4564 D. LANG AND W. KOHN 2 in( d)] may be taken to define a functional F that properties of n(x)are already close to those of the transforms one electron density into another true self-consistent solution. The phase difference n2(x)=FIn; x] between the oscillations in n(x)and n(x)was in fact found never to be more than 5(this reflects an The self-consistent solution to Eqs. (2. 16)is then accurate choice of trial density), and the oscilla n(x)=FIn; x]. The trial density no(x)employed infind- tion amplitude for n(x)[corresponding to 5=1 in ing this solution consisted of an exponential decay- Appendix A, since these oscillations are identical metal, matched to a linear combination of two ex- sons to be quite close to the correct value i.e, ing toward n in a Thomas- Fermi length inside the to those in n,(x)]is constrained for theoretical potentials with adjustable decay lengths outside the self-consistent s is close to unity (see Appen- A Gaussian (with its parameters adjustable)was dix A) added to simulate roughly the first large peak of the expected density oscillations, and the neutrality APPENDIX C: EVALUATION OF THE TERM So IN THE SURFACE ENERGY condition S.no(x)-n (x)]dx=0 was imposed Straight iteration was found not to be a conver It is useful for the following discussion to intro gent solution procedure, and so an analog of the duce the abbreviation E(S, s)to refer to the inter Newton-Raphson method, based on the use of the action energy of S and S per unit area normal to inear response function &F/6n, was employed the x-direction, where the symbols S and S may An no(x)was chosen and n,(x)=Fino x] evaluated be replaced by"*"(denoting a rigid, uniform A neutralizing charge distribution was added to n1 positive or negative background)or"lat"(denoting in the surface region, in order that dvet[n1i x// the lattice). E(S, S)is a self-energy. The charge xIxs. vanish, since the fact that no obeys distributions S and S' will be taken to occupy the eutrality condition does not guarantee that n, will half-space x<0, unless the subscript "inf"is af- y it also. The function n2(x)=FIn1: x] was then fixed to e, to indicate that they fill all of space [(i. e, that they refer to the uncleaved crystal The trial density no was readjusted until n, and cf. Fig. 5]. The classical cleavage energy dal n2 were close to one another implying that eac was near the true solution. ni was then corrected by the addition of a linear combination of functions 60el=E(lat, lat)+E(, lat)+E(, Ehs1auui(r) with the ar determined by the self-cor MEint (lat, lat)+Eint(-, lat)+Eint( The electron distribution of step 1 in Fig. 5 is n4x)+2a11(x)=n2(x)+∑a considered to interact with the lattice via a pur Coulomb potential, rather than via a pseudo tential. This distinction affects the value of bocl (),(1)22222x occur in the present calculations projected onto a set of M orthogonal functions The ur were taken to be the derivatives(so as to e Simple considerations of electrostatics lead to preserve charge neutrality) of the first M har 60 =E(lat, lat)+2 E(, lat) monic oscillator functions, with width and center chosen so as to localize them in the surface re -EInt(lat, lat)+2Eint(-, lat).(C1) gion, and Eq .(B1)was projected onto the oscilla tor functions themselves. The integral So(oF/ In evaluating this expression it is convenient to 8n,)ur(x)dx'was found by computation of the ex let APua(v) represent the Coulomb potential at pression x(F[n1+Aui; x]-n2(x))with Au,(x)<<n lattice site v due to all the ions of the semi-infi- Evaluating n(x)=Fli; x](which is the function ac nite lattice, minus that due to the ions of the infi tually given in Table I)provided a direct check nite (uncleaved)lattice. Here the semi-infinite on the self-consistency of n. The procedure de lattice is taken to be in the left half-space(x<O scribed here could be used repeatedly, but it as is site v, which implies that Aiat(v)is simply proved unnecessary to do so(with M= 8) the negative of the Coulomb potential t v due to a It will, of be recognized that the as semi-infinite lattice in the right half-space. It ptotic phase and amplitude of the Friedel oscil- is convenient in addition to introduce the symbol lations in n(x)are not affected by the addition of P ( v) to represent the corresponding difference unctions localized in the surface region the u(x)]. of potentials due to semi-infinite(x<0)and infi This is not important, however because these nite tive backgrounds. This difference is then