Exact "exact exchange" potential of two- and one-dimensional electron gases beyond the asymptotic limit

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by  V. U. Nazarov

Exact “exact exchange” potential of two- and one-dimensional electron gases beyond
the asymptotic limit
Vladimir U. Nazarov
Research Center for Applied Sciences, Academia Sinica, Taipei 11529, Taiwan∗
The exchange-correlation potential experienced by an electron in the free space adjacent to a
solid surface or to a low-dimensional system defines the fundamental image states and is generally
6 important in surface- and nano-science. Here we determine the potential near the two- and one-
1 dimensional electron gases (EG), doing this analytically at the level of the exact exchange of the
20 danendsiktFy-fiusntchteionFaerlmthieroardyiu(Ds,FtTh)e.pWoetefinntidalthoabte,yasttrh⊥e≫alrkeaF−d1y,wkhneorwenr⊥asiysmthpetodtiiscta−nec2e/frr⊥om, wthhieleEaGt
y r⊥ .kF−1,butstillinvacuum,qualitativeandquantitativedeviationsoftheexchangepotentialfrom
a theasymptoticlawoccur. Theplaygroundoftheexcitationstothelow-lyingimagestatesfallsinto
M the latter regime, causing significant departure from the Rydberg series. In general, our analytical
exchangepotentials establish benchmarksfor numericalapproaches in thelow-dimensional science,
where DFT is byfar themost common tool.
6

] PACSnumbers: 73.21.Fg,73.21.Hb
l
l
a
h I. INTRODUCTION behaviour of the xc potential, in this paper we are con-
- cerned with the potential in the whole space outside a
s
e 2(1)DEG.We solvethis problemexactly andanalytically
The image potential (IP) – a potential experienced by
m attheEXXlevelofDFTandfindthat,atr .k−1,but
a test charge outside a semi-infinite medium, a slab, or ⊥ F
. stillinvacuum,the potentialis qualitativelyandquanti-
t a system of a lower dimensionality – is a fundamental
a concept of classical electrostatics1. It is widely believed, tativelydifferentfromitsasymptoticform−e2/r⊥. How-
m ever, at larger distances r ≫ k−1, our potentials obey
although never proven2–4, that the exchange-correlation ⊥ F
- the correct asymptotic, which is known to be mando-
(xc)partoftheKohn-Sham(KS)potentialofthedensity-
d tary for slabs in general13. The non-asymptotic shape of
n functionaltheory(DFT)5mustasymptoticallyreproduce the potential in the r .k−1 region strongly affects ex-
o the classical IP at large distances from an extended sys- ⊥ F
perimentally important low-lying image states, causing
c tem. Muchefforthasbeenexertedoveryearstodescribe
[ significant deviations from the Rydberg series.
IP quantum-mechanically, both in order to account for Thispaperisorganizedasfollows: InSec.II wederive
3 the experimentally important image states at solid sur-
a closed-form EXX potentials for quasi-2(1)DEG with
v faces and at quasi-low-dimensional systems, and to gain
one filled subband. In Sec. III we take the full confine-
7 better understandingofthe non-trivialinterrelationsbe-
6 tween DFT and classical physics4,6–13. ment limit, obtaining analytical EXX potentials for 2D
0 and 1D electron gases. In Sec. IV we visualize and dis-
7 Regardless of the ultimate answer to the question of cuss the results. In Sec. V we derive further insights
0 whether or not the xc potential (which is not a physical from addressing the problem of quasi-2(1)DEG within
1. quantity) is equal in vacuum to the IP for a test charge the localized Hartree-Fock method. Section VI contains
0 (whichisaphysicalquantity)14,thedeterminationofthe conclusions. In Appendix A we discuss the classical im-
6 former is fundamentally important in quantum physics. age potential in two and one dimensions. In Appendix
1 Indeed, it defines the KS band-structure of the system B, finer details of the derivation of the main results are
v: of interest, which step, followed by a calculation of the given. Atomicunits(e2 =~=me =1)areusedthrough-
i system’s response within the time-dependent DFT15–18, out.
X
will produce excitations to image states (which arequite
r physical properties). While for semi-infinite media the
a
problem still remains highly controversial3,4, for slabs
II. EXX POTENTIAL OF QUASI-2(1)D
it has been firmly established9,12 that, on the level of ELECTRON GAS WITH ONE SUBBAND FILLED
the optimized effective potential-exact exchange (OEP-
EXX)19–21,theKSpotentialhastheasymptotic−e2/r ,
⊥ We start by considering a quasi d-dimensional (d =
validatlargedistancesr fromtheslab. However,apart
⊥ 2,1), generally speaking, spin-polarized EG. The posi-
from the thickness a, EG in the shape of a slab (quasi-
tively charged background with the d-dimensional den-
2D EG) or a cylinder (quasi-1D EG), when considered
sity n is strictly confined to the xy-plane and to the z-
quantum-mechanically, has a fundamental intrinsic pa-
axis, for d = 2 and d = 1, respectively. We are con-
rameter kF – the Fermi radius – and, therefore, even at cernedwith the KSproblem5 forspin-orbitalsψσ(r)and
a=0,twodifferentregimes,atr⊥ ≫kF−1 andr⊥ .kF−1, eigenenergies ǫσ, with the potential i
can be anticipated. i
At variance with a vast literature on the asymptotic vσ(r)=v (r)+v (r)+vσ (r), (1)
s ext H xc
2
where
Q2D Q1D
1
vs(r⊥) r⊥=z rk=z ρσ(rk)= Ω eikk·rk, (7)
|kkX|≤kFσ
r =(x,y) r⊥=(x,y)
k kσ being the Fermi radii for the corresponding spin ori-
F
entations. As will be seen below, the factorization (6)
vs(r⊥) is the key property of 2(1)DEG with only one subband
occupation,whichmakespossibletheexplicitsolutionto
the EXXproblemforthesesystems. Then,the exchange
energy of Eq. (3) can be written as
E =−1 |µσ0(r⊥)|2|µσ0(r′⊥)|2|ρσ(rk−r′k)|2drdr′
x 2 |r−r′|
σ Z
FIG. 1. Schematics of quasi-2D (left) and quasi-1D (right) X
electron gases and thenotations adopted. =−1 nσ(r)nσ(r′)|ρσ(rk−r′k)|2drdr′,
2 (nσ)2|r−r′|
σ Z
X
(8)
generally speaking, different for different spin orienta-
tions σ, where v (r), v (r)= n(r′)/|r−r′|dr′, and
ext H where the spin-density is
vxσc(r)= δδnEσRx(rc) (2) nσ(r)=ρσ(r,r)=|µσ0(r⊥)|2nσ, (9)
and nσ = nσ(r)dr is the d-dimensional uniform spin-
⊥
aretheexternal,Hartree,andxcpotentials,respectively,
density. Byvirtue ofEqs.(2), (8), and(9),the exchange
nσ(r) are spin-densities, n(r) = n↑(r) + n↓(r) is the potential eRvaluates explicitly to23.
