The derivative discontinuity in the strong-interaction limit of density functional theory
TThe derivative discontinuity in the strong-interaction limit of density functional theory
Andr´e Mirtschink, Michael Seidl, and Paola Gori-Giorgi Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling,FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands Institute of Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany (Dated: November 13, 2018)We generalize the exact strong-interaction limit of the exchange-correlation energy of Kohn-Shamdensity functional theory to open systems with fluctuating particle numbers. When used in theself-consistent Kohn-Sham procedure on strongly-interacting systems, this functional yields exactfeatures crucial for important applications such as quantum transport. In particular, the step-like structure of the highest-occupied Kohn-Sham eigenvalue is very well captured, with accuratequantitative agreement with exact many-body chemical potentials. Whilst it can be shown thata sharp derivative discontinuity is only present in the infinitely strongly-correlated limit, at finitecorrelation regimes we observe a slightly-smoothened discontinuity, with qualitative and quantitativefeatures that improve with increasing correlation. From the fundamental point of view, our resultsobtain the derivative discontinuity without making the assumptions used in its standard derivation,offering independent support for its existence.
First-principles calculations of many-electron systemssuch as solids, molecules, and nanostructures are based,to a very large extent, on Kohn-Sham (KS) [1] densityfunctional theory (DFT) [2]. KS DFT is, in principle, anexact theory, in which the ground-state energy and den-sity of an interacting many-electron system are mappedinto a problem of non-interacting electrons moving in theeffective one-body KS potential. In practice, KS DFTrelies on approximations for the exchange-correlation en-ergy that, although successful in very many cases, havestill deficiencies that hamper its overall usefulness [3].Exact KS DFT has many weird and counterintuitivefeatures often missed by the available approximations.One of the weirdest and most elusive of these featuresis the derivative discontinuity of the exact exchange-correlation energy functional at integer particle numbers N [4], which has been an incredibly long-debated issue[5–15] because its formal derivation relies on some (veryreasonable) assumptions. This discontinuity makes theexact KS potential “jump” by a constant ∆ xc when weadd to an N -electron system even a very tiny fraction η of an electron, aligning the chemical potential of thenon-interacting KS system to the exact, interacting, one,as schematically illustrated in Fig. 1. The derivative dis-continuity has crucial physical consequences. For exam-ple, it accounts for the difference between the KS andthe physical fundamental gaps [16–18], it allows a cor-rect KS DFT description of molecular dissociation [4, 19],and it should improve charge-transfer excitations in time-dependent DFT [20–22]. It also plays a fundamental rolein quantum transport, especially to capture the physicsof the Coulomb blockade and the Kondo effect [23–27].These are all cases in which the standard approximations,which miss this discontinuity, work poorly. Correctionsbased on the explicit enforcement of the discontinuityhave been often proposed as a practical solution (see,e.g., Refs. [28–31]). Recently, it has been shown [32, 33] that the exactstrong-interaction limit of DFT [34, 35] provides a non-local density functional, called “strictly-correlated elec-trons” (SCE) [34–36], which can be used to approximatethe exchange-correlation energy of KS DFT, capturingkey features of strong correlation such as charge localiza-tion in low-density quantum wires without any symmetrybreaking [32, 33]. This approximation becomes asymp-totically exact in the very low-density (or infinitely strongcorrelation) limit [32, 33]. The purpose of this Letter isto generalize the SCE functional to the case of fractionalparticle numbers, yielding an exchange-correlation func-tional for open systems that becomes more and more ac-curate as correlation increases, and that can be alreadyused for modeling nanotransport through low-densityquantum wires and dots. Our results also support theassumptions that were made to derive the existence ofthe derivative discontinuity [4], and provide new insightfor the construction of approximate functionals.The Letter is organized as follows. After briefly review-ing the essential background material, we present the rig-orous generalization of the SCE functional to fractionalelectron numbers. We then show that, without breakingthe spin symmetry, the self-consistent KS results withthe SCE functional display the right discontinuities atinteger electron numbers when correlation is very strong.This is obtained without making any additional assump-tion or imposing any ad hoc constraint. Finally, we drawour conclusions and discuss some perspectives. Hartree(effective) atomic units are used throughout. Fractional particle numbers in KS DFT–
