Study of surfaces and interfaces using quantum chemistry techniques
William A. Goddard Ill and T. C. McGill Arthur Amos Noyes Laboratory of Chemical Physics") and Harry G. Steele Laboratory of Electrical Sciences, California Institute of Technology, Pasadena, California 91125 (Received 23 April 1979; accepted 29 May 1979)
There are a number of difficulties in elucidating the microscopic details of the electronic states at surfaces and interfaces. The first step should be to determine the structure at the surface or interface, but this is difficult experimentally even for the clean, ordered surface and extremely difficult for cases with impurity atoms (e.g., nonordered oxide layers). The theoretical study of such geometries and energy surfaces is the subject of quantum chemistry. We present a review of some of the theoretical techniques from quantum chemistry that are being applied to surfaces. The procedure consists of treating a finite piece of the surface or interface as a molecule. Ab initio calculations are then carried out on the molecule using the generalized valence bond (GVB) method (with additional configuration interaction), thereby incorporating the dominant many-body effects. The reliability of these techniques is discussed by giving some examples from molecular chemistry and the surfaces of solids. The strengths and weaknesses of this approach are compared with more traditional band theory related methods and are illustrated with various examples. PACS numbers: 73.20.- r, 71.10. + x, 68.20. + t
I. INTRODUCTION of the theoretical methods. Section III illustrates some prob lems and successes of these methods while Sec. IV contains a A general goal of the surface scientist is to develop a micro summary. scopic description of the chemical processes, structure, and electronic states at surfaces and interfaces. In this endeavor II. THEORETICAL METHODS theory promises to play a key role since the experiments are frequently difficult to interpret unambiguously and theory Over the last two decades, methods have been developed is required to allow comparison between various experimental for accurate first principles calculations of the wavefunctions results. Further, many important phenomena on surfaces of molecules.l-7 These methods of quantum chemistry involve involve short-lived chemical intermediates or reactions at (i) exact evaluation of all one- and two-electron integrals defects or other centers present in small concentrations, involved in evaluating the energies of wavefunctions, and (ii) making experimental detection very difficult. explicit consideration of all electrons (core and valence). To be maximally useful in elucidating important surface Quantum chemical methods lead directly to total energies phenomena, the theory must be able to provide reliable an and hence can be used to obtain accurate geometries for swers concerning such questions as (i) the structure of surfaces, ground states and excited states. Indeed, these methods can (ii) the structure, site, and energy for chemisorbed species and be used to obtain complete potential surfaces (including ac reaction intermediates (including cases with broken bonds), tivation barriers) for reactions of molecules. In addition, these and (iii) the potential energy surfaces for the motion of the methods lead to direct evaluation of excitation energies, relevant atoms or molecules at the surface or interface. To ionization potentials, and electron affinities. provide a means of experimental verification of the theoretical Since the quantum chemical methods deal with explicit results, it is important for the theory to characterize those evaluation of the total energy of explicit wavefunctions, they properties of the surface or interface that might be observable are amenable to direct, sequential inclusion of electron cor experimentally. Thus the theory should predict atomic relation or many-body effects. This allows one to learn how spacings, vibrational frequencies, photoemission spectra, and sensitive various properties are to electron correlation and