N. D LANG AND KOHN where v[n;F]≡ =0,x>0. For orientation, we remark that a Thomas-Fermi Assuming that the form of Exon] and hence of calculation, f leads to an electron density distri vetrIn; F]is known, the solution of the following bution which decreases smoothly from its interior self-consistency problem gives the exact density value n to zero, over a distance of the order of the istribution of the system of n interacting ele Thomas-Fermi screening length[see Fig. 1(a) trons However, for quantitative purposes, such a calcu {-y2+le[n;F}如=∈;b lation is quite inadequate. It leads to a vanishing work function and negative surface energies, and n()=∑|(F)|2, does not exhibit the important Friedel oscillations of the electron density near the surface where the i are the N lowest-lying orthonormal oresented here uses the self-con- solutions of (2. 5a).The energy E,n] of the sys sistent equations of Kohn and Sham. These are tem is given by(2. 2),with based on the general theory of the inhomogenous electron gas, which includes exchange and corre- T[n]=∑∈;-∫ vet [ns;过]n()d (2.6) lation effects. We review these equations here It is convenient at this point to state a number of It is shown in Refs. 7 and 8 that the total elec facts Eqs. (2.7)-(2. 12) which are strictly cor tronic round-state energy of a many -elec rect for the present model [Eq.(2.1)], including system in an external potential v(r)can be written all many-body effects. Some of these statements in the following form are illustrated in Fig. 1 E!1在+/m件FmP The electrostatic potential energy difference of an electron between x=+o and x=-oo, the so +Tsn]+ExIn called electrostatic dipole barrier, which we de note by△q, is given by Here the functional Ts[n] is the kinetic energy of noninteracting electron system of density dist △q≡cp(+∞)-q(-∞) bution n(), and the functional E[n] represents 4 mLo dx fdx[n(x”)-n(1 the exchange and correlation energy(Hartree theo y corresponds to setting Exc=0). One then defines =4丌Cx{n(x)-n,(x)d (2.7) The chemical potential u of this system, de- ntm;=()+Fdy+m;1,(2.3) fined, as usual, as the ground-state energy differ ence of the N+ l and N electron systems(with the background charge fixed at Nlel )is given by (-∞)+以 Background, n+(x) where u is the intrinsic chemical potential of the Electrons, n(xI infinite system (relative to the electrostatic po tential in this system). 16 From its definition, u is given by where kF is the Fermi momentum of a degenerate electron gas of density n and ux(n)is the ex change and correlation part of the chemical po tential of an infinite uniform electron gas of den sity n. If the exchange and correlation energy per particle of such a gas is denoted by Ex (n),then from the definition of Exe[n] FIG. 1. Schematic representation of (a)density dis tributions and (b) various energies relevant to the metal The work function, defined as the minimum en ergy necessary to eject an electron, is