THEORY OF METAL SURFACES: CHARGE DENSITY the negative of the potential at v due to such a The integration range in Eq.(C5a)is then divided semi-infinite background in the right half-space into two segments at t=y, allowing Aouat(R)to be with△q()=△q1at(v)+△φ(u),Eq,(C1) implies written as a()=△qH/h)+△qa(k) (C6) 61=-ZL2△p(v)+∞) (C The symbol vt] stands for a summation over all lattice sites v of the semi-infinite(x<0)lattice X exp[-(2+m2+n2)(t+)]dt(C7a) which are in a volume bounded by y=L,z=墙L, The actual calculation of△q(v) and△a()=ΣF(,m,n,y),(cmb) will employ a technique for computing lattice sums discussed by Misra.and by van der Hoff and Benson, and will follow in outline the treat where F(x, y, a, y)=erfc[x(x2+y+22)1/1 ment of the Coulomb problem for the infinite lat ×(x2+y2+z2-l2 tice given by Coldwell-Horsfall and Maradudin. 39 For simplicity of exposition, the analysis pre The separation parameter y is chosen later to sented here will be confined to the case of a sim achieve equally rapid rates of convergence for all le cubic lattice of lattice constant a, cleaved of the series involved in Ap(). The special case along the(100)plane. The lattice planes are num bered by integers k, increasing from left to right, with k<0 designating planes in the left half space and k>0 designating those in the right half- space. Since all of the sites in a given lattice of Jacobis imaginary transformation for 0 func tions may be employed to put eg.(C7a)into the plane are equivalent, the label v used above ma m be replaced by the index k The two contributions to Ac(k), for k<0, △g()-2∑∑c(,m,n,y) be written (t-e)-12t-le-/tdt,(C8) with >mn extending over all integers, and where G(x,y,z,y)=y2+z2)-12(exp[2mx(y2+z2)12 △92-2dy erfc[my1(y2+2)1/2+x] (C4 +exp[-2mx(y2+221 The exponential convergence factor with E-O, is introduced for convenience in treating singula and where the prime on the summation sign signi terms; e will be dropped at any point in the analy fies omission of the man=0 term sis at which its omission does not give rise to a The Coulomb singularities(for E-o)in the first divergence. two terms of the sum The use of Eulers representation of the r func △)=△q、k)+△q/()+△p2?(k) then cancel in the following way △q(k)+ (t-∈)l/2tle2a permits writing these expressions in the form k)=a了 ≈2只H(,y)+n3/2c8|erte(j∈2) exp[-(j2+m2+n2)(+∈)]dt, J+1/2 △q/)=-2Y d k-1/2 (-∈) H( y)