then electronic excitation energies for the various species on the to use the appropriate methods in various applications. surface. These requirements place quite severe constraints upon the theory. Indeed, no theoretical methods currently A. Hartree-Fock satisfy all requirements; however, current quantum chemical 1. Basic equations techniques meet some of the important objectives and show promise of developments to satisfy the others. The most common approach to wavefunctions is the Har In this paper we review some of the methods that have been tree-Fock (HF) or molecular orbital method. For a simple applied in quantum chemistry to answer questions about molecule (referred to as closed-shell), this method involves chemical processes and structure. In Sec. II, we review some placing two electrons (one of each spin) in a Slater determi-
1308 J. Vac. Sci. Techno!., 16(5), Sept./Oct. 1979 0022-5355/79/051308-10$01.00 © 1979 American Vacuum Society 1308 1309 W. A. Goddard and T. C. McGill: Study of surfaces and interfaces 1309
nant of molecular orbitals and optimizing the shape of the where ]j(1) is the Coulomb potential at position T 1 due to 2 molecular orbitals to achieve the lowest energy. Thus the charge density I¢i 1 , wavefunction for anN electron (closed-shell) wavefunction is ]j(l) = JdsTz ¢j(2)¢i(2)' (9) \)/CS T12 (rJ,T2,. (1) and where Ki is the exchange operator defined so that where a is the antisymmetrizer or determinant operator and theN /2 spatial orbitals are taken as orthonormal, 2 2 3 Ki¢;(1) =
(7) That is, from a knowledge of orbital eigenvalues alone, one calculate the total energy. In addition to the one where cannot IE;l, must be able to separately evaluate the one- and two-electron Eo= L ZAZB, energy terms. Since total energies are required to determine A>B TAB geometries, one cannot calculate geometries without separate N/2 evaluation of the one- and two-electron energy terms. El = 2 L h;;, i=l 2. Basis sets and The HF operator HHF involves integral operators (Ki ), N/2 2 E2 = 2 L (2]ij- K;j) second-order partial derivatives ( -lf2 \7 ), and complicated i,j=l functions of electronic coordinates (]i ). As a result, a direct are the zero-electron, one-electron, and two-electron inter solution of these equations is practical only for atoms in which actions. symmetry can be used to remove all but the radial coordinates. Applying the variational principle to (7) leads to the nec For molecular systems (8) is solved by expanding each (un essary condition (the Euler-Lagrange equation) for the op known) molecular orbital¢; in terms of a known (finite) set timum orbitals, of basis functions
HHF¢; = €;¢; (i = 1,2, ... ,N /2), (8a) lxi';~I = 1,2, ... ,PI (14) where p ¢; = I: xi'CI'; i = 1,2, ... ,N /2. (15) N/2 1'=1 HHF = h + L (2]j -Kj), (8b) j=l This converts the coupled nonlinear integra-differential Eq.
J. Vac. Sci. Technol., Vol. 16, No.5, Sept./Oct. 1979 1310 W. A. Goddard and T. C. McGill: Study of surfaces and interfaces 1310
(8) to coupled nonlinear algebraic Eqs. (16), whole set of Gaussians combined with fixed relative coeffi 6 cients, •7·9 P P f.l = 1,2, ... ,P L H~vFCvi = L s!il' Cv;E; (16a) v=l v=l i = 1,2, ... ,N /2 Xcontracted = L d!iX!i,prim· (21) !i where Thus, in calculating the wavefunctions (e.g., HF), the con (16b) tracted basis functions are used, reducing the size of the ma and trices to be handled in the iterative calculations; however, integrals are evaluated in terms of primitive functions (20) H~! = J. Vac. Sci. Technol., Vol. 16, No. 5, Sept./Oct. 1979 1311 W. A. Goddard and T. C. McGill: Study of surfaces and interfaces 1311 TABLE I. Bond energies (eY). These values are from the bottom of the tion. Forcing ¢a to be equal to ¢h leads to the HF wave potential curve (D,) and hence the zero-point energy must be subtracted function so that HF is a special case of GVB. to yield the direct experimental value. All calculations use a DZd basis For a many-electron system, the GVB wavefunction can set. (See Ref. 13). generally be obtained from HF by replacing each electron HF GYB-Cl