J. Chem. Phys. 1978,69, 3325. (7) Fink, W. H.; Butkus, A. M.; Lopez, J. P. Int. J. Quantum Chem. Symp. 1979,13, 331. (8) Lopez, J. P.; Fink, W. H. J. Chem. Phys. 1981, 75, 2290. (9) Butkus, A. M.; Fink, W. H. J. Chem. Phys. 1980, 73, 2884. (10) Butkus, A. M.; Fink, W. H. J. Chem. Phys. 1980, 73, 2893. (11) Whitten, J. L. J. Chem. Phys. 1966, 44, 359. (12) Fink, W. H.; Allen, L. C. J. Chem. Phys. 1967, 46, 2261. (13) Roothaan, C. C. J. Reu. Mod. Phys. 1951,23, 69.
0 1982 American Chemical Society
Hydrogen Diffusion Across Diamond
a
P
L
m e 1. A schematic depiction of the idealized (100) face of diamond obtained by simple termination of the bulk structure. The heavy filled &des represent atoms in the surface layer; the open circles represent atoms in the next layer from the surface; and cross-hatched circles represent atoms in the third layer. Paths A, B, and C are possible straight line periodic diffusion paths away from an overhead site. Path A proceeds along the direction of highest solid atom density.
procedure has been previously presented.’* Secondly, the utilization of a projection technique15 applied to the variational function effects the localization. Repetition of these two steps in a cyclic manner (until convergence is obtained) leads to the best representation of a set of orbitals for the cluster in a truncated basis. The final set in the truncated basis maintains the maximum possible character (in an overlap sense) of the original cluster orbitals. The resulting localized representations are then transferred to locations in the hydrogenated cluster where a representation of the corresponding unit of a more extended solid is desired. After symmetry adaptation and symmetric orthogonalization of the transferred orbitals, they represent an electronic structure on the edges of the cluster which more closely simulate connection of these edges to an extended solid than would result if the electronic structure of the regions of the cluster were included in the variational step. Finally a wave function of the form fi = with now fixed to represent connection to an extended solid is again employed, and the electronic structure of the regions of the hydrogenated cluster contiguous with the absorbate is variationally determined. The virtues and deficiencies of a single-determinant wave function are well-known and it is worthwhile commenting on these within the context of the calculations presented below. Perhaps the most serious deficiency of a restricted Hartree-Fock-type wave function is its inability to describe chemical dissociation properly in many cases as, for example, the dissociation limit of H2 is a hypothetical average of neutral and ionic products instead of two neutral H atoms. Perhaps more pertinent to the present calculations is that ground-state CH dissociates improperly in the Hartree-Fock approximation. However, because of the reduced symmetry (C, or C,)of the cluster calculations employed in the work presented below, the single-determinant wave function correlates to the correct dissociation limits (e.g., ‘Al (C,J and 2S(H) for the 2Al state under C2J. Further, even when the single-determinant wave function dissociates improperly, it gives good results for equilibrium geometry and other properties such as force constants in the neighborhood of the equilibrium geometry. Bond distances are typically accurate to within a few hundredths of angstroms16and force constants to about (14)Fink, W. H.J. Chem. Phys. 1972,57,1822. (15)Fink, W. H.J. Chem. Phys. 1973,7,1045. (16)Pople, J. A. ‘A Priori Geometry Predictions” In Schaefer, 111, H. F., Ed., “Modern Theoretical Chemistry”; Plenum Press: New York, 1977;Vol. 4.
