The adiabatic strictly-correlated-electrons functional: kernel and exact properties

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by  Giovanna Lani

Theadiabaticstrictly-correlated-electronsfunctional: kernelandexactproperties
GiovannaLani,1 SimonediMarino,2 AugustoGerolin,3 RobertvanLeeuwen,4 andPaolaGori-Giorgi1
1Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling,
FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands
2LaboratoiredeMathe´matiquesd’Orsay,Univ. Paris-Sud,CNRS,Universite´ Paris-Saclay,91405Orsay,France.
3Dipartimento di Matematica, Univerisita´ di Pisa, Largo B. Pontecorvo, 56126 Pisa, Italy
4Department of Physics, Nanoscience Center, University of Jyva¨skyla¨,
40014 Jyva¨skyla¨, Finland; European Theoretical Spectroscopy Facility (ETSF)
(Dated:January18,2016)
Weinvestigateanumberofformalpropertiesoftheadiabaticstrictly-correlatedelectrons(SCE)functional,
relevantfortime-dependentpotentialsandforkernelsinlinearresponsetime-dependentdensityfunctionalthe-
6 ory. Amongtheformer, wefocusonthecompliancetoconstraintsofexactmany-bodytheories, suchasthe
1 generalisedtranslationalinvarianceandthezero-forcetheorem.Withinthelatter,wederiveananalyticalexpres-
0 sionfortheadiabaticSCEHartreeexchange-correlationkernelinonedimensionalsystems,andwecomputeit
2 numericallyforavarietyofmodeldensities. Weanalysethenon-localfeaturesofthiskernel,particularlythe

n onesthatarerelevantintacklingproblemswherekernelsderivedfromlocalorsemi-localfunctionalsareknown
a tofail.
J

5
1 I. INTRODUCTION In Eq. (1), vH([n],r,t) is the usual Hartree potential
computed with the time-dependent density n(r,t), and

] vxc([Ψ0,Φ0,n];r,t) is the exchange-correlation (xc) poten-
l
e While a considerable amount of work on the strictly- tial, depending also on the initial states Ψ0 and Φ0. Trad-
- correlated-electrons (SCE) formalism1–3 within the frame- ing the many-body TDSE for the one-particle TDKS equa-
r
t workofgroundstateKohn-Sham(KS)densityfunctionalthe- tionshasapricetopay,thatisthetime-dependentexchange-
s ory (DFT) has been carried out,4–8 the study of its perfor- correlation potential v ([n,Ψ ,Φ ];r,t) of TDDFT is an
. xc 0 0
at mancesinthetimedomainisjuststarting.9,10 Theaimofthis evenmorecomplexobjectthanthevxc forgroundstateDFT,
m work is to begin a systematic investigation of the SCE func- asitisafunctionalofthedensityatalltimest(cid:48) ≤ tand,ad-
tional in the context of time dependent problems, in order to ditionally, oftheinitialstateofboththeinteractingandnon-
-
d understand its fundamental aspects and its potential in tack- interacting systems. However, whenever the initial state for
n ling challenging problems for the standard approximations the evolution problem described by the TDKS equations is
o
employedintime-dependent(TD)DFT. chosen to be the ground state of the system, then the func-
c
tionaldependenceofthexcpotentialisontheelectronicden-
[
Wewillhencefocusonthosephysicalsituationsdescribed sityalone: sincethisscenariooccursnaturallyinmanyprob-

1 by an explicitly time-dependent Hamiltonian, and whose lems of interest, it doesn’t pose actual limitations and thus it
v dynamics is described by the time-dependent Schro¨dinger
isoftenadoptedinpracticalapplications.
7 equation (TDSE). Due to the existence of a time-dependent
7 density-potential mapping11–14 for interacting and non- Similarly to ground state DFT, in order to make use of
9 interacting systems, a time-dependent Kohn-Sham approach Eq. (1) one needs approximations for the exchange correla-
3
0 can be rigorously set up and employed to study the dynam- tion potential vxc. A first drastic approximation, which is
. ics of quantum systems at a manageable computational cost. usedinthelargemajorityofcasesinTDDFT,istheso-called
1 Choosingtheinitialnon-interactingwavefunctiontobeasin- adiabatic approximation, obtained by inserting in a ground-
60 gle Slater determinant of some spin orbitals ψj(x,t0), one stateapproximatevxc([n];r,t)theinstantaneousdensity, ig-
1 canreducetheTDSEtoasetofsingle-orbitalequations, the noring dependence on the density at earlier times. This ap-
: time-dependent Kohn-Sham (TDKS) equations, of the form proximation has a very specific range of validity – infinitely
v (inHartreeatomicunitsusedthroughout): slowly varying perturbations, such that the system is always
i
X in its ground state – but it is very often employed outside
ar i∂tψj(x,t)=(cid:16)− 12∇2+vext(r,t)+vH([n],r,t) (1) idte,pwenitdhinrgesuolntsththeatnactaunrevaorfythfreomprovbelreymsaatdisdfraecstsoerdy.toInpoceorr-,
(cid:17)
tain cases, it is still difficult to disentangle the errors due
+v ([Ψ ,Φ ,n];r,t) ψ (x,t),
xc 0 0 j
to the adiabatic approximation and the errors due to the ap-
with Ψ and Φ initial states of the true interacting and of proximation for the ground-state exchange correlation po-
0 0
tential, but considerable progress has been made in recent
the non-interacting KS system. The time-dependent density
years, by analysing, whenpossible, the “adiabaticallyexact”
is thus computed in the familiar way (for simplicity in this
potential.15–17Inothercases,itisinsteadwellestablishedthat
introductionweconsiderclosed-shellsystems)as:
neglecting all “memory effects” in v (or equivalently fre-
xc
N/2 quency dependence in the so called xc kernel, Fxc of linear
(cid:88)
n(r,t)=2 |ψ (r,t)|2. (2) response TDDFT), does not allow TDDFT to describe ex-
j
citations with a predominantly double character.18–22 In the
j=1
2
TDDFTframework,adiabaticityisthusequivalenttolocality tations. Finallywewillgiveourconclusionsandperspectives
intime. Themostcommonapproximationsusedtobuildthe forfuturework.