particle-density, and E is the xc energy. For the latter
xc
we use the EXX part, which can be written as22 vσ(r)= δEx =− 1 nσ(r′)|ρσ(rk−r′k)|2dr′
1 |ρ(r,r′)|2 x δnσ(r) (nσ)2 |r−r′|
E =− drdr′, (3) Z (10)
x 2Z |r−r′| =− 1 |µσ0(r′⊥)|2|ρσ(rk−r′k)|2dr′.
nσ |r−r′|
where Z
In Eq. (10) we easily recognize the Slater’s exchange
ρ(r,r′)= φi(r)φ∗i(r′) (4) potential22. This leads us to the important conclusion
iX∈occ that, for Q2(1)EG with only one subband filled for each
spin component, EXX and the Slater’s potentials coin-
is the density-matrix, the summation in Eq. (4) running
cideexactlyand,consequently,inthiscase,theEXXpo-
over the occupied states only.
tential can be expressed in terms of the occupied states
Let r and r be the coordinate vectors parallel and
k ⊥ only. Thelatterbecomeswrongwhenmoresubbandsare
perpendicular, respectively, to the extent of the EG. In
filled24.
other words, for d = 2, r is the radius-vector in the
k Evaluation of integrals in Eq. (7) is straightforward,
xy plane and r is a vector in the z-direction, while for
⊥ giving
d=1thisisviceversa,asschematizedinFig.1. Keeping
in mind the subsequent zero-thickness limit, we assume
obneeo,cactumpioesdt,fosrubebaachndspwinithditrheectwioanv.e-Tfuhnecrteifoonreµ,σ0a(llr⊥sp)into- ρσ(rk)= 2π1rk (2kFσsiJn1((kkFσFσrrkk)),, dd==21,, (11)
orbitals with the same spin orientation have the same
r -dependence whereJ (x)istheBesselfunctionofthefirstorder. With
⊥ 1
the use of Eqs. (11), the integral over r′ in Eq. (10) can
k
eikk·rk be taken in special functions, resulting in
φσ (r)= µσ(r ), (5)
kk Ω1/2 0 ⊥
F (kσ|r −r′ |)|µσ(r′ )|2
where Ω is the normalization area or length, in the 2D vxσ(r⊥)=− d F ⊥|r −⊥r′ | 0 ⊥ dr′⊥, (12)
and 1D cases, respectively. Then, by Eq. (4), for the Z ⊥ ⊥
density-matrix we can write
where
ρ(r,r′)= µσ0(r⊥)µσ0∗(r′⊥)ρσ(rk−r′k), (6) F (x)=1+ L1(2x)−I1(2x), (13)
2
σ x
X
3
1 1,1 where γ ≈ 0.5772 is the Euler’s constant. At large dis-
F (x)= G2,2 x2 2 , (14)
1 2π 2,4 1,1,−1,0 tances, at the both dimensionalities, the potential re-
(cid:20) (cid:12) 2 2 2 (cid:21)
(cid:12) spects the asymptotic −1/r⊥.
(cid:12)
L1(x) and I1(x) are th(cid:12)e first-order modified
Struve and Bessel functions, respectively, and 0
a , ..., a
Gm,n x 1 p is the Meijer G-function25,26.
p,q b , ..., b -0.1
1 q
(cid:20) (cid:12) (cid:21)
Function(cid:12)s F (x) are plotted in Fig. 2.
(cid:12) d -0.2
(cid:12) )
.
u
1 z)(a. --00..43 rs=13450 −|z1|
0.8 (x 2
v
-0.5
2D
0.6
) F (x) -0.6
x 2
F(d F1(x) -0.7
0.4 -15 -10 -5 0 5 10 15
z (a.u.)
0.2
FIG. 3. EXX potential of the spin-unpolarized 2DEG, ob-
tained with Eqs. (15) and (13), versus the distance from the
0 EGplanez,withthedensityparameterr =(πn)−1/2chang-
s
0 2 4 6 8 10 12 14
x ing from 10 a.u. (top) to 2 a.u. (bottom). The dotted lines
show theasymptotic −1/|z|.
FIG. 2. Functions F2(x) and F1(x) of Eqs. (13) and (14),
which determine the 2D and 1D exchange potential, respec-
tively.
