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PHYSICAL REVIEW A
VOLUME 57, NUMBER 1
JANUARY 1998
Approximate relativistic optimized potential method
T. Kreibich and E. K. U. Gross
Institut für Theoretische Physik, Universität Würzburg,
Am Hubland, D-97074 Würzburg,
Germany
E. Engel
Institut für Theoretische Physik, Universität Frankfurt, Robert Mayer Straße 8-10, D-60054 Frankfurt-am-Main, Germany
Received 28 July 1997
Approximate semianalytical solutions of the integral equation for the relativistic optimized potential are
constructed by extending a method recently proposed by Krieger, Li, and Iafrate Phys. Lett. A 146, 256
1990
to the relativistic regime. The quality of the approximation is tested in the longitudinal x -only limit
where fully numerical solutions of the relativistic optimized effective potential integral equation are available
for spherical atoms. The results obtained turn out to be in excellent agreement with the exact x -only values.
The proposed method provides significant improvement over the conventional relativistic local density ap
proximation and generalized gradient approximation schemes. S1050-2947 97 04912-3
PACS number s : 31.10. z, 71.10. w, 31.30.Jv, 31.15.Ew
e
RKLIf 3scheme is developed in Sec. IV before some limiting
cases are discussed in Sec. V. In Sec. VI, numerical results
0
of the ROPM and RKLI methods are presented and com4
pared to other RDFT methods.
I. INTRODUCTION
,
Since the seminal work of Hohenberg and Kohn 1 and
" #
&
' (
Kohn and Sham ! 2 $, density functional theory % DFT has
)
become a powerful tool for *ab initio +electronic
structure cal.
/
,
culations of atoms, molecules, and solids - 3–5 . The devel0
opment of more and more refined approximations of the
+
exchange-correlation 1 xc2 +energy functional E xc has led to
3
significant improvement
over the standard local density ap6
7
4
the so-called optimized
proximation 5 LDA9 . In :<particular,
> +
; =
4
potential
method
6,7
employing
explicitly orbital8 OPM
?
dependent
functionals
rather
than
the
traditional
density?
dependent
functionals
has
achieved
highly
accurate
results
@ A
7–2
0
B .
C
Most of the advances in DFT have been made in the conE
D
text of nonrelativistic physics. For high-Z atoms, however,
F
relativistic contributions have to be considered.
For example,
?
D
the ground-state energy of mercury G HgH decreases from its
I
nonrelativistic value J 18 408 a.u. to K 19 649 a.u. if relativistic effects are taken into account. Furthermore, even for
E
L
3
with moderate Z $, relativistic contributions to E M xc are
systems
N
larger than the differences between the currently best xc en+
ergy functionals O 21P . Until now, the calculation of such relaD
tivistic contributions has mostly been
based
on the relativisR
S
D
tic local density approximation Q RLDA . To go beyond the
of the OPM was formulated for
RLDA, an T x -only version
"
V
F
relativistic systems U 22,21 . As in the nonrelativistic case, the
3
solution of the resulting equations is a rather demanding task
and has been achieved so far only for systems of high symmetry, e.g., spherical atoms W 21,23–25X .
The purpose of the present paperR is to develop
a simpliY
[
fied version of the relativistic OPM Z ROPM 3scheme leading
D
to a generalization
of the approximation of Krieger, Li, and
_
D
Iafrate \ KLI]<^ 8 –10,26–36` to the realm of relativistic sysD
tems. This will be done for systems subject to arbitrary static
+
external four-potentials. The paper is organized as follows.
In Sec. II we give a brief review of the foundations of relaD
tivistic DFT a RDFTb . After that, in Sec. III, we develop the
R
F
ROPM generalizing
a nonrelativistic derivation of Görling
.
d D
and Levy c 37 to the relativistic domain. The relativistic KLI
1050-2947/98/57 1 /138 11 /$15.00
57
II. THEORETICAL BACKGROUND
9
On its most general
level, RDFT is based on quantum
h
i
electrodynamics g QED and thus contains not only relativisD
tic but also radiative effects. For a detailed derivation, also
j
including questions" of renormalization,
the reader is referred
l
D
to recent reviews k 23,38 . The central statement of RDFT —
D
D
the
relativistic version of the Hohenberg-Kohn m HKn theorem
o .
p
39 — can be stated in the following
way: The renormalized
r s t v
q
ground-state four-current j ( ur) of an interacting system of
Dirac particles uniquely determines, up to gauge transformar {}|
D
xzy j
tions, the external four-potential A w ext
as well as the
r }
q
ground-state wave function ~ j . As a consequence, any
0
observable of the relativistic many-body system under con3
is a functional of its ground-state four-current. As
sideration
j
in the nonrelativistic case, the exact ground-state four-current
including all quantum electrodynamical effects can in prin,
ciple be obtained from an auxiliary
noninteracting
system—
"