adiabaticv (andF inthelinearresponsecase)inTDDFT,
xc xc
arelocalandsemi-localfunctionals,whichthusaddtolocal-
ityintimelocalityinspaceaswell. Inthesecases,thetime-
dependentkernelFxc(r,r(cid:48),t,t(cid:48))isapproximatedas II. REVIEWOFTHESCEFORMALISM
δ2Eappr[n]
F ([n];rt,r(cid:48)t(cid:48))= xc , (3) The SCE formalism can be put in the DFT context start-
xc δn(r,t)δn(r(cid:48),t(cid:48))
ingwiththegeneralizationoftheHohenberg-Kohnfunctional
F[n]toscaledinteractions:
whereEappr[n]isevaluatedattheinstantaneousdensityand
xc
itisoftenalocalorsemi-localapproximatefunctional,which
F [n]= min(cid:104)Ψ|Tˆ+λVˆ |Ψ(cid:105) (4)
makesthekerneldifferentfromzeroonlyon(orverycloseto) λ ee
Ψ→n
thediagonalr = r(cid:48). Wehavealreadyhintedattheshortcom-
ings of the locality in time in this introduction, but also the where Tˆ and Vˆ are the familiar kinetic and two-body in-
ee
localityinspacehasseriouslimitations,anotoriousexample teraction operators, while λ is a parameter varying continu-
beingthedescriptionofexcitationswithalong-rangecharge ously from 0 to ∞, yielding different scenarios: F [n] =
λ=0
transfer (CT) character23,24. In the case of closed-shell frag- T [n] corresponds to the non interacting or Kohn-Sham sys-
s
ments, the introduction of a considerable portion of Hartree- tem, F [n] corresponds to the real physical system, while
λ=1
Fock exchange (often introduced at long-range only through F [n]definesthestrong-couplinglimit,1,2capturedbythe
λ=∞
range-separation) is able to fix the CT problem in linear re- strictly-correlated-electronfunctional
sponseTDDFT.25–29However,thissolutiondoesnotworkfor
theverychallengingcaseofhomolyticbondbreakingexcita- VSCE[n]≡ min(cid:104)Ψ|Vˆ |Ψ(cid:105). (5)
ee ee
tions, the prototypical example being the lowest excited sin- Ψ→n
gletstate1Σ+u oftheH2molecule.30,31Inthiscase,thekernel TheworkinghypothesistobuildtheminimizerofEq.(5)fora
shoulddivergeinordertocompensatethefactthatthisexcita- givendensityisthatinthislimitthemany-bodywavefunction
tionintheKSsystemgoestozeroasthebondisbroken.30,31 collapsesintoa3-dimensionalsubspaceofthefullconfigura-
In this context, we will show that, at least in a model one- tionspace,
dimensional case, the adiabatic SCE (ASCE) kernel shows a
verypromisingnon-localdivergingbehavior. 1 (cid:88)(cid:90) n(r)
In order to construct approximations both for potentials and |ΨSCE(r1,...,rN)|2 = N! dr N δ(r1−f℘(1)(r))
kernelsinTDDFT,onecanbeguidedbytryingtosatisfyexact ℘
propertiesandconstraintsofmany-bodytheories. Inaseries ×δ(r −f (r))···δ(r −f (r)), (6)
2 ℘(2) N ℘(N)
of works32–34 Dobson and Vignale devised a number of con-
straints(namedtheoremsafterwards)thatthetime-dependent where℘denotesapermutationof1,...,N,suchthatn(r)=
v shouldcomplyto, inordertoavoidunphysicalresultsor N(cid:82) |Ψ (r,r ,...,r )|2dr ···dr . The functional is
xc SCE 2 N 2 N
contradictionsinthetheory. Fromtheiranalysis, itappeared then specified in terms of the so-called co-motion functions
for the first time that the interplay between non locality in f ([n];r)thatdeterminethesetinwhich|Ψ |2 (cid:54)=0,
i SCE
space and non locality in time is a delicate issue in TDDFT
and this fact needs to be kept in mind when looking for ap- 1(cid:90) (cid:88)N 1
VSCE[n]= n(r) dr. (7)
proximations, making this task much more challenging than ee 2 |r−f ([n];r)|
i
ingroundstateDFT.Itisthusnaturaltoaskwhetherahighly i=2
non-localfunctionalsuchasSCEcansatisfytheseexactcon-
ThefunctionalderivativeofVSCE[n]definestheSCEpoten-
ditionswhenemployedintheadiabaticapproximation. ee
tial:
After briefly reviewing in Sec. II the basics ideas of the
δVSCE[n]
SCE formalism, we will show in Sec. III how the SCE po- vSCE([n];r)= ee , (8)
δn(r)
tential satisfies exact properties of many-body theories, such
asthezero-forcetheoremandthegeneralizedtranslationalin-
which can be computed via a rigorous and physically trans-
variance. Ouranalysiswillalsoshowhow,whilenon-locality
parentshortcut2astherepulsionfeltbyanelectroninrdueto
in time and non-locality in space have to go hand in hand,
theotherN −1electronsatpositionsr =f ([n];r),