IV. DISCUSSION
0
III. FULL 2(1)D CONFINEMENT LIMIT
-0.5
We now take the limit of the strictly d-dimensional
(zero-thickness) electron gas. Then |µσ(r )|2 = δ(r ),
0 ⊥ ⊥ ) -1
and Eq. (12) reduces to u. rs=10
. 5
vxσ(r⊥)=−Fd(kr⊥Fσr⊥). (15) vρ()(ax-1-.52 −ρ1 1234
We emphasize that Eq. (15) was obtained for the math-
1D
ematical idealization of an electron gas strictly confined -2.5
to a plane (a straightline), as a limiting case ofEq. (12)
for EG of a finite transverse extent. -3
The following zero-distance and asymptotic behaviour 0 1 2 3 4 5
ρ (a.u.)
can be directly obtained from Eqs. (15) and (13) for the
2D case
FIG. 4. EXX potential of the spin-unpolarized 1DEG, ob-
8kσ
vσ(z =0)=− F, (16) tained with Eqs. (15) and (14), versus the distance from the
x 3π EG line ρ, with the density parameter r =(2n)−1 changing
s
1 2 from 10 (top) to 1 a.u. (bottom). The dotted line shows the
vσ(z →∞)→− + +... (17)
x z πkσz2 asymptotic −1/ρ.
F
and from Eqs. (15) and (14) for the 1D case Itisknownthat,inthegeneralcase,EXX-OEPpoten-
tial cannot be expressed in terms of the occupied states
kσ
vσ(ρ→0)→ F[2log(kσρ)+2γ−3], (18) only,butall,theoccupiedandempty,statesareinvolved,
x π F
andacomplicatedOEPintegralequationmustbesolved
1 1
vσ(ρ→∞)→− + +..., (19) to calculate this potential19,20. It, therefore, may look
x ρ πkFσρ2 surprising that a drastic simplification can be achieved
4
in the case of 2(1)DEG with one subband filled, leading 0
to Eq. (10), the latter expressed in terms of the occu-
piedstatesonlyandnotinvolvingtheOEPequation. To -0.1 rs=5
clarify this point, in Appendix B1 we arrive at Eq. (10)
-0.2
following the traditional path of working out the EXX- ) 2
. =
OEP potential in terms of the eigenfunctions and the .u-0.3 rs Q1D
a
eigenenergiesofallthestates,seeingclearlyhowthesim- ( 1D
1
plifications arise due to the specifics of this system. ρ)-0.4 −
( ρ
In Figs. 3 and 4, the EXX potentials of 2DEG and x
v
1DEG, respectively, are plotted for a number of densi- -0.5
1D
ties as functions of the distance from the EG. An im-
-0.6
portant conclusion can be drawn from these figures to-
gether with the long-distance expansions (17) and (19):
-0.7
The more dense is the EG, the sooner the asymptotic is 0 5 10 15
approached with the increase of the distance. The char- ρ (a.u.)
acteristic distance scale is the inverse Fermi radius k−1,
F
which separatestwo distinct regions,the asymptotic one FIG. 6. The same as Fig. 5 but for 1D case.
realizingatr &k−1. Weemphasizethatinbothregions
⊥ F
there is no electron density, as is particularly clear with 0 3
the strictly 2D and 1D (zero-thickness) EG. In Figs. 7 ξ=1
2D
2.5
rs=5
0 )
-0.1 u.
-0.1 a.u.) ξ=0 3(a. 2
-0.2 rs=5 z)(-0.2 ×101.5
u.) v(x ξ=1 z) 1
a.-0.3 rs=2 Q2D n(
( -0.3 2D 0.5
z)-0.4 ξ=0
(
vx-0.5 Q22DD −|1z| -15 -10 -5z (a0.u.)5 10 15 0-10 -5 z (a0.u.) 5 10
2D
-0.6
FIG. 7. Left: Exchange potential as a function of the dis-
tance z from the plane of the possitive background of quasi-
-0.7
-15 -10 -5 0 5 10 15 2D(solidlines)and2D(dashedlines)EG,forthespin-neutral
z (a.u.) (ξ =0) and fully spin-polarized (ξ =1) cases at the density-
parameter r = 5. Right: The corresponding particle densi-
s
ties of the Q2D EG.
FIG.5. TheEXXpotentialofspin-neutralQ2DEGwithone
subbandfilled(solidlines)comparedtotheanalyticalsolution
forthezero-thicknesslimit(dashedlines)fortwovaluesofthe
density parameter rs = 2 and 5. The dotted lines show the Results for the EXX potential presented in Figs. 5 and
asymptotic −1/|z|. 6, for the 2D and 1D cases, respectively, show that the
zero-thickness limit of the EXX potential is a very good
and 8 we present the exchangepotential and the density approximation to that of the quasi-low-dimensional EG
forthe spin-neutralandfully spin-polarizedelectrongas, exceptforveryshortdistancesfromthesystem. Atthose
in the 2D and 1D cases, respectively. distances,theelectron-densityishigh(seeFigs.7and8),
A natural question arises: How relevant are the so- thedeviationinthisregionbeingnotsurprising,sincethe
lutions obtained for strictly low-dimensional case to potentialis notthat invacuum anymore. Results ofthe
the realistic quasi-low-dimensional EG. To answer this, calculations for the spin-polarized EG are presented in
we return to Eq. (12) and solve the KS problem self- Figs. 7 and 8. In Table I the low-lying eigenenergies of
consistently using27 the 2D and Q2D EG are presented. The deviation of
vext,H(r⊥)=vext(r⊥)+vH(r⊥)=2π× the EXX potential from −1/r⊥ in the non-asymptotic
region causes significant change in the spectra of the
∞
eigenenergies compared with the Rydberg series, − 1
(|z|−|z−z′|)n(z′)dz′, d=2, 2n2
−∞Z∞ρ′n(ρ′)log 2ρ2 dρ′, d=1. (20) ann=Hda1−v,i22n(,gn..−1.2f12o8)u.2n,dforth2eDaannadly1tiDcaclasEeXs,Xresppoetcetnivtieally,owfhtehree
ρ′2+ρ2+|ρ′2−ρ2|
Z0 strictly2(1)DEGtobegoodapproximationstothecorre-
5
0 4 Rydberg r =2 r =5
s s
1D n series 2D Q2D 2D Q2D
-0.1 rs=5 a.u.)3 ξ=1 1 0.500 0.360 0.511 0.164 0.204
) ( 2 0.125 0.161 0.196 0.092 0.103
u. 2
a. ξ=0 10 3 0.056 0.102 0.117 0.064 0.070
( ×2
vρ()x-0.2 ξ=1 nρ() 45 00..003210 00..006468 00..007532 00..004355 00..004387
Q1D ρ1 6 0.014 0.036 0.038 0.027 0.028
π
-0.3 1D 2 ξ=0
TABLEI.Thefirstsixeigenenergies(absolutevalues,a.u.) of
0
0 5 10 15 0 5 10 the EXX KS hamiltonian of 2D and Q2D EG for two values
ρ (a.u.) ρ (a.u.) of the density parameter r = 2 and 5, compared with the
s
Rydberg’sseries 1/(2n2).
