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Adiabatic Connection for Strictly-Correlated Electrons

Zhen-Fei Liu and Kieron Burke

Department of Chemistry, University of California, Irvine, California, 92697-2025, USA

(Dated: October 1, 2009)

Modern density functional theory (DFT) calculations employ the Kohn-Sham (KS) system of

non-interacting electrons as a reference, with all complications buried in the exchange-correlation

energy (EXC). The adiabatic connection formula gives an exact expression for EXC. We consider

DFT calculations that instead employ a reference of strictly-correlated electrons.

“decorrelation energy” that relates this reference to the real system, and derive the corresponding

adiabatic connection formula. We illustrate this theory in three situations, namely the uniform

electron gas, Hooke’s atom, and the stretched hydrogen molecule. The adiabatic connection for

strictly-correlated electrons provides an alternative perspective for understanding density functional

theory and constructing approximate functionals.

We define a

I. INTRODUCTION

For most modern calculations using density functional

theory (DFT) [1], the accuracy of results depends only on

approximations to the exchange-correlation functional,

EXC[n].An exact expression for EXC[n] is given by

the adiabatic connection formula [2, 3], in which EXC

is expressed as an integral over a coupling constant λ,

which connects the reference (Kohn-Sham system, λ = 0)

and the real physical interacting system (ground state,

λ = 1), keeping the density n(r) fixed. Study of the

adiabatic connection integral has proven very useful for

understanding approximate (hybrid) functionals [4, 5],

and is an ongoing area of research [6, 7, 8].

Almost all modern DFT calculations employ the Kohn-

Sham (KS) system [9] as a reference. The KS system is

defined as the unique fictitious system that has the same

density as the real system, but consists of non-interacting

electrons. The great practicality of KS DFT is due to the

relative ease with which the non-interacting equations

can be solved, with relatively crude approximations, giv-

ing KS DFT a useful balance between speed and accu-

racy.

However, DFT calculations could also be based on an-

other fictitious system which is known as the strictly-

correlated (SC) system [10]. The strictly-correlated sys-

tem has the same density as the real system (as does

the KS system), but the Hamiltonian consists of elec-

tron repulsion and external potential energy terms only.

In recent years, the pioneering work of Seidl and oth-

ers [10, 11, 12, 13, 14, 15, 16, 17] has led to substan-

tial progress in solving this problem exactly and effi-

ciently. The strictly-correlated electrons (SCE) ansatz

[14, 15, 16, 17] has been shown to yield the density and

energy of this system, going beyond the earlier point-

charge-plus-continuum (PC) model [12, 13]. They have

achieved great success in calculating spherical symmetric

systems with arbitrary number of electrons [15].

In this article, we look to the future and assume that

the strictly-correlated limit of any system can be cal-

culated with less difficulty than the original interacting

problem, and all our successive work is developed based

on this reference. We derive a new version of the adia-

batic connection formalism, which connects the strictly-

correlated system (fully interacting) and the physical sys-

tem. We also introduce a new coupling constant µ, and

a “decorrelation energy” EDC, the counterpart of EXCin

KS DFT, which must be evaluated to extract the true

ground-state energy from the calculation of the strictly-

correlated system. We argue that, as long as the strictly-

correlated system can be solved easily (just as the KS

case), one can develop another version of DFT based on

this system, a version that is better-suited to strongly

localized electrons.

Throughout this paper, we use atomic units (e2= ¯ h =

µ = 1), which means that if not particularly mentioned,

all energies are in Hartrees, and all lengths are in Bohr

radii, etc.

II. THEORY

In this section, we introduce the alternative adiabatic

connection formula, and relate its quantities to more fa-

miliar ones. All results here are formally exact.

A. Kohn-Sham Adiabatic Connection

In KS DFT, the total energy for the interacting

ground-state is expressed as:

?