Experimental pair CH3~CH3 3.13 4.07 4.19 ¢;(2i- l)¢;(2i) CH3~0H 2.73 4.25 4.28 of the HF wavefunction by the electron pair HO~OH 0.04 2.45 2.26 H2C=CH2 5.35 7.44 7.82 H2C=O 4.58 7.57 7.93 1Q2 4.26 4.23 leading to a total wavefunction of the form a[(¢Ia¢lb + ¢Ih¢Ia)(¢2a¢2b + ¢zh¢za) · · · (aj3aj3 .. . )]. II. D. Some surfaces [e.g., Si (lll), Si (lOO)]lead to electronic Note that although all orbitals are allowed to be different, the states similar to those occurring as chemical bonds are broken. total wavefunction is a proper eigenstate of total spin and As a result, HF theory is expected to lead to serious problems satisfies the Pauli principle (expansion of this wavefunction for such states, as discussed in Sec. III. would lead to numerous Slater determinants). For the systems discussed in this paper, the GVB orbitals B. Generalized valence bond are each primarily localized on a single center and hence the In the simple valence bond (VB) description, one starts with description is qualitatively as in VB. the proper atomic orbitals of the fragments and pairs these orbitals on different centers to form the bonds. For example, C. GVB-CI 14 for H2 the VB spatial wavefunction is In obtaining sufficient accuracy for assessing reaction J. Vac. Sci. Technol., Vol. 16, No.5, Sept./Oct. 1979 1312 W. A. Goddard and T. C. McGill: Study of surfaces and interfaces 1312 and ionic character. For the case of a strong bond, the overlap method19 in which a is adjusted by the total calculated pop of xe and Xr is large (say, S = 0. 7), leading to an overlap be ulation on a center is sometimes used. tween (renormalized) J. Vac. Sci. Technol., Vol. 16, No.5, Sept./Oct. 1979 1313 W. A. Goddard and T. C. McGill: Study of surfaces and interfaces 1313 A quite different approach to calculating wavefunctions 15 is the muffin-tin scattered-wave Xa approach.24·25 This is described in detail by Messmer25 and we will add only a few comments. The wavefunction is taken to be of the same form >ro as in HF, say (1) for a closed-shell wavefunction. However the ~ >- exchange term \..? 0:: N/2 w Vx =- L Kj ~ 5 j=l of Eq. (8b) is written as Vx :::;: -a(81/87r)l/3pl/3, where a ""' 0.7. In addition, the potential is generally taken FIG. l. Potential curves for low-lying states of 0 2 (from GVB-CI calcula to be of muffin-tin form, i.e., spherically symmetric within tions29). a sphere about each nucleus and constant in between the spheres. Although the wavefunction is of HF form as in (1), where the latter three orbital sets (with eight electrons) cor the proponents of this method generally argue that it includes respond closely to the eight electrons indicated in (31). In this some electron correlation effects. An advantage of this method case, dissociation leads to incorrect potential curves, as indi is that the two-electron interactions are expressed in terms of cated in Fig. 2. densities, allowing a heavy atom such as Pt to be studied as Using the GVB orbitals and carrying out a CI within these simply as a small atom such as Ni A serious problem is that orbitals leads to accurate potential curves; in Table II we because of the use of the muffin-tin approximation, the total compare bond distances, vibrational frequencies, and bond energies are very inaccurate, leading to very poor potential energies in order to provide some idea of the accuracy. surfaces. For example H20 is calculated26a to be linear in X a (the experimental bond angle is 104.5°), and NHs is calcula B. Ozone molecule ted26b to be planar (experimentally NH3 is pyramidal with a bond angle of 107.6°). Experimentally, the planar form of The VB description of ozone (Os) is obtained by combining NH3 is 0.2 eV higher in energy,ll whereas muffin-tin Xa an 0 atom to a diagram such as (31), leading to diagrams such leads to the planar geometry 1.4 eV lower than the pyram asl5a idal!26b On the other hand, application of the muffin-tin X a method to ozone27 led to better results than ab initio HF (but ~ ~w worse than ab initio GVB).28 I I (32) Ill. DISCUSSION We will illustrate some features and