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Title
Accurate spin-dependent electron liquid correlation energies for local spin density
calculations: a critical analysis
Permalink
https://escholarship.org/uc/item/23j4q7zm
Journal
Canadian Journal of Physics, 58(8)
ISSN
0008-4204 1208-6045
Authors
Vosko, S. H
Wilk, L.
Nusair, M.
Publication Date
1980-08-01
DOI
10.1139/p80-159
Peer reviewed
eScholarship.org Powered by the California Digital Library
University of California
1200
Accurate spin-dependent el
ec
tron liquid correlation energies for local spin density
calculations: a
cr
itical
an
alysis
1
S. H.
VOSK0
2
Departme11t
of
Physics, U11iversity ofToro11to, Toro11to,
011t
..
Canada
M5S
I A 7,
3
Departme11t of
Ph
ysics, University
of
California,
lr
v
i11e.
CA
.
U.S.A.
cmd
IBM.
Thomas
J. Watson
Resear
ch Ce11ter, Yo
rktow11
Heights
,
NY,
U.S.A
.
AND
L.
W I
LK
AND
M.
NUSAIR
Departme
nt
of
Physics. U11iversity
of
Toronto, Toronto,
0111.,
Ca11ada
M5S
/A7
Received March
19,
1980
We assess various approximate forms for the correlation energy
per
particle of the spin-
polarized homogeneous electron gas that have frequently been used in applications
of
the local
spin density approximation to the exchange-correlation energy functional. By accurately recal-
culating the RPA correlation energy as a function
of
electron density and spin polarization
we
demonstrate the inadequacies
of
the usual approximation for interpolating between the para- and
ferro-magnetic states and present an accurate new interpolation formula. A
Pade approximant
technique is used to accurately interpolate the recent Monte Carlo results (para and ferro)
of
Ceperley and Alder into the important range
of
densities for atoms. molecules. and metals. These
results can be combined with the RPA spin-dependence
so
as
to produce a correlation energy for a
spin-polarized homogeneous electron gas with an estimated maximum error
of
I mRy and thus
should reliably determine the magnitude
of
non-local corrections to the local spin density
approximation in real systems.
Nous evaluons
differentes formes
de
l'energie de correlation
par
particule dans un gaz
homogene d'electrons avec polarisation du spin qui ont
ete
utilisees dans
des
applications
de
('approximation densite locale de spin a
la
fonctionnelle
de
l'energie de correlation et d'echange.
En recalculant de
fayon precise l'energie de correlation RPA
en
fonction de la densite elec-
tronique
et
de la polarisation du spin nous montrons !es insuffisances
de
!'approximation usuelle
pour !'interpolation entre les
etats para- et ferromagnetiques
et
nous presentons une nouvelle
formule precise
d'interpolation
-.
Une technique d'appoximants
de
Pade
est
utilisee pour faire
une
interpolation precise
des
resultats Monte Carlo recents (para-et-ferro-)
de
Ceperley et Alder dans
l'intervalle important de densites pour les atomes, les molecules et les metaux. Ces resultats
peuvent
etre combines avec
la
dependance de spin RPA
de
fayon a
donner
une energie de
correlation pour un gaz homogene d'electrons avec polarisation
du
spin dont l'erreur maximum
estimee
est
de I mRy, determinant ainsi de fayon plus sure la grandeur des corrections non locales
a !'approximation de densite locale
de
spin dans les systemes reels.
Can. J. Phys .• 58,
1200
(
1980
)
1. In
tr
o
du
ction
The importance
of
the correlation energy for many-
electron systems has long been appreciated
and
a great
deal
of
effort has been directed towards its study.
In particular, the homogeneous electron gas has re-
ceived much attention for
near
ly 50 years (see Pines
(1) for a review
of
pre-1955 work) primarily as an
idealized model
of
a metal. More recently the develop-
ment and success
of
the Hohenberg, Kohn,
and
Sham
spin density functional
(SDF)
formalism (for recent
reviews see Lang (2), Rajagopal (3),
and
Mackintosh
and
Andersen (4))
and
especially the local spin
density approximati
on
(LSDA) to the exchange-
1
Supported in part by the
Natura
l Sciences
and
Engineering
Research Council
of
Canada.
2
Visiting Professor U
CI
1978-1979,
IBM
1979 Summer
Faculty Program.
3
Permanent address.
