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On an attempt to calculate the geometry of the (001) surface of bcc transition metals by a phenomenological approach
Surface Science 165 (1986) L85-L89 North-Holland, Amsterdam
L85
SURFACE SCIENCE LETTERS ON AN A T T E M P T T O CALCULATE T H E G E O M E T R Y OF T H E (001) SURFACE OF bcc T R A N S I T I O N M E T A L S BY A P H E N O M E N O L O G I C A L APPROACH S. PICK and M. TOM,~SEK J. Heyrovskj' Institute of Physical Chemistry and Electrochemistry, Czechosloval Academy of Sciences, Mhchova 7, 121 38 Prague 2, Czechoslovakia Received 11 June 1985; accepted for publication 3 October 1985
A phenomenological theory suggested recently by Finnis and Sinclair is used to compute the equilibrium (relaxed) geometries of the (001) surfaces of bcc transition metals. The geometry changes studied represent minor effects on the energetical scale, when compared with processes investigated originally in this theory. Although, e.g., surface contractions are obtained here for a number of metals, the agreement with experiment does not seem to be satisfactory in several respects. A discussion of this situation is presented.
In the last few years, one has observed a steadily growing interest in the theoretical description of various processes on transition metal surfaces. Among them, especially, semiempirical or phenomenological theories dealing mainly with chemisorption became popular. Let us mention only a few of the most recent ones [1-3]. In ref. [1], an LCAO formalism has been developed, which includes a number of physically relevant features and allows one to perform a fitting procedure by means of several available parameters. In refs. [2,3] (see also ref. [4]), on the other hand, an LCAO contribution to the energy appears at the end of a simplified density functional calculation. The method of refs. [2,3] relies on the recently formulated effective medium (quasiatom) approximation [5,6]. The papers [1-3] explain the qualitative aspects of chemisorption which have been observed experimentally, and are even succesful to get more or less correct quantitative results. Moreover, the discussion of ref. [3] signalizes that a number of general problems in surface physics and materials science can be investigated in a similar way. An analogous opinion has been expressed by the authors of ref. [7], where an essentially phenomenological theory stimulated by refs. [5,6] has been proposed to describe a wide spectrum of effects taking place in the bulk or on surface of transition metals. An analogous but more simplified phenomenological theory of bcc transition metals aiming to achieve similar goals appeared in ref. [8]. The latter theory is, 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
L86
S. Pick, M. Tomd,~ek / Geometry of (001) surface of transition metals
nevertheless, free from some serious drawbacks of earlier models. It is interesting to m e n t i o n that in ref. [7], surface relaxations were c o m p u t e d for Pd a n d Ni surfaces. The possibility of a similar calculation was suggested in ref. [8] for bcc transition metals. In the present paper, we c o n c e n t r a t e ourselves on the theory [8] which is applied to the (001) surface of bcc t r a n s i t i o n metals. Two p r o b l e m s were addressed: (1) The surface relaxation of the ideal (1 × 1) surface was calculated. Only the surface layer was allowed to relax. (The multilayer relaxation for W(001) was discussed in detail in ref. [9], cf. also ref. [10] for general i n f o r m a t i o n . ) Quite recently, an analogous investigation based on ref. [8] was p e r f o r m e d for surface relaxation [11]. A few m i n o r differences between the results presented here a n d those o b t a i n e d in ref. [11] are explained by the fact that in ref. [11], the subsurface layer was also allowed to relax. (2) The stability of the relaxed (1 × 1) surface with respect to high-symmetry d e f o r m a t i o n s (reconstructions) Fs~ X 3, M 1 a n d M5 was checked. Again, only the surface layer was allowed to reconstruct. For W, such a restriction does not c o n t r a d i c t e x p e r i m e n t a l observations [12]. T h e results, s u m m a r i z e d in table 1, deserve several e x p l a n a t i o n s a n d comments. (1) The relaxation given in this table represents the " b o n d length" change, i.e. the change of