The many-body exchange-correlation hole at metal surfaces

0.19 MB·English

by  Lucian A. Constantin J. M. Pitarke

The many-body exchange-correlation hole at metal surfaces
Lucian A. Constantin1 and J. M. Pitarke2,3
1Department of Physics and Quantum Theory Group,
Tulane University, New Orleans, LA 70118
2CIC nanoGUNE Consolider, Tolosa Hiribidea 76,
E-20018 Donostia - San Sebastian, Basque Country
3Materia Kondentsatuaren Fisika Saila (UPV/EHU),
DIPC, and Centro F´ısica Materiales (CSIC-UPV/EHU),
644 Posta kutxatila, E-48080 Bilbo, Basque Country
9
0 (Dated: January 25, 2009)
0
2 We present a detailed study of the coupling-constant-averaged exchange-correlation hole density
at a jellium surface, which we obtain in the random-phase approximation (RPA) of many-body
n
theory. We report contour plots of the exchange-only and exchange-correlation hole densities, the
a
integration of the exchange-correlation hole density over the surface plane, the on-top correlation
J
hole,andtheenergydensity. Wefindthattheon-topcorrelationholeisaccuratelydescribedbylocal
5
and semilocal density-functional approximations. We also find that for electrons that are localized
2
far outside the surface the main part of the corresponding exchange-correlation hole is localized at

theimage plane.
]
i
c
PACSnumbers: 71.10.Ca,71.15.Mb,71.45.Gm
s
-
l
r
t I. INTRODUCTION the corresponding (and sorrounding) coupling-constant-
m averagedxc hole density n¯ ([n];r,r′), as follows
xc
at. eleTcthreonesxycshtaemngeis-ctohrereolnaltyiodnen(sxitcy) fuennecrtgioynaolfthaatmhaasntyo- E [n]= dre (r)= 1 dr dr′n(r)n¯xc([n];r,r′),
m xc Z xc 2Z Z |r−r′|
be approximated in the Kohn-Sham (KS) formalism of
- density-functional theory (DFT).1 It is formally defined (3)
d where [see Eqs. (1) and (3)]:
by the following equation derived from the Hellmann-
n
o Feynman theorem:2 1 1
c n¯ ([n];r,r′)= dλρλ(r′,r)−n(r′), (4)
[ Exc[n]= 21Z drZ dr′Z 1dλρ|rλ2(−r′,rr′|) −U[n], (1) xc n(r)Z0 2
1 0 and e (r) is the xc energy density. The xc hole density
xc
v where n(r) is the density of a spin-unpolarized system n ([n];r,r′)istheresultofthreeeffects: self-interaction
xc
2 of N electrons, U[n]=(1/2) drn(r)n(r′)/|r−r′| is the correctionto the Hartreeapproximation,Pauliexclusion
7
Hartree energy, and ρλ(r′,r)Ris the reduced two-particle principle, and the electron correlation due to Coulomb
8 2
3 density matrix repulsion between electrons.
. The adiabatic-connection fluctuation-dissipation the-
1
ρλ(r′,r)=N(N −1) dr ...dr orem provides an elegant path to the exact coupling-
90 2 σ,σX′,...,σNZ 3 N constant-averaged xc hole density,3,4,5,6 which can be
written as follows7
0 ×|Ψλ(r′σ′,rσ,r σ ,...,r σ )|2. (2)
3 3 N N
v: 1 1 ∞ 1
Xi Htioenre,thΨaλt(ry1ieσl1d,s...t,hreNdσeNn)siitsytnhe(ra)natinsdymmmineitmriiczewsavthefeunexc-- n¯xc([n];r,r′)= n(r)[−π Z0 dωZ0 dλχλ(r,r′;ω)
ar pectation value of Tˆ +λVˆee, where Tˆ = − Ni=1∇2i/2 −n(r)δ(r−r′)], (5)
and Vˆee = 21 i j6=i |ri−1rj| are the kinetic ePnergy and where χλ(r,r′;ω) is the density-response function of the
theelectron-ePlectProninteractionoperators. Eq.(2)shows interactingsystematcouplingstrengthλandsatisfies,in
that ρλ2(r′,r)dr′dr is the joint probability of finding an the frameworkoftime-dependent density-functionalthe-
