Exchange-correlation functionals from the strongly-interacting limit of DFT: Applications to model chemical systems

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by  Francesc Malet

Exchange-correlation functionals from the strongly-interacting limit of DFT:
Applications to model chemical systems
Francesc Malet, Andr´e Mirtschink, Klaas J. H. Giesbertz, Lucas O. Wagner, and Paola Gori-Giorgi
Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling,
FEW,Vrije Universiteit,De Boelelaan 1083,1081HV Amsterdam, The Netherlands
We study model one-dimensional chemical systems (representative of their three-dimensional
counterparts) using the strictly-correlated electrons (SCE) functional, which, by construction, be-
comes asymptotically exact in the limit of infinite coupling strength. The SCE functional has a
highly non-local dependence on the density and is able to capture strong correlation within Kohn-
Sham theory without introducing any symmetry breaking. Chemical systems, however, are not
4 close enough to the strong-interaction limit so that, while ionization energies and the stretched H
2
1 molecule are accurately described, total energies are in general way too low. A correction based on
0 the exact next leading order in the expansion at infinite coupling strength of the Hohenberg-Kohn
2
functional largely improves the results.

n
a
J I. INTRODUCTION of non-interacting. The SCE functional has a highly

0 non-local dependence on the density, but its functional
3 derivative can be easily constructed,14,15 yielding a local
DespitetheenormoussuccessofKohn-Sham(KS)den-
sityfunctionaltheory(DFT)1 whenappliedtothestudy one-bodypotentialwhichcanbeusedintheKohn-Sham
]
h ofmanychemicalsystems,2therearestillimportantcases scheme to approximate the exchange-correlation term.
p for which standard approximate exchange-correlation The SCE functional tends asymptotically to the exact
m- functionals are inaccurate.2–4 In particular, systems in Hartree-exchange-correlation functional in the extreme
infinite correlation (or low-density) limit.
which rearrangement of electrons within some (near-
e
degeneracy) or many (strong correlation) partially filled Very promisingly, the first applications of this “KS
h
c levels is important, such as transition metals, stretched SCE” DFT approach, performed on one-dimensional
. bonds and Mott insulators, represent a big challenge for (1D) semiconductor quantum wires14,15 and on two-
s
c KS DFT.2,3 dimensional quantum dots,16 have shown that the
si SimilarlytotheunrestrictedHartree-Fockmethod,KS SCE exchange-correlation potential is able to describe
y DFT with approximate functionals tries to mimic the the physics of the strongly-correlated regime within
h physics of strong electronic correlation with symmetry the restricted Kohn-Sham scheme, truly making non-
p breaking, which, in many cases (but not always), yields interacting electrons behave as strongly correlated ones.
[ reasonable energies. In complex systems, however, sym- Itisthusnaturaltoaskwhetherwiththisformalismone
1 metry breaking can occur erratically and can be very can also cure the deficiencies of standard DFT approxi-
v sensitivetothefunctionalchosen.3 Whenmanydifferent mations in Chemistry.
2 brokensymmetrysolutionsarecompetingitbecomesdif- Thephysicsofstrongcorrelationencodedinthehighly
2
ficult to keep the potential energy surfaces continuous. non-localdensitydependenceoftheSCEfunctional,how-
8
The rigorous KS formulation is also partially lost, and ever, does not come for free: the SCE problem is sparse
7
. many properties are wrongly characterized.5,6 but nonlinear, and a general algorithm for its evaluation
1
It is important to keep in mind that Kohn-Sham DFT following the original formulation is still an open prob-
0
is, in principle, an exact theory, which should be able lem. Progress has been made recently17 by using the re-
4
1 to yield the right ground-state density and energy of formulation of the SCE functional as a mass transporta-
: strongly-correlated systems without resorting to symme- tion theory (or optimal transport) problem,18–20 which
v
try breaking. The quest for an approximate exchange- allows to evaluate the SCE functional and its functional
i
X correlation functional able to achieve this fundamental derivative by means of a maximization under linear con-
r goal is a very active research field,2,7–10 which aims at straints, although the procedure is still cumbersome and
a solvingwhatisarguablyoneofthemostimportantprob- needs further developments.17 Probably, it will be neces-
lems in electronic structure theory. sary to devise approximate ways to deal with the SCE
physics in the general three-dimensional case.