At zero tem-perature, open systems with fluctuating particle numbershave been first analyzed in the KS framework by consid-ering statistical mixtures [4]. When dealing with DFTfor quantum mechanical systems, we can work with purestates in which a degenerate system is composed by wellseparated fragments, and one focuses on the energy and a r X i v : . [ c ond - m a t . s t r- e l ] A ug exact&E N+1& (cid:1) &E N&& exact&E N& (cid:1) &E N+1&& Δ xc =I & (cid:1) &A && N& N+η&exact&E
N+1& (cid:1) &E N&& exact&E N& (cid:1) &E N+1&&
N& N+η&Δ xc&
N+electron&open&shell&N+electron&closed&shell&
FIG. 1: (color online). Schematic illustration of the spin-restricted (same potential for spin- ↑ and spin- ↓ electrons) KSspectrum when adding a tiny fraction η of an electron to an N -electron system. Top panel: when the KS N -electron sys-tem is open shell, the whole KS spectrum “jumps” by a pos-itive constant ∆ xc = I N − A N , which aligns the KS highestoccupied eigenvalue (HOMO) to minus the electron affinity − A N = E N +1 − E N . Bottom panel: when the KS systemis closed shell, the positive constant ∆ xc aligns the KS N -electron lowest unoccupied orbital (LUMO) to minus the ex-act affinity − A N . In both cases, in the exact KS system theHOMO is equal to the many-body chemical potential. density of one of the fragments alone [37–39]. A verysimple example is a stretched H +2 molecule, in which oneach proton we find, on average, electron [13, 19]. Akey point to prove the existence of the derivative disconti-nuity is the (empirical) observation that, for integer par-ticle numbers N , the energy E N of a N -electron systemin a given external potential is a concave-up function, i.e. E N ≤ ( E N +1 + E N − ). This implies that for fractionalelectron numbers Q = N + η (with 0 ≤ η ≤
1) the exactmany-electron ground-state energy E Q and density ρ Q ( r )lay on the line connecting the values at the two adjacentintegers: E N + η = (1 − η ) E N + η E N +1 (1) ρ N + η ( r ) = (1 − η ) ρ N ( r ) + η ρ N +1 ( r ) . (2)In KS DFT, one usually aims at obtaining the ex-act quantities of Eqs. (1)-(2) by means of a non-interacting system of Q = N + η electrons in the effectivesingle-particle KS potential v s ( r ; [ ρ Q ]), which necessarilychanges as we change Q . The energy is then minimizedby giving integer occupation to the N single particle KSspin-orbitals with lowest eigenvalues and fractional occu-pation η to the frontier orbital(s), often called HOMO(highest occupied molecular orbital) [40, 41].In the exact KS theory, the HOMO eigenvalue (cid:15) HOMO is the derivative of the total energy of Eq. (1) with respectto the particle number Q , ∂E Q ∂Q = (cid:15) HOMO [4, 40]. Thus,the exact (cid:15)
HOMO is constant between any two adjacentintegers (say, N and N + 1) and equal to the interactingchemical potential E N +1 − E N , jumping to a differentvalue when crossing an integer. This “step-like” behav-ior of the KS (cid:15) HOMO is not captured by the standardapproximate functionals (see, e.g., Refs. [38, 41, 42]),unless imposed a priori via additional constraints in aspin-unrestricted framework, as e.g., in Refs. [28, 29, 31]The alignment of the exact KS HOMO eigenvalue withthe interacting chemical potential implies that the ex-act KS one-body potential must jump by a constant ∆ xc (the derivative discontinuity) when crossing an integer, v s ( r ; [ ρ N + ]) − v s ( r ; [ ρ N − ]) = ∆ xc (see Fig. 1). Strong-interaction limit –
The strong-interaction limitof DFT is given by the SCE functional V SCE ee [ ρ ], definedas the minimum of the electronic interaction alone overall the wave functions yielding the density ρ [34, 36, 43], V SCE ee [ ρ ] = min Ψ → ρ (cid:104) Ψ | ˆ V ee | Ψ (cid:105) . (3)It can be shown [32, 44, 45] that in the low-density (orstrong-interaction) limit the exact Hartree and exchange-correlation functional E Hxc [ ρ ] of KS theory tends asymp-totically to V SCE ee [ ρ ].Physically, the functional V SCE ee [ ρ ] portrays the strictcorrelation regime, where the position r of one electrondetermines all the other N − r i through the so-called co-motion functions , r i = f i [ ρ ]( r ),some non-local functionals of the density [34, 35, 46, 47].Therefore, the net repulsion on an electron at position r due to the other N − r aloneand can be exactly represented [33, 34, 46, 47] by a localone-body potential, ∇ ˜ v SCE [ ρ ]( r ) = − N (cid:88) i =2 r − f i [ ρ ]( r ) | r − f i [ ρ ]( r ) | , (4)which is also the functional derivative of V SCE ee [ ρ ], δV SCE ee [ ρ ] δρ ( r ) = ˜ v SCE [ ρ ]( r ) [32, 33]. In terms of the co-motionfunctions we have [34] V SCE ee [ ρ ] = N − (cid:88) i =1 N (cid:88) j = i +1 (cid:90) d r ρ ( r ) N | f i [ ρ ]( r ) − f j [ ρ ]( r ) | . (5) SCE for fractional particle numbers –