[Traduit
par
le journal]
correlation (XC) energy functional has given the
correlation energy
of
the homogeneous electron gas
added significance for the study
of
real many-
electr
on
systems. Specifically, the correlation energy
per particle
of
the homogeneous electron gas,
i::
0
(r., I;), where r
5
and
I;
are the usual density and spin
polarization parameters, respectively (see Sect. 2), is
an
essential ingredient
of
the
LSDA
and
if known
accurately would hopefully improve agreement
between theory and experiment,
and
if
not, enable
one to assess unambiguously the importance
of
non-
local corrections to the LSDA in atoms, molecules,
and
metals.
The past several years have seen a very large com-
putational effort (4, 5) based on various approximate
forms for the fundamental quantity
i::
0
(r., I;).
In
our
view
, sufficient attention has
not
been given
to
the
accuracy
of
this quantity used in calculations,
0008-4204/80/081200-12$0
l.
00/0
©
1980
National Research Council
of
Canada/Conseil national
de
recherches du Canada
YOSKO
ET
AL.
1201
es~cially
with regard to its spin dependence. Since
various authors use different forms for e (r
Y)
in
1
.
c s•
'>
comp 1cated self-consistent calculations it
is
difficult
. '
to appreciate the meaning and significance
of
simi-
larities and differences in their results. Although there
have been many approximate determinations
of
the
correlation energy (i.e., beyond the
so-called random
phase approximation, hereinafter referred to as the
RP~)
for the paramagnetic state, Ec(r., 0), it was
not
until very recently (6,
7)
that
highly accurate results
were available for both the para-
and
ferro-maanetic
electron liquid states, albeit for a limited number
of
electron densities.
In
the present work we will compare
and
assess the
adequacy
of
the various approximate forms for
E~(ru
s)
that
have been frequently used in applica-
tions
of
the LSDA. Since the spin dependence
of
these approximate forms is based on a suagestion
of
vo~
Barth and
~edin
(8)
(to be referred to
as
vBH)
derived from their RPA calculation, we have carefully
re-examined the spin dependence
of
the RPA in
Sect. 3 and found the vBH form inaccurate for
de~sities
corresponding to r
5
=::;
6. A new parametri-
zation
of
the RPA is introduced which
is
accurate to
better
than
13 for all r,
and
S·
These new RPA
results
are
combined with Ceperley
and
Alder's
calcula~ions
by
~eans
of
a two-point Pade approxi-
mant mterpolat1on formula to produce a new
Ec(ru
s)
that
is
accurate for the important densities in
atoms, molecules, and solids, and thus should
provide a reliable means for judging the validity
of
the LSDA.
2. General Com
me
n
ts
and Cri
te
ria for Assessing
Co
rr
elation Energy
The
essential ingredients for applying the
SDF
formalism to
c~lcu
l
ating
the ground state properties
of
any system 1s the exchange-correlation function
al
E,c[n
1
,
ni), where nt/
1
is
the density for spin up/down
electrons, respectively.
The
LSDA
assumes E
[n
ni]
is
xc
t•
[2.1]
E,/·[np
11iJ = J di' n(l')E,c(np
11
1
)
where
Exc(n
1
,
11
1
)
is
the
XC
energy (per particle)
of
a
homogeneous electron gas with spin densities
n
and
+
F
d
. . . .
Il
l
n =
11
1
n
1
.
or
1scussing
Exe
1t
1s more convenient
to.use
the.sta~dard
variables r
5
ands
for density
and
spin polarization, respectively
(s
=
(11
1
- n
1
)/n).
It
is
well known (e.g., refs. 8 and 9)
that
the exchange
energy can be written in the form
(2.2)
Ex(ru
s)
=
E/(r
5
)
+
(E/(r
5
)
-
E/(r.)Jf(s)
=
e/(r.)
+
6e.(r.,
s)
where
P/F
stand for para/ferro-magnetic states,
respectively, with
and
(2
.3]
e/(rs)
= - 3/2mxr, =
E/(r.)
/2
1
1
3
ex
=
(4
/9rt)
1
'
3
ji(
Y)
= ((1 +
C)4
/3 +
(l
-
C)4
/3 -
2]
..,
2(2
1
'
3
-
1)
Unfortunately there
is
no analogous simple closed
form for the correlation energy
Ec(r
5
,
s).
Nonetheless
Ec(r.,
s)
can always be Written
as
(2.4] Ec(r.,
s)
S
E/(r
5
)
+ 6Ec(r
5
,
s)
Wh~re
E/(r,)
=
Ec(ru
0), thus defining
6eoCr.,
s)
which should not be confused with the von B
arth
and
Hedin suggestion for approximating
it
(see Sect. 3).