electron of arbitrary spin in dr′ at r′ and an electron of ory(TDDFT),thefollowingexactDyson-typeequation8
arbitraryspinin dr atr, assuming thatthe Coulombin-
teraction is λ/|r−r′|. In the case of noninteracting KS χ (r,r′;ω)=χ (r,r′;ω)+ dr dr χ (r,r ;ω)
electrons (i.e., λ = 0), ρλ=0(r′,r) is the exchange-only λ 0 Z 1 2 0 1
2
reducedtwo-particledensitymatrixthatisexpressiblein λ
× +f [n](r ,r ;ω) χ (r ,r′;ω). (6)
terms of KS orbitals. (Unless otherwise stated, atomic (cid:26)|r −r | xc,λ 1 2 (cid:27) λ 2
units are used throughout, i.e., e2 =~=m =1.) 1 2
e
Hence, the xc energy can be expressed as the elec- Here, χ (r,r′;ω) is the density-response function
0
trostatic interaction between individual electrons and of non-interacting KS electrons (which is exactly
2
known in terms of KS orbitals9) and f [n](r,r′;ω) k is the magnitude of the bulk Fermi wavevector. At
xc,λ F
is the Fourier transform with respect to time a jellium surface, the plane z = 0 separates the uniform
[f [n](r,r′;ω) = ∞ dteiωtf [n](r,t,r′,0)] of the positive background (z > 0) from the vacuum (z < 0),
xc,λ −∞ xc,λ
unknown λ-dependenRt xc kernel, formally defined by and the electrons can leak out into the vacuum. This
electron system is translationally invariant in the plane
δvλ [n](r,t) of the surface.
f [n](r,t,r′,t′)= xc , (7)
xc,λ δn(r′,t′) The exchange hole at a jellium surface was studied in
Ref. 22 (using a finite linear-potential model23), and in
where vλ [n](r,t) is the exact time-dependent xc poten- Refs. 24,25 (using the infinite barrier model (IBM)26).
xc
tial of TDDFT. When fxc,λ[n](r,r′;ω) is taken to be The behavior of the xc hole at a jellium surface was in-
zero, Eq. (6) reduces to the random phase approxima- vestigatedattheRPAlevelusingIBMorbitals.27 Hence,
tion (RPA). If the interacting density response function existing calculations of the exchange-only and xc hole
χλ(r,r′;ω)isreplacedbythe noninteractingKSdensity- at a jellium surface invoke either a finite linear-potential
response function χ0(r,r′;ω), then Eq. (5) yields the model or the IBM for the description of single-particle
exchange-only hole density. orbitals. An exception is a self-consistent calculation of
The scaling relation of the correlation hole density at the RPA xc hole density reported briefly in Refs. 28 and
coupling constant λ10,11 leads to the following equation 29, in which accurate LSDA single-particle orbitals were
for the coupling-constant-averagedcorrelation hole den- employed.
sity: In this paper, we present extensive self-consistent cal-
culationsoftheexact-exchangeholeandtheRPAxchole
n¯ ([n];r,r′)= 1dλ(λ)3nw([n ], λr, λr′), (8) at a jellium surface. We report contour plots of the cor-
c Z w c w/λ w w responding hole densities, the integration of the xc hole
0
density over the surface plane, and the on-top correla-
where 0 < w << 1 is a fixed constant, and nγ(r) = tion hole. We find that the on-top RPA correlation hole
γ3n(γr) is a uniformly-scaled density.12 Eq. (8) shows n¯ ([n];r,r) is accurately described by the on-top RPA-
c
that the whole many-body problem is equivalent to the based LSDA hole, in accord with the work of Perdew et
knowledge of the universal correlation hole density at a al.5,30,31
small, fixed coupling strength w.