The generaliza-tion of the SCE formalism to non-integer particle num-bers Q is not obvious, because in Eqs. (4)-(5) the sumruns over the integer number of electrons N . To pro-ceed in a rigorous way [3, 13, 19, 37, 39], we first ana-lyze a well separated Q -electron fragment inside a degen-erate system with total integer electron number M , asschematically illustrated in Fig. 2.Actually, from Eq. (5) it is not evident that V SCE ee [ ρ ]separates into the sum of contributions of the isolated total%%M=5%electrons%Q=2.5% Q=2.5% FIG. 2: (color online). Simple example of the analysis usedto deduce the SCE solution for fractional electron numbers:we considered a density with M = 5 electrons, made of twoseparated identical fragments (here modeled with two gaus-sians), and we have studied the exact SCE solution on eachfragment. The graphic shows the positions f i ( x ) of the other4 electrons as a function of the position x of the first electron.The two black circles represent the “local” SCE solution oneach fragment. fragments, because the interaction | f i [ ρ ]( r ) − f j [ ρ ]( r ) | − between two electrons on a given fragment may con-tribute significantly to the integral when r spans the re-gion of another fragment. However, we have recentlyshown [48] that another exact expression for V SCE ee [ ρ ] is V SCE ee [ ρ ] = 12 (cid:90) d r ρ ( r ) N (cid:88) k =2 | r − f k [ ρ ]( r ) | , (6)which separates into a sum of fragment contributions,showing explicitly that V SCE ee [ ρ ] is size consistent.We start from (quasi) one-dimensional systems (1D),for which we can construct the SCE solution analytically.The main findings and conclusions, however, should holdalso for two- and three-dimensional (3D) systems [49].Indeed, we have also performed a three-dimensional self-consistent calculation for a spherically-symmetric den-sity, for which we can deduce the fractional SCE solutionfrom our 1D construction, finding similar results.Figure 2 depicts the reasoning behind the constructionof the SCE functional for Q electrons. In the illustratedexample, we have solved the SCE problem for M = 5electrons, for a density made of two well-separated iden-tical fragments, each integrating to Q = 2 . N + 0 . f i ( x ) of the other 4 electrons is shownas a function of the position x of the first electron. Wefind that two adjacent strictly-correlated positions f i ( x ) and f i +1 ( x ) on the fragment always satisfy the conditionof total suppression of fluctuations [36], (cid:90) f i +1 ( x ) f i ( x ) ρ ( x (cid:48) ) dx (cid:48) = 1 , (7)so that there are values of x for which we have 3 electronsin the fragment, and values of x for which we find only 2particles (the third electron is in the other fragment).Similarly, we find that the general SCE solution for adensity integrating to Q = N + η electrons can be easilyobtained from Eq. (7) (see the Supplemental Material fora detailed derivation). From now on, we work with thefragment alone, considering that on our x -axis only thedensity ρ Q ( x ) is present. The co-motion functions read f i ( x ) = N − e [ N e ( x ) + i − x < a N +1+ η − i N − e [ N e ( x ) + i − N − x > a N +2 − i ∞ otherwise, (8)where the function N e ( x ) is defined via the density ρ Q ( x ), N e ( x ) = (cid:90) x −∞ ρ Q ( x (cid:48) ) dx (cid:48) , (9) a k = N − e ( k ), and i = 2 , ..., N + 1. Notice thateven if we have N + 1 co-motion functions, when x ∈ [ a N +1+ η − i , a N +2 − i ] one of the electrons stays at infinity,so that there are x -intervals for which we find N + 1 elec-trons in the density, and x -intervals for which we haveonly N electrons (one of the electrons cannot “enter” inthe density). Self-consistent KS SCE for N + η electrons – FromEqs. (8), (6) and (4) we can now construct the SCE func-tional V SCE ee [ ρ Q ] and its functional derivative ˜ v SCE [ ρ Q ]( r )for any one-dimensional density integrating to a non-integer particle number Q . [For Q = N + η electrons,the sum in Eqs. (6) and (4) runs up to N + 1.] We thenconsider Q electrons in the quasi-one-dimensional