Recently, analternativeapproachtothestandardway
of constructing functionals for KS DFT has been pro- Before adventuring into the challenging task of imple-
posed, in which the knowledge on the strong-interaction menting the SCE functional (or approximations thereof)
limit of DFT is used to build an approximation for the forgeneralthree-dimensionalsystems,wefeelitisimpor-
exchange-correlation energy and potential. The starting tant to understand whether this functional could play a
point is the so-called strictly-correlated-electrons (SCE) role for Chemistry. For this reason, we consider here
referencesystem,introducedbySeidlandcoworkers,11–13 a simple one-dimensional model that has recently been
which has the same density as the real interacting one, shown to be a useful laboratory to test functionals for
butinwhichtheelectronsareinfinitelycorrelatedinstead chemical problems,21,22 offering a reasonably close de-
2
scriptionofthethree-dimensionalcounterpartsandbeing
computationally much less demanding.
The paper is organized as follows. In Sec. II we briefly
describe the KS SCE approach, first at zeroth order of
approximationandlaterintroducinganhigherordercor-
rection properly renormalized. In Sec. III we present the
studied one-dimensional models and we describe the de- FIG. 1: Schematic illustration of the SCE reference system
tails of the actual calculations. The results are then pre- for a given density ρ(r) and N = 4 electrons. The empty
sented and discussed in Sec. IV. Finally, in Sec. V we circle represents the position of the reference particle, which
draw some conclusions and outlook for future works. isdifferentineachcase. Theotherelectronsmustadapttheir
relative positions in such a way that the superposition of all
thepossibleconfigurations(oneforeachr)yieldsthedensity
ρ(r).
II. THE KS SCE APPROACH
As a brief introduction to the KS SCE14,15 approach,
first consider the partitioning of the total energy den- perfectly) correlated: if one of the electrons, which can
sity functional for an N-electron system in the exter- be taken as reference and labeled as “1”, is found with
nal potential Vˆ = (cid:80)N v (r ) as E[ρ] = F[ρ] + some probability at a given position r, the other N 1
(cid:82) ext i=1 ext i −
ρ(r)v (r)dr, with the internal energy (kinetic plus electronswillbefound, withthesameprobability, atthe
ext
electron-electron repulsion) expressed as23,24 positions r f [ρ](r) (i = 2,...N).12,13,25 Since the f
i i i
≡
only depend on r, they are called co-motion functions.
F[ρ] min ΨTˆ+Vˆee Ψ , (1) They are highly non-local functionals of the density and
≡Ψ→ρ(cid:104) | | (cid:105)
satisfy, for each r, the set of differential equations25
where the search is only over ferrmionic wave functions.
It is the delicate interplay of the kinetic energy and the
ρ(r)dr=ρ(f (r))df (r) (i=2,...,N), (6)
electron-electron interaction that makes the evaluation i i
of this functional a daunting task. If the minimization
would only contain the kinetic energy, aswellasthefollowinggrouppropertiesthatensurethat
theN electronsareindistinguishable(sothatthereisno
T [ρ] min ΨTˆ Ψ , (2)
s ≡Ψ→ρ(cid:104) | | (cid:105) dependence on which electron is chosen as “electron 1”)
it is easy to determine its minimum. In particular, the
wave function Ψ which achieves this minimum is often a f1(r) r,
≡
single Slater determinant composed of orbitals that sat- f (r) f(r),
2
isfy the Kohn–Sham equations1 ≡
f (r)=f(f(r)),
3
(cid:18) 2 (cid:19) f (r)=f(f(f(r))),
∇ +v [ρ](r)+v [ρ](r) φ (r)=ε φ (r). (3) 4 (7)
ext Hxc i i i
− 2 .
.
.
The Hartree-exchange-correlation potential originates as
f(f(...f(f(r))))=r.
aLagrangemultipliertoensurethattheKohn-Shamsys- (cid:124) (cid:123)(cid:122) (cid:125)
tem yields the required density ρ and is related to the N times
remainderE [ρ] F[ρ] T [ρ]asitsfunctionalderiva-
Hxc s
≡ −
tive Figure 1 schematically illustrates the SCE reference sys-
tem for the case of 4 electrons in a given density ρ(r).