FIG. 8. The same as Fig. 7, butfor 1DEG.
DFT expression for energy
10 rsc 2 4 6 8 10 E = ǫσi − vxσc(r)nσ(r)dr+Exc
Xi,σ Xσ Z (21)
E
∆Fǫ 2D − 1 v (r)n(r)dr+ 1 n+(r)n+(r′)drdr′,
0.5 2 H 2 |r−r′|
u.)u.) Z Z
a.a. wherethe lastterminEq.(21)istheenergyoftheback-
((
ǫǫ∆∆ 0 ground positive charge n+(r), by virtue of Eqs. (8) and
,, (10), we have for the energy per particle
FF
EE 1D
0.5 1
ǫ=ǫ +ǫ − vσ(r )nσ(r )dr
K sub 2n x ⊥ ⊥ ⊥
Xσ Z (22)
00 rsc 2 4rs (a.u.)6 8 10 − 21n vext,H(r⊥) n(r⊥)+n+(r⊥) dr⊥,
Z
(cid:2) (cid:3)
FIG.9. DiagramsofstabilityofQ2D(upperpanel)andQ1D where the kinetic and the subband energies are
(lowerpanel)electrongaseswithonesubbandoccupied. The
Fermi energy EF = kF2/2 and the distance ∆ǫ between two 1 1+ξ2, d=2,
Alotwerst<subrbca(nsdhsadaeredsahroewasn, vrecrs≈us1t.h4e6daenndsit0y.7-p2,arianm2eDterarnsd. ǫK = 2rs2 ( π2 1+3ξ2 /48, d=1, (23)
s s s
1D cases, respectively), more than one subbands are filled, 1
ǫ = ǫ↑+ǫ(cid:0)↓+ξ ǫ↑(cid:1)−ǫ↓ , (24)
invalidating the results of this theory. The diagrams refer to sub 2 0 0 0 0
thespin-unpolarized case. h (cid:16) (cid:17)i
andξ =(n↑−n↓)/nisthespinpolarization. Inthezero-
thickness limit v =0, and the exchange energy per
ext,H
particle becomes ǫ = −23/2 (1+ξ)3/2+(1−ξ)3/2
x 3πrs
and ǫx = −∞, for 2D and 1Dh cases, in accord withi
sponding quasi-low-dimensional EG with one filled sub- Refs. 29 and 30, respectively.
band, we need to establish when the latter regime actu- Our solutions, in particular, show that, while, in the
ally holds. This question is answered in the diagrams of general case, EXX requires the knowledge of all, occu-
thestabilityoftheEGwithonesubbandfilled,presented pied and empty, states, and solving of the optimized ef-
in Fig. 9. For rs >rsc (rsc ≈1.46 and 0.72 in 2D and 1D fective potential(OEP)integralequationis necessary,in
cases, respectively) EF = kF2/2 < ∆ǫ, where ∆ǫ is the the case of quasi-2(1)DEG with only one subband occu-
subband gap, the state with one filled subband is stable. pied, it is possible to avoid those complications, still re-
Otherwise, at rs < rsc, it is energetically preferable to mainingwithintheexacttheory. Inthiscontext,wenote
start filling the second subband. The latter regions, cor- thatthelocalizedHartree-Fockpotential(LHF)31,which
responding to high electron densities and to which our hasprovenausefulconceptrequiringtheoccupiedstates
theory does not apply, are shaded at the diagrams. only andinvolving no OEP equationin the generalcase,
yieldsresultsveryclose(oftenindistinguishable)tothose
For the sake of completeness, we write down the total of EXX and Hartree-Fock(HF)31,32. Conceptually, LHF
energy in the one occupied subband regime. Using the potentialcanbeconstructedindependentlyfromHFand
6
EXX33, within the scheme of the optimized propagation V. EQUIVALENCE OF EXX AND LHF FOR
in time34, and it is one of the realizations of the “direct QUASI-2(1)DEG WITH ONE FILLED SUBBAND
energy” potentials35,36. This is, therefore, very instruc-
tive to learn that, for 2(1)DEG with one filled subband,
LHF coincides with both EXX and Slater’s potentials The purpose of this section is to prove that, within
exactly up to a constant, as we show in the next section. the one-filled-subband regime of 2(1)D electron gas, the
Here,itisinstructivetodrawtheanalogywithasinglet localizedHartree-Fock (LHF) method and and exact ex-
two-electron system with only one state filled, for which change (EXX) are exactly equivalent.
