1216 N. D. LANG ANd W. KOHN 3 for the principal faces of nine simple metals -Al Pb. Z metals Cu, Au, and Ag. Experimental data are generally available only for polycrystalline samples Background, n+(x] of unknown surface structure. This makes Electrons, n(x) tailed comparison between theory and experiment mpossible. Nonetheless, we can state the follow ng conclusions imple metals: The measured work functions range over 2.. 3 ev. with the possible exception of Li (where there is considerable uncertainty both in the experimental data and in the pseudopotential) agreement between the full theory and experiment is typically within 5-10%0. The ionic lattice contri- outions 8, which are characteristically of the or- der of 10% of the total work functions, contribute to establish this rather good agreement. Anisotro- evel (u) ies among the different faces are typically also of the order of 10% of the mean work function. In ac cordance with the arguments of Smoluchowski, w find the lowest work function to be associated wit FIG. 1. Schematic representation of (a) density dis- the least densely packed face among those consid- tributions at a metal surface and (b) various energies ered [(110) for icc, (111)for bcc]. relevant to a study of the work function. Noble metals: In view of the success with simple metals, we have applied the same technique to the noble metals to learn about the limits of validity of ative to the mean electrostatic potential there(see our theory. Here the experimental work functions Fig. 1). It is important to know if this expression range over 4.0-5 2 ev, and the calculated values includes properly all many-body effects, in par are 15-30% too low. It may be assumed that the ticular, the work done against the image force in presence of the filled d bands not far from the Fer- removing an electron from the metal. This is in mi level makes our highly simplified theory, based fact the case. As we have not found any rigorous on the inor ous-electron-gas model with demonstration of Eq .(2. 1)in the literature,we small pseudopotential corrections, much less ap propriate for these metals Since we are interested in removing one electron In summary, the theory we have outlined appears from the metal at K, we first develop a simpl to describe well the work functions of simple met- extension of the theory of Hohenberg and Kohn (HK)2 Is. Additional reliable experimental data for this to allow for a variable number of electrons, and class of metals would be highly desirable, partic- then use this theory in establishing the validity of ularly data on the work functions of single-crystal Eq.(2.1) ces. In the case of the noble metals. on the other In hk. it was shown that for a fixed number of hand, where, for metallurgical reasons, the ex- electrons N and arbitrary static external potential perimental data are much more consistent and re- v(r), there exists an energy expression liable, the present theory is less successful, and further theoretical work is needed. There is need also for additional theoretical studies on the transi- tion metals, which are not discussed in this paper. IL, RIGOROUS EXPRESSION FOR WORK FUNCTION +Gn], (2.2) alitative considerations, in the spirit of the with the following properties: (a)Gn] is a unive Sommerfeld electron theory of metals, strongly sal functional of n(r), not explicitly dependent suggest that the work function is given by the v(r), given by G[n]=(Vn, [T+U,) 1 n(r)n (2.1) Here Ao is the change in electrostatic potential where the wave function y, refers to the unique across the dipole layer created by the "spilling electron ground state with density n(r), and T and out"of electrons at the surface, and u is the chem- U are, respectively, the kinetic- and interaction- ical potential of the electrons in the bulk metal rel- energy operators; (b)eIn] is equal to the correct