4562 N. D. LANG AND W. KOHN 1 TABLE IV. Surface energies in the ion lattice model for eight simple metals. The table gives total surface energy o nt parts (a=Ou+dops +80e1) for the lattice structures and surface planes indicated, Units are ergs/em Also included are rs values, and values for the pseudopotential radius r (Ref. 3) Metal fcc(111) bcc(110) fcc(111) bcc(110) 2.07 1.12 408 2.65 30 0000 35 19 2.14 140 2.61 110 120 293 70 100 100 lations of Ashcroft and Langreth, with the excep typical error for these metals being about 25% tion of pb to which we shall return later. The We omitted the case of pb from the above consider calculations were carried out for mean densities tions. For this metal the measured surface ten n appropriate to the solid, using the values for the sion extrapolates to 620 ergs/ cm at zero temper pseudopotential radius re, given in Ref 3 ature, while our calculations gave a mean value of <a Since for given n, the lattice-plane spacings of of 1400 ergs/cm2. We have no real explanation of this large discrepancy at this time. It may be another, a difference which has been neglected noted, however, that ashcroft and langreth, in Oops, shown in column 5 of Table Iv, is the same their pseudopotential calculations of bulk energies, for the two types of faces. On the other hand, the also found rather less satisfactory agreement for classical cleavage energies shown in Pb than for the other metals. We also remark columns 6-9, differ rather considerably. This that, unlike the other metals considered, Pb is difference is reflected in the total surface ene tetravalent, and that furthermore, it has by far gies g listed in columns 8 and 9 the highest atomic number. We note that both 6pg and &]a are positive and In all the above described calculations the ion half-lattice was undistorted. In Appendix E, we Together they more than compensate for the neg how that allowing the surface plane of ions to ative values of ou. relax to a position of lower energy has a negligi Our final results are plotted in Fig. 4, ble effect on the calculated surface energies they are denoted"pseudopotential theory. "The results for fcc(111)and bcc (110)faces are IV CONCLUDING REMARKS joined by vertical lines. These lines may be re This paper reports the results of calculations garded as a rough measure of the uncertainties of the electron density distributions and surface introduced into our estimates for liquid-metal energies of simple metals. Many-body effects surface tensions by our present very incomplete are taken approximately into account by the use knowledge of the ionic configurations near a liquid of local effective exchange and correlation ener- surface. The same figure shows also the results gies [Eq.(2. 14)]. The interactions of electro of the uniform model and experimentally mea and ions are represented by pseudopotentials sured surface tensions. 31 It should be mentioned taken from theories of bulk metals. All electro- that particularly for Zn, there are still consider static energies, including an important classica able discrepancies among the data obtained by cleavage energy, are included different workers The results are compared with experimental We note that passing from the uniform to th data on surface tensions of eight liquid metals, ion lattice model leads to relatively small changes K, Rb CS, Mg, Zn, and Al, whose sur r the low-density alkali metals Cs, Rb and K, face electron densities vary by a factor of 20 and with an indication of slightly improved agreement whose surface energies range from 80 to 1000 with experiment. On the other hand for the der ergs/cm". For this entire set of metals, we find ser metals Na, Li, Mg, Zn, and Al, the differ- ood semiquantitative agreement, typical ences between the two models become progres- being about 25%. In the of the lower-density sively greater, and while the uniform model alkalis the agreement is especially close. For a lattice model ninth metal, Pb, the theoretical surface energy is follows the experimental trends quite well, a too high by a factor of about 2