allthe threepotentialscoincide32,33. Inthe presentcase,
although with an infinite number of electrons, the same We start by reminding the basic facts on the LHF
propertyholdsdueto(i)theseparationofthevariablesin potential31 within the framework of the optimized-
the two perpendicular directions and (ii) to the system propagation method (OPM)32–34. The LHF exchange
being uniform (the potential being flat) in the parallel potential v˜σ(r), experienced by electrons with the spin
x
direction. orientation σ, is a solution to the equation
1 ρσ(r,r′)ρσ(r′,r′′)ρσ(r′′,r)
v˜σ(r)nσ(r)= v˜σ(r′)− |ρσ(r,r′)|2dr′+ dr′dr′′, (25)
x x |r−r′| |r′−r′′|
Z (cid:20) (cid:21) Z
with the spin-resolved density-matrix defined as where the superscript at the sum means that only the
orbitals with the spin direction σ are included. For in-
tegegral number of particles, Eq. (25) determines the
potential up to the addition of an arbitrary constant
σ
ρσ(r,r′)= φ (r)φ∗(r′), (26) v˜σ(r)→v˜σ(r)+cσ.32 A relation
i i x x
i
X
1 |ρσ(r,r′)|2 1
v˜σ(r)nσ(r)dr+ drdr′ + v (r)n(r)dr=0 (27)
x 2 |r−r′| 2 H
σ (cid:20)Z Z (cid:21) Z
X
fixes this constant uniquely in the fully spin-polarized the sum of the single-particle eigenenergies32.
case, while in the presence of electrons of both spin ori- For 2(1)DEG with only one subband filled
entations,itmakesonlyoneoftheconstantsindependent
byfixingthequantityc↑N↑+c↓N↓,whereN↑ andN↓ are ρσ(r,r′)=µσ0(r⊥)µσ0∗(r′⊥)ρσ(rk−r′k), (28)
the number of electrons with spin up and spin down, re-
spectively. Inbothcasesthisfixesthetotalenergyofthe where ρσ(rk) is given by Eq. (7). Substituting Eqs. (28)
system, the electronic part of which is equal in OPM to and (9) into Eq. (25) and canceling out |µ (r )|2 from
σ ⊥
both sides, we can write
|µ (r′ )|2|ρσ(r −r′)|2
nσv˜σ(r )=− σ ⊥ k k dr′+ v˜σ(r′ )|µ (r′ )|2|ρσ(r −r′)|2dr′
x ⊥ |r−r′| x ⊥ σ ⊥ k k
Z Z (29)
|µ (r′ )|2|µ (r′′)|2ρσ(r −r′)ρσ(r′ −r′′)ρσ(r′′−r )
+ σ ⊥ σ ⊥ k k k k k k dr′dr′′,
|r′−r′′|
Z
where we have explicitly written that the potential is a r , and the apparent dependence on r is eliminated by
⊥ k
function of the perpendicular coordinate only. We no- the proper substitutions of the integration variables r′
k
tice,andthisiscrucialtoourderivation,thatthesecond and r′′. Recalling that Eqs. (25) are solvable up to ar-
k
and the third terms in the right-hand side of Eq. (29)
bitraryconstants only, we can, therefore,write by virtue
are constants: The terms in question do not depend on
7
of Eq. (29) where vσ(r ) is the EXX potential of Eq. (10). Substi-
x ⊥
tuting Eq. (30) into Eq. (27), we have
v˜σ(r )=vσ(r )+cσ, (30)
x ⊥ x ⊥
1 |ρσ(r,r′)|2 1
cσNσ =− vσ(r)nσ(r)dr+ drdr′ − v (r)n(r)dr. (31)
x 2 |r−r′| 2 H
σ σ (cid:20)Z Z (cid:21) Z
X X
As already mentioned above, within the framework of bythecomparisontotheresultsoftheself-consistentcal-
OPM, the electronic energy of a many-body system is a culations beyond the zero-thickness limit. They, conse-
sum of the eigenvalues of its single-particle LHF hamil- quently,areexpectedtoserveasefficientmeanstohandle
tonian(i.e.,itisa“directenergy”potential35,36). There- thegeneralproblemofthe exchangepotentialandimage
fore, we can write states in the low-dimensional science.
As a by-product, our solutions extend a short list of
σ 1 n+(r)n+(r′) analytical results known in the many-body physics.
E = ǫ˜ + drdr′, (32)
i 2 |r−r′|
σ i Z
XX
ACKNOWLEDGMENTS
wheretheelectrostaticenergyofthepositivebackground
was added, as in the main text. Since, due to Eq. (30),
the LHF Hamiltonian differs from the EXX one by the Support from the Ministry of Science and Technol-
constants cσ only, we can write from Eq. (32) ogy,Taiwan,GrantNo. 104-2112-M-001-007,isacknowl-
edged.
σ 1 n+(r)n+(r′)
E = ǫ + cσNσ+ drdr′,
i 2 |r−r′|
σ i σ Z Appendix A: Classical image potential of 2(1)D
XX X
(33)
conductor
whereǫ aretheEXXeigenenergies. Finally,substituting
i
Eq. (31) into Eq. (33) and comparing with Eq. (21), we
Let2(1)Dconductoroccupyaplane (line) r =0and
conclude that the LHF and EXX total energies exactly ⊥
letatestchargeQbepositionedatr =R andr =0.
coincide. ⊥ ⊥ k
The field of the test charge will cause a change in the
electron-density distribution, which we denote n (r ).