R

D
the relativistic Kohn-Sham RKS 3system 21,38,40,41 :
+
r
j r
t
r
ª

c 2

k
F

¯
k r
¡
k
r
r¢ £ j V¤¦¥ r§ $,
1©
¨
v
where j V« ( r) denotes the vacuum contribution to the fourt v
current. The four-component spinors ¬ k ( r) are solutions of
an effective single-particle Dirac equation in atomic units
®°¯ ± ² ³ ´
e m 1)
,
µ ¶
0
¸
ic
¹
r Ù}Ú t
Ö
)
ª
·
º¼»
¾ ¿ÁÀÃÂ
½ 2
c
r Ç}ÈÉ
A Ä SÅÆ j
rÊ
Ë k Ì
rÍÎÐÏ
Ñ Ò
k
k
rÓ $,
2Õ
Ô
v
with A Ä S×Ø j ( ur) being the effective four-potential, which can
be decomposed according to
Ö
138
Ä ÛÜ
r Ý}Þß u àá
AS j
r
Ö
u äå
A ext r
w âzã
æ ç3è é
d r
r êìë
j
ð
u ñ
r íïî
u òôó õ
r r
Ö
r ùûúü u ý
A xc
öø÷ j
r .
þ
. ÿ
3
© 1998 The American Physical Society

57
Here, Ö A w APPROXIMATE RELATIVISTIC OPTIMIZED . . .

t u v
E xL ,exact j
ext( r)
is a static but otherwise arbitrary external fourpotential,
the
second
on the right-hand side represents a
Hartree-like potential,term
and the last term, defined by
4
j r : A M xc
r
EM j
r xc j r
L
r
r
139

1
2
2
c
l , k ¡ e < F
=d > 3 r = d > 3 r ¢
£¥¤ ¦l§©¨ rªL« k ¬ ¸ rL® k±¯ ° r ²`³L´ ¦l µ r ¶`·
u ¹ u
r rº »
4
9
One major advantage of such an exact treatment of the exchange
energy lies in the fact that the spurious selfj
interactions contained in the Hartree energy are fully canL
,
celed. The price to be paid for the orbital dependence of E M xc
t v
¾ ( r) , defined
is that the calculation of the xc four-potential A M xc
)
by Eq. ¿ 4À $, is somewhat more complicated. It has to be deD
by an integral equation, as will be shown in the
(termined
following.
t u v
ÃÄ
( r) , Eq. Â 4 $, we can
Starting from the definition of A xc
Á
,
calculate the xc four-potential corresponding to an orbital?
xc energy functional by applying the chain rule
(dependent
for functional derviatives:
?
,
denotes the xc four-potential containing by construction all
nontrivial many-body effects.
Equations 1 – ! 4" represent the relativistic KS scheme
D
that has to be solved self-consistently. However, the calculat v
r
D
tion of the vacuum contribution j V# ( ur) to the four-current
requires the knowledge of an infinite number of $ 4positive and
I
negative energy% 3states, so that one would have to deal with
an infinite system of coupled equations. Since such a proce?
dure is highly impractical, we will, in the following, ignore
all vacuum contributions to the various energy components
and to the four-current, which is then given just by the first
D
term on the right-hand side of Eq. & 1' . This means that we
restrict ourselves to the calculation of relativistic effects and
neglect radiative corrections. Since we are aiming at elecD
tronic structure calculations for atoms, molecules, and solids,
ª
we expect the neglected terms to be small. If one were after
all interested in the radiative contributions, an *a posteriori
4
treatment should be sufficient and represents in
(perturbative
fact the standard approach.
ROPM Æ
Å rÇÈ
A M xc
É
Ê c 2 ËÌ k Í9Î
=d > 3 r Ï