In this equation, TS is non-interacting kinetic energy

of KS orbitals {ϕi} that are eigenfunctions of the non-

interacting KS equation, vext(r) is external potential (nu-

clear attraction in the case of atoms and molecules), U

is the Hartree energy defined as the “classical” Coulomb

repulsion between two electron clouds, and EXC is the

exchange-correlation energy [18]. The adiabatic connec-

tion integral [2, 3] then gives an exact expression for EXC:

E[n] = TS[n] +

d3rvext(r)n(r) + U[n] + EXC[n]. (1)

EXC[n] =

?1

0

dλWλ[n],

(2)

arXiv:0907.2736v2 [cond-mat.other] 1 Oct 2009

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2

where one can show that Wλ=< Ψλ|ˆVee|Ψλ> −U, in

which Ψλis the wavefunction that minimizesˆT + λˆVee

but has the same density as the real ground-state system

[19]. At λ = 0, one recovers the KS system, and at λ = 1,

one recovers the real interacting system. In this way, one

connects the KS system with the real interacting system

by changing λ from 0 to 1.

A cartoon of Wλversus λ is shown in the upper panel

of Fig. 1. By definition, we have W0= EXand the area

under the curve is EXC. We can also identify the kinetic

correlation energy TC= EXC− W1in this graph.

0

1

λ

0

EX

W1

Wλ

EX

EC

-TC

01

µ

2TC

0

Kµ

EDC

2W′∞

FIG. 1: Traditional (upper panel) and the new (lower panel)

adiabatic connection curves.

B.Strictly-Correlated Reference System

The KS wavefunction can be defined as the wavefunc-

tion that minimizes the kinetic energy alone, but whose

density matches that of the interacting system. Analo-

gously, the SC wavefunction is found by minimizing the

electron-electron repulsion energy alone, subject to re-

producing the exact density. In practice, there might be

multiple degeneracies, so it is best defined in the limit as

the kinetic energy goes to zero.

Then, using the strictly-correlated (SC) system as the

reference, the energy of the true interacting ground state

is:

?

where Usc =< Ψ∞|ˆVee|Ψ∞>. In KS DFT quantities,

Usc= U + W∞[see Eq. (2)].

Just as we separate the Hartree energy from the total

energy in KS DFT [Eq. (1)], here in Eq. (3) we sepa-

rate TS[n] from the total energy in SC DFT. There are

a variety of algorithms that can be used to extract this

quantity accurately for any given density, effectively by

inverting the KS equations [20]. We label the remainder

as the “decorrelation energy”, EDC[n]. The reason we call

it “decorrelation energy” is that, if we consider the elec-

trons in the reference system “strictly correlated”, with

energy Usc[n], the electrons in the real system are less

correlated than in the reference system. We will see the

physical meaning explicitly very soon.

So far, we have defined our reference, and next we de-

duce an exact expression for the newly-defined “decor-

relation energy” EDC[n] with the adiabatic connection

formalism, just as one does for EXC[n] [10, 12] in the KS

DFT [Eq.( 2)].

E[n] = Usc[n]+

d3rvext(r)n(r)+TS[n]+EDC[n], (3)

C.Strictly-Correlated Adiabatic Connection

Formula

We denote Ψµas the wavefunction minimizingˆHµ=

µ2ˆT +ˆVee+ˆVµ

the strictly-correlated system, and for µ = 1, we recover

the real system. For each value of µ, there is a corre-

sponding unique external potential yielding the correct

density, vµ

extwith density n(r). For µ = 0, we recover

ext(r). So we have:

Eµ= ?Ψµ|µ2ˆT +ˆVee+ˆVµ

Using Hellmann-Feynman theorem [21], we have:

?

dµ

ext|Ψµ?

(4)

dEµ

dµ

=Ψµ

?????

dˆHµ

?????Ψµ

?

=

?

Ψµ

?????2µˆT +dˆVµ

ext

dµ

?????Ψµ

?

.

(5)

Integrating and cancelling the external potential terms at

both sides, we recognize the left hand side is just TS[n]+

EDC[n]. Thus:

EDC[n] =

?1

0

dµ2µ

?