1 ⊥
Thelatterisdeterminedbytheconstancyofthepotential
VI. CONCLUSIONS
at the plane (line) of the conductor
We have obtained explicit solutions to the problem Q − n1(r′k) dr′ =0, (A1)
oqfuatshie-twexoa-catnedxcohnaen-gdeim–enospitoinmailzeedlecetffreocntigvaespesotwenitthiaolnoef R⊥2 +r2k Z |r′k−rk| k
q
subbandfilledintermsofthedensity. Ithasbeenproven bywhichwealsofixthepotentialorigin. TakingFourier-
that the EXX potential, the localized Hartree-Fock po- transform with respect to r we have
k
tential, and the Slater’s potential all coincide with each
v (R )
otherexactly(upto anarbitraryconstant)forthesesys- n (q )=Q qk ⊥ , (A2)
tems. 1 k vq (0)
k
By taking the limit of the zero-thickness [full 2(1)D
where
confinement] of the respective electron gases, we have
found exactanalytical solutions to the static EXX prob- v (r )= e−iqk·rk dr . (A3)
lem for 2(1)DEG. We have identified a non-asymptotic qk ⊥ r2+r2 k
Z k ⊥
regime, which realizes in the free space at distances less
q
orcomparabletotheinverseFermiradiusoftheelectron In real space Eq. (A2) yields for the density
gas. While our solutions reproduce the before known
asymptotic of the exchange potential at large distances, n1(rk)= (2Qπ)d vvqk(R(0⊥))eiqk·rkdqk (A4)
at shorter distances the variance of the exchange poten- q
Z k
tialfromitsasymptoticstronglyaffectsthelow-lyingex- and for its induced potential
cited states, causing departure from the Rydberg series.
gaWsesi,thoiunraanwaildyeticraalngpeotoefnttihaelsdaecncsuitriaetseloyfatphpereolxeicmtraotne φind(rk,r⊥)=−(2Qπ)d vqk(Rv⊥q)(v0q)k(r⊥)eiqk·rkdqk.
Z k
those of realistic quasi-2(1)D systems, as demonstrated (A5)
8
The total energy of the system, which, being the work of particles) do not affect the the first order variation
requiredtomovethetestchargefromR toinfinityand, of E , the latter taken at the ground-state of the den-
⊥ x
therefore, is the image potential, is sity. Therefore, with respect to finding the ground-state
exchange potential for our systems, we need to consider
E =Qφ (r =0,R )+ 1 n1(rk)n1(r′k)dr dr′, the variations of the density which depend on r⊥ only.
ind k ⊥ 2 |r −r′| k k By the chain rule, this leads to the variation of the
Z k k
KS potential v as a function of r only, too. Since the
(A6) s ⊥
variables r and r separate in KS equations and since
which, by Eqs. (A4) and (A5) can be rewritten as ⊥ k
theground-stateoccupiedspin-orbitals(5)factorizewith
Q2 vq2 (R⊥) Q2 the same perpendicular part for all of them, it is obvi-
E =− k dq + ous,that,uponavariationofvσ(r), thechangedorbitals
(2π)d v (0) k 2(2π)d s
Z qk (A7) remain of the same form (5) with one and the same,
vq2 (R⊥) Q2 vq2 (R⊥) changed, perpendicular part. In other words, through
× k dq =− k dq .
v (0) k 2(2π)d v (0) k the variational procedure, the system remains that with
q q
Z k Z k only one subband filled.
2D-case
1. Proof of Eq. (10) in terms of the orbital
wave-functions and eigenenergies
ByEq.(A3),vqk(R⊥)= 2qπke−qkR⊥,andEq.(A7)eval-
uates to
Here we give an alternative proof of Eq. (10) which
Q2 showshowthegeneralEXXformalismleadstotheSlater
E =− , (A8) potential in the case of quasi 2(1)DEG with only one
4R
⊥
subband filled. We can write
which coincides with the result for a semi-infinite metal.
δE δn(r′) δE
x = x dr′ = χσ(r,r′)vσ(r′)dr′,
δvσ(r) δvσ(r)δnσ(r′) s x
s Z s Z
1D-case (B2)
wherewehaveusedthechainrule,thedefinitionsofvσ(r)
x
and of the KS spin-density-response function χσ(r,r′),
By Eq. (A3), vqk(R⊥) = 2K0(qkR⊥), where Kn(x) is andthesymmetryofthelatterinitstwocoordinastevari-
the modified Bessel function of the second kind. Since
ables. The explicitexpressionofχσ(r,r′)interms ofthe
K (0)=∞, Eq. (A7) evaluates to s
0
KS wave-functions and eigenenergies is
E =0, (A9)
ψσ∗(r)ψσ(r)ψσ∗(r′)ψσ(r′)
χσ(r,r′)= i j j i +c.c. (B3)
in accordance that the notion of the strictly 1D electron s ǫ −ǫ
i j
i∈occ
gas is inherently inconsistent and a finite width must be X
j6∈occ
introducedformeaningfulresults30. Thesecomplications
are not, however,relevantto the purposes of the present In the case of 2(1)D EG with the flat in-plane potential,
work. the variables r and r separate in the KS equations, i
k ⊥
(j) becomes a composite index i=(k ,n ), where k is
ki i k
the parallel momentum, n indexes the transverse bands,
Appendix B: Further particulars of the derivation of and
Eq. (10)
k2
ǫσ = ki +λσ , (B4)
The derivationofEq.(10)has beenassuming thatthe i 2 ni
systemremainsthat withone subbandfilled[and, there-
fore,Eq.(8)holding]throughoutthe variationalprocess.
