THEORY OF METAL SURFACES: CHARGE DENSITY 4563 In spite of the unexplained failure for pb, we Taking proper account of these o(x"2)correc believe that we have come a substantial step tions was found to be important in the actual cal- closer to a quantitative theory of the electronic culation. The computations implied by Eqs (2.16)are carried into the metal to a point xmls, Effects which we have presumed to be small but at which p and the wave-function renormalization which need to be further examined include second constant are presumably chosen so that vett e order pseudopotential terms and contributions to xmin]=-2kp and a ()matches a pure sine wave the surface energy from changes of the zero point of unit amplitude. In order that these choices lattice vibrations 35 represent an adequate approximation to the More fundamental remaining questions concern the use of local exchange and correlation ener irst be separated out from the computed values gies; the use of pseudopotentials, designed for of vott [n; x] and th (x). Making I xmin I so large bulk metals, in the surface region; and the simu- that these terms become unimportant leads to lation of a liquid-metal surface by an appropriate face of a solid metal. It is hoped that additional good experimental data for both liquid and solid metals will become The condition that the friedel oscillations in available to allow a wider test of the theory and n(x)be self-consistent in the asymptotic region perhaps to suggest necessary modifications leads to the requirement that the s of Eq.(Al) Te would like to thank J. Rudnick for helpful 多=[1+krpk()/(2n2)]1 discussions. The assistance of the staff of the This result is obtained by substituting the form UCSD Computation Center is also very much given for y,x)in Eq. (A3)into Eq.(2. 16d).The appreciated parameter s is found to increase from 1.004 at APPENDIX A: FRIEDEL OSCILLATIONS rs=2 to 1. 07 at rs=6 (using Wigner's formula for ex to obtain u ) The fact that the oscilla General Forms: Importance in Numerical Calculations tions in the rs=2 curve of Fig. 2 are so much maller than those at r.=5 is clear Substitution of the sine-wave form of a, (x)into counted for by this small variation in 5.The Eq.(2. 16d)yields the well-known result reason for the difference is rather as follows From Eq. (3. 16)of Ref. 36, it is seen that qu (Al) tum density oscillations of the type analyzed here are reduced in general by inverse powers of the where t=l and yp =y r).Use of this form for integral n(x)in Eq.( 2. 16b)then gives k2 (which tends toward Ixl as x -oo). Here xo is the i +o (A2) tr[n; xo]=H=0). If vet is take exhibit simple exponential decay toward-tkF to with u()the derivative with respect to density the left of xo, with decay length A, then x=lx-xpI2 of the correlation part of uxe n). The oscillatory for large Ixl, with xp=21n2+xo erms in the electrostatic and exchange potentials In the present calculation, xo (equal to 1.4 at r it is found, cancel each other exactly in the 6 and 2.4 at rs=2)and the characteristic length asymptotic region. The solution ,x)of Eq a over which vett varies, increase slowly as rs de 16a)for the potential of Eq. (A2), in turn, also creases. This means that in terms of the wave exhibits a correction of o(x-2) length of the Friedel oscillations(/kp),xF,the effective origin of x, rapidly moves further and further to the right as r decreases. This implies, vR)- sin [kx-y(e) in turn, that the first oscillations to appear to the left of x=0 become smaller and smaller relative to i as n increases )sin[(+2kr)x-y()-2r] APPENDIX B: SELF-CONSISTENCY PROCEDURE sin[(-2kp)x-y()+2yE (A3) The set of equations(2. 16)in the order (c),(b) (a),(d)[with n replaced by n, in(a),(b),(c)and by