The Journal of Physical Chemistry, Vol. 86, No. 15, 1982 2851
10%. As the diffusive motion of H atom along (100) diamond is most concerned with geometries and force constants, the results presented below can be expected to be reasonably reliable, subject to the limitations of modeling chemisorption with only a finite cluster of atoms. The present approach is most closely related to ab-initio cluster calculations simulating electronic structure at surfaces.1%4 The model approach offers 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 wave function $q, thereby reducing edge effects. Although the formalism within which this model approach has been developed7J4J5is very different from the Green’s function formalism of the embedded cluster a p p r ~ a c h there ~ ~ - ~is~an almost identical physical content. In that perspective, the model calculations can be considered as the first ab-initio implementation of the embedded cluster approach, albeit with a severely restricted representation of the defective substrate. Other ways of attempting to reduce edge effects in cluster calculations are to employ high spin unrestricted Hartree-Fock wave functions for the ~ l u s t e rand ~ ~ to ,~~ surround the cluster with hydrogen atoms.27v30-33The methods we have employed are somewhat more difficult to implement, but do recognize the extended nature of the problem at the outset. Each carbon atom in the bulk of the diamond crystal is linked tetrahedrally to four equivalent neighbors; the resulting space group is Fd3m with a lattice parameter at 20 “C of 3.566 79 The structure of the (100) face has been assumed to be an extension of the above bulk structure. Although some experiments have shown that surface reconstruction on diamond is not easily
(17)Pulay, Peter “Direct Use of the Gradient for Investigating Molecular Energy Surfaces” In Schaefer 111, H. F., Ed.,“Modern Theoretical Chemistry” Plenum Press: New York, 1977;Vol. 4. (18)Stoll, H.;Preuss, H. Phys. Status Solidi 1972,B53, 519. (19)Kunz, A. B.; Mickish, D. J.; Dentrach, P. W. Solid State Commun. 1973,13,35. (20)Stoll, H.;Preuss, H. Phys. Status Solidi 1974,B64, 103. (21)Bauschlicker, C.W.; Kislow, D. H.; Bender, C. F.;Schaefer 111, H.F.J . Chem. Phys. 1975,62,4815. (22)Brewington, R. B.; Bender, C. F.; Schaefer, 111, H.F.J. Chem. Phys. 1976,64,905. (23)Bauschlicker, C. W.; Bender, C. F.; Schaefer, 111, H. F. Chem. Phys. 1976,15,227. (24)Marshall, R. F.; Blint, R. J.; Kunz, A. B. Phys. Reu. B 1976,13, 3333. (25)Goldstein, S.;Curtiss, L. A.; Euwema, R. N. J. Phys. C 1976,9, 4131. (26)Wood, J. Chem. Phys. Lett. 1977,50,129. (27)Snyder, L.C.;Wasserman, Z. Surf. Sci. 1978,71,407. (28)Stoll, H.;Preuss, H. Surf. Sci. 1977,65,229. (29)Cox, B. N.; Bauschlicker, C. W. Surf. Sci. 1981,108,483. (30)Redondo, A,; Goodard, 111, W. A.; McGill, T. C.; Surratt, G. T. Solid State Commun. 1976,20,733. (31)Goddard, 111, W. A,; Rodondo, A.; McGill, T. C. Solid State Commun. 1976,18,981. (32)Goddard, 111, W. A.; Barton, J. J.; Redondo, A.; McGill, T. C. Vac. Sci. Technol. 1978,15, 1274. (33)Kenton, A. C.; Ribarsky, M . W. Phys. Rev. B 1981,23, 2897. (34)Cox, B. N.; Bauschlicker, C. W. Surf. Sci. In press. (35)Grimley, T.B.;Pisani, C. J.Phys. C 1974,7,2831. (36)Pisani, C.Phys. Reu. B 1978,17,3143. (37)Pisani, C.; Ricco, F. Surf. Sci. 1980,92,481. (38)Wyckoff, R. W. G. “Crystal Structures”;Interscience: New York, 1963;2nd ed, p 35. (39)Lurie, P. G.; Wilson, 3. M. Surf. Sci. 1977,65,453.1977,65,476.
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The Journal of Physical Chemistry, Voi. 86, No. 15, 1982
C
C
/ \
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Lopez and Fink 150-
A’
/ \
\ / C
\
C
A”
90
E rr
n
n
L\*/L\n/L L L
60 30
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0
0 Flgure 2. The two cluster models used in this study. The upper structure is designated (3S,28) corresponding wlth three surface and two bulk atoms in the cluster. Calculations on this cluster should be most reliable when the hydrogen atom is close to 0 . The lower structure is designated (2S, 3B) corresponding with two surface and three bulk atoms in the cluster. Calculations on this cluster should be most reliable when the hydrogen atom is close to a.