We will begin with a review
of
recent calculations
of
the correlation energy
and
present some simple
criteria for judging their accuracy. These results will
provide a standard for
our
new estimates in Sect. 4.
Our
main emphasis will be
on
its spin dependence
and
the importance
of
calculating it in a consistent
manner.
The
reason for this is
that
E P(r )
and
E/(r.)
are
of
the same sign (both being
ne~ati~e)
and
le/I
> I
E/I
for important densities.
On
the
other
ha~d
~E._(r
5
,
s)
is positive while 6E,(r.,
s)
is negative
(this
1s
JUSt
the well-known fact
that
correlation
~nhibits
ferromagnetism (10)) and although l
6EcCi'.,
s)I
IS
smaller than l6E,(r., s)I, it
is
of
much more similar
magnitude especially in the range
'•
;;;::;
3, so
that
any
error
in 6£c(r
5
,
s)
is
magnified in 6E
(r
Y)
XC
S'
'-,
•
There have been numerous studies
of
E/(r.)
and
recently Ceperly and Alder (7) have also studied
E/(r.)
= Ec(r,, 1) for several values
of
r
5
,
but the
s-dependence over the whole range has been calcu-
lated only. in
the.~PA
by von Barth
and
Hedin (8)
for metallic densities
and
more recently Dunaevskii
(11) has attempted to use the method
of
Singwi et al.
(12) to study 6ec(r
5
,
s).
We do not believe
that
his
results
are
correct since his 6Ec(r., 1) conflicts with
the known high density behaviour (see below and
Fig. 2).
The
factorization
of
the correlation energy in
[2.4] has more significance than might
appear
at
first
glance. Firstly, it emphasizes the independence
of
the
spin polarization
part
from the paramagnetic
'background'
and
accentuates
that
·
E/(r
5
)
and
AEc(rs,
s)
can be taken from different sources with the
condition
that
6Ec('s•
0)
= 0, to obtain
'a
more
accurate
Ee('~·
s)
than
from, say, an RPA calculation.
Secondly..
_(2.4]
s~resses
that
when calculating
6Ec(r.,
s)
It
IS essential to use the Same approximation
for the
E/(
rs)
and
the
Ec(r
5
,
s)
parts, i.e., include the
same Feynman graphs.
In
fact there are strong
reasons for believing
that
6£cCrs,
s)
converges more
1202
CAN. J. PHYS. VOL. 58.
1980
TABLE
1.
Recent calculations
of
-E/(r,)
in mRy. Quantities in parentheses were obtained by
interpolation from values
at
different r,'s.
'Av'
denotes the average value
of
-E
0
P(r,) not
including
RPA,
H-67,
KW, and
MRT
-E/(r,)
(mRy)
r, = 1
1·,
= 2 r, = 3 r, = 4 r, = 5 r, = 6
RPA 157.6 123.6 105.5
93.6 85.0 78.2
H-67
(23)
92
STLS
(24)
125
97
SSTL
(12)
124
92
KW
(25)
76.3 70.6
vs
(21)
130
98
LB
(26)
125
91.9
cw
(27)
(117)
91.3
F
(16)
118
88.4
MRT
(28)
137
103
C-
78
(6)
122
87.4
Av
123
92.3
rapidly
than
E/(r
5
).
For
example, the leading correc-
tion to the
RPA correlation energy
ERPA(r
5
,
t;)
at
high
density is the second-order exchange
eb
<
2
> = 48.36
mRy (13, 14)
and
is
the same order
of
magnitude
as
ERPA(r.,
t;)
(see Table 1). However, it is simple
to
show
that
Eb
C
2
J is independent oft; as well
as
1'
5
(15)
so
that
the two leading terms
of
t.Ec(r., t;) are given
exactly by the
RP A in the high density limit. Recall
that
eb
c
2
J contains two bare coulomb interactions,
thus graphs
of
this structure will become t;-dependent
when screening is included. Freeman (16) has calcu-
lated the effect
of
replacing one bare coulomb line in
the second-order exchange
graph
by a dynamically
screened line for
t;
= 0
and
finds a weak r
5
depen-
dence (he denotes this contribution by
t.Ec/).
How-
ever, it should be pointed
out
that
Freeman does
not
include all the graphs which give corrections
of
the
order
r.
In
r,
and
r,
to
Eb
C
2
J which must be grouped
together ( 17) to obtain the correct coefficients
of
these terms. Generalizing DuBois (17) to allow for
spin polarization, we will denote the combination
of
these terms
as
E
1
,(r.,
t;).