δE [ρ]
δρH(xrc) ≡vHxc[ρ](r). (4) The figure shows three configurations, each of them cor-
responding to the reference electron being at a differ-
Seidl and co-workers have demonstrated that the func- ent position r (represented by an empty circle). The
tional F[ρ] can be also explicitly evaluated if it would other three electrons adapt their positions, given by the
only contain the electron-electron interaction11,25,26 co-motion functions (represented by solid symbols), in
order to minimize the total Coulomb repulsion and un-
VeSeCE[ρ]≡Ψm→inρ(cid:104)Ψ|Vˆee|Ψ(cid:105). (5) der the constraint that the superposition of all the pos-
sible configurations (one for each r) yields the density of
This strictly-correlated-electrons (SCE) functional is the the physical system ρ(r). The SCE system thus repre-
strongly-interactinglimitofF[ρ],describingthesituation sents a smooth N-electron quantum-mechanical density
in which the kinetic energy is negligible. The SCE func- by means of an infinite superposition of classical config-
tional is the natural counterpart of the Kohn–Sham ki- urations, which fulfill Eq. (6) for every r. The square
neticenergyasitdefines,insteadofanon-interactingref- modulusofthecorrespondingSCEwavefunction(which
erencesystem,oneinwhichtheelectronsareinfinitely(or becomes a distribution in this limit18,19) can be written
3
as Equivalently, comparing Eqs. (4) and (12), we see that
the KS SCE uses v to approximate the Hartree and
Ψ (r ,r ,...,r )2 = 1 (cid:88)(cid:90) drρ(r)δ(r f (r)) exchange-correlationSCpEotentials,
SCE 1 2 N 1 ℘(1)
| | N! N −
℘ v (r) v (r). (15)
Hxc SCE
δ(r f (r)) δ(r f (r)), (8) (cid:39)
2 ℘(2) N ℘(N)
× − ··· −
NoticethatEq.(11)impliesthatv asanapproximate
SCE
where ℘ denotes a permutation of 1,...,N, such that v has the right asymptotics in the limit r .
ρ(r) = N(cid:82) ΨSCE(r,r2,...,rN)2dr2 drN. The SCE HExcquation (14) shows that the KS SCE DF| T|→ap∞proach
| | ···
systemcanthusbevisualizedasa“floatingWignercrys- treats both the kinetic energy and the electron-electron
tal” describing the density ρ(r). interaction on the same footing, letting the two terms
Theco-motionfunctionsarethekeyobjectfortheSCE compete in a self-consistent way within the Kohn–Sham
functional, analogously to the Kohn–Sham orbitals for scheme. Furthermore, the method becomes asymptoti-
thenon-interactingkineticenergyfunctionalTs[ρ]. They cally exact both in the very weak and very strong cor-
can be used to express the functional VeSeCE[ρ] explicitly relation limits.14,15 At intermediate correlation regimes,
as25,27 howeveritcanseriouslyunderestimatethetotalenergy.15
(cid:90) ρ(r)N(cid:88)−1 (cid:88)N 1
VSCE[ρ]= dr
ee N fi(r) fj(r) A. Higher-order corrections to zeroth-order KS
i=1 j=i+1| − |
(9) SCE
(cid:90) N
1 (cid:88) 1
= drρ(r) .
2 r fi(r) In order to discuss corrections to KS SCE, it is useful
i=2 | − |
to rewrite the approximation of Eq. (13) in the language
An important property of the SCE system is the follow- of the usual adiabatic connection (coupling-constant in-
ing one: since the position of one electron at a given tegration)ofDFT.28,29 TheHKfunctionalofEq.(1)and
r determines the other N 1 electronic positions, the theKSkineticenergyfunctionalofEq.(2)canbeseenas
−
net Coulomb repulsion acting on a certain position r be- thevalueatλ=1andλ=0ofamoregeneralfunctional
comes a function of r itself. As a consequence, this force F [ρ], in which the electronic interaction is rescaled by a
λ
can be written in terms of the negative gradient of some coupling strength parameter λ,
one-body local potential v (r),15 such that
SCE (cid:68) (cid:69)
F [ρ]= min ΨTˆ+λVˆ Ψ . (16)
λ ee
v [ρ](r) F (r)=(cid:88)N r−fi[ρ](r) . Ψ→ρ | |
−∇ SCE ≡ Coulomb i=2 |r−fi[ρ](r)|3 By denoting Ψλ[ρ] the minimizing wave function in
