VOLUME 77, NUMBER 18 PHYSICAL REVIEW LETTERS 28OCTOBER 1996
can guide
GGA’s c
but fail fo
fsn
"
, n
#
,0
A rst-pr
[13] by
expansion
the electr
cutting o
rules on t
[14] func
designed
ifference
6 prob-
ple new
han those
ough the
atures of
he result-
form
, z , tdg,
(3)ations and formal properties of various GGA’s
a rational choice among them. Semiempirical
an be remarkably successful for small molecules,
r delocalized electrons in the uniform gas [when
,0d fi ne
unif
XC
sn
"
, n
#
d] and thus in simple metals.
inciples numerical GGA can be constructed
starting from the second-order density-gradient
for the exchange-correlation hole surrounding
on in a system of slowly varying density, then
ff its spurious long-range parts to satisfy sum
he exact hole. The Perdew-Wang 1991 (PW91)
tional is an analytic t to this numerical GGA,
to satisfy several further exact conditions [13].
sion in the slowly varying limit makes little d
to the energies of real systems. We solve the
lems above with a simple new derivation of a sim
GGA functional in which all parameters [other t
in e
unif
XC
sn
"
, n
#
d] are fundamental constants. Alth
derivation depends only on the most general fe
the real-space construction [13] behind PW91, t
ing functional is close to numerical GGA.
We begin with the GGA for correlation in the
E
GGA
C
fn
"
, n
#
g ­
Z
d
3
rnfe
unif
C
sr
s
, z d 1 Hsr
sGeneralized Gradient App
John P. Perdew, Kieron Bu
Department of Physics and Quantum Theory Group,
(Received 21
Generalized gradient approximations (GGA’s) fo
the local spin density (LSD) description of atom
derivation of a simple GGA, in which all param
constants. Only general features of the detailed
(PW91) GGA are invoked. Improvements over PW
response of the uniform electron gas, correct behavi
[S0031-9007(96)01479-2]
PACS numbers: 71.15.Mb, 71.45.Gm
Kohn-Sham density functional theory [1,2] is widely
used for self-consistent- eld electronic structure calcula-
tions of the ground-state properties of atoms, molecules,
and solids. In this theory, only the exchange-correlation
energy E
XC
­ E
X
1 E
C
as a functional of the electron
spin densities n
"
srd and n
#
srd must be approximated. The
most popular functionals have a form appropriate for
slowly varying densities: the local spin density (LSD) ap-
proximation
E
LSD
XC
fn
"
, n
#
g ­
Z
d
3
rne
unif
XC
sn
"
, n
#
d, (1)
where n ­ n
"
1 n
#
, and the generalized gradient approxi-
mation (GGA) [3,4]
E
GGA
XC
fn
"
, n
#
g ­
Z
d
3
rfsn
"
,n
#
,=n
"
,=n
#
d. (2)
In comparison with LSD, GGA’s tend to improve total
energies [4], atomization energies [4{6], energy barriers
and structural energy differences [7{9]. GGA’s expand
and soften bonds [6], an effect that sometimes corrects
[10] and sometimes overcorrects [11] the LSD prediction.
Typically, GGA’s favor density inhomogeneity more than
LSD does.
To facilitate practical calculations, e
unif
XC
and f must
be parametrized analytic functions. The exchange-
correlation energy per particle of a uniform electron gas,
e
unif
XC
sn
"
, n
#
d, is well established [12], but the best choice
for fsn
"
, n
#
, =n
"
, =n
#
d is still a matter of debate. Judging
the deriv0031-9007y96y77(18)y3865(4) 10.00roximation Made Simple
rke,* Matthias Ernzerhof
Tulane University, New Orleans, Louisiana 70118
May 1996)
r the exchange-correlation energy improve upon
s, molecules, and solids. We present a simple
eters (other than those in LSD) are fundamental
construction underlying the Perdew-Wang 1991
91 include an accurate description of the linear
or under uniform scaling, and a smoother potential.
PW91 incorporates some inhomogeneity effects while
retaining many of the best features of LSD, but has its
own problems: (1) The derivation is long, and depends on
a mass of detail. (2) The analytic function f, tted to the
numerical results of the real-space cutoff, is complicated
and nontransparent. (3) f is overparametrized. (4) The
parameters are not seamlessly joined [15], leading to
spurious wiggles in the exchange-correlation potential
dE
XC
ydn
s
srd for small [16] and large [16,17] dimension-
less density gradients, which can bedevil the construction
of GGA-based electron-ion pseudopotentials [18{20].
(5) Although the numerical GGA correlation energy func-
tional behaves properly [13] under Levy’s uniform scaling
to the high-density limit [21], its analytic parametrization
(PW91) does not [22]. (6) Because PW91 reduces to the
second-order gradient expansion for density variations
that are either slowly varying or small, it descibes the
linear response of the density of a uniform electron gas
less satisfactorily than does LSD [20,23].