non-localityinspaceandlocalityintimecancoexistwithout i i
violatingtheabovementionedproperties. InSec.IVwewill
N
derive an analytical expression for the SCE kernel for one- ∇vSCE([n];r)=−(cid:88) r−fi([n];r) . (9)
dimensionalsystems,andthencomputeitnumericallyforvar- |r−fi([n];r)|3
i=2
iousdensityprofiles. Wewillcompletethesectionwithadis-
cussiononsomegeneralfeaturesofthekernel,pinpointingat All the f [n](r), whose physical meaning is to give the posi-
i
thosewhicharisefromitshighlynon-localnatureandwhich tions of all the others N − 1 electrons once the position of
couldbepromisingforthedescriptionofbond-breakingexci- areferenceelectronhasbeenfixedinr,satisfythefollowing
3
non-lineardifferentialequation: dependent density, the time-dependent xc potential is rigidly
translatedbythesamequantity. Wewillrefertothisproperty
n(fi([n];r))dfi([n];r)=n(r)dr i=2,..,N −1 (10) as generalized translational invariance (GTI), since it holds
alsoforcoordinatesframeswhichareacceleratedwithrespect
which shows their non-local dependence on n(r). Further- totheoriginalone.43
more the co-motion function obey (cyclic) group properties2 ThusingeneraltheGTIcanbeformalizedasfollows: given
whichensurethattheelectronsareindistinguishable. an arbitrary (be or not time-dependent) shift of the density
Intherecentyears,ithasbeenrealizedthattheproblemde- R(t):
finedbytheminimization(5)isequivalenttoanoptimaltrans-
portproblemwithCoulombcost.35,36 Sincethen,theoptimal n(cid:48)(r,t)=n(r−R(t),t) (11)
transport community has been able to prove several rigorous
results. Inparticular,theSCEstate(6)hasbeenproventobe thexc potentialassociated withthis densityhas totransform
the true minimizer for any number of particles N in one di- accordinglyto:
mensional(1D)systems37andinanydimensionforN =2.35
Formoregeneralcases,ithasbeenshownthattheminimizer vxc([n(cid:48)];r,t)=vxc([n];r−R(t),t) (12)
mightnotbealwaysoftheSCEform38. Eveninthosecases,
however,SCE-likesolutionsseemtobeabletogoveryclose
tothetrueminimum,39 andinseveralcasesitisstillpossible
A. PropertiesoftheadiabaticSCEpotential
toproveEq.(9).39
Inthelow-densitylimit(orstrong-couplinglimit)theexact
WewillnowshowexplicitlyshowthattheASCEcomplies
Hartree andexchange-correlation (Hxc)energy functionalof
KS DFT tends asymptotically to VSCE[n],36,40 Thus, in the tothisrequirement.
ee
followingwedenotevSCE([n];r)ofEq.(9)asvSCE([n];r),to We begin by observing that in the SCE limit, upon the shift
Hxc ofthedensity,alltherelativedistancesbetweentheelectrons
stressthatthispotentialisthestrong-couplingapproximation
tothestandardHxcpotentialofKSDFT.4,40 havestilltobethesame, thustheco-motionfunctionstrans-
formas:
Now that the basics of the SCE formalism at the ground
state level have been reviewed, we can move to the time-
f ([n(cid:48)];r)=f ([n];r−R(t))+R(t). (13)
dependentdomain. i i
In the one dimensional case, where the co-motion functions
can be expressed in terms of a simple one-dimensional inte-
III. EXACTPROPERTIESFROMMANY-BODY gral,onecanshowexplicitlythattheaboverelationholds,see
THEORIES App. A for details. Substituting the transformed co-motion
functionsintotheexpressionfortheSCEpotentialgives:
Since the success of TDDFT relies heavily on the avail-
N
abilityandthequalityoftheapproximationsforvxc([n];r,t) ∇vSCE([n(cid:48)];r,t)=−(cid:88) r−fi([n];r−R(t))−R(t)
and for the linear response exchange-correlation kernel Hxc |r−fi([n];r−R(t))−R(t)|3
i=2
F ([n],r,r(cid:48),t,t(cid:48)), there have been intense research efforts
xc =∇vSCE([n];r−R(t),t). (14)
towards better approximations. As already mentioned in the Hxc
introduction,awaytoguidesuchapproximationsistoresort IntegrationandsubtractionoftheHartreepotential(whichsat-
to the compliance to exact constraints from many-body the- isfytheGTIstraightforwardly)yields:
ories, similarly to what has been done extensively already in
ground state DFT. A first exact condition is given by scal- vSCE([n(cid:48)];r,t)=vSCE([n];r−R(t),t) (15)
xc xc
ing relations,41,42 a second one by a sum rule for the time-
dependent exchange-correlation energy41 and just like in the whichistherelationwewantedtoprove.