modelquantum wire of Refs. [50, 51], in which the effectiveelectron-electron interaction is obtained by integratingthe Coulomb repulsion on the lateral degrees of freedom[50, 52], and is given by w b ( x ) = √ π b exp (cid:16) x b (cid:17) erfc (cid:0) x b (cid:1) .The parameter b fixes the thickness of the wire, set to b = 0 . x ) is the com-plementary error function. As in Ref. [51], we consideran external harmonic confinement v ext ( x ) = ω x inthe direction of motion of the electrons, which for small ω yields bound states with very low density, where thefunctional V SCE ee [ ρ ] becomes closer and closer to the ex-act E Hxc [ ρ ]. This is crucial to show that when E Hxc [ ρ ] → V SCE ee [ ρ ] we recover the exact features of Fig. 1.We have then performed self-consistent, spin-restricted(same KS potential for ↑ and ↓ spins), KS calculationswith the SCE potential, as in Refs. [32, 33], this timeby giving fractional occupation to the KS HOMO orbital e HOMO e HOMO e HOMO
Q"KS"LDA"KS"LDA"KS"LDA" KS"SCE"KS"SCE"KS"SCE"CI"CI" CI"L=1"L=70"L=150" 1 2 3 41 2 3 41 2 3 4 ε H O M O " ε H O M O " ε H O M O " FIG. 3: (color online). Self-consistent results for the spin-restricted KS HOMO eigenvalue (in Hartrees) as a functionof the particle number Q for a quasi-1D quantum wire withharmonic confinement along the direction of motion of theelectrons, v ext ( x ) = ω x , and ω = L . The KS SCE resultsare compared with the standard KS LDA and with the exactchemical potential E N +1 − E N from full CI calculations [33]. [40, 41, 53]. The SCE potential ˜ v SCE [ ρ Q ]( x ), represent-ing the approximate Hartree plus exchange-correlationpotentials in our KS SCE calculations, is obtained byintegrating the 1D analogue of Eq. (4), with boundarycondition ˜ v SCE [ ρ Q ]( | x | → ∞ ) = 0, using the co-motionfunctions of Eq. (8).In Fig. 3 we show our self-consistent KS SCE re-sults for the HOMO eigenvalue as a function of theparticle number, comparing with the KS LDA values(the 1D LDA functional is from Ref. [54]), and withthe full configuration-interaction (CI) results [33] for thechemical potential E N +1 − E N . We have considered amoderately-correlated case, L = 1, and two strongly-correlated cases, L = 70 and L = 150, where L is an effec-tive confinement-length such that ω = L . For moderatecorrelation, L = 1, we see that LDA, having ∆ xc = 0,shows a discontinuity in the HOMO value only when fill-ing a new shell (even N ; case illustrated in the secondpanel of Fig. 1), while KS SCE shows a small verticalchange also when the N system is open shell (at odd N ).When correlation becomes stronger ( L = 70 and 150), Q" KS"SCE" ω =10 *5" exact"0.5" 1.0" 1.5" 2.0" ε HOMO" (10 *1" mH )" FIG. 4: (color online). Self-consistent results for the spin-restricted KS HOMO eigenvalue as a function of the particlenumber Q for a three-dimensional electronic system in theexternal harmonic potential, v ext ( r ) = ω r , with ω = 10 − .The KS SCE results are compared with the exact chemicalpotential E N +1 − E N [55, 56]. the KS SCE self-consistent HOMO approaches more andmore the exact step structure, with very good quantita-tive agreement with the full CI chemical potentials. Forsuch cases, KS LDA yields essentially a continuous curve,since the single particle energies are all very close.In Fig. 4, we show the result for a 3D system, in whichthe electron-electron interaction is Coulombic and theexternal potential is harmonic (“Hooke’s atom”). The co-motion functions can be deduced from Eq. (8), as detailedin the Supplemental Material. For small ω , we find thatthe self-consistent KS SCE HOMO approaches again theexact step structure, which becomes sharper and sharperas correlation increases ( ω decreases) [59], similarly to the1D results of Fig. 3. Notice, again, that for 0 ≤ Q ≤ Q . Concluding remarks and perspectives–
The discontinu-ity in the HOMO for open shell systems (case in the upperpanel of Fig. 1) from the self-consistent KS SCE is notjust a unique result, but also an independent proof thatthe exact KS formalism should have this feature. In fact,we have only used the exact V SCE ee [ ρ Q ] in the KS self-consistent procedure, without imposing any other condi-tion on our functional. Until now, this feature has onlybeen captured in the context of lattice hamiltonians [57],or by imposing it a priori in a spin-unrestricted frame-work [60], as, e.g., in Refs. [28, 29, 31].From a practical point of view, our results could al-ready be used to model transport through a correlatedquantum wire or quantum dot, going beyond the lat-tice calculations of Refs. [25, 27]. 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