(10) Eq. (16), and by defining
From the above equation one can see that for a finite
(cid:68) (cid:69)
systeminthelimit r thepotentialvSCE[ρ](r)goes Wλ[ρ] Ψλ[ρ]Vˆee Ψλ[ρ] EHartree[ρ], (17)
| |→∞ ≡ | | −
as
N 1 one obtains the well-known exact formula28 for the
vSCE[ρ](|r|→∞)= r− . (11) exchange-correlation functional Exc[ρ],
| |
(cid:90) 1
Furthermore, it can be shown that it satisfies the impor-
E [ρ]= W [ρ]dλ. (18)
tant exact relation15 xc λ
0
vSCE[ρ](r)= δVδeSρeC(rE)[ρ] , (12) tTehrme fuinnctthieoneaxlpVaneSesCioEn[ρo]f−WEH[ρar]trweeh[ρen] iλs the ze.roTthhe-onrdexert
λ
leading term in the series is given by26,30→ ∞
thusprovidingapowerfulshortcuttotheconstructionof
the functional derivative of the SCE functional. W(cid:48) [ρ]
The zeroth-order “KS SCE” approach15 approximates Wλ→∞[ρ] = W∞[ρ]+ √∞λ +O(λ−p) (19)
the Hohenberg-Kohn (HK) functional of Eq. (1) as
W [ρ] = VSCE[ρ] E [ρ] (20)
F[ρ] min(cid:68)ΨTˆ Ψ(cid:69)+ min(cid:68)ΨVˆee Ψ(cid:69) W∞∞(cid:48) [ρ] = VeeZeePE[ρ],− Hartree (21)
≈Ψ→ρ | | Ψ→ρ | |
=T [ρ]+VSCE[ρ], (13) whereZPEstandsfor“zero-pointenergy”, andp 5/4–
s ee seeRef.26 forfurtherdetails. Physically,thezeroth≥-order
yielding a rigorous lower-bound to the exact energy,14,15 term VeSeCE[ρ] in the expansion (21) corresponds to the
since interactionenergywhentheelectronsare“frozen”inthe
strictly-correlated positions of the SCE floating Wigner
T [ρ]+VSCE[ρ] F[ρ]. (14) crystal. The ZPE term in the series takes into account
s ee ≤
4
small vibrations of the electrons around their SCE posi-
tions,anditisgivenby(forelectronsinD dimensions)26 �=1 �
!1
VZPE[ρ]= 1(cid:90) drρ(r)D(cid:88)N−D ωn(r) . (22)
ee 2 N 2
n=1 W0
W +
1
The ω (r) are the zero-point-energy vibrational frequen- 1 p�
n
cies around the SCE minimum,26 given by the square ESCE
xc
root of the eigenvalues of the Hessian matrix entering
W
the expansion up to second order of the potential energy
1
of the electrons in the SCE system.26
We see that the KS SCE approximation of Eq. (13)
corresponds to setting W [ρ] = W [ρ] in the integrand �=1 �
λ ∞ !1
ofEq.(18),asschematicallyshownbytheredareainthe
upper panel of Fig. 2. The exact correction to KS SCE
would be tThe[ρs]u=m oΨf the[ρk]inTˆetΨic co[rρre]latioTn[ρe]n,ergy, (23) Ex excess W1+ Wp1�0
c λ=1 λ=1 s
(cid:104) | | (cid:105)−
and of the electron-electron decorrelation energy,12,31 Tc+Vede W�
W
Vd[ρ]= Ψ [ρ]Vˆ Ψ [ρ] VSCE[ρ]. (24) 1
ee (cid:104) λ=1 | ee| λ=1 (cid:105)− ee
If we insert the expansion of Eq. (19) into Eq. (18) we
obtain FIG. 2: Schematic representation of the functional W [ρ] of
λ
Eq.(17)asafunctionofλ. TheKSSCEapproximationcor-
Tc[ρ]+Vede[ρ]≈2VeZePE[ρ]. (25) responds to setting the exchange-correlation energy equal to
theredareaintheupperpanel. Simplyaddingthezeropoint
Thiscorrection,however,isingeneralwaytoolarge,asit term, corresponds to overcorrecting the KS SCE energy, by
includes the positive contribution to the integrand com- including also the positive blue area in the upper panel. The
ing from the integrable divergence λ−1/2, as repre- simple approximation of Eqs. (26)-(28) removes the excess
sented by the blue area in the upper∝panel of Fig. 2. In energy by shifting the value of Wλ[ρ] at λ=0.
order to get a more realistic correction, we consider here
a simplified interaction-strength-interpolation (ISI)29,32
which sets the value of W [ρ] at λ=0 equal to its exact ordertoavoidthedivergenceoftheCoulombinteraction
λ
value, the exchange energy E [ρ], inonedimension,weconsidertheelectronsandthenuclei
x
to interact via a soft-Coulomb potential given by
W(cid:48) [ρ]
WisiZP[ρ] = W [ρ]+ ∞ , (26)
λ ∞ (cid:112)λ+a[ρ] v (x)= q1q2 , (29)
soft
(cid:18) W(cid:48) [ρ] (cid:19)2 √1+x2
a[ρ] = ∞ . (27)
Ex[ρ]−W∞[ρ] where q1 and q2 are the charges of the nuclei/electrons.
