Accuracy of density functionals for molecular electronics: the Anderson junction

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by  Zhenfei Liu

Accuracy of density functionals for molecular electronics: the Anderson junction
Zhen-Fei Liu, Justin P. Bergfield, and Kieron Burke
Departments of Chemistry and of Physics, University of California, Irvine, California 92697, USA
Charles A. Stafford
Department of Physics, University of Arizona, 1118 East Fourth Street, Tucson, Arizona 85721, USA
(Dated: January 9, 2012)
The exact ground-state exchange-correlation functional of Kohn-Sham density functional theory
yields the exact transmission through an Anderson junction at zero bias and temperature. The
exact impurity charge susceptibility is used to construct the exact exchange-correlation potential.
2 Weanalyzethesuccessesandlimitationsofvarioustypesofapproximations,includingsmoothand
1 discontinuous functionals of the occupation, as well as symmetry-broken approaches.
0
2
Since the pioneering experiments of Reed and Tour tentials with incorrectly positioned KS eigenvalues, both
n
a on dithiolated benzene [1], there has been tremendous occupied and unoccupied. These errors become severe
J progress in the ability to both create and characterize when the molecule is only weakly coupled to the leads

5 [2] organic molecular junctions. But accurate simulation [22]. Calculated transmission can be too large by several
of these devices remains a challenge, both theoretically orders of magnitude due to this incorrect positioning of

]
l and computationally [3]. The essential physics has been the levels. Recent calculations [23] using beyond-DFT
l
a well understood since the ground-breaking work of Lan- techniques to correctly position the levels show greatly
h dauer and Bu¨ttiker [4, 5] in the context of mesoscopic improved agreement with experiment.
- devices, including both Coulomb blockade and Kondo But this progress returns us to the earlier concern:
s
e effects [6, 7]. Calculations with simple model Hamilto- Evenwithanexactground-stateXCfunctional,arethere
m
nians demonstrate such effects at a qualitative level [8]. XCcorrectionstotheLandauer-Bu¨ttikerresult? Thean-
. Ontheotherhand,organicmoleculesconnectedtometal swer appears to be yes in general [15], but in a previous
t
a leads [9] require hundreds of atoms and thousands of ba- work [24] we argued that, under a broad range of condi-
m
sisfunctionsforasufficientlyaccuratecalculationoftheir tions applicable to typical experiments, such corrections
- total energy, geometry, and single-particle states. Such can vanish. This result was shown by exact calculations
d
conditionsareroutineformoderndensityfunctionalthe- on an impurity model (Anderson model) employing the
n
o ory (DFT) calculations [10], but the ability of present exactXCfunctional. Inthepresentwork,weanalyzedif-
c functional approximations to predict accurate currents ferent approximate treatments, applied to the Anderson
[
remains an open question [11]. junction, andcalculatetheirerrors. Theimplicationsfor

1 The standard DFT method for calculating current DFT calculations of transport in general are discussed.
v throughsuchadeviceistoperformaground-stateKohn- The Anderson model [25] is a single interacting site
0 Sham(KS)DFTcalculation[12]onasystemuponwhich (C) connected to two non-interacting electrodes (L,R).
1
a difference between the chemical potentials of the left The Hamiltonian of the system is H=H +H +H .
3 C T L,R
andrightleadshasbeenimposed(theappliedbias), and Each lead is represented by a non-interacting Fermi gas:
1
(cid:80)
1. calculate the transmission through the KS potential us- HL,R = kσ∈L,Rεkσnˆσ, with chemical potential µ and
ing the Landauer-Bu¨ttiker formula. But there is nothing thecentralinteractingsiteis: H =ε(nˆ +nˆ )+Unˆ nˆ ,
0 C ↑ ↓ ↑ ↓
2 in the basic theorems of DFT that directly implies that where nˆσ = d†σdσ is the number operator for spin σ and
1 such a calculation would yield the correct current, even U isthechargingenergyrepresentingon-siteinteraction.