Ψµ???ˆT

???Ψµ?

− TS[n].

(6)

This is our adiabatic connection formula for strictly-

correlated electrons. Finally, since TC[n] = T[n] − TS[n],

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3

and TS[n] is independent of µ:

EDC[n] =

?1

0

dµKµ[n],

(7)

where

Kµ[n] = 2µTµ

C[n].

(8)

This is the SC doppelganger of Eq. (2). We plot a car-

toon of the integrand Kµ vs. µ in the lower panel of

Fig. 1, identifying the area below the curve as EDC, and

noting that K1= 2TC.

D. Relation to Kohn-Sham DFT

From a formal viewpoint, what we derived here is not

new, but simply another way to describe the real inter-

acting system. Thus we can relate all quantities defined

here, such as EDC[n] and Kµ[n], to quantities defined in

the traditional KS DFT. Since both Eq. (3) and Eq. (1)

are exact for the real system, and if we use the expression

of Usc[n] in KS language [see discussion below Eq. (3)],

we find:

EDC[n] = EXC[n] − W∞[n].

(9)

Thus EDC[n] defined in our theory is just the difference

between the usual exchange-correlation energy of the real

system, EXC[n], and the potential contribution to the

exchange-correlation energy of the strictly-correlated sys-

tem, W∞[n].

We can also deduce an expression for Kµin terms of

Wλ. Since ΨµminimizesˆHµ= µ2ˆT +ˆVeewhile yielding

n(r), and ΨλminimizesˆT +λˆVee, we deduce Ψ1/µ2= Ψλ.

Now, from the scaling properties of KS DFT [22], we

know:

Tλ

C= Eλ

C− λdEλ

C

dλ.

(10)

If we write Eλ

tribtion to Eλ

C= Tλ

C, Uλ

C+ Uλ

C= λ(Wλ− EX), we have:

dTλ

C

dλλ

C, i.e., Uλ

Cis the potential con-

=Uλ

C

−dUλ

C

dλ.

(11)

Integrating over λ from 0 to 1/µ2, and using the defini-

tion of Wλin Eq. (2) and that Eλ

we can express Tµ

X= λEXby scaling [22],

C[n] in terms of Wλ[n], we find:

Kµ[n] = 2µ

?1/µ2

0

dλ?Wλ[n] − W1/µ2[n]?.

(12)

From this relation, we can generate the new adiabatic

connection curve, as long as we know the integrand of

the KS adiabatic connection, i.e. Wλ[n] for λ = 1 to ∞.

E.Exact conditions

Many of the well-established exact conditions on the

correlation energy can be translated and applied to the

decorrelation energy. In particular, the simple relations

between scaling the density and altering the coupling con-

stant all apply, i.e.,

Eλ

C[n] = λ2EC[n1/λ],

(13)

where nγ(r) = γ3n(γr). Thus, in terms of scaling:

Kµ[n] =

2

µ3TC[nµ2].

(14)

Note that, as µ → ∞, Kµ → 0, while Kµ=0 = 2W?

where W?

[12]:

∞,

∞is defined in the expansion of Wλas λ → ∞

W?

∞= lim

λ→∞

√

λ(Wλ− W∞).

(15)

Thus the SC energy is found from solving the strictly-

correlated system, while Kµ=0is determined by the zero-

point oscillations around that solution. Both are cur-

rently calculable for spherical systems [15, 17].

The most general property known about the correla-

tion energy [22] is that, under scaling toward lower densi-

ties, it must become more negative. In turn, this implies

that Wλis never positive. Using the definition of Tµ

changing variable λ = 1/µ2in Eq. (11), we find:

C and

dTµ

dµ

C

=

2

µ5

dWλ

dλ

< 0,

(16)

then using Kµ = 2µTµ

find:

C and the fact that Kµ > 0, we

d

dµlnKµ< 0.

(17)

Also, because Tµ

an integration of Kµ, is always positive.