For the calculations presented here we have considered
In this way, we remove the excess positive contribution,
nuclear charges q =1, 2, 3 and 4, corresponding to the
1
as shown in the lower panel of Fig. 2. We then obtain
elements H, He, Li and Be, respectively. In each case we
the renormalized correction
have also studied different ionic species. Finally, for the
(cid:16)(cid:112) (cid:112) (cid:17) H molecule, wehaveconsideredinteratomicseparations
T [ρ]+Vd[ρ] 2VZPE[ρ] 1+a[ρ] a[ρ] . (28) 2
c ee ≈ ee − RH−H in the range between 0 and 20 atomic units.
Thiscorrectionissizeconsistentonlywhenasystemdis-
sociates into equal fragments. A full size-consistent ap-
proximationwouldrequirealocalinterpolationalongthe A. Calculation of the co-motion functions and of
adiabatic connection, as discussed in Ref.27. the SCE potential
In one dimension, the co-motion functions f (x) can
i
III. DETAILS OF THE CALCULATIONS FOR be calculated analytically by integrating Eqs. (6) for a
ONE-DIMENSIONAL MODEL SYSTEMS given density ρ(x),11,18,33 choosing boundary conditions
that make the density between two adjacent strictly-
We have considered one-dimensional (1D) models for correlated positions always integrate to 1 (total sup-
different atoms and ions and for the H molecule. In pression of fluctuations),11 as schematically illustrated
2
5
B. Calculation of the Zero-Point Energies (ZPE) in
1D
To compute the ZPE we start from the classical en-
ergy expression of the SCE system as a function of the
coordinates of the individual electrons. Since the kinetic
1.0 1.0 energyintheλ limitbecomesinfinitelysmall, only
0.4 0.6
→∞
the potential parts remain: the interaction between the
x f (x) f (x) electrons and the SCE potential
2 3
N−1 N N
(cid:88) (cid:88) (cid:88)
E (x ,...,x )= w(x x ) v (x ).
pot 1 N i j SCE i
| − | −
FIG. 3: Schematic illustration of the co-motion functions in i=1 j=i+1 i=1
1D.Twoadjacentstrictlycorrelatedpositionsarealwayssep- (34)
arated by a distance such that the density between them in- Notice that the SCE potential counteracts exactly the
tegrates exactly to one electron. repulsive forces due to the interaction, making E
pot
stationary on the 1D subspace of RN parametrized by
f(x) x = f (x) = x,x = f (x),...,x = f (x) .
1 1 2 2 N N
≡ { }
in Fig. 3, The diagonal contributions to the Hessian are readily
evaluated to be
(cid:90)fif(ix+)1(x)ρ(x(cid:48))dx(cid:48) =1, (30) ∂x2iEpot(cid:0)f(x)(cid:1)=(cid:88)N w(cid:48)(cid:48)(|fi(x)−fk(x)|)ρρ(cid:0)(cid:0)ffki((xx))(cid:1)(cid:1) (35)
k(cid:54)=i
and ensuring that the f (x) satisfy the required group and the off-diagonal elements become (i=j)
i
(cid:54)
properties of Eq. (7).11,18,25 This yields ∂ ∂ E (cid:0)f(x)(cid:1)= w(cid:48)(cid:48)(f (x) f (x)). (36)
xi xj pot − | i − j |
(cid:40)N−1[N (x)+i 1] x a In the special case of a two-electron system, the matrix
fi(x)= Ne−1[Ne(x)+i−1 N] x≤>aN+1−i, (31) elements of the Hessian simplify to
where the funectioneN (x) i−s de−fined as N+1−i ∂x21Epot(cid:0)x,f(x)(cid:1)=w(cid:48)(cid:48)(|x−f(x)|)ρ(cid:0)ρf((xx))(cid:1),
e
(cid:0) (cid:1)
(cid:90) x ∂2 E (cid:0)x,f(x)(cid:1)=w(cid:48)(cid:48)(x f(x))ρ f(x) , (37)
N (x)= ρ(x(cid:48))dx(cid:48), (32) x2 pot | − | ρ(x)
e
−∞ ∂ ∂ E (cid:0)x,f(x)(cid:1)= w(cid:48)(cid:48)(x f(x)),
x1 x2 pot − | − |
and ak =Ne−1(k). Equation (10) becomes in this case whereweusedf1(x)=xandhavesetf(x)=f2(x). The
Hessian is readily diagonalized and gives the zero-point
N frequency
(cid:88)
v(cid:48) [ρ](x)= w(cid:48)(x f (x))sgn(x f (x)), (33) (cid:118)
SCE i=2 | − i | − i ω(x)=(cid:117)(cid:117)(cid:116)w(cid:48)(cid:48)(cid:0)x f(x)(cid:1)(cid:32) (cid:0)ρ(x)(cid:1) + ρ(cid:0)f(x)(cid:1)(cid:33), (38)
| − | ρ f(x) ρ(x)
where w(x) denotes the interaction between the parti-
cles, which will be the soft-Coulomb interaction (29) in which can be used in the ZPE expression (22) to calcu-
our case, and w(cid:48)(x) is its derivative. Notice the highly late the ZPE correction. The other eigenvalue is zero as
non-localdependenceoftheco-motionfunctionsonρ(x), expected, since the SCE system corresponds to a float-
as clearly shown by Eqs. (31)-(32), and the great simpli- ing Wigner crystal, and E should be degenerate on
pot
ficationofthefunctionalderivativeoftheSCEfunctional f(x) x R . Hence, the energy surface is flat in this
{ | ∈ }
provided by Eq. (33). direction which gives a vanishing eigenvalue in the Hes-
To perform practical calculations, one must proceed sian of the classical potential energy.