the distance b e t w e e n two nearest neighbours, one of them being at the surface a n d the other one in the subsurface layer. The m i n u s sign denotes the surface (bond) contraction, whereas the positive sign is associated with the surface expansion. The c o r r e s p o n d i n g values of the relative i n t e r p l a n a r c o n t r a c t i o n s (expansions) are three times higher. (2) The F s, X3, M 1 a n d M 5 modes represent the d e f o r m a t i o n s which are physically most interesting. The M 5 mode is the well k n o w n zig-zag reconstruction observed for W and M o [12-15]. The l o n g i t u d i n a l i n - p l a n e mode "X3 was studied by several authors [12-15] (in ref. [13], the n o t a t i o n L z was used). A renewed
Table 1 Results of the present calculation are summarized for (001) surfaces of bcc transition metals; 3b is the surface bond contraction ( < 0) or expansion ( > 0), and 3EI, is the corresponding energy change per surface atom, both for the ideal (1 × 1) surface; energy barriers for particular high symmetry deformations (with the amplitude 0.01a) of the relaxed surface are given in the last four columns; all energy values are given in eV Metal
8b (%)
6Eh
6E(T'5)
6E(N, ~)
3E(M1)
3E(Ms)
V Nb Ta Cr Mo W Fe
- 2 - Y5 - 1 0~5 - 1 - 0.5 1
- 0.017 - 0.057 - 0.010 - 0.003 - 0.007 - 0.001 -0.003
_<0.001 - 10 4 0.002 0.001 0.003 0.003 0.001
0.002 0.002 0.004 0.006 0.008 0.010 0.002
0.002 0.002 0.004 0.003 0.004 0.006 0.002
0.002 0.002 0.004 0.006 0.008 0.010 0.003
s. Pick, M. Tomhgek / Geomet~ of (O01) surface of transition metals
L87
interest to the buckled mode M 1 [12-15] arose owing to the experiments of ref. [16[. Finally, the surface rigid shift _Fswas proposed in ref. [17] and discussed in refs. [12,15]. For the X 3, M 5 and F 5 modes, two independent polarizations are possible, parallel to [1,0] or [1,1]. However, for the small deformations studied here, the results in table 1 are polarization-independent. For any deformation or relaxation, the corresponding change of the total energy per one surface atom (more rigorously, per set consisting from one surface, one subsurface and one second subsurface layer atom) is given. The positive value of the energy change corresponds to energetically unfavourable deformations. The stability of the surface is shown for small deformations, the amplitude of which at surface atoms amounts to 0.01a (a is the lattice constant). For some selected larger values of the deformations we have checked, however, that the energetical barrier grows quickly. (3) Finally, let us note that a modified parametrization was used in ref. [8] for Cr and Fe. This fact should be kept in mind when comparing the results obtained for different metals. One can see from table 1 that for all bcc transition metals, moderate surface relaxations are predicted. The contraction is most serious for N b and it is small for Mo, W. All the surfaces appear to be stable with respect to the deformations considered. Whereas the predictions agree with the fact that m a n y (001) surfaces of bcc transition metals contract, the quantitative comparison with experiment is less satisfactory. The measurements show large surface contraction [18] and also the reconstruction [12] for the Mo and W(001) surfaces. Also for Ta [19] and V [20] (001) surfaces, considerable contractions are reported. (Actually, a good agreement with experiment is achieved for V.) It would be interesting to check whether a surface-phonon softening can be obtained for some modes (we are indebted to the referee for the suggestion to discuss this point). An explicit evaluation of the surface-phonon spectrum would need more elaborated calculations than done in this work. Nevertheless, a simple guess indicates the absence of such a softening. Let us consider the case of W(001). The analysis of experimental data shows that softened phonons of X 3 or M1, 5 symmetry are to be looked for in the frequency range ~0 < 4 T H z [21]. In the harmonic approximation, the corresponding potential energy change per surface atom reads ~ E ~- M J x 2 / 2 . For 0~ ~ 4 THz, x ~ 0.01a, one finds 6 E - 10 4 eV. (The elasticity theory, with elastic constants from ref. [8], gives the same guess without referring to the phonon spectrum.) The energy changes r E , listed in table 1 for W, are essentially higher ( = 10 -2 eV) and consequently, no surface phonon softening is expected. A possible explanation of these facts is the following. The theory [8] fits the bulk distribution of the charge. In fact, during the surface formation and reconstruction (relaxation), also the charge redistribution can be important, corresponding to the sp ~ d charge transfer, or caused by a redistribution of electrons within the quasidegenerate d-orbitals or within the delocalized spstates [22]. To explain these effects in an intuitive way, let us note that sp- and