Ñ
L
Ò
=d > 3 r Ð
F
ÓxÔtÕ inÖ × Ý±Þ ß r à`á
k
±Ø Ù Ú ur Û`Ü â Ö A Ä ãEä uråEæ©ç
k
S
E ROPM
xc
,
è í Ar ÄSîéEï ê rëEì
j urð
c.c.
)In order to derive a relativistic generalization of the OPM
integral equation, we start out from the total-energy functional of a system of interacting Dirac particles * neglecting
+vacuum contributions, 3subject to a static external fourv
4
- t( r):
potential A w ext
0 r 1243
=d >3 r ¯? @ rACBED
5
k
6 c 2 798 k :; <F
O =d >3 èr j PQ urR Ö Aw SUT urV
¸
ic
F GIH
½ 2
c
1
2
=d > 3 r = d > 3 r X
r
j
YZ r[ r j \] r ^`_
f
a
r b r ced
/
ROPM
A M xc
r
c 2 k
L
lnm .
op q
$
_ 8
,
=d > 3 r F

i!#"
' u ()
k r
/
*&+
Ö

, r -.
01 u 2354
k
AÄS r
,
:
c.c. 6 Ä S 97 18 ; r< $, r= .
>
?
@
9
_
B
D
Acting with the response operator A 8 0on Eq. C 9 and using
the identity
5
D
)In
contrast to the ordinary RDFT approach, the xc energy
functional is given here as/ an explicit
functional of the RKS
vxwty
four-spinors rts iu . Still, E M ROPM
izn{ represents a functional
xc
0
of the density:
Via the HK theorem applied to noninteracting
r | t v
3
systems, j ( r) uniquely determines the effective potential
r
A Ä S}~ j . With this very potential, the Dirac equation 2 is
r
3
solved to obtain the set of single-particle orbitals t i j
ª
which are then used to calculate the quantity
L / ROPM t
r e
E xc
. Therefore every functional, depending ±exi j
¸
0
plicitly
on RKS spinors, is an implicit functional of the den3
sity, provided the orbitals come from a local potential. This
allows us to use the exact expression for the longitudinal
+
exchange energy, i.e., the relativistic Fock term
=d > 3 r ROPM E xc
$&%
i
A S r Ä
so that Eq. 9 ,can be rewritten as
JLK k M rN
gihkj
E M ROPM
xc
j
3
ext
W
ü r j ý þ urÿ
õ Ä kö ÷Eø r,$ r ù`ú : û
S
D
/
ñA ò
7
j
III. RELATIVISTIC OPTIMIZED POTENTIAL METHOD
ROPM
E . tot
j
ô
.
The last term on the right-hand side of Eq. ó 7 is readily
identified with the inverse of the static response function of a
3
system of noninteracting Dirac particles
A
j
¼ =6½
.
=d > 3 r E
Ä
I 1 J
FHG rK $, rLM Ä P
N ORQ r,$ r STUWVYX[Z\V^] r_a`
S
S
r bc
d 10e
we obtain f after rearranging the indicesg
ª
j
=d> 3 r h A/M ROPM
i
r klm
xc
t
D
v c
u
q
w x
2

k
yz
Ä nHoqp
S
r r $, rs
=d > 3 r {
F
|
L
&
}~
E ROPM

i#
xc
k r
Ö Ä q u ,c.c. 11
& u
AS r
k r
To further evaluate this equation, we note that the first
funcj
tional derivative on the right-hand side of Eq. 11 is readily
140
57
T. KREIBICH, E. K. U. GROSS, AND E. ENGEL
j
L
,
IV. TRANSFORMATION OF THE ROPM
INTEGRAL EQUATIONS
m
AND THE RELATIVISTIC KLI APPROXIMATION
computed once an expression for E ROPM
in terms of singlexc
particle
spinors
is
given.
The
remaining
functional
derivative
j
is calculated by using standard first-order perturbation
D
theory, yielding
4
&

Ö
In order to use the ROPM equations derived in the preceding section, we have to solve Eq. n 15o for the xc four4
Unfortunately, there is no known analytic solution
(potential.
Ö ROPMt u v
for A xc
( r) depending explicitly on the set of singlep
4
particle spinors qsr it . We therefore have to deal with Eq. u 15v
I
numerically, which is a rather demanding task. Thus a simÖ ROPMt u v
4
plified scheme for the calculation of A xc
w ( r) appears
highly desirable.
To this end we shall perform a transformation of the
R
integral equations similar to the one recently introROPM
?
duced by KLI in the nonrelativistic domain x 29,31y . This will
lead to an alternative but still exact form of the ROPM inteq
equation which naturally lends itself as a starting point
(gral
for systematic approximations. We start out by defining
,
© ¦ ª u «¬
ur
l r
¯ ¦ ² u a³ ´¶µ^· k ¸ ur¹ .
u 5¡£¢ ¤