: if the exact ground-state functional were used. H is the tunneling between leads and the central site.
v T
i Thelimitofweakbiasismoreeasilyanalyzedthanthe The tunneling width Γ is a constant in the broad-band
X
generalcase,becausetheKubolinearresponseformalism limit. A schematic is shown in Fig. 1. Real molecules
r applies [13, 14]. In that case one finds that, in princi- can be mapped onto the Anderson model [26, 27].
a
ple, there are exchange-correlation (XC) corrections to Inapreviouswork[24],wecalculatedtheexactrelation
the current in the standard approach [15], but little is betweenoccupancyonthecentralsiteandon-siteenergy
known about their magnitude [16, 17]. Even without εforanAndersonjunction,usingtheBetheansatz(BA)
these corrections, one can ask if the standard approx- [28]. We showed that exact KS DFT yields the exact
imations used in most ground-state DFT calculations transport at zero temperature and in the linear response
(i.e., generalized gradient approximations [18] and hy- regime, although the KS spectral function differs from
brids of these with Hartree-Fock exchange [19, 20]) are the exact one away from the Fermi energy. This is be-
sufficiently accurate for transport purposes. The answer causetheAndersonjunctionhasonlyonesiteandtrans-
appearsdefinitivelyno! Becauseofself-interactionerrors, mission is a function of occupation number due to the
such approximations are well-known [21] to produce po- Friedel-Langreth sum rule [29, 30]. Thus, for this sim-
2
2ε+U
exact
1 [5,6]-Pade
RHF
Γ Γ UHF
0.8
μ
ε
1) 0.6
(c
χ
U
0.4
0.2
FIG. 1. A cartoon for Anderson model. The model consists
of two featureless leads and a central region with on-site in-
teractionU. Γisthetunnelingwidth. Twomany-bodylevels 0
0 0.2 0.4 0.6 0.8 1
of the central region are shown.
y
ple model, all failures of approximate XC calculations of FIG. 2. Dimensionless susceptibility χ˜ = Uχ (1) as a func-
c c
transmission can be attributed to failures to reproduce tion of y =U/Γ for the Anderson junction [exact, [5,6]-Pad´e
the exact occupation number, i.e., there are no XC cor- fit (see text), RHF and UHF].
rections to the standard practice of applying KS DFT to
the ground-state and finding transmission through the
single-particle potential. On the other hand, the stan-
dardapproximationsinuseinDFTcalculationsoftrans- the Kondo plateau at zero temperature and weak bias
porthaveavarietyofshortcomings. Themostprominent [24, 32].
one, as we shall see, is the lack of a discontinuity in the As shown in Ref. [24], the XC potential can be very
XC potential with particle number [31]. accurately parametrized with the form:
Beforestudyingapproximations,werefineourprevious
(cid:20) (cid:18) (cid:19)(cid:21)
numerical fit of BA results, using analytic results from ε α 2 1−(cid:104)n (cid:105)
XC = 1−(cid:104)n (cid:105)− tan−1 C (4)
many-body theory. We re-introduce [25] reduced vari- U 2 C π σ
ablesy =Γ/U,whichmeasurestheratiooflead-coupling
to the onsite Coulomb repulsion, while x=(µ−(cid:15))/U is The tan−1 term jumps by π as (cid:104)n (cid:105) passes through 1,
C
the difference between the leads’ chemical potential and leadingtodiscontinuousbehaviorwithoccupation. Thus
theonsitelevelenergy, inunitsofU. For x<0, thecen- σdeterminesthewidthofthisregion,whileαdetermines
tral site is above the chemical potential, at x = 0 they itsstrength. Bothσ andα arefunctionsofy =Γ/U and
match. were extracted numerically by fitting to the exact solu-
The occupation in the KS system is given by self- tion,andwereroughlyfitbysimplePad´eapproximations
consistent solution of the KS equation for occupation: there.
