THEORY OF METAL SURFACES: CHARGE DENSITY 4557 更=φ(+∞)-=△q-μ (2 relation energy give, within a few percent, the same results In the interior of the metal, Vett approaches a constant value [see(2. 3),(2. 4), and(2. 8c)] We can now rewrite the self-consistency problem (2. 5)in a form specific to the present problem: Ue→q(-∞)+μi) Hence the eigenfunctions of (2. 5a)can be labeled 2 dx tuer In;x]vo2(x)=2 by the quantum numbers k, k ku, with the fol where has the asymptotic form(2. 11b).vett is aP*, R,, k, =P, (x)exp[i(R, y +ksz)] (2.11a given by where,forx→-∞ e;x=4团n]-4Cax":dx W(x)=sink -y(k)] [n(x”")-n,、(x)+μn(x),(2.16b) Here y(k)is the phase shift which is uniquely de termined by the conditions that y(o)=0 and that th型n]=△n]-a r(k)be continuous. The eigenvalues of(2 a are The density is in turn given by then [from(2. 10) 2广a (2.16d) ∈,A,An=p(-∞)+μx如)+k2+k2+k2),(2.12a) If for convenience we choose the zero of energy The numerical solution of these equations requires careful treatment of quantum oscillations which are present in the density and potential(see Appen 0 (2.13) dix A1). Details concerning the method of solution then by(2.8),q(-∞)+μ)=-k2,and(2.12a) are given in Appendix B B 2+k2+k2-12) (2.12b) The self-consistent system of Eqs.(2. 16)was In order now to make practical use of the theory solved for the bulk metallic densit embodied in Eqs. (2. 2)-(2. 6), some approximate 2-6 at intervals of 0.5. The degree of self- form of the exchange and correlation energy func- consistency achieved in n(x)varied from 0.08% tional is required. For a system with very slowly (for rs=2)to 0. 7%(for rs=6)of the asymptotic density En[n]=∫∈min(式)n(F)d式 (2.14) Table I gives n(x)for rs=2, 2 displays nx)for rs=2 and 5. It will be observed with errors proportional to the squares of the that for the low mean density corresponding to density gradients. Following Refs. 8 and 18, we rs=5, there are sizeable Friedel oscillations shall use this form for the present problem, even ng an ot of n by 120 though in the surface region of a typical metal the hand, at the high mean density corresponding to density varies quite rapidly. a"control"calcula s=2, the density distribution begins to resemble tion, to be described below, and the fact that the the monotonically decreasing form of the Thomas final results are in rather good agreement with Fermi theory (ef. discussion in Appendix A2) experiment suggest that the errors introduced by Figure 3 shows the electrostatic potential ener approximation(2. 14)are not too serious. This o()and the effective potential vern; x]for question is discussed again later on in the present r.=5. It will be noticed that the electrostati section, and in the concluding remarks arrier△φ=q(∞)-q(-∞) is very small, but For ex(n), the exchange and correlation energy that in the vicinity of the surface, o(x)exhibits per particle of a uniform electron gas, we use the substantial oscillation. The corresponding oscil approximation due to wigner. In atomic units, it lation in vet is considerably smaller. This can be explained by the fact that, for large negative 0.458 x, the oscillatory terms of and of the exchange ∈x(n)=- part ofμ cancel exactly( Appendix A).Bothφ and vett are given in Table I for integral rs values where r (n)is defined by from 2 to 6 (4丌/3)[yam)3=1/mn Approximation (2. 14)for Exen] is based on the assumption of a nearly uniform gas. It leads to Other more recently suggested forms of the co an effective exchange and correlation potential