capabilities of some ~ of these theoretical methods with several examples. The set of low-lying states so obtained is shown in Fig. 3 and denoted as 47r, 57r, or 61r to indicate how many electrons are in p orbitals perpendicular to the molecular plane. In each A. Oxygen molecule case there is a singly-occupied orbital on each terminal atom. As discussed in more detail elsewhere,29-31 the VB de In the ground state these singly-occupied orbitals are sin scription of 0 2 leads to configurations such as glet-paired, leading to a weak bonding effect (the doubly occupied p1r orbital on the central 0 plays a major factor in ~ ~ ± 00 u--0 (31) Q) ~ ~ ~--~ 0 ± ..r:: ~ ~ ~ -149.60 0a::: where only the p orbitals are shown (both directions per w z pendicular to the axis), and the dots indicate the number of w electrons in each orbital (the 1s and 2s pairs have been ...J -149.70 f:! omitted). As the molecule is pulled apart, the wavefunction 0 changes smoothly to that of two oxygen atoms, as shown in t- Fig. I.29 -149.80'-----:-'-:-----"---=-'-:-----'----L::-----L---::"-::----'--::-'. The HF wavefunction for 0 2 has the configuration 2.0 3.0 4.0 5.0 6.0 INTERNUCLEAR DISTANCE (bohr) (1 0" gls )2(1 O"u 1s )2(2rrg2s )2 FIG. 2. HF potential curve for the ground state (21:;) of 02 compared with X (2rru2s )2(3rr g2p )2(17r u2P )4(17r g2p )2, the GVB-CI potential curve for the lowest two states.29 J. Vac. Sci. Technol., Vol. 16, No.5, Sept./Oct. 1979 1314 W. A. Goddard and T. C. McGill: Study of surfaces and interfaces 1314 TABLE II. Spectroscopic Constants for 0 2. Excitation Vibrational Bond energies< Bond distance frequency energy< (cY) (A) (meY) (eY) Calculatedd Experimental" Calculatedd Experimental" Calculated<.d Experimental" Calculatcdct Experimental" '2:t 10.19 1.655 81 0.92b 'Au 8.79 1.648 78 0.56b 832:;; 6.31 6.17 1.625 1.604 8,1 88 0.81 b 1.01 A3~t 4.25 4.39 1.528 1.522 104 99 0.63 0.82 C3Au 4.17 4.31 1.522 106 0.70 0.91 c'2:;; 3.94 4.10 1.525 1.517 103 98 0.94 1.12 h'2:i 1.69 1.64 1.260 1.227 187 17S 3.19 3.58 a'D.g 1.09 0.98 1.249 1.216 198 ~187 3.79 4.23 X32:;- 0 0 1.238 1.208 210 196 4.88 5.21 a P. Krupenie, J. Phys. Chern. Ref. Data 1, 423 ( 1972). b These states dissociate to an excited stale on one of the 0 atoms. c Evaluated at the minimum on the potential curve. d Reference 29. coupling the singly occupied orbitals, an effect analogous to not. As a result, HF interchanges the states. As indicated superexchange coupling in antiferromagnetic transition metal below, some surfaces lead to analogous electronic states and oxides). hence similar problems. Quantitative studies based on the GVB wavefunctions have Given the serious problems that HF has with ozone, one explained some confusing spectroscopic results and guided would suspect that muffin-tin X a would lead to similar experimental probes on the excited states. 15a·28•31 As indicated problems (since the form of the wavefunction is similar). In in Table III, such theoretical calculations can provide a high fact, however, the X a calculations lead to the correct ground level of accuracy. state, 27 providing evidence that these calculations do include In Table IV we compare the excitation energies from some electron correlation effects (as often suggested by X a GVB-CI calculations with those from HF calculations. It is proponents). Even so, there are serious discrepancies between apparent that there are very serious errors in the HF de the Xa results and the accurate GVB-CI calculations, scription. Most serious is that although ozone is known to have suggesting that X a is missing some very important electron a singlet ground state, HF predicts a ground state triplet (l correlation effects. A serious problem at this point is how to 3B2) and puts the lowest singlet state at over 2 eV above this proceed from X a to accurate results. If part of the electron 3 triplet state (another triplet state 2 B2 is also placed below the correlation is included in X a, how does one include the first singlet). The problem with HF is obvious from the VB missing electron correlation effects? An advantage of ab initio diagram (32). The two singly occupied orbitals are separated methods such as GVB is that there is a well-defined procedure by a large distance and have a low overlap. This is analogous for adding additional correlation effects, eventually leading to an H2 molecule at largeR. The triplet coupling of the or to essentially exact results. 