ANG AND W. KOHN and the barrier potential Ao. (Significant out ACKNOWLEDGMENTS ward movement of the last layer of atoms would reduce A and thus further increase the error. 3) One of us (W. K)wishes to thank the Hebrew It is not whether more sophisticated pseudo- University in Jerusalem for their hospitality during potential ry would remove most of the discre a sabbatical leave. We would also like to express ancy or whether these metals, with hybridized s-d our appreciation to S. Berger for assistance with bands, require a different approach a part of the numerical work. *supported in part by the National Science Foundation using an artificial small 6v()localized near the sur- and doing a direct self-cor stent calculation of Ad pResent address: IBM Thomas J. Watson Research and hence of =Ao-F for this case. The result for 64 Center, Yorktown Heights, New York 10598 agreed within a few percent with the perturbation result Permanent addre D. Lang and W. Kohn, Phys. Rev. B 1, 4555 16The surface-charge distributions n,(x)also play an (1970); this paper will be denoted as LK-L. important role in theories of the image force and of 2P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 small-gap condensers, which will be discussed in a forth (1964) coming paper Sw. Kohn and L, J. Sham, Phys. Rev. 140, A1133 IThe case re >ld(d is the spacing of the lattice planes (1965) which did not occur in LK-l, does occur in some of the A preliminary account of this part of the calculation present calculations as been given in N. D. Lang, Solid State Commun N. W. Ashcroft and D. C. Langreth, Phys. Rev. 1047(1969) 682(1967) J. R, Smith, Phys. Rev. 181, 522(1969) IN. W. Ashcroft and D. C.Langreth, Phys. Rev 661(1941) 500(1967) An example of a semiempirical study of the work func 2tof these three faces, the(111) face is the most this paper is given by D. Steiner and E. P. Gyftopoulos, closely packed and the (110)face the least closely packed in Proceedings of the Twenty-Seventh Annual Conference for the fcc structure; for the bcc structure the roles a n Physical Electronics, Massachusetts Institute of reversed. Technology, Cambridge, Mass. 1967(unpublished Al [(111),(100)1: B. Sieroczynska-Wojas, Fiz. Tverd The extension of HK to the grand canonical ensemble Tela 10, 693(1968)[Soviet Phys, Solid State 10,544 at arbitrary temperatures has already been given by 1968); Zn (0001): L. P. Mosteller, T. Huen, and F. Phys,Rev,137,A1441(1965).[ On the Wooten,Phys.Rev.184,364(969);K[(110,(100 ght-hand sides of Eg (9) and the secon numbered C. Richard and P. Saget, J. Phys. Suppl. 31, C1 quation of this paper the term -un(r)dr must be 155(1970). Professor G -A. Boutry (private communi- added. I We present de novo the specially simple case of cation) has described current work on the (110) faces of zero temperature K, Rb, and Cs. Single-crystal data on the noble metals ' e use atomic units, with lelam=h=1. In this Cu, Au, Ag [(111),(100)] is given by P. system the unit of energy is 27.2ev. Coquet, and E. Boursey, Czech. J. Phys 10The first wave-mechanical computation of the work (1969); see also, for Cu, P. Kohler, z Phys function using this model was given by J. Bardeen, Phys 21191(1966) Rev.49,653(1936) 2Note, however, the exceptions to this trend for the For this corrected form, see Eg. (3. 58)of D. Pine 100) faces of Al and Pb. Elementary Excitations in Solids(Benjamin, New York, 24 For example, W, Mo: O. D. Protopopov, E. V. Mi 1963 heeva, B. N. Sheinberg, and G. N. Shuppe, Fiz. Tverd. ingwi, A, Sjolander Tela 8, 1140(1966)[Soviet Phys, Solid State 8, 909 and R. H. Land, Phys. Rev. B 1, 1044(1970) (1966)]; Ni: A, Kashetov and N. A. Gorbatyi, Fiz. Tverd 1 One can easily show that in the high-density limit Tela 10, 2135(1968)[Soviet Phys. Solid State 10, 1673 he leading terms of F and Ao are both I*p; they diverge (1969)] dividually but cancel each other in (3. 6). We conjee- 25A1: J. C. Riviere, Proc. Phys. Soc. (London)B70 ture that in the uniform model the work function tends to 676(1957) erected in Ref. 28) a finite limit (in the vicinity of 4 ev) as the interior ik 158, 607(1960)(as quoted in Ref, 28); Zn: mann and G. Wedler, Z. Angew. Phys. 14, 70 1In connection with the material of this section we (1962); Mg: R. Garron, C. R. Acad. Sci 258, 1458 (1964): Na: R. L. Gerlach and T. N. Rhodin, Surface e double-layer moment has beer i.19,403(1970), and priv nett and c uke, Phys. R 1060(1969) 15 The values of were chosen large enough that n,(x) for Na deposited on three different faces of Ni); K, Rb Cs: G.-A. Bountry and H. Dormount, Philips Tech. Rev. as well above the "noise level" associated with lack of 30,225(1969) tency, and small enough to be in th P.A. Anderson, Phys. Rev. 75, 1205(1949)(as sponse region. These conditions were checked by re- erected peating the calculations with the values of 8 doubled 2A. P O ikov and B. M. Tsarev, Fiz. Tverd. The entire theory of this section was also checked by 9,3512 [Soviet Phys. Solid State 9, 2766