Based on these properties of Kµ, a crude approxima-

tion to Kµcan be a simple exponential parametrization,

using K0and the derivative of lnK at µ = 0 as inputs:

C > 0, so Kµ= 2µTµ

C > 0, and EDC, as

K = K0e−γµ,γ = −

d

dµlnK

????

0

.

(18)

III. ILLUSTRATIONS

In this section, we illustrate the theory developed

above on three different systems, to show how Kµ be-

haves for very different systems, and where the adia-

batic connection formula might be most usefully approx-

imated.

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4

A.Uniform Electron Gas

For a uniform electron gas, we assume we know the cor-

relation energy per particle, ?C, accurately as a function

of rs= (3/4πn)1/3. In order to apply Eq. (12) to calcu-

late Kµ[n], we use ?λ

into Eq. (10), changing variables λ = 1/µ2, and using

Kµ= 2µTµ

find:

= −2

µ3

drs

C(rs) = λ2?C(λrs) [22]. Substituting

C = Nκµ, with N the number of particles, we

κunif

µ

d

(rs?C(rs))|rs/µ2.

(19)

Using Eq. (9) and the definition of W∞, we find:

?unif

DC = ?C+d0

rs,

(20)

where d0 is defined below and d0 = 0.433521. In the

large-rslimit or the low-density expansion [23]:

?C(rs) = −d0

rs

+

d1

r3/2

s

+d2

r2

s

+ ···

(21)

where d2= −3.66151 from data of Ref. [23]. Substitut-

ing this expansion into Eq. (19), we find:

κunif

µ

=

d1

r3/2

s

+ 2µd2

r2

s

+ ···

as µ → 0.

(22)

Thus κµis expected to have a well-behaved expansion in

powers of µ for small µ.

Using Perdew and Wang’s [23] parametrization of the

correlation energy of the uniform gas, we plot κµvs. µ

for rs= 1 in Fig. 2, and find ?DC= 0.374 at rs= 1.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

κ(µ)

µ

PW

exponential

FIG. 2: Exact adiabatic connection curve κµfor uniform elec-

tron gas (rs = 1) and a simple exponential parametrization.

Using the exact curve for rs= 1 in the simple expo-

nential parametrization [Eq. (18)], we find κ0= 1.44073

and γ = 5.0826. We plot the exponential parametriza-

tion in Fig. 2 and we can see that it decays much faster

than the exact curve, producing a ?DCthat is too small

by about 25%, which means about 150% larger in |?C|

[see Eq. (9)].

We calculated ?DC/|?C| for different values of rs and

plot the curve in Fig. 3. At small rs, ?DC? |?C|, which

suggests that the KS reference system is a better starting

point, as a smaller contribution to the energy needs to be

approximated. At large rs, |?C| > ?DCso ?DCis a smaller

quantity and may be better approximated. Under such

circumstances, the strictly-correlated system might serve

as a better reference. For the uniform gas, the switch-

over occurs at about rs= 16, which is at densities much

lower than those relevant to typical processes of chem-

ical and physical interest. However, as we show below,

for systems with static correlation, this regime can occur

much more easily.

8

6

4

2

0

20 1612840

EDC/|EC|

rs

FIG. 3: ?DC/|?C| for different rs for uniform electron gas.

B.Hooke’s Atom

As we pointed out, as long as we have an approximate

Wλ[n] for λ between 1 and ∞, we can substitute it into

Eq. (12) to get the new adiabatic connection formula for

the decorrelation energy. Of course, most such formulas

focus on the shape between 0 and 1, since only that sec-

tion is needed for the regular correlation energy. But any

such approximate formula can be equally applied to Kµ,

yielding an approximation for the decorrelation energy.

Peach et al. [5] analyze various parametrizations for Wλ,

and the same forms can be used as well to parametrize

Kµ, based on the similar shape of Wλand Kµcurves. In

general, application to Kµ will yield a distinct approx-

imation to the ground-state energy, with quite different

properties.