3 bital is well described by HF (leading to the 1 B2 state of 1 ozone), but the singlet coupling (leading to the 1 A1 state) is C. The ( 111) surface of Si The unreconstructed (1 X 1) surface of Si contains an array OZONE EXCITED STATES of singly-occupied orbitals (dangling-bond orbitals) separated from each other by (at least) one subsurface atom. Consider 70 just one pair of such dangling bond orbitals, 60 50 ~~::~==· > .. w ---\ ~ 4.0 \ ' '- \2 3Bz TABLE Ill. Spectroscopic constants for ozone." GVB-Cl Experimental 20 ~<~ ~~~ {\ A 1 ,'/ \-2 A1 0-0 bond length (A) 1.299 A 1.278 A 10 \:=::: ~ 47 -\\ 000 bond angle (degrees) 116.0° 116.8° v 1 (meY) 153 138 v3(meY) 87.7 87.4 VB CONFIG VERTICAL ADIABATIC FIG. 3. The electronic excited states of ozone. 28•32 a Reference 28. J. Vac. Sci. Technol., Vol. 16, No. 5, Sept./Oct. 1979 1315 W. A. Goddard and T. C. McGill: Study of surfaces and interfaces 1315 TABLE IV. Excitation energies (eV) for ozone (based on ground state but the g states are different. Substituting Eq. (33) and Eqs. geometry).< (42) and (45) leads to GVB-Cia HP Xa-SWb 1'lfif = ¢g(1)¢g(2)- A¢u(l)¢u(2), I 1A1 41T -1.60 +2.16 -1.65 1 'l'i'~ = ¢u(1)¢u(2) + A¢g(l)¢g(2), I 382 41T 0 0 0 I 3A2 51T 0.49 2.95 -0.53 where A- 1 as the overlapS- 0. I 3B1 51T 0.41 2.74 -0.43 The resulting spectrum of states for the VB and HF de I 1Az 51T 0.74 3.41 0.82 scription is given in Fig. 4. Thus, although the singlet and JIB! 51T 0.81 3.58 1.00 triplet states should be nearly degenerate, HF necessarily leads 2 3B2 61T 1.57 1.64 1.66 2 1A1 61T 1.98 5.88 2.77 to a triplet ground state. I 182 41T 4.52 6.01 1.34 The implications of these results for the Si( 111) surface (unreconstructed) are as follows. 34 Normal closed-shell HF a Reference 28. calculations will lead to a band of states (corresponding to ¢g b Reference 27. 3 and ¢u of the two-orbital system) which are half occupied. c The states are referenced with respect to I 8 2 since all wavefunctions lead to a good description of this state. Thus the HF results would suggest that the surface states are metallic. However, carrying out open-shell HF calculations, one would find that the ground state has each dangling-bond in which two singly-occupied orbitals are separated by an orbital spin-paired to form a ferromagnetic spin state (cor intervening atom so that their overlap is small. responding to the 3'lfMO state of the two-orbital system). On In the molecular orbital description the orbitals are14 the other hand, a proper treatment of the finite surface would lead to a ground state (analogous to 1'1'?) having singly ¢g =X£+ Xr occupied orbitals on each surface Si spin-paired into an overall ¢u =X£- Xr, (33) singlet state with low-lying excited states having these same orbitals paired to higher spins (analogous to 3 '1'~ 8 ). For the with a small separation in the orbitals (to be specific, we will perfect surface at 0 K, these surface states would not be con assume ¢g to be lower than ¢u). The molecular orbital ducting. wavefunctions for the two-electron system are Now consider distortions of the surface to form the 2 X 1 l'lf~? = ¢g(1)¢g(2)[a,B- ,Ba] (34) reconstructed surface. The closed-shell HF wavefunction (¢g¢g or ¢u¢u) of the unreconstructed surface has equal 1 '1'~2° = ¢u(1)¢u(2)[a,B- ,Ba] (35) amounts of ionic and covalent character. Thus, bad descrip 3'l'tt0 = [¢g(1)¢u(2)- ¢u0 )¢g(2)]aa (36) tions result because HF cannot describe a singlet pair of radical centers. If the geometry is allowed to readjust, the HF 1'l'tt0 = [¢g(l)¢u(2) + ¢u(l)¢g(2)][a,B- ,Ba]. (37) wavefunction will be biased toward geometries which do not Substituting (33) into (34)-(37) and omitting the spin factors lead to radical