static case, the time-dependent xc potential should be self- A second important constraint is that the xc potential can-
interactionfree. notexertanetexternalforceonthesystem,whichisnothing
In addition to the constraints enumerated above, a very im- elsethanthecompliancetoNewton’sthirdlawofmotion. In
portant condition on approximate xc potentials is that they DFTthispropertygoesunderthenameofzeroforcetheorem
shouldbeGalileaninvariant,asaconsequenceofthefactthat (ZFT)andin33 itwasshownhowitisautomaticallysatisfied
the TDSE itself exhibits this symmetry. This condition was fortranslationallyinvariantxcpotentials. Onemaythinkthat
firstinvestigatedbyVignale,33asageneralizationofanearlier this is a trivial requirement to be satisfied, but in practice it
work by Dobson32 on the so called harmonic potential theo- isn’t. Forexamplein44 itwasdemonstratednumericallythat
rem (HPT), which states that upon the application of a time- computingthedipolemomentofsmallNa andNa+clusters,
5 9
dependent field to a many-body system confined by an har- via the exact exchange Krieger-Li-Iafrate approximation to
monicpotential,itstime-dependentdensityisrigidlyshifted. v ,yieldedanincreasedamplitudeinthedipoleoscillations,
xc
In33 it was demonstrated that the HPT is automatically sat- mostlikelyduetospuriousinternalforcesappearingasacon-
isfiedwheneverthetime-dependentxcpotentialobeysapre- sequence of the violation of the ZFT. The ZFT not only has
ciseconstraint,thatisuponarigidshiftofthesystem’stime- implicationsfortheapproximationstothetime-dependentxc
4
potential,butalsoonanotherkeyquantityofTDDFT,namely B. PropertiesoftheadiabaticSCEkernel
the exchange correlation kernel. In34 Vignale showed how a
frequency dependent (thus non local in time) F cannot be
xc Let’snowturntotheASCEkernel,
local in space, in order to satisfy the ZFT. A notable exam-
ple of a kernel which violates the ZFT and the HPT too, is δ2VSCE[n]
theGross-KohnFxc45 whichindeedisfrequencydependent, FHAxScCE([n];rt,r(cid:48)t(cid:48))= δn(r,te)en(r(cid:48),t(cid:48))δ(t−t(cid:48)), (21)
butlocalinspace,asitisbasedonthehomogeneouselectron
gas. This peculiar issue in TDDFT is commonly known as OnceagainweresorttoanexpansionforthedensityinR(t),
ultra non-locality problem and makes particularly challeng- thatisn(r−R(t),t)≈n(r,t)−R(t)·∇n(r,t),combining
ingtheconstructionofapproximatefrequencydependentker- thiswithEq.(15)andinvokingthearbitrarinessofR(t)and
nels. AdiabaticFxc, derivedfromfullylocalfunctionals, do thedefinitionofASCExckernel,weobtain:
not violate the ZFT. It is legitimate to ask if an adiabatic but
(cid:90)
highly non local functional like the ASCE, does violate the
dr(cid:48)FASCE([n];rt,r(cid:48)t(cid:48))∇n(r(cid:48))=δ(t−t(cid:48))∇vSCE([n];r,t),
ZFT.Strictlyspeakingwealreadyknowthatitdoesn’t,sinceit Hxc Hxc
respectstheGTI,butinthefollowingwewillexplicitlyshow
whichshowsthattheASCEkernelindeedsatisfiestheZFTin
thatwhilenonlocalityintimerequiresnonlocalityinspace,
theconverseisnottrue. thelinearresponseregime.
Let’s consider once again a shift in the density: n(cid:48)(r) =
n(r−R). Observing that the generalized HK energy func-
tionalFλ[n]istranslationallyinvariant(sincebothTˆandVˆee C. Propertiesoftheco-motionfunctions
are)onehas:
Atthispointitseemsnaturaltoalsoinvestigatesomeprop-
F [n]=F [n(cid:48)]. (16)
λ λ erties of the co-motion functions. Combining again the ex-
pansionforthedensityofEq.(17)withEq.(13)oneobtains:
ExpansionofthedensityinpowersofRgives:
n(cid:48)(r)=n(r−R)=n(r)−R·∇n(r)+O(R2), (17) (cid:90) dr(cid:48)δfi,α([n];r) ∂ n(r(cid:48))= ∂ f ([n];r)−δ (22)
δn(r(cid:48)) ∂r(cid:48) ∂r i,α αβ
β β
andexpandingbothsidesofEq.(16)yields:
where α,β run over Cartesian indices x,y,z. Eq. (22) is a
(cid:90) δF
0= dr λ (−R·∇n(r)) (18) sumrulethatcanbewrittenalsoforadiabatictime-dependent
δn(r) co-motion functions and the static density and may be em-
ployed as constraint to devise approximate co-motion func-
whichisvalidforanyarbitraryshiftR.
tions. Furthermoreitcanbeusedtoverify(particularlyinthe
ThecaseF = VSCE correspondstotheSCEfunctional,
λ=∞ ee easier one-dimensional case) the functional variation of the
hence:
co-motionfunctionswithrespecttothedensity.
(cid:90)
0= drvSCE([n];r)∇n(r)
(cid:90)
=− dr∇vSCE([n];r)n(r) (19) IV. SCEHARTREE-EXCHANGECORRELATION
KERNELFORONE-DIMENSIONALSYSTEMS
which shows that the SCE potential does indeed satisfy the
ZFTforstaticdensities.Additionally,sincethedifferentiation In the one-dimensional case with convex repulsive inter-
above is completely general and holds for any density, even particleinteractionw(|x|), theSCEsolution1 isknowntobe
time-dependentones,onehas: exactforanynumberofelectronsN,37 andcanbeexpressed
inarathersimpleformintermsofthefunctionN ([n];x),
(cid:90) e
0= dr∇vSCE([n];r,t)n(r,t), (20) (cid:90) x
N ([n];x)= n(y)dy, (23)
e
−∞
whichshowsthattheASCExcpotentialsatisfiestheZFTfor
time-dependentdensitiesaswell. andofitsinverseN−1([n];x):
e
f ([n];x)=f+([n];x)θ(x−a [n])+f−([n];x)θ(a [n]−x)
i i k i k
whereθ(x)istheusualHeavisidestepfunction,and
f+([n];x)=N−1([n];N ([n];x)+i−1) (24)
i e e
f−([n];x)=N−1([n];N ([n];x)+i−1−N),
i e e
witha [n]=N−1([n];k),andk =N +1−i.