L88
S. Pick, M. Torna~ek / Geometry of IO01) surface of transition metals"
d-electrons act often in the opposite way [22]. For t r a n s i t i o n metals with partly filled d - b a n d s , chemical ideas based on the b o n d i n g picture m a y be useful. Generally, this picture favours the surface c o n t r a c t i o n [22] a n d u n d e r special conditions, surface reconstruction [15]. For noble metals, the delocalized sp-electrons seem to be decisive, a n d surface e x p a n s i o n is possible as well [22,23]. Naturally, also other factors (the simple form of the theory, the choice of p a r a m e t e r s fitted) can influence the results of the present paper. It is i m p o r t a n t to stress that the energies given in table 1 are essentially smaller than those studied in ref. [8] (cf. also ref. [24]). This is a m o n g other reasons due to the well k n o w n partial c o m p e n s a t i o n of attractive a n d repulsive c o n t r i b u t i o n s to the total energy. Our results indicate that it is difficult to predict reliably the fine geometry changes (relaxation, reconstruction) caused by a defect (surface). The same is true for small energies associated with such changes. I n the terminology of ref. [11], these energies represent the difference b e t w e e n the relaxed a n d the unrelaxed configuration energies. O n the other h a n d , no conclusions on the possibility to assess gross features (large energies) associated with the defect formations themselves follow from our study. The p r e s e n t authors believe that the ideas of ref. [8] have a firm physical f o u n d a tion. However, there is little experience yet with theories of similar nature, be it the explicit f o r m u l a t i o n or the range of applicability. A further investigation of these questions is necessary in the future.
References [11 [2] [3] [4]
C.M. Varma and A.J. Wilson, Phys. Rev. B22 (1980) 3795. P. Nordlander, S, Holloway and J.K. Norskov, Surface Sci. 136 (1984) 59. B. Chakraborty, S. Ho/loway and J.K. Norskov, Surface Sci. 152/153 (1985) 660. B.I. Lundqvist, in: Many-Body Phenomena at Surfaces, Eds. D. Langreth and H. Suhl (Academic Press, Orlando, FE, 1984) p. 93. [5] J.K. Norskov and N.D. Lang, Phys. Rev. B21 (1980) 2131. [6] M.J. Stott and E. Zaremba, Phys. Rev. B22 (1980) 1564. [7] M.S. Daw and M.I. Baskes, Phys. Rev. B29 (1984) 6443. [81 M.W. Finnis and J.E. Sinclair, Phil. Mag. A50 (1984) 45. [9] C,L. Fu, S, Ohnishi, E. Wimmer and A.J. Freeman, Phys. Rev. Letters 53 (1984) 675. [10] U. Landman, R.N. Barnett, C.L. Cleveland and R.H. Rest, J. Vacuum Sci. Technol. A3 (1985) 1574. [11] C.C. Matthai and D.J. Bacon, Phil. Mag. A52 (1985) 1. [121 D.A. King, Phys. Scripta T4 (1983) 34. [131 A. Fasolino, G. Santoro and E. Tosatti, Phys. Rev. Letters 44 (1980) 1684; Surface Sci. 125 (1983) 317. [141 1. Terakura, K. Terakura and N. Hamada, Surface Sci. 111 (1981) 479. [15] M. Tomggek and S. Pick, Surface Sci. 140 (1980) L279: Czech. J. Phys. B35 (1985) 768; Physica 132B (1985) 79. [16] R.T, Tung, W.R. Graham and A.J. Melmed, Surface Sci. 115 (1982) 576. [17] J.A. Walker, M.K. Debe and D.A. King, Surface Sci. 104 (1981) 405. [181 See, e.g.M.A. Van Hove, Surface Sci. 80 (1979) 1.
S. Pick, M. Tomhgek / Geometry of (001) surface of transition metals [19] [20] [21] [22]
L89
A. Titov and W, Moritz, Surface Sci. 123 (1982) L709. V. Jensen, J.N. Andersen, H.B. Nielsen and D.L. Adams, Surface Sci. 116 (1982) 66. S. Pick and M. Tom~gek, Czech. J. Phys. B34 (1984) 1235, and references given therein. V. Heine and L.D. Marks, Competition between Pair-Wise and Multi-Atom Forces at Noble Metal Surfaces, to be published. [23] L.D. Marks, V. Heine and D.J. Smith, Direct Observation of Elastic and Plastic Deformations at Au (111) Surfaces, to be published. [24] C.L. Fu, S. Ohnishi, H.J.F. Jansen and A.J. Freeman, Phys. Rev. B31 (1985) 1168.