(cid:18) (cid:19)
1 1 µ−ε ((cid:104)n (cid:105)) However,wecangreatlyimprovethefitofσ. Acentral
(cid:104)n (cid:105)= + arctan S C . (1)
C 2 π Γ objectintheAndersonjunctionisthechargesusceptibil-
ity, χ ((cid:104)n (cid:105)) = d(cid:104)n (cid:105)/dµ. At half-filling, this is known
where the KS level is written as c C C
analytically[33, 34]:
U
ε ((cid:104)n (cid:105))=ε+ (cid:104)n (cid:105)+ε ((cid:104)n (cid:105)), (2)
S C 2 C XC C 1(cid:114)2(cid:90) ∞ e−πyt2/2
withthesecondtermbeingtheHartreecontributionand χ˜ = dt , (5)
the third being the XC contribution (in fact, only cor- c π y −∞ 1+(cid:16)(2y)−1+t(cid:17)2
relation, as exchange is zero for this model), which is a
functionoftheoccupation. Consideredinreverse, thisis
where χ˜ = Uχ (1) is dimensionless, and is plotted in
c c
a definition of the exact ε , if the occupation is known,
XC Fig. 2. Thiscurvecanbereadilyfittoa[5,6]Pad´eform:
as it is from the BA solution. The KS transmission is
then
5 6
T(E)E=µ =sin2(cid:16)π2 (cid:104)nC(cid:105)(cid:17), (3) χ˜mc od(y)=(cid:88)akyk/(cid:88)bkyk, (6)
k=1 k=0
and matches the true transmission in the many-body
system, by virtue of the sum-rule. The exact ground- whose 11 independent coefficients are chosen to recover
state functional yields the exact transmission, including the Taylor-expansion around y = 0 (strongly-correlated
3
limit)1 exactly to 5 orders, around y → ∞ to 6 or- the (cid:104)n (cid:105) vs. (µ−(cid:15))/U curve (see Figs. 3 and 4). It has
C
ders,andaregiveninTableI.Theweak-correlationlimit a maximum at about y =0.291, and as y varies from ∞
can also be extracted via the Yosida-Yamada perturba- (weakly-correlatedlimit)to0(strongly-correlatedlimit),
tive approach [37–39]. The quantity χ˜ has the physical the slope at the symmetric point increases at first. Be-
c
meaning of the slope at particle-hole symmetry point in yond the maximum value, the slope decreases, and the
Coulomb blockade pleateau gradually develops.
TABLE I. Coefficients in the [5,6]-Pad´e approximation [Eq. (6)].
k a b
k k
0 − π3(π6+6π4−225π2+675)
1 8π2 −12π2(π6+54π4−945π2+3105)
2 −576π(8π4−120π2+405) 12π(π8−30π6+555π4−6525π2+29700)
3 64(π8−36π6+153π4+135π2+8910) 96(π8−80π6+975π4−2925π2+1350)
4 256π(4π6−204π4+1530π2+945) 48π(π8−30π6−225π4+3375π2+8100)
5 128π2(π6−42π4+315π2+135) 576π2(π6−50π4+375π2+225)
6 − πa /2
5
BytakingderivativesonbothsidesofEq. (4),thetwo
coefficients α and σ are constrained by χ˜ : 1
c µ 0.8 U=Γ
2α E= 0.6
σ = π(2/χ˜ −yπ+α−1). (7) T(E) 0.4
c 0.2
Retaining the simple form of Ref. [24], a [0,1] Pad´e, α= 0
2
1/(1+5.68y), we determine σ from the Pad´e fit to the
1.5
susceptibility and Eq. (7). This yields highly accurate
>
occupations, KS potentials, and transmissions, including nC 1
<
theKondoplateau. Itagreesverywellwiththenumerical 0.5
fit to the BA results of Ref. [24], and matches more 0
-1 -0.5 0 0.5 1 1.5 2
closely than the simpler analytic fit used there. x
1
We now move on to the central topic of this work, exact
0.8 HF
which is the accuracy of approximate functional treat- U 0.6 disc
ments. In such treatments, ε is approximated as a ε/S 0.4
XC 0.2
function of (cid:104)n (cid:105) in Eq. (2), and the resulting Eq. (1) is
C 0
solved self-consistently for (cid:104)n (cid:105). The simplest such ap-
C 0 0.5 1 1.5 2
proximation is to simply set ε = 0, i.e., Hartree-Fock <nC>
XC
(HF), and should be accurate when correlation is weak.