Page 5

5

To give just one example, one of the earliest sensible

smooth parametrizations is the [1,1] Pade of Ernzerhof

[24]:

?1 + bλ

One can imagine applying it with inputs of e.g., EX,

W?

from the SC limit. It yields a sensible approximation to

Wλin the 0 to 1 range, but because it was not designed

with the strictly-correlated limit in mind, the formula

itself is not immediately applicable to the decorrelation

energy, since, e.g., Kµ=0vanishes. However, much more

sensible is to make the same approximation directly for

Kµinstead, if one is doing an SC calculation, i.e.,

?

Wλ= a

1 + cλ

?

.

(23)

0given by G¨ orling-Levy perturbation theory, and W∞

Kµ= ˜ a

1 +˜bµ

1 + ˜ cµ

?

,

(24)

whose inputs could be K0, K?

Kµ=1. This is then a very different approximation from

the same idea applied to the usual adiabatic connection

formula.

On the other hand, there are several approximations

designed to span the entire range of λ, the most famous

being ISI (interaction strength interpolation) model [13]

developed by Seidl et al. This model uses the values and

the derivatives of Wλ at two limits, namely the high-

density limit (KS system, λ = 0) and the low-density

limit (strictly-correlated system, λ → ∞), to make an

interpolation. Another approximation to Wλ is devel-

oped in our previous work [25], which employs W0,W∞

and W?

obtained from the two models.

Hooke’s atom is a two-electron system (i.e., with

Coulomb repulsion) in a spherical harmonic well [26].

Using the accurate values W0

−0.743,W?

= −0.947µ + 1.029Aµ −0.336

where A =?1 + 0.653/µ2and B = A−0.263. With the

= −0.228

?1 + 0.354/µ2. We plot the two forms of

taken from Ref. [8]. We compare three quantities in

Table I. Although KISI

µ

contains a spurious µlnµ term as

µ → 0 [15, 17, 25], it nonetheless yields accurate results.

The simple model, applied with the usual inputs, is less

accurate pointwise, but integrates to an accurate value.

We can try the simple exponential parametrization Eq.

(18) for Kµagain for Hooke’s atom. Because we do not

know the value of d/dµlnKµat µ = 0 exactly, instead

0, and, e.g., a GGA for

0as inputs. We compare the approximate Kµ’s

=

−0.515,W∞

=

0= −0.101 [8], and W?

∞= 0.208 [27], we find:

KISI

µ

µB

+ 0.270µlnB, (25)

same data substituted in Wsimp[25], we find:

Ksimp

µ

α4µ(α3− α2+ 1) + 1.287µ(α − 1),

(26)

where α =

Kµ in Fig. 4. The exact curve (down to µ = 0.5) is

TABLE I: Comparison of several quantities for three approx-

imations to Kµ. Note: ISI uses K0 as an input. The exact

values are taken from Ref. [8].

exactISI simp exponential

K0 0.416 0.416 0.383

K1 0.058 0.054 0.059

EDC 0.189 0.191 0.190

0.456

0.058

0.193

we do an exponential fitting using the method of least

squares, with the exact Kµvalues (between µ = 0.5 and

1) taken from Ref. [8]. We plot Kµvs. µ in Fig. 4 and

compare several quantities in Table I.

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

K(µ)

µ

exact

exact K0

ISI

simp

exponential

FIG. 4: Adiabatic connection curves for Hooke’s atom. The

exact curve (down to µ = 0.5) is taken from Ref. [8].

C.H2 Bond Dissociation

Bond dissociation of the H2molecule produces a well-

known dilemma in computational chemistry [28, 29, 30,

31]. In the exact case, as the bond length R → ∞, the

hydrogen molecule should dissociate to two free hydrogen

atoms, with the ground state always a singlet and spin-

unpolarized. However, spin-restricted, e.g., restricted

Hartree-Fock or restricted Kohn-Sham DFT, give the

correct spin multiplicity, i.e.