In
the high density limit
[2.5]
E1x<rs.
0)
=
eb(l)
+ r.(A1x
In'
· + Cix)
+ O(rs2)
The
constants A£,
and
C
1
x have been evaluated by
Carr
and Maradudin (18) to be
13
and -
21
mRy,
respectively.
For
graphs
of
this structure Misawa (19)
has pointed
out
a scaling relation between
t;
= 0 and
1,
namely e
1
x(/'
5
,
1) = e
1
.(r./2
413
, 0). (This relation is
consistent with
eb
C
2
J being constant.)
It
should be
contrasted with
RPA scaling (10, 19), i.e.,
ERPA(r.,
1)
= teRPA(r,/2
413
,
O).
The
t;-dependenceofhigher order
graphs has not been investigated; however, there is
good evidence (Ceperley and Alder (7), see Sect.
4)
that
t.ec(r., 1) differs from
t.eRPA(r.,
1) by
at
most
75 64
80
70
63
57
75
64
56
50
60.3 52.3 46.0 40.7
81
70
62
56
74.2 62.5 54.4
(76.5)
67.9
(61.
O)
(56.
5)
73.3 63.6 56.7 51.5
85
74
65
59
72.2
62.4 55.0
(49.
8)
76.0 65.8
58.3 53.5
113
while
e/(r.)
differs from the RPA value by 25
to
303
for metallic densities.
Janak, Moruzzi, and Williams'
(20) (hereinafter
JMW)
effort to improve
on
the RPA is in the spirit
of
[2.4]. They took
E/(r.)
from the calculation
of
Vashishta and Singwi (21); however, their procedure
for estimating
t.ec(r., t;) suffers in two aspects (which
in fact do compensate for some values
of
r,
and
t;,
see Sect. 5): (i)
ecF
(r
5
)
was determined from
e/(r.)
by
RPA scaling which clearly does
not
satisfy the above
criterion for consistency and
(ii) the
von
Barth and
Hedin suggestion for expressing
t.ec(r., t;) as
[E/
(r.) -
e/(r.)]/(t;)
is
inaccurate (see Sect. 3).
Table 1 contains a summary
of
recent calculations
ofe/(r.).
Hubbard's
classic 1957 work (22) has been
omitted since it has been superceded by his 1967
work (23). Also, Ceperley and Alder (7) (hereinafter
to be referred to as CA) have improved Ceperley's (6)
calculations for a
few
values
of
r
5
(2,
5,
10
, 20,
50,
and
100). Since only two values are in the range
considered in Table 1
we
delay discussion
of
their
work until
Sect. 4.
It
is very difficult to judge the
accuracy
of
any one
of
these calculations, especially
in the range 2
~
r.
~
6.
In
particular, even
at
as high
a density
as
!'
5
= 1 the maximum disagreement is
50
mRy (i.e.,
-11
to
+40%
of
the average) while
at
'•
= 6
the
extreme values differ from the average
by approximately
203.
It
is
not
difficult
to
establish
criteria for assessing the validity
of
these calculations,
at
least in the high density limit. Recall t
hat
the
correlation energy can be written as a
sum
of
Feyn-
man graphs, the dominant contribution being the
RPA which contains the
In
r
5
singularity.
Thus
it
is
useful to write
VOSKO ET AL.
1203
80
70
>.
60
Q:
.5
.
50
~
~
c.a::
Iii
4
I
•
~
Q.u
Iii
30
20
0
[44j\
.,.,,
,,
F
SSTL
0
...
,,
ms
~-
0
,.__
.
.._
-
VS
---.,.
____
•
..
_____
o-----·x-----~
• • x -
~
. . .
.
MRT
2 3
4 5 6
expect
that
its
maximum
error
is
a few millirydbergs.
Thus
combining the average value
of
e:/(r
5
)
given in
Table
1 with
accurate
RPA
calculations
for
t.e:c(r.,
()
will result in
an
E:c(r.,
s)
which has a maximum
error
of
10% for metallic densities
and
for the
important
transition
metals where r.
~
2 the
error
will
be
even
less.
Since
there
have
been
very few calculations
of
e:/(r
s)
it
is even
more
important
to
have criteria
for
assessing
t.e:c(r
5
,
1).