Here we show that this, indeed, is the case. Let us write φσ(r)= eikki·rkµσ (r ), (B5)
the variation of the exchange energy i Ω1/2 ni ⊥
where µσ (r ) are the wave-functions of the (3 − d)-
δEx = vx(r⊥)δn(r)dr, (B1) dimensionnial⊥motion in the potential vsσ(r⊥), and λni
Z are the corresponding eigenenergies. We substitute
where,duetothesymmetryoftheproblem,wehaveused Eqs.(B3)-(B5)into Eq.(B2) andnote that, by the sym-
thefactthatv isafunctionoftheperpendicularcoordi- metry of the problem, the left-hand side of Eq. (B2) de-
x
nateonly. Therefore,thevariontionsofthedensitywhich pendsonr⊥ onlyand,therefore,wecanaverageEq.(B2)
avarage out to zero in the parallel to the EG direction inrkoverΩwithoutchangingit. Doingthisandintegrat-
(which is necessary for the conservation of the number ing overr′ onthe right-handside with accountof vσ(r′)
k x
9
being a function of r′ only, we see that only k = k Then, using the chain rule,
⊥ kj ki
contribute, leading to
δE δE δφσ (r′) δE δφσ∗(r′)
δvδsσE(xr) =nσZ χσs(r⊥,r′⊥)vxσ(r′⊥)dr′⊥, (B6) δvsσ(xr)=mX∈occZ (cid:20)δφσm(xr′) δvmsσ(r) +δφσm∗(xr′) δvmsσ(r) (cid:21)dr′.
(B9)
where
∞ µσ∗(r′ )µσ(r′ )µσ(r )µσ∗(r )
χ (r ,r′ )= 0 ⊥ n ⊥ 0 ⊥ n ⊥ +c.c., Due to Eq. (B8)
s ⊥ ⊥ λσ −λσ
n=1 0 n
X
(B7) δE φσ(r′′)φσ∗(r′′)φσ∗(r′)
andwehavetakenaccountofthefactthatonlyonestate x =− i m i dr′′,(m,σ)∈occ.
δφσ(r′) |r′−r′′|
with the wave-function µσ0(r⊥) is occupied in the trans- m (i,σX)∈occZ
verse direction. (B10)
On the other hand, we can evaluate δE /δvσ(r) di- By the use of the perturbation theory, we can write
x s
rectly. By Eqs. (3) and (4)
δφ (r′) φσ(r′)φσ (r)φσ∗(r)
1 φσ(r)φσ∗(r)φσ(r′)φσ∗(r′) m = l m l . (B11)
E =− i j j i drdr′. δvσ(r) ǫσ −ǫσ
x 2 |r−r′| s l6=m m l
Xσ (i,σX)∈occZ X
(j,σ)∈occ
(B8) SubstitutionofEqs.(B10)and(B11)intoEq.(B9)gives
δE φσ(r′′)φσ∗(r′′)φσ∗(r′)φσ(r′)φσ(r)φσ∗(r)
x =− i m i l m l +c.c. dr′dr′′. (B12)
δvσ(r) |r′−r′′|(ǫσ −ǫσ)
s (m,Xσ)∈occ lX6=mZ (cid:20) m l (cid:21)
(i,σ)∈occ
Further, substituting Eqs. (B4) and (B5) into Eq. (B12), we have
δE 1
x
=−
δvσ(r) Ω3
s |kkiX|≤kFσ (kkl,nlX)6=(kkm,0)
|kkm|≤kFσ
(B13)
ei(kki−kkm)·r′k′µσ0(r′⊥′)µσ0∗(r′⊥′)µσ0∗(r′⊥)ei(kkl−kki)·r′kµσnl(r′⊥)ei(kkm−kkl)·rkµσ0(r⊥)µσn∗l(r⊥) +c.c.dr′dr′′,
k2 k2
Z km +λσ − kl −λσ |r′−r′′|
 2 0 2 nl 
 (cid:18) (cid:19) 
 
where, again, it has been taken into account that the occupied orbitals have only one transverse band µσ(r ).
0 ⊥
Similartotheabove,weaverageEq.(B13)overr withoutchangingit. Thisleadstoonlythetermswithk =k
k kl km
remaining, and gives
δEx =− 1 ∞ |µσ0(r′⊥′)|2µσ0∗(r′⊥)µσn(r′⊥)µσ0(r⊥)µσn∗(r⊥)ei(kkm−kki)·(r′k−r′k′) +c.c. dr′dr′′, (B14)
δvσ(r) Ω3 (λσ−λσ)|r′−r′′|
s |kkiX|≤kFσ nX=1Z " 0 n #
|kkm|≤kFσ
or, according to the definition (7),
δEx =−1 ∞ |µσ0(r′⊥′)|2µ0σ∗(r′⊥)µσn(r′⊥)µσ0(r⊥)µσn∗(r⊥)|ρσ(r′k−r′k′)|2 +c.c. dr′dr′′, (B15)
δvσ(r) Ω (λσ−λσ)|r′−r′′|
s n=1Z " 0 n #
X
or, by Eq. (B7) and integrating over r′,
k
δEx =− dr′ χσ(r ,r′ ) |µσ0(r′⊥′)|2|ρσ(r′k−r′k′)|2dr′′. (B16)
δvσ(r) ⊥ s ⊥ ⊥ |r′−r′′|
s Z Z
10
Combining Eqs. (B6) and (B16), we have
|µσ(r′′)|2|ρσ(r′ −r′′)|2
dr′ χσ(r ,r′ ) nσvσ(r′ )+ 0 ⊥ k k dr′′ =0. (B17)
⊥ s ⊥ ⊥ x ⊥ |r′−r′′|
" #
Z Z
For Eq. (B17) to hold, the expression in the square brackets must be a constant, which retrieves the exchange
potential of Eq. (10) up to an arbitrary constant.
∗
[email protected] (1984).