There is a ”Jacob’s ladder”13 classification (in RPA
andbeyondRPA)ofnonempiricalapproximationstothe
II. THE EXACT-EXCHANGE HOLE AND THE
angle-averagedxc hole density
RPA XC HOLE AT A JELLIUM SURFACE
1
n¯ ([n];r,u)= dΩn¯ ([n];r,r′), (9)
xc xc Letusconsiderajelliumsurfacewiththesurfaceplane
4π Z
atz =0. Usingitstranslationalinvarianceinaplaneper-
where dΩ is the differential solid angle around the di- pendicular to the z axis, the coupling-constant-averaged
rection of u = r′ − r. The simplest rung of the lad- xc hole density of Eq. (5) can be written as follows29
der is the local spin density approximation (LSDA) of
the xc hole density n¯xc(n↑,n↓;u) that has as ingredi- n¯xc([n];r,z,z′)=−21π 0∞dq|| q||J0(q||r)[πn1(z) 01dλ 0∞dω
ents only the spin densities. (For the RPA-based LSDA
R R R
×χλ(q ,z,z′,ω)−δ(z−z′)], (10)
xc hole and for the LSDA xc hole, see Refs. 14,15 ||
and Refs. 14,16,17, respectively.) The next rung is
where r = |r −r′|, and q is a two-dimensional (2D)
the generalized gradient approximation (GGA) xc hole || || ||
density n¯ (n ,n ,∇n ,∇n ,u). (See Ref. 14 for the wavevector. χλ(q ,z,z′,ω) represents the 2D Fourier
xc ↑ ↓ ↑ ↓ ||
smoothed GGA exchange hole model, Ref. 18 for the transformof the interacting density response function of
PBE-GGA19 correlation hole, and Ref. 15 for the RPA- Eq.(6),whichintheRPAisobtainedbyneglectingthexc
based GGA hole model. For a GGA xc hole constructed kernel f . The exact-exchange hole density is obtained
xc
for solids, see Ref. 20.) The third rung on this lad- bysimplyreplacinginEq.(10)χλ(q ,z,z′,ω)bythecor-
||
der is the non-empirical meta-GGA xc hole density21 responding KS noninteracting density response function
n¯ (n ,n ,∇n ,∇n ,τ ,τ ,u)thatdependsonspinden- χ0(q ,z,z′,ω).
xc ↑ ↓ ↑ ↓ ↑ ↓ ||
sities and their gradients, as well as the positive KS ki- For the evaluation of Eq. (10), we follow the method
neticenergydensitiesτ andτ ,andthatwasconstructed describedinRef.7. Weconsiderajelliumslab,andweas-
↑ ↓
to satisfy many exact constraints. (For an RPA-based sumethattheelectrondensityn(z)vanishesatadistance
meta-GGA xc hole model, see also Ref. 21.) z = 2λ (λ = 2π/k is the bulk Fermi wavelength)
0 F F F
Jellium is a simple model of a simple metal, in which fromeitherjelliumedge.32 Weexpandthe single-particle
the ion cores are replaced by a uniform positive back- wave functions entering the evaluation of χ0(q ,z,z′,ω)
||
ground of density n¯ = 3/4πr3 = k3/3π2 and the va- inasineFourierrepresentation,andthedensity-response
s F
lence electrons in the spin-unpolarized bulk neutralize functions χ0(q ,z,z′,ω) and χ (z,z′;q ,ω) in a double-
|| λ k
this background. r is the bulk density parameter and cosine Fourier representation. We also expand the Dirac
s
3
delta function entering Eq. (10) in a double-cosine rep-
2
resentation (see Eq. (A2) of Ref. 7). We take all the
1.5
occupied and unoccupied single-particle orbitals and en-
ergiestobetheLSDAeigenfunctionsandeigenvaluesofa 1
KSHamiltonian,as obtainedbyusing the Perdew-Wang
0.5
parametrization33 oftheCeperley-Alderxcenergyofthe
F
uniform electron gas.34 λ 0
In the calculations presented below, we have consid- r/||
-0.5
ered jellium slabs with several bulk parameters r and a
s
thickness a = 2.23λ for the positive background. For -1
F
r =2.07,suchslabcorrespondstoaboutfouratomiclay-
s -1.5
ersofAl(100)andit wasusedin the wavevectoranalysis
of the RPA35 and beyond-RPA20,36 xc surface energy. -2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
In Figs. 1 and 2, we show contour plots for the exact- z/λ
F
exchangeholedensityandtheself-consistentRPAxchole
density, respectively. In the bulk, both the exchange-
2
only hole and the xc hole are spherical and the xc hole
is more localized, as in the case of a uniform electron 1.5
gas. Near the surface, both the exchange-only hole and 1
the xc hole happen to be distorted, the center of gravity
0.5
being closer to the surface when correlation is included.