Using
the
fact
that
e:b
<
2
l is inde-
pendent
of
rs
and
s.
the
analogue
of
[2.8] is
that
(2.10]
01t.E:c(rs,
1)
=
~E:c(r.,
1) -
~E:RPA(rs,
1)
must
go
to
zero
as r
5
-
0 with a positive value having
an
infinite slope
at
rs
= 0 (see (2.11]). By using
the
scaling relation
for
£ 1 .(r., s) given
above
to
obtain
the analogues
of
A
1
x
and
C
1
x
for
s = 1 we
note
that
(2.11] 0
2
t.E:c(r
5
,
1) =
0
1
~£c(r
5
,
1)
's
- rs{t.Aix ln
rs
+
~Cix}
FIG.
J. A comparison
of
results for e. '"(r,) - eRPAP(r, ) (see where
Table J for notation). Equation (4.4] refers
to
our
Pade fit to ( I )
Ceperley and Alder (7) as described
in
the text (see Table 5).
~A
1 x =
2
4 / 3 - I A 1 x
where
oe:c(r.,
s) is sma
ll
er
than
the
other
two
terms
for the
range
of
r
5
of
interest
and
is a
smooth
function
of
r •.
In
the high density
limit
,
according
to
[2.5],
it
is
(2.7]
OE:c(r
5
,
0) = r
5
(A
1
x
In
rs + C
1
.)
+
Dr
5
+
O(r/)
The
constant
D has
not
been evaluated (17, 18).
Thus
if
one considers
(2.8)
01E:c{I'.,
S)
=
E:c(l's,
S)
-
E:RPA(rs,
S)
it
should
tend
smoothly
to
e:b
<
2
l as
rs
- 0. A
more
refined criterion
for
e:
0
P(r.) is provided by the
quantity
(2.9) 0
2
E:c(r
5
,
0) = 0
1
e:c(t·
5
,
0) -
E:b
<
2
l
- r
5
(A
1
x
In
rs
+
C1x)
which
should
approach
zero
linearly
as
r
5
- 0.
Either
of
these criteria immediately
throws
doubt
on
the validity
of
the Keiser
and
Wu
(KW)
(25)
and
Mandal,
Rao,
and
Tripathy
(MRT)
(28) results
(see
Fig.
1). Also, a
smooth
extrapolation
of
the
Chakravarty
and
Woo
(CW)
(27) results
for
rs
= 0.565
and
1.13 does
not
appear
to
be
approach-
ing
the
correct
limit.
Furthermore,
a similar
plot
of
criterion [2.9) suggests
that
the
Vashishta
and
Singwi
(21) calculations
are
inaccurate
for
r
5
near
1.
In
an
effort
to
present
an
unbiased view we
have
used all
the results
in
Table
1 except
RPA,
KW,
MRT,
and
H-67
to
arrive
at
the average presented there.
We
and
~Cix
= (
2
L3
- I
)cix
-
(!Ai.~
In
2)/2
4
1
3
must
tend
to
zero
linearly with r
5
•
For
the
level
of
accuracy presently available for
t.e:
0
(r., 1) criterion
[2.10) is
most
useful. Figure 2
contains
plots
of
0
1
t.e:
0
(r., 1) for the calculations
of
Dunaevskii (11),
the
parametrized
forms
of
Gunnarsson
and
Lundq-
vist (9) (hereinafter
GL)
and
JMW
used in
LSDA
calculations, the result
from
Freeman
's (16) calcu-
lation using
the
scaling relation
for
£ l
.(r
s, s),
Perdew's
parametrization
of
Ceperley's
old
calcula-
tion
(6),
and
our
result
obtained
by
interpolating
CA's
(see Sect. 4)
most
recent calculations. Clearly
the
results
from
Dunaevskii,
GL,
and
JMW
have the
wrong
behaviour
for
rs
- 0. (Recall
that
0
1
t.s
0
(r., 1)
must
be positive in the vicinity
of
r
5
= 0.)
The
values
derived
from
Freeman's
and
CA
's
calculations agree
with this
rigorous
result.
For
rs
> 1 we
favour
our
result (as derived
from
CA
in Sect. 4)
for
two
rea-
sons:
li)
Freeman's
calculation is
not
really consis-
tent
in retaining all
graphs
of
a given
order
in
r.
(see
above)
and
(ii) even
if
the er
ro
rs estimated
by
CA
are
off
by
a
factor
of
2
or
3,
o
1
~e:
c
(rs,
1) would
remain positive
and
not
change
significantly.
Thus
we
must
conclude
that
the
GL
and
JMW
values
of
t.e:
c
(r
5
,
1)
are
in
error
by
approximately 10
mRy
in
the
metallic density range.
It
should
be emphasized
that
this will
not
necessarily give
an
error
of
10
mRy
per
spin
since s is usually less
than
one
.