In Fig. 3, we plot several quantities for U =Γ, both ex- FIG. 3. Upper panel: transmission as a function of x =
actly and in HF, showing that HF is very accurate here. (µ−(cid:15))/U;middlepanel: occupationasafunctionofx;lower
We find [25]: panel: KS potential as a function of occupation. Results are
shown for Bethe ansatz or exact KS DFT (exact), Hatree-
2
χ˜HF = , (8) Fock(HF),anddiscontinuousapproximation[disc,Eq. (10)].
c 1+yπ U =Γ in all cases.
which is correct to leading order in y−1:
χ˜ →2/(πy)−2/(πy)2+2γ/(πy)3+··· y →∞, (9)
c
where γ = 3 − π2/4 exactly, but γ = 1 in HF. Thus
we regard U (cid:46) Γ as the weakly correlated regime. On
1 In Refs. [35] and [36], this expansion was reported incorrectly,
the other hand, in Fig. 4, we show the same plots for
with minus sign on the second term. We believe it is corre-
sponding to Wilson ratio R = 1, however this is not true in U = 10Γ. Now, in the exact occupation, the slope near
strongly-correlatedlimit. x=0.5ismuchweaker,leadingtoatransmissionplateau
4
tion (BALDA) [40] and variations [41] have been used
1 to impose discontinuous behavior on the levels. For sim-
µE= 00..68 U=10Γ ple models, all of these can be considered as LDA+U-
T(E) 0.4 like. The methodology of LDA+U [42] has become in-
0.2 creasingly popular in recent years, especially for those
0
focused on moderately correlated systems such as tran-
2
sition metal oxides, for which LDA and GGA often have
1.5
> zero KS band gap. In some fashion, a Hubbard U is
C 1
<n added to some orbitals of a DFT Hamiltonian. Some-
0.5
times U is regarded as an empirical parameter, while
0
-1 -0.5 0 0.5 1 1.5 2 others have found self-consistent prescriptions. In any
x
event, despitenotfittinginthestrictDFTframework, it
1
0.8 eRxHacFt is a method borne of practical necessity for many situa-
U 0.6 UHF tions [43].
ε/S 0.4 disc
To gain a qualitative understanding of the effects of
0.2
0 such models, we define a very simple XC potential that
0 0.5 1 1.5 2 has a discontinuity. To do this, we simply take the
<n >
C
Hartree form, symmetrize it around the half-filled point,
and replace U by a screened U˜. We find that a sim-
FIG. 4. Upper panel: transmission as a function of x = ple fit U˜ =U/(1+0.25/y) works well. U˜ being different
(µ−(cid:15))/U;middlepanel: occupationasafunctionofx;lower
fromU andparticle-holesymmetryguaranteeanexplicit
panel: KS potential as a function of occupation. Results are
derivative discontinuity of ε with respect to occupation
shown for Bethe ansatz or exact KS DFT (exact), restricted S
Hatree-Fock (RHF), unrestricted Hartree-Fock (UHF), and number. This yields
discontinuous approximation [disc, Eq. (10)]. U =10Γ in all
(cid:20) (cid:21)
cases. 1 1
ε [n]= U˜nθ(1−n)+ U + U˜(n−2) θ(n−1), (10)
S 2 2
where θ(x) is the Heaviside theta function and for sim-
(the Kondo plateau) for 0 ≤ x ≤ 1. The plateau effect plicity, n is just (cid:104)n (cid:105).