F
For an electron that is localized far outside the surface, λ 0
thecorrespondingexchange-onlyholeandxcholeremain r/||
-0.5
localized near the surface; Figs. 1 and 2 show that the
introductionofcorrelationresultsinaflatterhole,which -1
in the case of an electron that is infinitely far from the
-1.5
surface becomes completely localized at a plane parallel
to the surface. This is the image plane. We recall that -2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
theRPAxcholedensityisexactinthelimitoflargesep- z/λ
arations (where u = |r−r′| → ∞), and yields therefore F
the exact location of the image plane.
2
The integration of the xc hole density over the whole
surface plane, 1.5
∞ 1
b ([n],z,z′)= dr n¯ ([n];r,z,z′), (11)
xc Z xc 0.5
0
F
λ 0
represents a quantity of interest for a variety of the- r/||
oretical and experimental situations (see for example -0.5
Refs. 37,38). Below we show that b ([n];z,z′) repre-
xc -1
sents a suitable quantity to describe the behavior of the
xc hole corresponding to a given electron located at an -1.5
arbitrary distance from the surface. In Fig. 3, we plot
-2
this quantity, versus z′, for r = 2.07 and a given elec- -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
s
tron located at z = 0.5λF, z = 0, z = −0.5λF, and z/λF
z = −1.5λ . We see from this figure that (i) correla-
F
tion damps out the oscillations that the exchange hole 2
exhibits in the bulk part of the surface, and (ii) in the
1.5
case of a given electron located far from the surface into
the vacuum the main part of the exchange-only and the 1
xc hole is found to be near the surface (see also Figs. 1 0.5
and 2), although the exchange-only hole appears to be F
λ 0
muchmoredelocalizedwithaconsiderableweightwithin r/||
the bulk. -0.5
Letusnowfocusontheon-topxchole. TheLSDAac-
-1
curately accounts for short wavelength contributions to
the xc energy;30 thus, all the nonempirical approxima- -1.5
tions ofthe xc hole have been constructedto recoverthe
-2
LSDA on-top xc hole n¯LSDA(r,r). The slowly-varying -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
xc
z/λ
F
FIG. 1: Contour plots of the exchange hole density
n¯x(r||,z,z′) for several fixed values of the electron posi-
tion: z = 0.5λF (inside the bulk), z = 0 (on the surface),
z=−0.5λF (inthevacuum)andz=−1.5λF (faroutsidethe
4
2
1.5 0.005
0
1
-0.005 correlation
0.5 -0.01
λ/r||F 0 (z,z’) -0-0.0.0125
-0.5 bxc -0.025 exchange
-1 -0.03 rs = 2.07
-0.035 z/λF = 0.5 xc
-1.5
-0.04
-2 -0.045
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1
z/λ z’/λ
F F
2
1.5 0.005
correlation
0
1
-0.005
0.5
F ) -0.01
λ/r|| 0 (z,z’ -0.015
-0.5 xc
b -0.02 exchange
-1 -0.025 rs = 2.07
-1.5 -0.03 z/λF = 0 xc
-2 -0.035
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1
z/λ z’/λ
F F
2
1.5 0.005
correlation
1
0
0.5
λ/r||F 0 (z,z’) -0.005 exchange
-0.5 bxc -0.01
-1
r = 2.07
-0.015 s
-1.5 z/λ = -0.5 xc
F
-2 -0.02
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1
z/λ z’/λ
F F
2
1.5 0.002
0.001
1 correlation
0
0.5 -0.001
λ/r||F 0 (z,z’) --00..000032
-0.5 bxc -0.004 xc
-1 -0.005 rs = 2.07 exchange
-0.006 z/λF = -1.5
-1.5
-0.007
-2 -0.008
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1
z/λ z’/λ
F F
FIG. 2: Contour plots of the RPA coupling-constant- FIG.3: bxc(z,z′)ofEq. (11)versusz′/λF forthesameposi-
averagedxcholedensityn¯x(r||,z,z′)forseveralfixedvaluesof tionsoftheelectron asinFigs. 1and2. Thebulkparameter
the electron position: z =0.5λF (inside the bulk), z =0 (on is rs=2.07 and thejellium surface is at z=0.