self-consistentlyinthreesteps: i)generatetheco-motion For N > 2 we have diagonalized numerically the Hes-
functions via Eqs. (31)-(32) for a given density ρ(x); ii) sian matrix on our grid. This needs to be done just in
calculatevSCE(x)byintegratingEq.(33)withthebound- one interval between two adjacent ai = Ne−1(i), e.g. for
aryconditionvSCE[ρ](x )=0; iii)usetheapproxi- x [a1,a2],becausethepropertiesoftheco-motionfunc-
| |→∞ ∈
mationv (x) v (x)tosolvetheKohn-Shamequa- tions ensure that
Hxc SCE
≈
tions (3). The total energy is then obtained by adding (cid:90) ∞ ρ(x)N(cid:88)−1 (cid:90) ai+1 N(cid:88)−1
the external potential contribution to Eq. (13). As said, dx ω (x)= dxρ(x) ω (x).
n n
we work in the original spin-restricted KS framework, in −∞ N n=1 ai n=1
which each spatial orbital is doubly occupied. (39)
6
A warning should be added because the soft Coulomb
TABLE I: Total energies obtained with the different ap-
interaction w(x) = v (x) of Eq. (29) is not convex for
soft proaches. The DMRG data are from Ref.22
x<2−1/2. AsitisevidentfromtheformulasforN =2of
KS SCE DMRG LDA LSDA KS SCE + isiZPE
Eq.(38),theZPEbreaksdownwhenw(cid:48)(cid:48)(x f(x))<0.
| − | H -0.67 -0.67 -0.60 -0.65 -0.67
For all the systems studied, the minimum possible SCE
H− -0.89 -0.73 - - -0.75
electron-electron distance is always larger than 2−1/2, so
He -2.38 -2.24 -2.20 -2.20 -2.25
thatthenonconvexityofw(x)doesnotposeanyproblem. He− -2.42 - - - -2.22
He+ -1.48 -1.48 -1.41 -1.45 -1.48
Li -4.43 -4.21 -4.16 -4.18 -4.22
C. Comparison with other approaches Li− -4.51 - - - -4.19
Li+ -4.02 -3.90 -3.85 -3.85 -3.92
We validate our results by comparing them with those Li2+ -2.34 -2.34 -2.26 -2.30 -2.34
Be -7.12 -6.79 -6.76 -6.76 -6.79
obtained from the density matrix renormalization group
(DMRG) as described in Ref.22. For the H poten- Be+ -6.65 -6.45 -6.39 -6.41 -6.47
2 Be2+ -5.72 -5.62 -5.56 -5.56 -5.64
tial energy curve, we have also carried out full CI
Be3+ -3.21 -3.21 -3.13 -3.18 -3.21
calculations on a numerical grid. We have also per-
formed, for comparison, calculations with the Kohn-
Sham local-density-approximation in both the spin-
TABLEII: SameasTableIfortheionizationenergies(using
restricted (LDA) and spin-unrestricted (LSDA) formu-
the HOMO eigenvalue for the DFT approaches)
lations. The parametrization of the L(S)DA exchange-
−εKSSCE DMRG −εLDA −εLSDA
correlation functional with soft-coulomb interaction is HOMO HOMO HOMO
H 0.67 0.67 0.35 0.41
taken from Ref.21.