the possibility of adsorption-induced reconstruction must be kept in mind. As the calculations presented here ignore any geometry modification on the surface, the results obtained must be considered a first approximation to the problem. The diffusion path can be studied by making a cluster with the five carbon atoms on the diagonal A of Figure 1. This cluster will be denoted C5(zig-zag) or C5zz. The cluster with the hydrogen atom will be denoted C5Hzz. The hydrogen atom is initially positioned 2.0 bohrs above the central surface carbon (the equilibrium position found for the overhead site)’ and is moved in a straight line until it reaches a position 3.5 bohrs above the second layer carbon (the equilibrium position found for the bridge site).’ The hydrogen atom moving along this line is interacting with three surface carbons and two second-layer carbons (these second-layer carbons are taken as bulk carbons in the model calculations); this cluster will be symbolized as C5Hzz (3S, 2B). Further examination of Figure 1shows that a five carbon zig-zag chain fragment of the diffusion path A permits another model for the diffusion. If instead of taking the central atom of the figure and the two atoms on either side of it to produce the C5Hzz (3S, 2B) cluster, one advances along the path to just above a second layer carbon and takes that atom and two atoms on either side of it, one obtains a cluster which will be called C5Hzz (2S, 3B). This designation is to indicate that the hydrogen atom now interacts with two surface and three second-layer atoms. Both of these clusters are shown in Figure 2. The length of the line segment connecting the overhead with the bridge site is 2.390 bohrs (1.265 A). This line is almost parallel to the diamond surface; the inclination angle is about 4 O . The two clusters belong to the C, point group (except for the overhead position of C5Hzz (3S, 2B) and for the bridge position of C5Hzz (2S, 3B) where they transform according to C2J. With the clusters oriented in the xy plane, S and P,, P, orbitals of the basis will transform under irreducible representation a’ and the P, orbitals will transform under a”. For the model calculations only s-type orbitals are used to define the fixed space. In both clusters a total of eight electrons are in the fixed space, giving an occupation (4a’)2. The fixed space of C5Hzz (3S, 2B) contains the bulk or(40) Marsh, J.
B.; Farnsworth, H. E. Surf. Sci. 1964, I, 3.
114
112
3/4
I
Flgure 3. Calculated relative energies as the hydrogen atom moves from the overhead site at 0.0 to the bridge slte at 1.0 along the path depicted for cluster (35,28) in Figure 2. The upper cwve for each state Is obtalned with a completely variational SCF calculation for the cluster while the lower curve for each state is obtained with the model calculatlon slmulating connectlon to a more extended solid. Occupations of the A” and A’ states are given in the text.
L 112 314
0
114
I
Flgure 4. Calculated relative energies as the hydrogen atom moves from the overhead site at 0.0 to the bridge site at 1.0 along the path depicted for cluster (2S, 38) in Figure 2: upper curve, SCF; lower curve, model as in Figure 3. These results should be more reliable at 1.0 than is Figure 3 and vlce versa at 0.0.
bitals 1s(2), ls(3), 2s(2),and 2s(3). For C5Hzz (2S,3B) the fixed space is made up of the bulk orbitals 1s(5), ls(6), 2s(5), and 246). The expansions for the bulk orbitals are those obtained in ref 8.
Results The potential energy curves for the SCF and model calculations are shown in Figure 3 for C5Hzz (3S, 2B) and in Figure 4 for C5Hzz (2S, 3B). The distance is measured in units of a (2.390 b o b , the length of a repetitive segment between the overhead and bridge sites). The origin of this segment is placed at the top of the overhead site for both clusters. The minimum energy along the repetitive segment of the A” state is obtained when the hydrogen atom is at the origin of this segment (the overhead position). Thus, this energy has been chosen as the zero-reference point for the potential energy of both clusters. The occupation obtained for both clusters was of the form (la”)2(14a’)2(X)1where X = 15a’ or 2a”. The results on both clusters show the same general trend: A” is the ground state; the potential energy increases monotonically from the overhead to the bridge position for the A” state. But the excited state A’ is differently described by the two clusters. The A’ state of C5Hzz (3S, 2B) presents a minimum at about a/4 whereas this minimum is absent for the C5Hzz (2S, 3B) cluster. While the curves in Figures 3 and 4 have been calculated and presented over the full length of diffusion, they are not equally reliable over this entire range. Because the
Hydrogen Diffusion Across Diamond