L85
SURFACE SCIENCE LETTERS ON AN A T T E M P T T O CALCULATE T H E G E O M E T R Y OF T H E (001) SURFACE OF bcc T R A N S I T I O N M E T A L S BY A P H E N O M E N O L O G I C A L APPROACH S. PICK and M. TOM,~SEK J. Heyrovskj' Institute of Physical Chemistry and Electrochemistry, Czechosloval Academy of Sciences, Mhchova 7, 121 38 Prague 2, Czechoslovakia Received 11 June 1985; accepted for publication 3 October 1985
A phenomenological theory suggested recently by Finnis and Sinclair is used to compute the equilibrium (relaxed) geometries of the (001) surfaces of bcc transition metals. The geometry changes studied represent minor effects on the energetical scale, when compared with processes investigated originally in this theory. Although, e.g., surface contractions are obtained here for a number of metals, the agreement with experiment does not seem to be satisfactory in several respects. A discussion of this situation is presented.
In the last few years, one has observed a steadily growing interest in the theoretical description of various processes on transition metal surfaces. Among them, especially, semiempirical or phenomenological theories dealing mainly with chemisorption became popular. Let us mention only a few of the most recent ones [1-3]. In ref. [1], an LCAO formalism has been developed, which includes a number of physically relevant features and allows one to perform a fitting procedure by means of several available parameters. In refs. [2,3] (see also ref. [4]), on the other hand, an LCAO contribution to the energy appears at the end of a simplified density functional calculation. The method of refs. [2,3] relies on the recently formulated effective medium (quasiatom) approximation [5,6]. The papers [1-3] explain the qualitative aspects of chemisorption which have been observed experimentally, and are even succesful to get more or less correct quantitative results. Moreover, the discussion of ref. [3] signalizes that a number of general problems in surface physics and materials science can be investigated in a similar way. An analogous opinion has been expressed by the authors of ref. [7], where an essentially phenomenological theory stimulated by refs. [5,6] has been proposed to describe a wide spectrum of effects taking place in the bulk or on surface of transition metals. An analogous but more simplified phenomenological theory of bcc transition metals aiming to achieve similar goals appeared in ref. [8]. The latter theory is, 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
L86
S. Pick, M. Tomd,~ek / Geometry of (001) surface of transition metals
nevertheless, free from some serious drawbacks of earlier models. It is interesting to m e n t i o n that in ref. [7], surface relaxations were c o m p u t e d for Pd a n d Ni surfaces. The possibility of a similar calculation was suggested in ref. [8] for bcc transition metals. In the present paper, we c o n c e n t r a t e ourselves on the theory [8] which is applied to the (001) surface of bcc t r a n s i t i o n metals. Two p r o b l e m s were addressed: (1) The surface relaxation of the ideal (1 × 1) surface was calculated. Only the surface layer was allowed to relax. (The multilayer relaxation for W(001) was discussed in detail in ref. [9], cf. also ref. [10] for general i n f o r m a t i o n . ) Quite recently, an analogous investigation based on ref. [8] was p e r f o r m e d for surface relaxation [11]. A few m i n o r differences between the results presented here a n d those o b t a i n e d in ref. [11] are explained by the fact that in ref. [11], the subsurface layer was also allowed to relax. (2) The stability of the relaxed (1 × 1) surface with respect to high-symmetry d e f o r m a t i o n s (reconstructions) Fs~ X 3, M 1 a n d M5 was checked. Again, only the surface layer was allowed to reconstruct. For W, such a restriction does not c o n t r a d i c t e x p e r i m e n t a l observations [12]. T h e results, s u m m a r i z e d in table 1, deserve several e x p l a n a t i o n s a n d comments. (1) The relaxation given in this table represents the " b o n d length" change, i.e. the change of the distance b e t w e e n two nearest neighbours, one of them being at the surface a n d the other one in the subsurface layer. The m i n u s sign denotes the surface (bond) contraction, whereas the positive sign is associated with the surface expansion. The c o r r e s p o n d i n g values of the relative i n t e r p l a n a r c o n t r a c t i o n s (expansions) are three times higher. (2) The F s, X3, M 1 a n d M 5 modes represent the d e f o r m a t i o n s which are physically most interesting. The M 5 mode is the well k n o w n zig-zag reconstruction observed for W and M o [12-15]. The l o n g i t u d i n a l i n - p l a n e mode "X3 was studied by several authors [12-15] (in ref. [13], the n o t a t i o n L z was used). A renewed
