3 THEORY OF METAL SURFACES: WORK FUNCTION ground-state energy for a given v(r)when the cor- rect density n(r), corresponding to u, is used on 小()≡v()+ the right-hand side of the equation; (c)the first variations of En] about this density, consistent is the total electrostatic potential. Now consider with the restriction first a particle-conserving, but otherwise arbi ≡∫6n()dF=0 (2. 4) trary variation On. Equation(2. 11)then gives vanIs where u 'is some constant, independent of r. 6En]=0 (2. 5) Next, consider a particle nonconserving variation Now consider an ensemble of from(2.11)and (2. 13)we then see that tronic systems at the absolute zero of tempera μ=μ=中()+6G[n]/6n() (2.14) ture, specified by the external potentials v(r)and the chemical potentials u. The subsidiary cond The work function is, by definition tion( 2. 4)will no longer be imposed on the density 更=[中(∞)+El]-E variations of interest. We define R, In]=[n]-ufn(r)dr (2.6) where (oo)is the total electrostatic potential far from the slab considered above and eu is the Clearly, for a first variation of the density n,(F) ground-state energy of the slab with M electrons which does satisfy the condition(2. 4) (but still with N units of positive charge).[Both 中( 613nu]=51En]=0 ( 2. 7) but the combination(2. 15)does not. ]Using the definition of the chemical potential and Eq . (2. 14) Now let nu,(F)and nu+ u, (r)be the correct elec- this can also be written as tron densities corresponding to a given u(r)and respectively,toμandμ+bμ, These two chemi- 中=(∞)-以=[(∞)-列 cal potentials describe two systems whose total mumbers of particles differ by 8N: 6N=∫[n,(式)-n,(6)1d=∫b6n2(F)dF (2.8) 乒=μ-可=(6G[n/6m(F》 (2.17) The corresponding first-order change of &p, u is Here◇ denotes an average over the metal.μis given by the bulk chemical potential relative to the mean interior potential; its independence of this poten 2202, [n]=Emn+6u, u]-Elnm, J]-u6N=0 tial may be verified from the definition (2. 3)of (2.9) GLn]. Equation (2. 16)is equivalent to the postu- where the vanishing follows from the thermody (2.1) namic definition of the chemical potential at T All many-body effects are contained in K. Since an arbitrary small variation on is a change and correlation contributions to u and in (unique)sum of variations of the types 6n, and &n2 their effect on the barrier potential As. In par it follows that, in general, ticular, the image-force effect on p may be re- garded as contained in the disappearance of part (2. 10) of the correlation energy when one electron is We now apply this theory to the work function. moved away from the metal surface We consider a neutral slab of metal. all of whose IlL. UNIFORM-POSITIVE-BACKGROUND MODEL dimensions are macroscopic but whose surface consists overwhelmingly of two parallel faces the We consider in this section a model of a metal ork function of which we wish to consider(the surface in which the positive ions are replaced by physical properties of these two surfaces are taken a uniform positive charge background filling the to be identical). Let n(f)be the correct electron half-space x<0. 10 The electron density in thi density corresponding to the given nuclear poten- model is shown schematically in Fig. 1 (a) tial, the chemical potential u, and a total number We consider first the quantity u in Eq.(2. 1) of electrons N. By(2.6),(2. 2), and(2. 10)w Since deep in the metal interior the electron den have, for a small variation in density on(F) sity has a constant value n, u takes on the simple ∫(,c) 正=k+2 (3.1) where Here kp=(372n1/3 is the bulk Fermi wave numb