centers. This can be achieved by distorting one leads to Si atom so as to stabilize a positive center (moving the Si down) 1 and the adjacent Si so as to stabilize a negative center (moving 1 VB WF VB WF HF WF 'l'i'f = (xcXr + XrXc)[a,B- ,Ba] (42) IN TERMS OF V 8 IN TERMS OF H F IN TERMS OF ATOMIC ENERGY MOLECULAR ENERGY MOLECULAR ORBITALS LEVELS ORBITALS LEVELS ORBITALS 3 '1'~ 8 = (xcXr- XrXc)aa (43) POSITIV!:_{ IX1 -X,I~_r-¢, ---~ 1 8 ION IX +X,I _/-·'--¢g ---~ '1'~ = (xcxc- XrXr)[a,B- ,Ba] (44) 1 , S'NGcET (X X +X,X,) -"_/- i¢g¢g + ¢,¢,I --- 1 1 1 'l'i'~ = (XcXc + XrXr)[a,B- ,Ba], (45) SINGLET(XX-XX)-/-'..__,"'"- +" "1-_/-'.._(-"- "-+"- "-1 2 ':l r r ''f'q'+'u 'r'u '1-'g '+'g '~"'u '+'u '~-'g with low-lying covalent states (42) and (43) and high-lying NEUTRALc- ¢, ¢g I'- 1¢9 J. Vac. Sci. Technol., Vol. 16, No.5, Sept./Oct. 1979 1316 W. A. Goddard and T. C. McGill: Study of surfaces and interfaces 1316 Recent experimental studies by Monch35 indicate that in IV. SUMMARY AND CONCLUSIONS the 2 X 1 structure of Si(lll), one Si moves down by 0.16 A The techniques of quantum chemistry have been developed with respect to the unreconstructed surface, while the other to provide structure, bond energies, and reaction pathways moves down by 0.00 A. In comparison we have shown30•36 that for molecules. Those techniques based upon a single-particle a normal dangling bond Si relaxes down by 0.08 A, the positive approximation to the wavefunction (e.g., HF) have varying ion relaxes down by 0.38 A, and the negative ion relaxes up success in calculating accurately various properties. For ex by 0.17 A. Thus the average displacement of the surface Si is ample, HF is known to give rather good geometries. However, just that expected of a dangling bond orbital, and the indi it gives rather poor values for bond energies and describes vidual displacements are much less than that expected for the certain biradical states very poorly. To obtain more reliable ionic species. However, accurate HF calculations for the 2 X values for bond energies and to obtain a valid description of 1 structure of Si(111) would probably lead to distortions of the orbitals on radicals, one must employ schemes containing approximately -0.38 and +0.17, as expected for the ion electron correlation effects. One such technique suitable also forms. for studies on surfaces is GVB. The GVB method contains the dominant electron-electron correlation effects and provides D. Si(100) orbitals suitable for a limited CI wavefunctions yielding ac As an additional example, consider the reconstruction of curate values for the bond energies, reaction surfaces, and the Si(100) surface. Each surface Si is bonded to two subsur structure. face Si, leading to Since the primary questions about surfaces and interfaces are often analogous to the ones quantum chemistry answers on molecular systems, we suggest that these techniques will ® ~ ~ ~ (46) be very useful in the field of surfaces and interfaces. The J~:;:. J%, J~ major deficiency in ab initio methods that include electron l~ "" correlation effects is that it is not yet possible to study infinite but bonding these surface atoms in pairs37 leads to systems. Consequently, it is necessary to model the surface with a finite clusters of atoms. This approach often appears particularly inappropriate to the solid state physicist who is (47) trained to think about the importance of long-range order and delocalized states. However, processes involving reactions have long been adequately described by chemists in terms of That is, one can form one strong bond between each Si, but localized models, and even on surfaces it is plausible that the this leaves a radical (dangling bond) orbital on each Si that has chemistry should be dominated by local effects. Since many only a small overlap with other dangling bond orbitals. The of the important questions about surfaces and interfaces in ground state of this system would be a nondegenerate (non volve reactions or chemistry, such approaches should provide conducting) singlet. 