4560 N. D. LANG AND W. KOHN introduced by the approximate treatment of corre- lations are not significant IIL ION LATTICE MODEL In the present section, we shall calculate the sur face energy on the basis of a model in which the ice, are represented by appropriate pseudopoten- tials. Such a model is known to be quantitatively successful for simple bulk metals in which the con- duction band has s-p character and is adequatel (bcc loL separated from d-like states. POTENTIAL For these metals, the difference between the Ifcc lll) total pseudopotential and the potential due to the -- UNI FORM POSITIV uniform charge background is small. Therefore, BACKGROUND MODEL taking advantage of the stationary propertyof ex- pression(2. 2)for E, n], we shall calculate all energies in the present model using the electronic density distributions n(x)of the uniform background model. In this way, we avoid the much more diffi- FIG. 4. Comparison of theoretical values of the sur- cult problem of solving truly 3-dimensional Schr face energy with zero-temperature extrapolations of ex- perimental results for liquid-metal surface tensions We adopt here the local ion pseudopotential pro (open circles). Dashed curve gives the surface energy for the positive background model. Vertical lines give osed by Ashcroft, which has the form eoretical values corrected for the presence of the lat (F) y≤c tice: The lower end point gives the value appropriate to point that appropriate to (3.1) bcc lattice. In both cases, the surface plane is taken to plane which is most densely pac where Z is the ionic charge and r is a cutoff ra- the alkali metals of lower density, the lines are con dius which has been determined for each metal to tracted almost to point give a good description of the bulk properties This is equivalent to representing each ion by an tive purposes. Finally, we note that, particularly effective charge distribution nion (r)which gives at higher densities, there are large cancellations rise to the potential(3.1 between positive and negative terms, making the Since in the present model, the electron densi- final results rather sensitive to small errors in ties n(x)of the uniform background case are em the individual terms. Over the range rs=3-5, ployed, the intrinsic electronic energies TIn]and covering the alkali metals Li, Na, K, for which Exc[n] are the same as before. The difference of Smith gives calculated surface energies, our re- the surface energies in the two models sults exceed his by about 50% In Fig. 4, the calculated surface energies are 6G=0-0g (3.2) compared with linear extrapolations to zero tem is therefore entirely due to the differences in elec perature of measured liquid-metal surface ten- trostatic interaction energies of all positive and sions. The agreement between theory and experi- is fair for the lo but for higher-density metals the measured surface tensions increase rapidly with density, while the calculated surface energies decrease towards larg negative values. This basic shortcoming of the uniform background model, which all previous cal culations have also encountered, will be corrected in the following section by going over to a model in which the positive ions are more realistically treated. The correlation contribution to the calcu lated o(Table Ii)is never more than about 15% of FIG. 5. Two steps for calculating the electrostatic imental value, indicating that contributions to the surface energy