Vacuum, asymptotic structure

For the self-consistent orbitals, we have determined1"4 the exact analytical asymptotic structure in the classically forbidden vacuum region of (i) the Slater potential VJ (r), (u) the functional derivative (exchange potential)... 

We refer the reader to the original literature for details of the derivation of these results. What the derivations all show, however, is that the asymptotic structure in the vacuum region is governed by and arises entirely from the orbitals deep in the metal interior where the structure of 4>k(x) = sin[kx + 5(k)]. The orbitals in the regions at and outside the surface do not contribute to the structure far in the vacuum. 

In Fig. 5 the correlation-kinetic potential component Wt (z) is plotted. For these densities, the potential is entirely positive, possesses the correct asymptotic structure of Eq. (45) in the vacuum, and exhibits the Bardeen-Friedel oscillations. Once again, thepotential w[ z) is an order of magnitude smaller than the Pauli component Wx (z). For higher density metals (rs < 2), the correlation-kinetic contribution to vx(z) will be less significant. It will vanish entirely for the very slowly varying density case for which33,34 vx(z) = Wx (z). 

The physical interpretation of the functional derivative vx(r) shows that it is comprised of a term Wx (r) representative of Pauli correlations, and a term wj (r) that constitutes part of the total correlation-kinetic contribution Wt (r). cThe exact asymptotic structure of these components in the vacuum has been determined and shown to also be image-potential-like. Although the structure of vx(r) about the surface and asymptotically in the vacuum and metal-bulk regions is comprised primarily of its Pauli component, the correlation-kinetic contribution is not insignificant for medium and low density metals. It is only for high density systems (rs < 2) that vx(r) is represented essentially by its Pauli component Wx (r). Thus, we see that the uniform electron gas result of -kF/ir for the functional derivative vx(r), which is the asymptotic metal-bulk value, is not a consequence of Pauli correlations alone as is thought to be the case. There is also a small correlation-kinetic contribution. The Pauli and correlation-kinetic contributions have now been quantified. 

As discussed in the text, the asymptotic structure of vxc(r) in the vacuum can be thought of as arising from that part of the Coulomb hole localized to the... 

For the nonuniform electron gas at a metal surface, the Slater potential has an erroneous asymptotic behavior both in the classically forbidden region as well as in the metal bulk. In the vacuum region, the Slater potential has the analytical [10] asymptotic structure [35,51] V r) = — Xs(p)/x, with the coefficient otsiP) defined by Eq. (103). In the metal bulk this potential approaches [35] a value of ( — 1) in units of (3kp/27r) instead of the correct Kohn-Sham value of ( — 2/3). Further, in contrast to finite systems, the Slater potential V (r) and the work W,(r) are not equivalent [31, 35, 51] asymptotically in the classically forbidden region. This is because, for asymptotic positions of the electron in the vacuum, the Fermi hole continues to spread within the crystal and thus remains a dynamic charge distribution [34]. 

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