H− 0.089 0.06 - -
He 0.72 0.75 0.48 0.48
He+ 1.48 1.48 1.12 1.18
IV. RESULTS Li 0.32 0.31 0.14 0.17
Li+ 1.50 1.56 1.24 1.24
A. 1D atoms and ions Li2+ 2.34 2.34 1.95 2.00
Be 0.34 0.34 0.16 0.16
Be+ 0.81 0.83 0.60 0.63
TableIshowsthetotalenergiesfordifferentatomicele-
Be2+ 2.34 2.41 2.06 2.06
ments,comparingtheresultsobtainedwiththeKSSCE,
Be3+ 3.21 3.21 2.81 2.86
DMRG and KS L(S)DA approaches. The renormalized
ZPEcorrection“isiZP”ofEq.(28)hasbeenaddedatthe
postfunctional level to the KS SCE self-consistent ener-
asymptotic behavior of the SCE potential.
gies. We see that KS SCE largely underestimates all the
total energies (except for the N = 1 systems for which
it is exact), providing a lower bound that it is not very
B. 1D H molecule
tight, and resulting in an accuracyworse thanthe one of 2
L(S)DA. The isiZP correction improves the results con-
sistently, getting much closer to the DMRG calculations Figure4showsthedissociationenergycurvesobtained
than L(S)DA. from the various methods. One can see that whereas the
Particularlyinterestingisthecaseofthenegativeions, KS LDA, KS LSDA and restricted Hartree-Fock (HF)
that are a notorious problem in approximate KS DFT. energies are relatively close to the CI values near equi-
Similarly to the 3D case, the anions are all not bound in librium, the KS SCE approach yields a large error due
L(S)DA, while KS SCE overbinds, yielding a bound sys- toitsoverestimationoftheelectroniccorrelation. Asthe
temalsoforHe− andLi−,whichareunboundinDMRG. interatomicdistanceincreases,however,onecanseethat
The isiZP correction, however, correctly predicts H− to while the spin-resticted LDA and HF energies as usual
be bound (with a rather good energy) and He− and Li− becometoopositive,theKSSCEresultbecomesnowin-
to be unbound (as their energy becomes higher than the creasingly more accurate, tending to the exact curve in
one of the corresponding neutral system). Notice that in the dissociation limit. The ability to correctly describe
Ref.21 the 1D He− and Li− were predicted to be bound, this limit is remarkable in a spin-restricted formalism.
although the value reported for the energy of Li− was In the figure we also show the energy curve obtained
higher than the one of Li. With DMRG we found these whenthefullZPEcorrectionandtherenormalizedisiZP
two 1D anions to be unbound. corrections are added, at a postfunctional level, to the
Table II is the analogue of Table I comparing the neg- zeroth-order KS SCE energies. One can see that, as
ative of the highest occupied (HOMO) KS eigenvalues expected from the discussion in Sec. IIA, the addition
with the ionization energies obtained from DMRG. We of the bare ZPE gives energies way too high. The in-
see that the KS SCE HOMO yields quite accurate es- clusion of the isiZP, instead, gives very good results for
timates of the ionization energies, thanks to the right R (cid:46) 4 a.u, but displays a “bump” in the potential
H−H
7
−1.0 tion of the correlation. The LDA approach shows a very
0th-order KS-SCE + ZPE goodagreementwiththeexactresult. Astheinteratomic
−1.1 separation increases, here with RH−H = 5, the LDA
HF
KS LDA largelyoverestimatesthedensityatthemidpoint,clearly
reflecting its inability to properly describe the molecular
−1.2
dissociation process. The KS SCE approach, instead,
KS LSDA
u.] shows an improving tendency in the direction of the ex-
y [a.−1.3 act result, in accordance with the energy curve of Fig. 4.
erg Finally, in the dissociation limit, represented in the fig-
n
e−1.4 ure by RH−H = 15, the agreement between the KS SCE
CI
and CI densities is excellent, whereas the spin-restricted
−1.5 0th-order KS-SCE + isiZPE LDA density is too delocalized.
The rapid decrease of the exact density at the mid-
−1.6 0th-order KS-SCE bondastheinteratomicseparationRH−H increasesisre-
lated to the existence of a barrier in the corresponding
Kohn-Shampotential,34,35 which weshow inFig.6. The
0 2 4 6 8 10 12 14
RH-H [a.u.] exact barrier has a component that is known to saturate
for large internuclear distances R with a height de-
H−H
FIG. 4: Dissociation energy curves for H corresponding to termined by the ionization potential.35 This component
2
the different approaches. is due to the kinetic correlation energy34,35 and is thus
not captured by the SCE functional that lacks the ki-
netic energy contribution. The KS SCE barrier, thus,
decreases when R increases, and becomes small at
0.6 H−H
KS-SCE
large R . However, at large internuclear distances,
KS LDA R =1.5 H−H
0.4 CI H-H the energetic contribution of the barrier is negligible so
that even a very small barrier (as the one obtained in
0.2 KS SCE) is enough to get an accurate localized density
and the correct energy at dissociation. When the confin-
0 ing potential is harmonic, as in the quantum wires and
quantum dots studied in Ref.14–16, the barriers remain
0.3 finite in the KS SCE potential also at very low density.