k e
5
InthiscasetheSCEpotentialissimplygivenby thekernelisgivenby
N (cid:90) ∞ FSCE([n];x,x(cid:48))=(G(x(cid:48))−G(x)+G(0))θ(x(cid:48)−f(x))
(cid:88) Hxc
vHSCxcE([n];x)=− w(cid:48)(|y−fi([n];y)|)× for x>0, x(cid:48) <0. (32)
i=2 x
sgn(y−fi([n];y))dy. (25) TheresultingSCEFSCE([n];x,x(cid:48))forthiscaseisplottedin
Hxc
the first panel of Fig.1: as it is evident from Eq. (29), the
TheSCEkernelisthenequaltothevariationoftheSCEpo- kernel has in the first and third quadrants (x,x(cid:48) > 0 and
tentialwithrespecttotheelectrondensity, x,x(cid:48) < 0) the same value as along the diagonal (x = x(cid:48)),
while in the second and fourth quadrants (Eq. (32)) the ker-
δvSCE([n];x)
FSCE([n];x,x(cid:48))= Hxc , (26) nelisdifferentfromzeroonlyintheregiondelimitedbythe
Hxc δn(x(cid:48)) x,x(cid:48) axesandtheco-motionfunctionx(cid:48) =f(x). Thebehav-
ior of an adiabatic kernel local in space, such as the ALDA,
whichcanbecarriedout(thedetailsofthederivationaregiven
isinsteadradicallydifferent: theFALDA(n];x,x(cid:48))hasanon-
in Appendix B-C), yielding (for densities supported on the xc
zero component only along the diagonal, δ(x−x(cid:48)), and the
wholerealline)thecompactexpression
Hartreecomponent,equaltow(|x−x(cid:48)|),hasamaximumon
FSCE([n];x,x(cid:48))=(cid:88)N (cid:90) ∞ w(cid:48)(cid:48)(|y−fi([n];y)|)] (27) thediagonal,decayingas1/|x−x(cid:48)|outsideit.
Hxc n(f ([n];y))
i=2 x i
×[θ(y−x(cid:48))−θ(f ([n];y)−x(cid:48))]dy. B. ModelHomonuclearmolecule
i
Fromthisexpression,itisnotevidentthatEq.(27)satisfiesthe
Thesecondtypeofdensityconsideredisamodel2-electron
symmetry requirement FSCE([n];x,x(cid:48)) = FSCE([n];x(cid:48),x).
Hxc Hxc densitywhichresemblestheoneofahomonuclearmolecule,
AnexplicitproofofthissymmetryisgiveninAppendixD.
a(cid:16) (cid:17)
n(x)= e−a|x−R2|+e−a|x+R2| , (33)
2
A. AnalyticalExample where a = 1 and where R, the distance between the two
nuclei, canbeincreasedarbitrarilytosimulatethemolecular
bond stretching. For this case we numerically computed the
WebeginbyconsideringN =2electronsintheLorentzian
SCEkernelfordifferentvaluesofR,toobtaininsightsonhow
densityprofile,
ahighlynon-localkernelbehavesforaproblemwhichbears
2 1 aresemblancetotheH2 dissociation. TheresultsforR = 3,
n(x)= , (28) R = 8 and R = 12 are presented respectively in panels (b),
π1+x2
(c)and(d)ofFig.1.
forwhichtheco-motionfunctionissimplyf (x) ≡ f(x) = Aside from the peak in the origin, a very interesting fea-
2
−1.FromthegeneralexpressionofEq.(27),weobtaininthe ture displayed by the SCE kernel is the appearance, as R is
x
firstquadrant increased,oftwoplateaux,eachoccupyingalargesquarere-
gion (of size ≈ R×R) of the first and the third quadrants.
FSCE([n];x,x(cid:48))=G(−max{x,x(cid:48)}) for x>0, x(cid:48) >0, As in the case of the lorentzian density, the SCE kernel has
Hxc
(29) alsonon-zerocomponentsinthesecondandfourthquadrants,
wherewehavedefinedthefunctionG(x) but they are now much smaller than the ones in the I and III
quadrants.
(cid:90) x w(cid:48)(cid:48)(|y−f([n];y)|)] The height of the plateaux increases as R increases. A
G(x)= dy, (30)
n(f([n];y)) closer analysis of the function G(x) defined by Eq. (30) for
−∞
the case of the density (33) (see Appendix E) shows that the
which in this case, and with e-e interaction w(|x|) = 1 heightandsizeoftheplateauxareapproximatelygivenby
|x|
(sinceintheSCEwavefunctiontheparticlesnevergetontop
of each other, the 1/x divergence at x = 0 in 1D does not 1
FSCE([n];x,x(cid:48))≈
poseanyproblem),isequalto Hxc n(0)(R−1/a)2
1 1 1 1
(cid:40) π 1 x≤0 for (cid:46)|x|(cid:46)R− , and (cid:46)|x(cid:48)|(cid:46)R− ,
G(x)= 21+x2 (31) a a a a
π211++2xx22 x>0. withx,x(cid:48) ≥0 or x,x(cid:48) ≤0 (34)
Sinceourdensitysatisfiesn(−x)=n(x),inthiscasetheker- As an example, we show in Fig. 2 the SCE kernel along the
nel in the third quadrant (x < 0 and x(cid:48) < 0) is equal to the diagonalforR=8,12and20,multipliedbyn(0)(R−1/a)2,
oneinthefirstquadrant.Inthesecondquadrant–andbysym- where n(0) = e−aR/2. The value of the SCE kernel on the
metrythefourth,sinceFSCE([n];x,x(cid:48))=FSCE([n];x(cid:48),x)– diagonalalsodefinesthevalueofthekernelinthewholefirst
Hxc Hxc
6
andthirdquadrants,seeEq.(29). Thus,weseethattheSCE
e-R2(R-1)2FSCE(x,x)
Hxc
1.5
R=20
1.0
(a)SCEkernelforaLorentziandensity.
0.5 R=12
R=8
-20 -10 10 20 x
Figure2. TheSCEkernelforthe“homonucleardimer”densityof
Eq.(33)witha = 1anddifferentinternuclearseparationsRalong
thediagonalx = x(cid:48). Thekernelhasbeenmultipliedbyn(0)(R−
(b)SCEkernelfora“homonucleardimer”
1/a)2,toshowitsscalingwithR. Thevaluealongthediagonalis
densitywithR=3
exactlythesameasthevalueofthekernelinthewholefirstandthird
quadrant,seeEq.(29)andFig.1.
kerneldevelopsplateauxregionswhoseheightdivergesexpo-
nentiallyasRisincreased.
Thisdivergenceisverypromisingtocapturebond-breaking
excitations, because it makes diverge matrix elements of the
kernel between atomic orbitals centered on the same site.