the surface), z = −0.5λF (in the vacuum) and z = −1.5λF
5

0.002 0.003
0 RPA-based LSDA + 0.002
ole -0.002 gradient correction 0.001
h
n -0.004 0 rs=6
o
correlati --00-0..00.000186 RPA-based LSDAbc(z,z) r=2.07 b(z,z’)c ---000...000000321 rs=3 rs=4
n-top --00..001142 s --00..000054 rs=2.07 rs=5
o z/λ = -0.5
-0.016 exact RPA -0.006 F
-0.018 -0.007
-0.6 -0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1
z/λ z’/λ
F F
FIG. 4: On-top coupling-constant-averaged correlation hole FIG. 5: The correlation hole bc(z,z′) of an electron at po-
n¯c(r,r) at a jellium surface. Also shown is bc(z,z) of Eq. sition z = −0.5λF for several values of the bulk parameter
(11). Thebulkparameterisrs=2.07andthejelliumsurface rs=2.07,3,4,5, and 6. The jellium surface is at z=0.
is at z=0.
5showsthe integratedcorrelationholeofEq.(11)foran
electron gas was treated within RPA by Langreth and
electron at the vacuum side of the surface, at the posi-
Perdew5. For a spin-unpolarized system, the gradient
tion z = −0.5λ and for several values of the electron-
F
correction to the LSDA on-top correlaton hole density
density parameter r : 1.5, 2.07, 3, 4, 5, and 6. In the
is31 s
bulk, the correlation hole exhibits damped oscillations
|∇n|2 with rs-dependent amplitude and a periodthat does not
n¯GEA(r,r)=n¯LSDA(r,r)+ . (12) depend onthe electrondensity andis close to the period
c c 72π3n2
(∼0.56λ )ofthecorrespondingoscillationsexhibitedby
F
InFig. 4,weshowthe on-topcorrelationholeforthe ex- the exchange-only hole.
act RPA, the RPA-based LSDA (see Ref. 15) and the Finally, we look at the xc energy density e de-
xc
RPA-based GEA of Eq. (12). We see that for a jel- fined in Eq. (3). We note that adding to the ac-
lium surface the RPA-based LSDA on-top correlation tual e of Eq. (3) an arbitrary function of the posi-
xc
hole nearly coincides with the corresponding exact RPA tion r that integrates to zero yields the same total xc
on-top correlation hole; this is in contrast with the case energy.43 The Laplacianofthe density ∇2n integratesto
ofstronginhomogeneoussystems(e.g.,Hooke’satom).30 zero for finite systems, it plays an important role in the
The gradientcorrectionofEq.(12) improvesthe already gradient expansion of the kinetic-energy density,44,45,46
accurateRPA-basedLSDAon-topcorrelationholeinthe and it is an important ingredient in the construction of
slowly-varyingdensity region,butis inacurateinthe tail density-functional approximations for the kinetic energy
of the density. Fig. 4 also shows that the integrated density44,45 and the xc energy.45
b (z,z)of Eq. (11) is more (less) negative in the vacuum We define the simplest possible Laplacian-level RPA-
c
(bulk) than the actual on-top correlation hole. basedLSDA(theRPA-basedL-LSDA)xcenergydensity:
At this point, we would like to emphasize that while
the RPA on-top correlationhole in the bulk is too nega- eL−LSDA−RPA(r)=eLSDA−RPA(r)−C∇2n(r), (14)
xc xc
tivebutfinite,theon-topcorrelationholedivergesinthe
bulk within a TDDFT scheme that uses a wavevector where C is a constant parameter which we find by mini-
and frequency independet xc kernel like in the adiabatic mizing the difference between eRxcPA−L−LSDA and eRxcPA.
local-density approximation (ALDA) We find C = 0.3 for a jellium slab with rs = 2.07, and
its value gets larger as r increases.