1 L. D. Landau and E. M. Lifshitz, Electrodynamics of con- 17 E. K. U. Gross and W. Kohn, “Local density-functional
tinuous media (Pergamon Press, New York and London, theoryoffrequency-dependentlinearresponse,”Phys.Rev.
1960). Lett. 55, 2850–2852 (1985).
2 N. D. Lang and W. Kohn, “Theory of metal surfaces: 18 E. K. U. Gross and W. Kohn, “Local density-functional
Chargedensityandsurfaceenergy,”Phys.Rev.B1,4555– theoryoffrequency-dependentlinearresponse,”Phys.Rev.
4568 (1970). Lett. 57, 923–923 (1986).
3 J. F. Dobson, in Density Functional Theory, edited by 19 R.T.SharpandG.K.Horton,“Avariationalapproachto
E. K. U. Gross and R. M. Dreizler (Plenum, New-York, the unipotential many-electron problem,” Phys. Rev. 90,
1995) p. 393. 317–317 (1953).
4 L. A. Constantin and J. M. Pitarke, “Adiabatic- 20 J. D. Talman and W. F. Shadwick, “Optimized effective
connection-fluctuation-dissipation approach to long-range atomic central potential,” Phys.Rev.A 14, 36–40 (1976).
behaviorofexchange-correlation energyatmetal surfaces: 21 A.G¨orlingandM.Levy,“ExactKohn-Shamschemebased
A numerical study for jellium slabs,” Phys. Rev. B 83, onperturbationtheory,”Phys.Rev.A50,196–204(1994).
075116 (2011). 22 J. C. Slater, “A simplification of the Hartree-Fock
5 W. Kohn and L. J. Sham, “Self-consistent equations in- Method,” Phys.Rev.81, 385–390 (1951).
cludingexchangeandcorrelation effects,”Phys.Rev.140, 23 In Eq. (10), the 3D spin-density nσ(r) is varied at a fixed
A1133–A1138 (1965). valueofthed-dimensionalspin-densitynσ (and,therefore,
6 L. J. Sham, “Exchange and correlation in density- kσ), which amounts to the conservation of the total num-
F
functional theory,”Phys. Rev.B 32, 3876–3882 (1985). ber of particles. In Appendix B further particulars of the
7 M.K.HarbolaandV.Sahni,“Asymptoticstructureofthe derivation are given. For an alternative proof of Eq. (10)
Slater-exchangepotentialatmetallicsurfaces,”Phys.Rev. in terms of the KS eigenfunctions and eigenenergies, see
B 36, 5024–5026 (1987). AppendixB1.
8 A. G. Eguiluz, M. Heinrichsmeier, A. Fleszar, and 24 Inparticular,thisdoesnotholdforametalsurface,where
W. Hanke,“First-principles evaluation of the surface bar- theSlater and OEP potentials differ7.
rier for a Kohn-Sham electron at a metal surface,” Phys. 25 A.P.Prudnikov,O.I.Marichev, andYu.A.Brychkov,In-
Rev.Lett. 68, 1359–1362 (1992). tegralsandSeries,Vol.3: MoreSpecialFunctions(Gordon
9 C.M.Horowitz,C.R.Proetto, andS.Rigamonti,“Kohn- and Breach, Newark, NJ, 1990).
Sham exchange potential for a metallic surface,” Phys. 26 Wolfram Research, Mathematica, version 10.3 ed. (Cham-
Rev.Lett. 97, 026802 (2006). paign, Illinios, 2015).
10 C.M.Horowitz,C.R.Proetto, andJ.M.Pitarke,“Local- 27 Although, separately the external and Hartree potentials
ized versusextendedsystemsindensityfunctionaltheory: are infinite, theirsum of Eq. (20) is finite.
Some lessons from the Kohn-Sham exact exchange poten- 28 G. Q. Hassoun, “One- and two-dimensional hydrogen
tial,” Phys. Rev.B 81, 121106 (2010). atoms,” American Journal of Physics 49, 143–146 (1981).
11 Z. Qian, “Asymptotic behavior of the Kohn-Sham ex- 29 B.TanatarandD.M.Ceperley,“Groundstateofthetwo-
change potential at a metal surface,” Phys. Rev. B 85, dimensional electron gas,” Phys. Rev. B 39, 5005–5016
115124 (2012). (1989).
12 E. Engel, “Exact exchange plane-wave-pseudopotential 30 A.GoldandA.Ghazali,“Exchangeeffectsinaquasi-one-
calculations for slabs,” The Journal of Chemical Physics dimensional electron gas,” Phys. Rev. B 41, 8318–8322
140, 18A505 (2014). (1990).
13 E. Engel, “Asymptotic behavior of exact exchange poten- 31 F.Della Sala andA.G¨orling, “Efficient localized Hartree-
tial of slabs,” Phys.Rev.B 89, 245105 (2014). Fock methods as effective exact-exchange Kohn-Sham
14 See Appendix A for the classical image potential of methods for molecules,” The Journal of Chemical Physics
2(1)DEG. 115, 5718–5732 (2001).
15 A.ZangwillandP.Soven,“Density-functionalapproachto 32 V. U. Nazarov and G. Vignale, “Derivative discontinu-
local-field effectsinfinitesystems: Photoabsorptioninthe ity with localized Hartree-Fock potential,” TheJournal of
rare gases,” Phys. Rev.A 21, 1561–1572 (1980). Chemical Physics 143, 064111 (2015).
16 E. Runge and E. K. U. Gross, “Density-functional the- 33 V. U. Nazarov, “Time-dependent effective potential and
oryfortime-dependentsystems,”Phys.Rev.Lett.52,997 exchangekernelofhomogeneouselectron gas,”Phys.Rev.