Table 1 Results of the present calculation are summarized for (001) surfaces of bcc transition metals; 3b is the surface bond contraction ( < 0) or expansion ( > 0), and 3EI, is the corresponding energy change per surface atom, both for the ideal (1 × 1) surface; energy barriers for particular high symmetry deformations (with the amplitude 0.01a) of the relaxed surface are given in the last four columns; all energy values are given in eV Metal
8b (%)
6Eh
6E(T'5)
6E(N, ~)
3E(M1)
3E(Ms)
V Nb Ta Cr Mo W Fe
- 2 - Y5 - 1 0~5 - 1 - 0.5 1
- 0.017 - 0.057 - 0.010 - 0.003 - 0.007 - 0.001 -0.003
_<0.001 - 10 4 0.002 0.001 0.003 0.003 0.001
0.002 0.002 0.004 0.006 0.008 0.010 0.002
0.002 0.002 0.004 0.003 0.004 0.006 0.002
0.002 0.002 0.004 0.006 0.008 0.010 0.003
s. Pick, M. Tomhgek / Geomet~ of (O01) surface of transition metals
L87
interest to the buckled mode M 1 [12-15] arose owing to the experiments of ref. [16[. Finally, the surface rigid shift _Fswas proposed in ref. [17] and discussed in refs. [12,15]. For the X 3, M 5 and F 5 modes, two independent polarizations are possible, parallel to [1,0] or [1,1]. However, for the small deformations studied here, the results in table 1 are polarization-independent. For any deformation or relaxation, the corresponding change of the total energy per one surface atom (more rigorously, per set consisting from one surface, one subsurface and one second subsurface layer atom) is given. The positive value of the energy change corresponds to energetically unfavourable deformations. The stability of the surface is shown for small deformations, the amplitude of which at surface atoms amounts to 0.01a (a is the lattice constant). For some selected larger values of the deformations we have checked, however, that the energetical barrier grows quickly. (3) Finally, let us note that a modified parametrization was used in ref. [8] for Cr and Fe. This fact should be kept in mind when comparing the results obtained for different metals. One can see from table 1 that for all bcc transition metals, moderate surface relaxations are predicted. The contraction is most serious for N b and it is small for Mo, W. All the surfaces appear to be stable with respect to the deformations considered. Whereas the predictions agree with the fact that m a n y (001) surfaces of bcc transition metals contract, the quantitative comparison with experiment is less satisfactory. The measurements show large surface contraction [18] and also the reconstruction [12] for the Mo and W(001) surfaces. Also for Ta [19] and V [20] (001) surfaces, considerable contractions are reported. (Actually, a good agreement with experiment is achieved for V.) It would be interesting to check whether a surface-phonon softening can be obtained for some modes (we are indebted to the referee for the suggestion to discuss this point). An explicit evaluation of the surface-phonon spectrum would need more elaborated calculations than done in this work. Nevertheless, a simple guess indicates the absence of such a softening. Let us consider the case of W(001). The analysis of experimental data shows that softened phonons of X 3 or M1, 5 symmetry are to be looked for in the frequency range ~0 < 4 T H z [21]. In the harmonic approximation, the corresponding potential energy change per surface atom reads ~ E ~- M J x 2 / 2 . For 0~ ~ 4 THz, x ~ 0.01a, one finds 6 E - 10 4 eV. (The elasticity theory, with elastic constants from ref. [8], gives the same guess without referring to the phonon spectrum.) The energy changes r E , listed in table 1 for W, are essentially higher ( = 10 -2 eV) and consequently, no surface phonon softening is expected. A possible explanation of these facts is the following. The theory [8] fits the bulk distribution of the charge. In fact, during the surface formation and reconstruction (relaxation), also the charge redistribution can be important, corresponding to the sp ~ d charge transfer, or caused by a redistribution of electrons within the quasidegenerate d-orbitals or within the delocalized spstates [22]. To explain these effects in an intuitive way, let us note that sp- and