Considerthebasicexampleofthelowestexcitedsingletstate
1Σ+oftheH molecule,30,31wherewehaveamatrixelement
u 2
(c)SCEkernelfora“homonucleardimer” ofthekind
densitywithR=8
(cid:90) (cid:90)
dx dx(cid:48)σ (x)σ (x)FSCE([n];x,x(cid:48))σ (x(cid:48))σ (x(cid:48)),
g u Hxc g u
(35)
whereσ = c (φ ±φ ),withφ theatomicorbitals
g,u u,g A B A,B
centeredinthetwoatoms,andc anormalizationconstant.
u,g
IntheTDDFTlinearresponseequationsthismatrixelementis
multipliedbythecorrespondingKSorbitalenergydifference
(cid:15) −(cid:15) , which goes to zero as R → ∞, so that the kernel
u g
matrix element must diverge in order to keep the excitation
finite (as it is in the exact system).30,31 We see that for the
termscenteredonthesameatomAappearingin(35)wehave,
(d)SCEkernelfora“homonucleardimer”
densitywithR=12 forlargeR,
Figure 1. The SCE Hartree-exchange-correlation kernel for (cid:90) (cid:90)
N = 2 electrons for the Lorentzian density profile of dx dx(cid:48)|φA(x)|2FHSCxcE([n];x,x(cid:48))|φA(x(cid:48))|2
Eq. (28) (a) and for “homonuclear dimer” densities n(x) =
(cid:16) (cid:17) 1
1 e−|x+R/2|+e−|x−R/2| , with R = 3 (b), R = 8 (c) and ≈ . (36)
2 n(0)(R−1/a)2
R = 12 (d). The FSCE([n];x,x(cid:48)) displays, in all the four cases,
Hxc
apeakstructureincorrespondenceoftheverticalasymptotesofthe Equation(36)holdsbecausetheproductoftheatomicorbitals
co-motionfunctions. Twosymmetricquasi-plateaux,inthefirstand withthesamecenter, |φ (x)|2|φ (x(cid:48))|2, issignificantlydif-
A A
thirdquadrants,appearforthestretched“homonucleardimer”den-
ferent from zero only in one of the two plateau regions of
sities: theybecomegreaterinheightandflatterasthevalueofRis
Eq.(34),wherewecanapproximatetheSCEkernelwiththe
increased.
constantvalue 1 ,whichdivergesexponentiallyas
n(0)(R−1/a)2
R→∞. Althoughamorecarefulanalysisisneededtoverify
if this divergence is really able to compensate the vanishing
7
oftheKSexcitation(cid:15)u−(cid:15)g comingfromaself-consistentKS AppendixA:Explicitcalculationoftheshiftedco-motion
SCE calculation, we see that the FSCE([n];x,x(cid:48)) embodies functionsin1D
Hxc
the right physics: its very non-local dependence on the den-
sitymakesitdivergeintheatomicregion,onlywhenanother Let us consider the negative semi-axis (the positive one
distantatomispresent. gives an analogous result). The function N ([n];x) for a
e
Finally, the height of the peak in the origin can be easily shifteddensityn(cid:48)reads:
obtained from the properties of the function G(x) (see Ap-
pendixE),anditcanbeshowntobealwaysequalto
x x−R(t)
(cid:90) (cid:90)
(cid:18) R R(cid:19) 2 Ne([n(cid:48)];x)= n(cid:48)(y)dy = n(y−R(t))dy
FSCE([n];0,0)=2FSCE [n]; , ≈ .
Hxc Hxc 2 2 n(0)(R−1/a)2 −∞ −∞
(37) =N ([n];x−R(t)), (A1)
e
andtakingitsinverse,
V. CONCLUSIONSANDPERSPECTIVES
N−1[N ([n(cid:48)];x)]=x
e e
N−1[N ([n];x−R(t))]+R(t)=x. (A2)
InthisworkwehaveexploredtheSCElimitinthecontext e e
of time-dependent problems, focusing on the formal proper-
Combiningtheaboverelationswehave:
tiesoftheadiabaticSCE(ASCE)functional. Wefirstexam-
ined some properties of the ASCE time-dependent potential, f ([n(cid:48)];x)=N−1[N ([n(cid:48)];x)+i−1] (A3)
i e e
inparticularthecompliancetoconstraintsofexactmany-body =N−1[N ([n];x−R(t))+i−1]+R(t)
theories, such as the generalized translational invariance and e e
=f ([n];x−R(t))+R(t)
the zero-force theorem, and showed that the ASCE satisfies i
both. Whileitiswellknownthatnon-localityintimerequires
whichisthe1DversionofEq.(13).
nonlocalityinspace,wehaveshownthattheconverseisnot
trueusingtheexampleoftheASCE.
Inthesecondhalfofthepaperwederivedananalyticalex-
pressionfortheSCEHartreeexchange-correlationkernelfor AppendixB:Functionalvariationofthe1Dco-motionfunctions
withrespecttothedensity
one-dimensional problems, and we have computed it numer-
ically for various density profiles. In particular, we have an-
alyzedthecaseofamodelhomonuclear2-electronmolecule LetnbeadensityofameasureinRsuchthat(cid:82) n(x)dx=
R
asthebondisstretched,findingthattheSCEkerneldisplaysa N. Then, the co-motion functions (or the optimal transport
verypromisingdivergingbehaviorthatcouldtackletheprob- maps) are given explicitly by Eqs. (24)-(25), which corre-
lemofhomolyticbond-breakingexcitations. spondtothecondition(cid:82)fi(x)n(y)dy ∈{i,−N+i},depend-
x
In future works we will implement the whole linear re- ingonwhetherx≤a ornot.