eigenfunction ofˆS2, but produce an overestimated to-

tal energy, much higher than that of two free hydrogen

atoms. Spin-unrestricted, e.g., unrestricted Hartree-Fock

or unrestricted Kohn-Sham DFT, give a fairly good total

energy, but the wavefunction is spin-contaminated, i.e.,

the deviation of <ˆS2> from the exact value is signifi-

cant. This is known as “symmetry breaking” in H2bond

dissociation.

the wavefunction is an

Page 6

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Fuchs et al. [32] argued that DFT within RPA (ran-

dom phase approximation) gives a correct picture of

the H2 bond dissociation within the spin-restricted KS

scheme. They also gave highly-accurate adiabatic con-

nection curves for ground-state H2at bond length R =

1.4˚ A and stretched H2 at bond length R = 5˚ A. The

curves were interpolated particularly between λ = 0 and

1, shown as the difference of the integrand, ∆Wλ, be-

tween the stretched H2molecule and two free H atoms

(Fig. 1 and 3 of Ref. [32]).

For R = 1.4˚ A and R = 5˚ A, if we use an interpolation

(see Ref. 63 in Fuchs paper [32]) to estimate ∆Wλ, we

find reasonable values ∆W∞= −7.00 and ∆W∞= 0.13,

respectively. Using Eq. (9), we find ∆EDC= 4.96 and

∆EDC= 0.69, respectively. We compare ∆EDCand ∆EC

values in Table II. The comparison shows a physical ex-

ample where the strictly-correlated system is a better

starting point in the calculation.

TABLE II: Comparison of several quantities for stretched H2

at different bond lengths. The values for ∆EX and ∆EXC are

taken from Ref. [32]. All values are in eV.

bond length 1.4˚ A

∆EX

∆EXC

∆W∞

∆EC

∆EDC

5˚ A

∞

-0.98 5.85 8.5

-2.04 0.82 0.0

-7.00 0.13 0.0

-1.06 -5.03 -8.5

4.96 0.69 0.0

We can see that at the equilibrium bond length, |∆EC|

is much smaller than ∆EDC, presumably making it eas-

ier to approximate the ground-state energy starting from

the KS reference system. This is typical at equilibrium.

However for stretched bonds, ∆EDCis much smaller than

|∆EC|, and so ∆EDCinstead may be better accurately ap-

proximated in the calculation and the strictly-correlated

system should be a better reference.

strong static correlation, such as Cr2and O3, might fall

somewhere inbetween.

Molecules with

IV.CONCLUSION

In this paper, we constructed an adiabatic connection

formalism for the strictly-correlated system. We found

that this adiabatic connection formula and curve can be

well defined with respect to this new reference. Our for-

mula connects the strictly-correlated system and the real

system, by Eq. (7). We also defined the quantity for this

new integral “decorrelation energy” and related this with

the usual KS adiabatic connection. We illustrated how

the decorrelation energy behaves, using the uniform elec-

tron gas, Hooke’s atom, and stretched H2as examples.

We emphasize again that a real application of this

theory is only possible when the reference, i.e., the

strictly-correlated system, can be routinely calculated.

At present, one can calculate quantities such as Uscex-

actly only for spherical symmetric systems [14]. However,

nonempirical approximations to EXCof Kohn-Sham the-

ory can be employed to estimate W∞with useful accu-

racy [33]. The computation of this quantity may become

much easier in the future [34]. If this is so, then based

on the properties discussed here, the strictly-correlated

reference may be preferable in cases that are difficult for

standard KS DFT calculations with standard approxi-

mations to EXC.In fact, a recent work [34] indepen-

dently shows progress using exactly the formalism dis-

cussed here and suggests approximations to EDC. In any

event, the advent of strictly-correlated calculations opens

up a whole new alternative approach to DFT calculations

of electronic structure, and only experience can show how

and when this will prove more fruitful than the tradi-

tional (KS) scheme.

V. ACKNOWLEDGEMENT

We thank John Perdew, Michael Seidl and Paola Gori-

Giorgi for kind discussions.

by the National Science Foundation under No. CHE-

0809859.

This work is supported

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