L88
S. Pick, M. Torna~ek / Geometry of IO01) surface of transition metals"
d-electrons act often in the opposite way [22]. For t r a n s i t i o n metals with partly filled d - b a n d s , chemical ideas based on the b o n d i n g picture m a y be useful. Generally, this picture favours the surface c o n t r a c t i o n [22] a n d u n d e r special conditions, surface reconstruction [15]. For noble metals, the delocalized sp-electrons seem to be decisive, a n d surface e x p a n s i o n is possible as well [22,23]. Naturally, also other factors (the simple form of the theory, the choice of p a r a m e t e r s fitted) can influence the results of the present paper. It is i m p o r t a n t to stress that the energies given in table 1 are essentially smaller than those studied in ref. [8] (cf. also ref. [24]). This is a m o n g other reasons due to the well k n o w n partial c o m p e n s a t i o n of attractive a n d repulsive c o n t r i b u t i o n s to the total energy. Our results indicate that it is difficult to predict reliably the fine geometry changes (relaxation, reconstruction) caused by a defect (surface). The same is true for small energies associated with such changes. I n the terminology of ref. [11], these energies represent the difference b e t w e e n the relaxed a n d the unrelaxed configuration energies. O n the other h a n d , no conclusions on the possibility to assess gross features (large energies) associated with the defect formations themselves follow from our study. The p r e s e n t authors believe that the ideas of ref. [8] have a firm physical f o u n d a tion. However, there is little experience yet with theories of similar nature, be it the explicit f o r m u l a t i o n or the range of applicability. A further investigation of these questions is necessary in the future.
References [11 [2] [3] [4]
C.M. Varma and A.J. Wilson, Phys. Rev. B22 (1980) 3795. P. Nordlander, S, Holloway and J.K. Norskov, Surface Sci. 136 (1984) 59. B. Chakraborty, S. Ho/loway and J.K. Norskov, Surface Sci. 152/153 (1985) 660. B.I. Lundqvist, in: Many-Body Phenomena at Surfaces, Eds. D. Langreth and H. Suhl (Academic Press, Orlando, FE, 1984) p. 93. [5] J.K. Norskov and N.D. Lang, Phys. Rev. B21 (1980) 2131. [6] M.J. Stott and E. Zaremba, Phys. Rev. B22 (1980) 1564. [7] M.S. Daw and M.I. Baskes, Phys. Rev. B29 (1984) 6443. [81 M.W. Finnis and J.E. Sinclair, Phil. Mag. A50 (1984) 45. [9] C,L. Fu, S, Ohnishi, E. Wimmer and A.J. Freeman, Phys. Rev. Letters 53 (1984) 675. [10] U. Landman, R.N. Barnett, C.L. Cleveland and R.H. Rest, J. Vacuum Sci. Technol. A3 (1985) 1574. [11] C.C. Matthai and D.J. Bacon, Phil. Mag. A52 (1985) 1. [121 D.A. King, Phys. Scripta T4 (1983) 34. [131 A. Fasolino, G. Santoro and E. Tosatti, Phys. Rev. Letters 44 (1980) 1684; Surface Sci. 125 (1983) 317. [141 1. Terakura, K. Terakura and N. Hamada, Surface Sci. 111 (1981) 479. [15] M. Tomggek and S. Pick, Surface Sci. 140 (1980) L279: Czech. J. Phys. B35 (1985) 768; Physica 132B (1985) 79. [16] R.T, Tung, W.R. Graham and A.J. Melmed, Surface Sci. 115 (1982) 576. [17] J.A. Walker, M.K. Debe and D.A. King, Surface Sci. 104 (1981) 405. [181 See, e.g.M.A. Van Hove, Surface Sci. 80 (1979) 1.
S. Pick, M. Tomhgek / Geometry of (001) surface of transition metals [19] [20] [21] [22]
L89
A. Titov and W, Moritz, Surface Sci. 123 (1982) L709. V. Jensen, J.N. Andersen, H.B. Nielsen and D.L. Adams, Surface Sci. 116 (1982) 66. S. Pick and M. Tom~gek, Czech. J. Phys. B34 (1984) 1235, and references given therein. V. Heine and L.D. Marks, Competition between Pair-Wise and Multi-Atom Forces at Noble Metal Surfaces, to be published. [23] L.D. Marks, V. Heine and D.J. Smith, Direct Observation of Elastic and Plastic Deformations at Au (111) Surfaces, to be published. [24] C.L. Fu, S. Ohnishi, H.J.F. Jansen and A.J. Freeman, Phys. Rev. B31 (1985) 1168.