Adiabatic connection for strictly correlated electrons.

Abstract

Modern density functional theory (DFT) calculations employ the Kohn-Sham system of noninteracting electrons as a reference, with all complications buried in the exchange-correlation energy (E(XC)). The adiabatic connection formula gives an exact expression for E(XC). We consider DFT calculations that instead employ a reference of strictly correlated electrons. We define a "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 DFT and constructing approximate functionals.

Full text

Page 1

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

Page 2

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],

Page 3

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.

Page 4

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