R =5
x) H-H In LDA we see that at large RH−H there is a barrier lo-
(ρ0.2 calizedontheatomsratherthaninthemidbond,leading
to overestimation of the charge in the bond.
0.1
0
V. CONCLUSIONS AND PERSPECTIVES
0.3 R =15
H-H
The strictly-correlated electrons functional provides
0.2
an alternative route to the construction of approxima-
0.1 tions for KS DFT. Instead of following the standard
path of including more and more ingredients (“Jacob’s
0
0 4 8 12 ladder”)36,37 such as the local density, the local density
x
gradients,theKohn-Shamlocalkineticenergies,etc.,the
new ingredient here is the non-locality encoded in the
FIG.5: DensitiesfortheH2 moleculecorrespondingtodiffer- SCE functional and higher-order corrections.
entinteratomicseparationsobtainedwiththeKSSCE,LDA,
While for low-dimensional nanodevices, which can
and CI approaches.
reach very low densities because of their stronger
confining potentials, the KS SCE approach is very
accurate,15,16 chemical systems are in general not close
energy curve, which now tends from above to the exact enough to the strong-interaction limit, so that total en-
dissociation limit, reached only at RH−H (cid:38)20. ergies are too low. We see, however, that the SCE
Figure 5 shows the electronic densities obtained with functional is able to capture the strong correlation of
the KS SCE, KS LDA, and CI approaches for differ- a stretched bond, and, with a correction renormalized
ent interatomic separations R . One can see that for with exact exchange, to yield accurate results for total
H−H
R = 1.5, which corresponds to a near-equilibrium energies, predicting the delicate physics of negative ions.
H−H
configuration, the KSSCEdensity isslightlyless peaked Thanks to the right asymptotic properties of the SCE
at the midbond due to the above-mentioned overestima- potential, the KS SCE HOMO also yields an accurate
8
ent approach17). However, it should be possible to build
0.8 R =0
H H-H approximateco-motionfunctionsor,moregenerally,non-
2 RH-H=2
R =5 localfunctionalsinspiredtotheSCEmathematicalstruc-
e H-H
vsc0.4 RH-H=10 ture.
R =15
H-H It seems also clear that the accuracy of SCE is some-
R =20
H-H whatcomplementarytotheonesofstandardfunctionals,
sothatcorrectionstoSCEwhicheitherincludeexactex-
0.8
change (like the simple one presented here), or based on
ct standardapproximationscanbepursuedinthefuture. In
a
x general,givenanapproximateexchange-correlationfunc-
ec 0.4 tional Exacpprox[ρ] it is possible to extract from it a cor-
hx rectiontoKSSCEbyusingthescalingproperties38,39 of
v
DFT.Bydefining,forelectronsinDdimensions,ascaled
density ρ (r) γDρ(γr) with γ >0, we have39
γ
≡
LDA 0.8 Tc[ρ]+Vede[ρ]≈Exacpprox[ρ]−γli→m0γ1Exacpprox[ργ]. (40)
c
hx0.4 ThiswayofconstructingcorrectionstoKSSCEhasbeen
v
tested by using the LDA functional in Refs.15 for quan-
tum wires and very recently in Ref.40 for the anions of
0 5 10 15
x theHeisoloectronicseries. Whileintheformercasethey
gave very disappointing results, in the latter they im-
proved the results considerably, showing that for chemi-
FIG. 6: Exact Kohn-Sham potential (by inversion), Hatree- cal problems this could be a good way to proceed.
exchange-correlation potential from KS LDA, and v po-
SCE
tential for different interatomic separations R for the 1D
H−H
H molecule.
2
Acknowledgements
estimate of ionization energies. P.G-G. and K.J.H.G. acknowledge support from the
Overall, it seems promising to use the SCE physics NetherlandsOrganizationforScientificResearch(NWO)
as an ingredient to build approximate functionals. An through a VIDI and a VENI grant, respectively. This
exact evaluation of the co-motion functions in the gen- research was also supported by a Marie Curie Intra Eu-
eral 3D case might turn out to be too demanding (al- ropean Fellowship within the 7th European Community
though progress has been done recently through a differ- Framework Programme (F.M.).
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