i
sponse TDDFT equations for one-dimensional problems us- Weconsidernε(x)=n(x)+ε(n˜(x)−n(x))andwewant
ingtheSCEkernel,analysingifitsdivergingbehaviorisable to determine the corresponding co-motion functions fε. For
i
toopenthegapinamodelMottinsulator,madeofachainofH every x ∈ R we define x as the point such that f (x ) =
ε i ε
atoms. Workonbond-breakingexcitationsinrealtimeprop- fε(x),andthenwenoticethat
i
agationwiththeASCEpotentialhasalsoshownverypromis-
ingresultsinthissense,andiscurrentlyinpreparation.10Last (cid:90) fi(xε) (cid:90) fiε(x)
but not least, we will use our insights to design approximate n(y)dy− nε(y)dy ∈{−N,0,N},
kernelsbasedontheSCEform. xε x
(cid:90) fε(x) (cid:90) fε(x) (cid:90) fε(x)
i i i
n(y)dy− n(y)−ε (n˜(y)−n(y))dy
ACKNOWLEDGMENTS xε x x
∈{−N,0,N},
(cid:90) x (cid:90) fε(x)
It is our pleasure to dedicate this paper to Evert Jan i
n(y)dy−ε (n˜(y)−n(y))dy ∈{−N,0,N}.
Baerends,whohasbeenapioneerinunderstandingandusing
xε x
linear response TDDFT, and a wonderful mentor to some of (B1)
us. WethankAndre´Mirtschink,KlaasGiesbertzandMichael
Seidlformanyusefuldiscussionsandforacriticalreadingof SincethislastquantityisforsurelessthanN initsabsolute
the manuscript. This work was supported by an IEF fellow- value,thenitmustbeequalto0andso
shipfromtheMarieCurieprogram(n.629625)andfromthe
European Research Council under H2020/ERC Consolidator (cid:90) x (cid:90) fiε(x)
n(y)dy =ε (n˜(y)−n(y))dy; (B2)
Grant“corr-DFT”(GrantNo.648932).RvLlikestothankthe
AcademyofFinlandforsupport.AGacknowledgesCNPqfor xε x
thefinancialsupportofhisPh.D.,whenthisworkwasdone. thisimpliesforsurethatifn>0everywherethenx −xisof
ε
8
orderεandtheneitherf −fεisoforderεor(whenx=a ) theregularpart
i i i
f ∼ ±∞andfε ∼ ∓∞. Inthefirstcaseitisclearthatthe
i i
right-handsideof(B2)canbeapproximatedby(cid:82)xfi(x)(n˜−n) K (x,x(cid:48))=(cid:88)N (cid:90) ∞w(cid:48)(cid:48)(|y−f ([n];y)|)·
losingalowerorderterm;infactthisistruealsointhesecond R i
case, using the fact that n˜ −n has null total integral. Now, i=2 x
definingthequantity θ(y−x(cid:48))−θ(fi([n];y)−x(cid:48))dy. (C1)
n(f (y))
i
(cid:90) fi(x)
αi(x)= (n˜(y)−n(y))dy (B3) The singular part is more delicate as the classical chain rule
x doesnotapplyanymore: wefindthatδf /δnisproportional
i
andusing(B2), thelastobservation, andthecontinuityofn, to(fi)(cid:48). [Thechainruleforderivativeswhenwehaveastep
wecaninferthat functionisnottrivial:justtaketheexampleofg(θ),forwhich
thederivativeis(g(1)−g(0))δ andnot(g(cid:48)◦θ)·θ(cid:48) =g(cid:48)(0)δ .]
0 0
α (x)
x =x−ε i +o(ε). (B4) In this case, the term coming from the jump, noticing that
ε n(x) sgn(a −f±([n];a ))=∓1,hastheform
i i i
Thismeansthatwehave
N
(cid:88) (cid:16)
fiε(x)−fi(x) = fi(xε)−fi(x) ∼−(f (x))(cid:48)αi(x), (B5) KS(x,x(cid:48))= θ(ai−x)· w(cid:48)(|ai−fi+([n];ai)|)+
ε ε i n(x) i=2
(cid:17) α (a )
w(cid:48)(|a −f−([n];a )|) · i i . (C2)
wherethederivativeisadistributionalderivative,whichtakes i i i n(a )
i
into account also the jumps. Away from x = a , which is
i
the only point where the jump can occur, we can make a In particular, whenever n(x) > 0 on the whole real line we
simplification, as we have fi(cid:48)(x) = n(x)/n(fi(x)), and so would have f±([n];ai) = ±∞ and this term becomes 0 if
(f (x))(cid:48)αi(x) = αi(x) . For the jump we will get an addi- w(cid:48)(∞) = 0whichisthecasefortheCoulombinteractionor
i n(x) n(fi(x)) anysituationinwhichtheforce tendsto0 withthedistance.
tionaltermintheformofadeltafunction,
FordensitiessupportedonthewholeRwehaveonlytheterm
α (x) (C1),whichcoincideswithEq.(27).
