Analytical asymptotic structure of the Kohn-Sham exchange potential at a metal surface

Abstract

In a recent paper we presented the analytical asymptotic structure of the Kohn-Sham exchange potential in the classically forbidden region at a metal-vacuum interface. This result is valid for self-consistently determined orbitals of the semi-infinite jellium and structureless-pseudopotential models of a metal surface. In this paper we provide the details of our derivation. The correctness of the analytical derivation is further substantiated through numerical work. ͓S0163-1829͑97͒09231-X͔ In a recent paper, 1 we presented the analytical asymptotic structure of the exchange potential x (r) component of the Kohn-Sham 2 ͑KS͒ density-functional theory 3 exchangecorrelation potential xc (r) in the classically forbidden region of a metal-vacuum interface. The potentials xc (r) andx (r) are defined as the functional derivatives ␦E xc KS ͓͔/␦(r) and ␦E x KS ͓͔/␦(r) of the KS theory exchange-correlation E xc KS ͓͔ and exchange E x KS͓͔ energy functionals of the density ͑r͒, respectively. The asymptotic structure of x (r), valid for the self-consistent orbitals of both the semi-infinite jellium 4,5 and structureless-pseudopotential 4,6 ͑stabilized-jellium͒ models, is image-potential-like of the form Ϫ␣ KS,x (␤)/x, where x is the distance from the surface. The coefficient ␣ KS,x (␤) depends upon the metal properties through the parameter ␤ ϭ(W/⑀ F ) 1/2 , where W is the surface-barrier height and ⑀ F the Fermi energy. For metallic densities corresponding to a Wigner-Seitz radius of r s ϭ2 -6, the coefficient ␣ KS,x (␤) varies from 0.195-0.274. For ␤ϭ2 1/2 , the coefficient ␣ KS,x (␤) is exactly 1 4 , thereby leading to the classical imagepotential structure for x (r). The contrast of this result with the work of others 7-10 is discussed elsewhere. 1,11 We present in this paper only our derivation of the asymptotic structure of the potential x (r).The relationship between density-functional theory and many-body perturbation theory as established by Sham 3,7 is via the integral equation relating xc (r) to the nonlocal exchange-correlation component ⌺ xc (r,rЈ;) of the selfenergy ⌺͑r,rЈ;͒. This equation is ͵ drЈ xc ͑ rЈ͒ ͵ d⑀ G s ͑ r,rЈ;⑀ ͒G͑ rЈ,r;⑀ ͒ ϭ ͵ ͵ drЈdrЉ ͵ d⑀ G s ͑r,rЈ;⑀͒⌺ xc ͑ rЈ,rЉ;⑀ ͒G͑ rЉ,r,⑀ ͒, ͑1͒where G(r,rЈ;⑀) is the one-particle Green function and G s (r,rЈ;⑀) the KS Green function. From this equation Sham 7 derived the asymptotic structure of xc (r) to bewhere the electron is at the Fermi level ⑀ F . The asymptotic structure of the exchange component x (r) is obtained by substituting the self-energy ⌺ x (r,rЈ)ϭϪ␥ s (r,rЈ)/2͉rϪrЈ͉ into the above equation. Here ␥ S (r,rЈ)ϭ2⌺ k ⌿ k *(r)⌿ k (rЈ)is the idempotent density matrix constructed with the KS orbitals ⌿ k (r). The resulting expression is recognized to be the orbital-dependent potential 12 x,k (r) defined asx,k ͑ r͒ϭ ͵ x,k ͑ r,rЈ͒ ͉rϪrЈ͉ drЈ, ͑3͒due to the orbital-dependent Fermi hole x,k (r,rЈ) of Hartree-Fock theory which in turn is defined as ͑4͒For both jellium and structureless-pseudopotential models of a metal surface, there is translational symme...

Localized versus extended systems in density functional theory: Some lessons from the Kohn-Sham exact exchange potential

“…The correct asymptotics ͑at z → ϱ͒ of the Slater potential V x S ͑z͒ have been reported for a SI jellium [12][13][14] and for jellium slabs, 10 with the result that V x S ͑z͒ decays in both cases as −␤ / z but with a coefficient ␤ that in the case of a SI jellium is electron-density-dependent while for jellium slabs ␤ =1. Hence, here we focus on the remaining contributions: V x ⌬ ͑z͒ and V x Shift ͑z͒.…”

Quantum mechanical image potential theory

“…The claim that v x (z) contributes to the asymptotic structure of v xc (z) was confirmed by Solomatin and Sahni ͑SS͒, 19 who showed analytically via the OPM integral equation that v x (z→ϱ)ϭϪ␣ KS,x /z. They also explained the work of Eguiluz et al by showing analytically that for jellium-slab metal, v x (z) must decay as Ϫ1/z 2 .…”

“…It is well known 12,15,19 that only k,kЈ ϳk F region in the above integral contributes to ⌫ x (1) (r 1 ,r 2 ) at large z 1 , z 2 . Therefore, we can put x 1ʈ ϭx 2ʈ in Eq.…”

Structure of the Pauli and Correlation-Kinetic Components of the Kohn–Sham Exchange Potential at a Metal Surface

“…The proof of the fact that the above expression is valid for self-consistently determined orbitals is essentially the same as that given for the Kohn Sham potential in Ref. [18]. Once again, one can divide up the z axis into three parts: a metal bulk region, a surface region with a finite effective width, and the vacuum region.…”