hydrogen approaches a cluster edge as it moves away from o in Figure 3 and as it moves away from a in Figure 4 edge effects will become increasingly important. Thus Figure 3 is most reliable in the vicinity of o and Figure 4 is most reliable in the vicinity of a. Both curves do establish that there is a monotonic increase in energy between the overhead and bridge sites for the ground state and therefore the diffusion barrier is simply the difference in energy between these two sites. Using the most sophisticated calculations available for both of these sites yields a barrier height of 66 kcal/mol for the cluster SCF calculation and 80 kcal/mol for the model calculation.’ Returning to the A’ state we see that a minimum occurs in the curve for C5Hzz (3S,2B) at a/4, but no such minimum occurs for C$zz (2S, 3B). If we recall the discussion above regarding the regions along the diffusion path where them two clusters have their respectively greater reliability, the curve possessing the minimum should be the more reliable. Thus, the minimum for the A’ state must be inferred to occur for a position displaced from the directly overhead site. This displaced position will be called the shifted overhead site. Of course, the A’ state represents an electronic excitation of the hydrogen adsorbed on diamond system and consequently will have little interest for the observed ground-state chemistry, nevertheless its properties are worth discussing. This shifted overhead site will have two different barriers to diffusion corresponding to diffusion in the direction of the overhead site (smaller barrier) and in the direction of the bridge site (larger barrier). The SCF and model give 24 and 22 kcal/mol, respectively, for the smaller barrier and 90 and 105 kcal/mol, respectively, for the larger barrier. The appearance of the shifted overhead site can be understood by examining the energy values of the C5Hzz (3S,2B) cluster for the Al and B2 states under the point group C%. The energy separation between them is small,’ so when they mix together into the A’ state a strong interaction could be expected. This interaction would lower the energy of one of the states below the value for A2 level as soon as the symmetry is no longer C2”. Here the hydrogen diffusive motion reduces the symmetry to C, with the resulting immediate decrease in the energy. Another interesting aspect to be noticed is that the Cl-H distance is still 2.0 b o b s when the hydrogen atom is at the shifted-overhead site. But the lateral motion about this new site is more restricted (greater force constant) than for the A” state around the overhead site. The vibrational frequency of the lateral oscillatory motion around the overhead site for the A” ground state has been estimated from the C5Hzz (3S, 2B) potential energy curves. Harmonic oscillations for small displacementa from the equilibrium position have been assumed. Thus, a least-squares fit of the data with the parabolic equation V = 1/2kx2 has been performed. For the SCF
The Journal of Physical Chemistry, Vol. 86,No. 15, 1982 2853
a value of 0.965 mdyn/A was calculated for the force constant. The fundamental vibrational frequency is esThe reduced mass p was timated from uo = 1/27r(k/~)’/~. taken as equal to the hydrogen atom mass. With this, the calculated value of vo was 1274 cm-’. The same procedure applied to the A” state of the model calculation yields a force constant of 0.84 mdyn/A and a vibrational frequency equal to 1190 cm-’. The errors in the parabolic fit were also e ~ t i m a t e d . The ~ ~ ~final ~ ~ results are (1274 f 40) cm-’ for SCF (1190 f 85) cm-’ for model The uncertainties reflect only the quality of data fitting to a parabola and should not be taken in the same sense as experimental bars. These numbers seem reasonable since they are a little below the H-C bending vibration in alkanes which is in the range 1350-1470 cm-’. The calculated vibrational frequency for the A’ state is about 2350 cm-’, which is below but closer now to the stretching vibrational frequency of C-H in alkanes (2850-2960 cm-’).
Conclusion The calculations give a very high activation energy for diffusion along the chosen .crystallographic orientation. There is some indirect evidence that this barrier could be high; specifically, experiments of hydrogen diffusion in pyrolytic carbona give a barrier of about 100 kcal/mol. An interesting outcome of the calculations is the appearance of a new adsorption site for hydrogen atom. This “shifted-overhead”site is only present for the excited state A’. Vibrational frequencies for the lateral motion around the overhead site of the ground state A” have been calculated. From the point of view of the calculations, the difference in the correlation energy between different positions of the hydrogen atom along the repetitive segment is probably small, since the frequency is calculated for small displacement from the equilibrium position. Thus, it is reasonable to expect that the shape of the curve in the vicinity of the overhead site would not be too much different from one obtained by taking into account the correlation effects. Another problem is how well the clusters are representing the diffusion process on the surface of the real crystal. Hopefully, the model which simulates connection to the bulk gives better results. The model has been constructred to simulate connection to the bulk and, consequently, should give better results. This would be reflected in the slightly smaller vibrational frequency given by the model. (41) Margenau, H.;Murphy, G. “The Mathematics of Physics and Chemistry”;Van Nostrand: New York, 1962. (42) Birge, R. T. Rev. Mod. Phys. 1947, 19, 298. (43) Causey, R.A.;Ellemand, T. S.;Verghese, K. Carbon 1979,17,323.