C
is missed entirely by HF, because of the too-smooth de- While this model does contain a discontinuity, and
pendence (in fact, linear) of its KS level on occupation yields the exact result as y → 0, curing the worst de-
(see bottom panel). Note that at temperatures equal fects of HF, it misses entirely the finite slope of the KS
to or above the Kondo temperature, the Kondo effect potential at half-filling for finite U, which is determined
is destroyed, and the central plateau in transmission is by the susceptibility. The explicit derivative discontinu-
replaced by two Hubbard peaks around x = 0 and 1. ity is exact in the strongly-correlated limit with infinite
Then the behavior of the HF curve is exactly as qualita- U/Γ, but should be “rounded” in finite U/Γ [24, 36], or
tively predicted in Ref. [15], smearing out the two sharp infinitetemperature[44]. Toseethisforfinite(butvery
featuresintoonepeakmidwaybetweenthem. Thisisbe- large) U/Γ, in Fig. 5, we show similar results as in Figs.
cause the KS level shifts linearly with occupation in HF, 3and4,butwithU =100Γandweonlyshowtheregion
instead of more suddenly with occupation in the exact around (cid:104)n (cid:105)=1 at x=0, where the rounded derivative
C
solution. More generally, all smooth density function- discontinuity occurs. The transmission is accurate both
als, such as the local density approximation [12] and the forweakandstrongcorrelation,butisnotsoeverywhere
generalized gradient approximation [18], suffer from the inbetween. Inparticular,itisoverestimatedfor(cid:104)n (cid:105)just
C
same qualitative failure, and so would produce incorrect above 0 (and just below 1) for U = 10Γ because of this
peakscenteredatx=0.5. Alltheseerrorsarisefromthe lack of a finite slope. This is where we expect the great-
approximationstothefunctional; theexactground-state est errors in such models, but the region of inaccuracy
functional reproduces the exact occupation by construc- (on the scale of x) shrinks as U/Γ→∞.
tion, and so yields the exact transmission. Finally, we discuss a different class of approximations.
Therehavethusbeenseveralsuggestions[40]toincor- A well-known (and much debated) technique for mim-
porate the discontinuous behavior with occupation into icking strong correlation is to allow a mean-field calcula-
approximations in transport calculations. At the prac- tion to break symmetries that are preserved in the exact
tical level, Toher et al. [22] showed in a model calcula- calculation. Perhaps the most celebrated prototype of
tion how self-interaction corrections would greatly sup- such a calculation is for HF applied to an H molecule
2
press zero-bias conductance in local density approxima- with a large bond distance. At a crucial value of the
tion calculations for molecules weakly coupled to leads. bond distance (called the Coulson-Fischer point), an un-
More recently, Bethe Ansatz Local Density Approxima- restricted calculation, i.e., one that allows a difference
5
in spin occupations, yields a lower energy than the re- calculation,butnottheindividualspin-densities. Infact,
stricted one. This remains the case for all larger separa- an alternative approach is to interpret another variable,
tions, and the unrestricted solution correctly yields the such as the ontop pair density, as being accurately ap-
sumofatomicenergiesasR→∞,whereastherestricted proximated in such treatments [45].
Hartree-Fock (RHF) solution dissociates to unpolarized WeapplythesamereasoningtotheAndersonjunction,
H atoms with the wrong energies. This is the celebrated just as was done by Anderson when creating the model
symmetry dilemma: with a mean-field approximation, we are using [25]. The symmetries are different, but the
forlargeseparations,onecaneithergettherightsymme- principleisthesame. Weallowthemean-fieldcalculation
try(RHF)ortherightenergy[unrestrictedHartree-Fock to break spin-symmetry if this leads to lower energy on
(UHF)], but not both. The same issues arise in approx- the central site, with spin equations:
imate DFT treatments of this problem [45]. Of course,
(cid:18) (cid:19)
the exact functional manages to get the correct energy (cid:104)n (cid:105)= 1+1 arctan µ−ε−U(cid:104)n↓(cid:105)−εXC((cid:104)n↑(cid:105),(cid:104)n↓(cid:105)) ,
with the correct symmetry, and there have been many ↑ 2 π Γ
attempts to reproduce this with various more sophisti- (11)
cated approximations. But a more pragmatic approach and reverse for (cid:104)n↓(cid:105), and (cid:104)nC(cid:105)=(cid:104)n↑(cid:105)+(cid:104)n↓(cid:105). Again, the
istoaccepttheresultsastheyare, interpretingthegood simplest calculation is UHF, where ε = 0. The solu-
XC
energeticsastheresultofapplyingtheapproximatefunc- tions are identical to those found in the original problem
tional to a frozen fluctuation of the system. The true by Anderson [25]. For y >1/π, i.e., U <πΓ, there is no
ground-state wavefunction fluctuates between configura- spontaneous symmetry-breaking, and UHF=RHF. But
tions with one spin and then the other (left and right beyondthatcriticalvalue,thespin-densitydifferencebe-
localized for stretched H ), and the true ground-state comesfinite,andtheunrestrictedsolutiondiffers. Define
2
density has unbroken symmetry. But the approximate thedensitydifferenceas(cid:104)m(cid:105)=(cid:104)n↑(cid:105)−(cid:104)n↓(cid:105)inUHF,which
functionals give most accurate energies when applied to satisfies:
thefrozenfluctuations. Thus,onecaninterpretboththe
(cid:16)π (cid:17) (cid:104)m(cid:105)
total density and energy as being accurate from such a tan (cid:104)m(cid:105) = . UHF (12)
2 2y
For y >1/π, (cid:104)m(cid:105)=0, but otherwise a solution with (cid:104)m(cid:105)
finite exists. In all cases, we take only the total density
1 exact fromtheUHFcalculation,andweknowthetrue(cid:104)m(cid:105)=0
µE= 00..68 UdHisFc U=100Γ always. In particular, as y → 0 (strong correlation),
T(E) 0.4 UHF(SB) χ˜c →0 with the correct linear term:
0.2
0 χ˜ →(8/π)y+(96γ/π2)y2+··· y →0, (13)
c
1
0.8 1 where γ =1 in the exact solution, but γ =1/3 in UHF.
<n>C 00..46 <m> 0000....2468 So UHF recovers the leading term. The green curve in
0.2 0 0 0.2 0.5 Fig. 2showstheUHFvalueofχ˜c,demonstratingbothits
0 accuracy for both strong and weakly correlated systems,
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
x and the discontinuous change at 1/π.
1
RHF Even beyond the “Coulson-Fisher point” of 1/π, the
0.8
U 0.6 symmetry-breaking only occurs for 0 ≤ (cid:104)m(cid:105) ≤ 1, i.e.,
ε/S 0.4 outside this region, the UHF solution is that of RHF, as
0.2
canbeseenintheinsetofmiddlepanelinFig. 5. Butthe
0
densityisveryaccuratelygivenbyUHF(consideringthe
0.8 0.85 0.9 0.95 1 1.05 1.1
<nC> scale of horizontal axis), and the KS potential develops
the correct derivative discontinuity as y →0.
FIG. 5. Upper panel: transmission as a function of x = Todemonstratethisaccuracy, weplotthecorrespond-
(µ−(cid:15))/U;middlepanel: occupationasafunctionofx;lower ing transmissions in Fig. 4, using Eq. (3). The fig-
panel: KS potential as a function of occupation. Results are ure shows how the transmission using (cid:104)n (cid:105) from UHF is
shownforBetheansatzorexactKSDFT(exact),unrestricted C
almost exact (considering the scale of horizontal axis).
Hartree-Fock (UHF),anddiscontinuousapproximation[disc,
To demonstrate the error in ignoring the fact that the
Eq. (10)]. Also shown in the upper panel is UHF results for
UHF produces incorrect spin densities, we also plot the
transmission using (incorrect) spin densities [UHF(SB), with
symmetry breaking], and (cid:104)m(cid:105) = (cid:104)n (cid:105)−(cid:104)n (cid:105) for UHF as a transmissionthroughsuchasolution,whichiscompletely
↑ ↓
functionofxasaninsetinthemiddlepanel. U =100Γinall wrong (see dark red curve in upper panel of Fig. 5, only
cases, and only the region near (cid:104)nC(cid:105)=1 and x=0 is shown. one peak is present because only region near x = 0 and
(cid:104)n (cid:105)=1isshownthere). Ourresultsareconsistentwith
C
6
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