This last problem illustrates a fact which is often over-
looked: The semilocal form of Eq. (2) is too restrictive
to reproduce all the known behaviors of the exact func-
tional [13]. In contrast to the construction of the PW91
functional, which was designed to satisfy as many exact
conditions as possible, the GGA presented here satis es
only those which are energetically signi cant. For exa-
mple, in the pseudopotential theory of simple metals, the
linear-response limit is physically important. On the other
hand, recovery of the exact second-order gradient expan- 1996 The American Physical Society 3865

VOLUME 77, NUMBER 18 PHYSICAL REVIEW LETTERS 28OCTOBER 1996
3 GGAwhere r
s
is the local Seitz radius (n ­ 3y4pr
3
s
­ k
F
y
3p
2
), z ­ sn
"
2 n
#
dyn is the relative spin polarization,
and t ­ j=njy2fk
s
n is a dimensionless density gradi-
ent [13,14]. Here fsz d ­ fs1 1zd
2y3
1s12zd
2y3
gy
2 is a spin-scaling factor [24], and k
s
­
p
4k
F
ypa
0
is the
Thomas-Fermi screening wave number (a
0
­ h¯
2
yme
2
).
=z corrections to Eq. (3), which are small for most pur-
poses, will be derived in later work. We construct the
gradient contribution H from three conditions:
(a) In the slowly varying limit (t ! 0), H is given by
its second-order gradient expansion [24]
H ! se
2
ya
0
dbf
3
t
2
, (4)
where b . 0.066 725. This is the high-density (r
s
! 0)
limit [25] of the weakly r
s
-dependent gradient coef cient
[26] for the correlation energy [with a Yukawa interaction
se
2
yud exps2kud in the limit k ! 0], and also the coef-
cient which emerges naturally from the numerical GGA
[13] discussed earlier.
(b) In the rapidly varying limit t ! `,
H ! 2e
unif
C
, (5)
making correlation vanish. As t ! ` in the numerical
GGA, the sum rule
R
d
3
un
C
sr,r 1 ud ­ 0on the corre-
lation hole density n
C
is only satis ed by n
C
­ 0. For
example, in the tail of the electron density of a nite sys-
tem, the exchange energy density and potential dominate
their correlation counterparts in reality, but not in LSD.
(c) Under uniform scaling to the high-density limit
[nsrd ! l
3
nslrd and l ! `, whence r
s
! 0 as l
21
and
t ! ` as l
1y2
], the correlation energy must scale to a
constant [21]. Thus [27] H must cancel the logarithmic
singularity of e
unif
C
[28] in this limit: e
unif
C
sr
s
, z d !
se
2
ya
o
df
3
fg lnsr
s
ya
0
d 2vg, where g and v are weak
functions [12] of z which we shall replace by their
z ­ 0 values, g ­ s1 2 ln 2dyp
2
. 0.031 091 and v .
0.046 644,so
H!se
2
ya
0
dgf
3
ln t
2
. (6)
Conditions (a), (b), and (c) are satis ed by the simple
ansatz
H ­ se
2
ya
0
dgf
3
3 ln
Ω
1 1
b
g
t
2
∑
1 1 At
2
1 1 At
2
1 A
2
t
4
∏æ
, (7)
where
A ­
b
g
fexph2e
unif
C
ysgf
3
e
2
ya
0
dj 2 1g
21
.
(8)
The function H starts out from t ­ 0 like Eq. (4), and
grows monotonically to the limit of Eq. (5) as t ! `; thus
E
GGA
C
# 0.(Happears as one of two terms in the PW91
correlation energy, but with g ­ 0.025.) Under uniform
3866scaling to the high density limit, E
C
tends to
2
e
2
a
0
Z
d
3
rngf
3
3 ln
∑
1 1
1
xs
2
yf
2
1 sxs
2
yf
2
d
2
∏
, (9)
where s ­ j=njy2k
F
n ­ sr
s
ya
0
d
1y2
ftyc is another
dimensionless density gradient, c ­ s3p
2
y16d
1y3
.
1.2277, and x ­ sbygdc
2
exps2vygd.0.721 61. The
correlation energy for a two-electron ion of nuclear
charge Z ! ` is 2` by LSD, 1` by PW91, 20.0482
Hartree by Eq. (9), and 20.0467 exactly [29]. For a nite
system, s cannot vanish identically, except on sets of
measure zero, so Eq. (9) is nite; for an in nite jellium, s
vanishes everywhere, and Eq. (9) reduces to 2` as GGA
reduces to LSD.
The GGA for the exchange energy will be constructed
from four further conditions:
(d) Under the uniform density scaling described along
with condition (c) above, E
X
must scale [30] like l. Thus,
for z ­ 0 everywhere, we must have
E
GGA
X
­
Z
d
3
rne
unif
X
sndF
X
ssd, (10)
where e
unif
X
­ 23e
2
k
F
y4p . To recover the correct uni-
form gas limit, F
X
s0d ­ 1.
FIG. 1. Enhancement factors of Eq. (15) showing GGA non-
locality. Solid curves denote the present GGA, while open
circles denote the PW91 of Refs. [4,13,14].