s
fALDA[n](r,r′,ω)= dvxλc,unif[n(r)]δ(r−r′), (13) In Fig. 6, we show ∆exc(z) = eRxcPA(z) − eaxpcprox(z)
xc,λ dn(r) versusz/λF for a jellium slabwith rs =2.07andseveral
RPA-based approximations for eapprox(z). The RPA-
xc
or the energy-optimized local-density approximation of based PBE15 improves considerably the behavior of the
Ref. 39. (See the discussion after Eq. (3.9) of Ref. 39). RPA-based LDA. The ARPA-GGA47 is a GGA func-
Here, vλ,unif[n(r)] is the xc potential of a uniform tional that fits the RPA xc energy density of the Airy
xc
electron gas of density n(r). An xc kernel borrowed gas and is remarkably accurate for jellium surfaces. The
from a uniform-gas xc kernel that has the correct large- RPA-based GGA++ is the RPA versionof the GGA++
wavevectorbehavior(see,e.g.,the xckernelsofRefs.40, of Ref. 38. (eRPA−GGA++ = eRPA−LSDAF (l), where
xc xc xc
41,42) would yield a finite on-top correlation hole. Fig. l = r2∇2n/n is a reduced Laplacian and F (l) is de-
s xc
6
density at a metal surface. When the electron is in the

vacuum, its hole remains localized near the surface (its
8e-05
RPA-based LSDA minimumisontheimageplane)andhasdampedoscilla-
7e-05 RPA-based PBE
tionsinthebulk. Wefindthattheon-topcorrelationhole
6e-05 r=2.07 ARPA
s RPA-based GGA++ is accurately described by local and semilocal density-
5e-05
RPA-based L-LSDA
4e-05 functionalapproximations,asexpectedfromRef.5 . We
(z) 3e-05 also find that for an electron that is localized far outside
c
ex 2e-05 the surface the main part of the corresponding xc hole
∆
1e-05 is completely localized at a plane parallel to the surface,
0 which is the image plane.
-1e-05 Because of an integration by parts that occurs in the
-2e-05 underlying gradient expansion, a GGA (or meta-GGA)
-3e-05
hole is meaningful only after averaging over the elec-
-1 -0.5 0 0.5 1 1.5
z/λ tron density n(r).18,20 This average smooths the sharp
F
cutoffs used in the construction of the angle-averaged
GGA xc hole density. The wavevector analysis of the
FIG. 6: ∆exc(z) = eRxcPA(z)−eaxpcprox(z) versus z/λF at a jeliumxcsurfaceenergyisanimportantandhardtestfor
surfaceofajelliumslab,forseveralxcapproximations: RPA- the LSDA,GGA,andmeta-GGAangle-averagedxc hole
based LSDA15, RPA-based PBE15, ARPA GGA47, RPA-
densities, showing not only the accuracy of the xc hole
based GGA++38, and RPA-based L-LSDA (Eq. (14) with
but also the error cancellation between their exchange
C = 0.3). The bulk parameter is rs = 2.07, and the edge of and correlation contributions. Thus, Refs. 35 and20,21
thepositive background is at z=0.
haveshownthattheTPSSmeta-GGA21 andthePBEsol
GGA20 xc hole densities improve considerably the ac-
curacy of their LSDA and PBE counterparts at jellium
finedinEq.(3)ofRef. 38.) Althoughthe GGA++func-
surfaces, both within RPA and beyond RPA.48
tionalwasconstructedfortheSicrystal,weobservethat
The exchange energy density does not have a gradient
the RPA-based GGA++ improves over the RPA-based
expansion49,asdoesthekineticenergydensity. However
LSDA in the bulk near the jellium surface showing that
the existenceofgradientexpansionofthe xcenergyden-
itcanbeagoodapproximationforsystemswithsmallos-
sity is still an open problem. We use our RPA xc hole
cillations. (Inthebulk,closetothejelliumsurface,there
density to compare the xc energy densities of severalap-
are Friedel oscillations as well as quantum oscillations
proximations. The most accurate ones are ARPA GGA
due to the finite thickness of the jellium slab). We note
finallythateRPA−L−LSDA significantlyreducesthelocal of Ref. 47 and RPA-based L-LSDA of Eq. (14).
xc
erroroftheRPA-basedLSDAnearthejelliumsurface,al-
though by construction ERPA−L−LSDA =ERPA−LSDA.
xc xc
Acknowledgments
III. CONCLUSIONS We thank J. P. Perdew and J. F. Dobson for many
valuable discussions and suggestions. J.M.P. acknowl-
We have presented extensive self-consistent calcula- edges partial support by the Spanish MEC (grant No.
tions of the exact-exchange hole and the RPA xc hole FIS2006-01343and CSD2006-53)andthe EC 6th frame-
at a jellium surface. workNetworkofExcellenceNANOQUANTA.L.A.C.ac-
WehavepresentedadetailedstudyoftheRPAxchole knowledges NSF support (Grant No. DMR05-01588).
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