34 useful information. Thus, although such approaches may be Now consider the results of a normal (closed-shell) HF unsuccessful at predicting those features of the electronic calculation. The symmetrical structure of (4 7) would lead to spectrum arising from the infinite size of the system, we be equal amounts of radical and ionic character and hence to a lieve that application of these theoretical quantum chemical narrow surface band that is half filled (as discussed in previous methods will lead to genuine insights into the phenomena at reactions), that is, metallic surface states. Allowing the surfaces and interfaces. Even so, the theoretical challenge is structure to relax with one Si of each pair moving down (sta to extend these methods to very large clusters or also to find bilizing a positive center) and one Si moving up (stabilizing some method of modeling the electronic wavefunction of the a negative center) cluster with the properties of the semi-infinite system in which it should be imbedded. ACKNOWLEDGEMENT This work was supported in part by a grant from the Na tional Science Foundation (No. DMR74-04965). should remove the biradical character and hence a much lower energy for the HF wavefunction. Indeed, approximate HF calculations on a surface model •)Contribution No. 6017. 1 like (47) led to semimetallic surface states.38 Since experi Modern Theoretical Chemistry: Methods of Electronic Structure Theory, 39 edited by H. F. Schaefer III (Plenum, New York, 1977), Vol. III. mental evidence is that the surface is not metallic, this result 2Modern Theoretical Chemistry: Applications of Electronic Structure 39 was used to argue against a pairing model such as (47). Theory, edited by H. F. Schaefer III (Plenum, New york, 1977), Vol. Calculations on a structure (48) were found to remove this IV. metallic character, and it was suggested that the Si(lOO) sur 3F. W. Bobrowicz and W. A. Goddard III, Ref. 1, pp. 79-128. 4 face might involve geometries such as (48). A. C. Wahl and G. Das, Ref. 1, pp. 51-78. 5I. Shavitt, Ref. l, pp. 189-276. The point in this analysis is that such results are peculiar to 6J. W. Moskowitz and C. C. Snyder, Ref. l, pp. 387-412. defects in the HF wavefunction and need not be relevant to 7J A. Pople, Ref. 2, pp. l-28. the real surface. 8 We use Hartree atomic units which are defined so that h = l, m, = 1, Ie I J. Vac. Sci. Technol., Vol. 16, No.5, Sept./Oct. 1979 1317 W. A. Goddard and T. C. McGill: Study of surfaces and interfaces 1317 = 1. Thus the unit of energy is the hartree = 27.2116 eV, the unit of length 23D. J. Chadi, Phys. Rev. Lett. 41, 1062 (1978). is the bohr= 0.529177 A, and the unit of velocity is c/137.036. In these 24J. W. D. Connolly, in Modern Theoretical Chemistry: Semiempirical units, kinetic energy is -%\12. Methods-Techniques, edited by G. A. Segal (Plenum, New York, 1977), 9(a) T. H. Dunning, Jr., J. Chern. Phys. 53,2823 (1970); (b) T. H. Dunning, pp. 105-132. Jr., and P. J. Hay, Ref. l, pp. 1-28. 25R. P. 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Hay and W. A. Goddard III, Chern. Phys. Lett. 14,46 (1972); P. J. Hay, Chern. Res. 6, 368 (1973); (b) W. J. Hunt, P. J. Hay, and W. A. Goddard and T. H. Dunning, Jr., and W. A. Goddard III, Chern. Phys. Lett. 23,457 III, J. Chern. Phys. 57,738 (1972). (1973); J. Chern. Phys. 62, 3912 (1975). 16In both HF and GVB, several basis functions would be used (rather than 33 1 '1t~r as written is not correct. It overlaps 1 '1t~r and hence should be or just X£ and Xr) However, the features described here would remain sub thogonalized to ~~~r. stantially the same. 34Quantitative results are presented in A. Redondo, Ph. D. Thesis, (California 17R. Hoffmann, J. Chern. Phys. 39, 1397 (1964). Institute of Technology, 1976). 18S. T. Pantelides, J. Vac. Sci. Techno!. 16, 1349 (1979). 35W. Monch, R. Feder, and P. P. Aver. J. Vac. Sci. Techno!. 16, 1238 19R. Rein, N. Fukuda, H. Win, and G. A. Clarke, J. Chern. Phys. 45,4743 (1979). (1966). 36A. Redondo, W. A. Goddard III, T. C. McGill, and G. T. 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