Extended-Hückel study of the (111), (100), and (110) surfaces of copper

We have found that the energy bands of bulk copper can be fit equally well with a set of Hamiltonian parameters $H_{\mathrm{ij}}$ and overlap parameters $S_{\mathrm{ij}}={\delta }_{\mathrm{ij}}$ corresponding to Wannier basis functions or by a set ($H_{\mathrm{ij}}^{\prime }, S_{\mathrm{ij}}^{\prime }$) corresponding to atomic basis functions or by any of a continuum of sets ($H_{\mathrm{ij}}^{\prime \prime }, S_{\mathrm{ij}}^{\prime \prime }$) between ($H_{\mathrm{ij}}, S_{\mathrm{ij}}$) and ($H_{\mathrm{ij}}^{\prime }, S_{\mathrm{ij}}^{\prime }$). Using the Wannier parameters in thin-film calculations we previously obtained surface-charge deficits of $-0.198e$, $-0.269e$, and $-0.354e$ for the (111), (100), and (110) films, respectively. Using atomic parameters we obtain a surface-charge surplus of $0.256e$ for a (111) film. Using a set of intermediate parameters we obtain deviations from surface-charge neutrality of $0.0079e$, $-0.0013e$, and $-0.0452e$ for the (111), (100), and (110) films. We give a physical explanation of why one would expect a set of intermediate parameters to yield approximate surface-charge neutrality on all three surfaces. Exact surface-charge neutrality can be obtained by making small surface-parameter shifts. We discuss the effect of the new parameters on the surface states, the most interesting of which is the (111) state in the $L_{2^{\prime }}-L_1$ gap. It is shifted downward but still lies 0.1 eV above $E_F$ rather than the 0.4 eV below $E_F$ needed if it is to be used to explain the photoemission data of Gartland and Slagsvold.