(f+([n];a )−f−([n];a ))δ(x−a )· i . (B6)
i i i i i n(x)
However, if the support of n is compact, denoting
Intheend,noticingthat−αi(x) = (cid:82)Rδn(x(cid:48))·(cid:0)θ(x−x(cid:48))− by n−,n+ the extremes of the support, we would have
θ(fi([n];x)−x(cid:48))(cid:1)dx(cid:48),weget f±([n];ai)=n±andthecontributionwouldbenonzero. We
canwriteitinaclearerform(usingθ(f+([n];a )−x(cid:48))=0)
i i
δfi([n];x) =(cid:0)θ(x−x(cid:48))−θ(f ([n];x)−x(cid:48))(cid:1)·
δn(x(cid:48)) i N
(cid:88) (cid:16) (cid:17)
(cid:18) 1 f+([n];a )−f−([n],a ) (cid:19) KS(x,x(cid:48))= θ(ai−x)· w(cid:48)(|ai−n+|)+w(cid:48)(|ai−n−|) ·
n(f ([n];x) + i in(a )i i ·δ(x−ai) i=2
i i θ(a −x(cid:48))
(B7) i . (C3)
n(a )
i
AppendixC:FunctionalvariationfortheSCEpotential(kernel)
Now we want to use the results of the previous section
AppendixD:SymmetryoftheSCEkernelin1D
on the variation of the co-motion functions to derive an
expression for δvSCE(x)/δn(x(cid:48)), where vSCE(x) is given
Hxc Hxc
in Eq. (25). We can write FSCE(x,x(cid:48)) = K (x,x(cid:48)) +
Hxc R
KS(x,x(cid:48)),whereKR andKS are,respectively,thecontribu- The symmetry in x and x(cid:48) of the singular term KS(x,x(cid:48))
tionduetotheregularandsingularpartof δfi(x). Whenthe ofEq.(C3)isobvious,whileforK (x,x(cid:48))ofEq.(C1)thisis
δn(x(cid:48)) R
co-motionfunctionsdonothavejumps(thatis,wheny (cid:54)=a ) subtler. In order to prove it we compute ∂2 K (x,x(cid:48)): if
i ∂x∂x(cid:48) R
wecansimplyapplythechainruleandfindanexpressionfor wecanprovethatthisquantityissymmetric,thenthesymme-
9
tryofthewholekernelisautomaticallyproven: AppendixE:PropertiesofthefunctionG(x)forthe
homonuclear1Ddensity
∂2
K ([n];x,x(cid:48))=
∂x∂x(cid:48) R Forthe2-electrondensityofEq.(33)witha = 1itiseasy
(cid:88)N w(cid:48)(cid:48)(|x−f ([n];x)|)δ(x(cid:48)−x)−δ(x(cid:48)−fi([n];x)) toshowthattheco-motionfunctionsatisfies
i n(f ([n];x)) (cid:18) 2 (cid:19)
i=2 i f([n];x→0+)=ln(x)−R+ln , (E1)
1+e−R
=(cid:88)N w(cid:48)(cid:48)(|x−fi([n];x)|)δ(x(cid:48)−x)−
n(f ([n];x)) yielding
i
i=2
(cid:88)N w(cid:48)(cid:48)(|x−x(cid:48)|)δ(x(cid:48)−fi([n];x)). (D1) n(f([n];x→0+))=xe−R/2. (E2)
n(f ([n];x))
i=2 i The case x → 0− can be obtained from f([n];−x) =
−f([n];x) and n(−x) = n(x). Inserting these expansions
For the first term we will use the fact that h(x)δ(x(cid:48) −x) =
inthedefinitionofthefunctionG(x)ofEq.(30)weseethat
h(x(cid:48))δ(x−x(cid:48)), while for the second term we will use also
itsderivativeisgivenby
thatδ(g(x)−g(x(cid:48)))= 1 δ(x−x(cid:48)). Inparticularwehave
g(cid:48)(x)
x(cid:48) =f (f (x(cid:48))),andso,sincef(cid:48)([n],y)= n(y) : 2eR/2
i N−i+2 i n(fi([n];y) G(cid:48)(x→0+)= (cid:16) (cid:17) , (E3)
x|R−ln 2 −ln(x)|3
δ(x(cid:48)−f ([n];x))=δ(f (f (x(cid:48)))−f (x)) 1+e−R
i i N−i+2 i
n(f ([n];x)) showing that G(x) has an infinite slope in x = 0. Further-
=δ(f ([n];x(cid:48))−x) i . (D2)
N−i+2 n(x) more,wealsohave,fromthepropertiesoftheco-motionfunc-
tionandfromthesymmetryofthedensityn(−x)=n(x)that
Plugging in this expression and using again the fact that
h(x)δ(y−x)=h(y)δ(x−y)wefindthat G(−x)=2G(0)−G(x) (E4)
G(f(x))=G(x)−G(0) forx>0 (E5)
∂2 G(f(x))=G(x)+G(0) forx<0. (E6)
K ([n];x,x(cid:48))=
∂x∂x(cid:48) R
Since f([n];−R/2) = R/2, for x = −R/2 both properties
(cid:88)N w(cid:48)(cid:48)(|x−fi([n];x)|)δ(x(cid:48)−x) (E4)and(E6)musthold, implyingthatG(0) = 2G(−R/2),
n(fi([n];x)) whichisEq.(37).
i=2
When R is large, if x is well inside one of the atomic re-
−(cid:88)N w(cid:48)(cid:48)(|x−x(cid:48)|)δ(x(cid:48)−fi([n];x)) gions then f∓([n];x) ≈ x ± R, yielding the constant dis-
i=2 n(fi([n];x)) tance|x−fi([n];x)| ≈R,producingtheplateauxregionsin
thekernel. Thisbehaviorholdsuntiltheelectroninf([n];x)
=(cid:88)N w(cid:48)(cid:48)(|x(cid:48)−fi([n];x(cid:48))|)δ(x−x(cid:48)) approachestheoriginandstartsto“see”theseconddensityin
n(f ([n];x(cid:48))) theoverlapregionpresentinthemidbond.Thishappenswhen
i
i=2
x≈±(R−1/a). Atthispoint,thelargenegativexbehavior
−(cid:88)N w(cid:48)(cid:48)(|x−x(cid:48)|)δ(fN−i+2([n];x(cid:48))−x) ofG(x)startstoappear,
n(x)
i=2 1
G(x→−∞)= , (E7)
=(cid:88)N w(cid:48)(cid:48)(|x(cid:48)−fi([n];x(cid:48))|)δ(x−x(cid:48)) n(0)x2
n(f ([n];x(cid:48)))
i
i=2 yieldingtheplateauvalueofEq.(34).
−(cid:88)N w(cid:48)(cid:48)(|x−x(cid:48)|)δ(fN−i+2([n];x(cid:48))−x)
n(f ([n];x(cid:48)))
N−i+2
i=2
∂2
= K ([n];x(cid:48),x), (D3)
∂x∂x(cid:48) R
whereinthelaststepwejustrelabeledthesecondsum.
10
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