1220 N. D LANG AND W. KOHN 3 Now in the (N-1)-electron ground state, associ ted with Ex, the electric field inside the metal is zero, and outside each of the two parallel faces is given by &=4/2A (directed along the outwar normal), with 2A the total surface area. Let us take the x axis to have its origin in one of th e Iaces and point outwards [see Fig. 1(a). Then n, is a unction of x only. Further, let us denote the y-z average of &0(r)by Bu(x)=j. v(x, y, 2)dv dz/,dy dz (4.7) cE atomic units Then, using (4.6), Eq.(4. 4)reduces to Bulk density corresponds to fs"2 1.00 x io-3 atomic units 6=A.6(x)(x)dx =-∠。6(x)nx)dx/n(x)dx Js 5v(x)no()dx/no(x)dx where a is a length large compared with a scre ing length but small compared with the thickness the slab. In the last step both numerator and de- nominator have been reduced by a factor of 2. Since (4. 8)is homogeneous in no, no(x) can now be taken to be the surface-charge density in the semi-in finite uniform-background model induced by an ar Bulk densify corresponds to rs =4 bitrary weak electric field perpendicular to the s-3.00x 10-4 atomic units) B. Calculations To evaluate (4. 8)we require n(x) and 5v(x). na(x) was determined as the difference between two self consistent densities: one corresponding to a neu tral metal with vanishing field outside, and the other corresponding to a metal with a small surface charge and finite field outside. The method used was the same as that described in LK-I, and n,(x) was evaluated for r between 2 and 6.15 The func tions n,(x)for rs=2, 4, and 6 are shown in Figs. 3(a),(b), and (c), respectively. They are rather wide density distributions, which outside the metal DISTANCE (atomic units) drop to half their maximum value at approximately 2-3 (or 1.0-1.5 A)from the nominal surface. 18 is. 3.00x 10". atomic units The perturbation potential 6v(x), Eq.(4.7),was obtained, as explained in LK-I, Appendix D, using seudopotentials of the form employed by ashcroft FIG. 3. The e n(x) in the electron-density dis and Langreth8-20 tribution nx) of the uniform-background model, induced oy a weak external electric field along the outward nox 0f=(4) along +x). Curves are shown for(a)rs=2,(b) nd (c)r=6, n is the These authors have determined the radius r fo electron density deep in the metal. each metal to give a good description of bulk prop the cubic crystals the calculations were done for the(111),(100), and (110)crystal faces, and for Our calculated results for u, 8p, and =u+6p hep crystals, for the(0001)face. In the are given in Table II for nine simple metals. For which Ashcroft and Langreth give two possible