® ¯ ¦ ± l r
°
Ä
q

¥l
AS r
k
l
¦ §¨

k
º 12»
l k
equation also enables us to give an explicit expression
(This
for the response function
Ä
¼ ÄP
½ ¾À¿ $
S r, r ÁÂ : Ã
Å
Ê
A Ä S ÆÇ r ÈÉ
Ë c 2 ÌÍ Î Ï
k
<F
¯ k Ñ rÒÓ^Ô^Õ k Ö r×
Ø
Ð
13Ù
in terms of the RKS spinors:
Ú ÄP
Û ÜÀÝ u Þ $ u ßà
S r ,r
â c
,

á
ãqä
2

çè
k
åæ
ý
<F ¦ é ê ¥l

k
þ°ÿ ¦
L
l
14
3
D
3
d r

c 2 k
r !
¯
k r A M ROPM
xc j
F
ROPM
¶ 1 23 u 506 , 7 8
+
xc
u - u .0/ 0
"$%'
# E
4 r
c.c. 0,
& ( u )* G Ä Sk , r $, r
k
r
k
+
¯ k ur ¢¡¤£
<F
t
<
G Ä Sk ? ur @ $, urA : B$C D
¦
EF
¥l

©
18ª
¶ 19·
15=
Since the RKS spinors ¸s¹ iº 3span an orthonormal set, one
readily proves the orthogonality relation
3è » }
¾ ¿ À ur0Á Â 80.
d r k¼ ½ urL
k
v
+
¥ ur0¦ § ,c.c. ¨ 80,
k
¶
¬ u ®
'
² ³ u ´ µ 0 .
¯
k r : ¯±° k r
where G Ä Sk ( ur > $, ur) is defined as
ª

t v
where the adjoint spinor ¯
« k ( ur) is defined in the usual way,
i.e.,
ª
L
9;: 80,1,2,3
17
such that the ROPM integral equations can be rewritten as
Finally, putting Eqs. 11 $, 12 $, and 14 together leads to the
R
ROPM integral equations for the local xc four-potential
Ö ROPMt u v
A xc ( r):
c 2 k ROPM
+
xc
u u
$'
E
u G Ä Sk r $, r $,
k r
l k
c.c.

3 è ¯ u Ö ROPM u
d r
A xc
r
k r
z }
{ |u~
k r :
ë ì
¦ò
û
¯
¯¦ ö
k r íîaï¶ð^ñ l r óôPõ l r÷aø^ùú k rü
G ¦ H ur IJLK ¦ M'N urO
l
l
.
P QSR ¦
k
l
Å
+
20
We now use the fact that G Ä Sk ( ur Æ $, ur) is the Green function of
thet RKS
equation projected onto the subspace orthogonal to
Ç ( urv) , i.e., it satisfies the equation
k
T 16U
D
l k
V
Now the ROPM scheme is complete: For a given approximation of the xc energy, the ROPM integral equations have
D
ROPMt v
W ( r) simultaneously with the RKS
to be solved for A M xc
" Y Z
+
X
equation
2 until self-consistency is achieved. Note that Eq.
[
?
t u v
Ö
15\ determines the xc four-potential A xc
] ( r) only up to an
arbitrary constant, which can be specified by requiring
ROPMt v
^ ( r) to vanish asymptotically _ for finite systems` .
A M xc
To conclude this section, we note that exchange and corF
relation contributions can be treated separately within the
ROPM scheme. This is most easily seen by starting out with
0
only
the
four-potential,
defined
as
t v ced
f g r exchange
h t v
E a x / j ( r) , instead of Eq. i 4j and repeating the
A a xb ( r)
3
steps which lead to Eq. k 15l and likewise for the correlation
4
potential.
Ã" Ä
+
t
v
Í
G Ä Sk È ur É $, urÊÌË hˆ DÎ}Ï ur0Ð ÑSÒ k Ó ÔÖÕÖ×ÙØÚ ur ÙÛ Ü urÝ0Þàß

k
á ur â ãLä k'
å æ urçéè .
ê
21ë
ì t v
The operator hˆ í Dî ( r) denotes the Hermitian conjugate of the
R
RKS Hamiltonian,
ìˆ
h Dï}ð urñ : ò±ó
¶
ô
0 ¸
ic
õ öø÷úù
Ö
c û±üý A Ä S þ ÿ u r $ ,
½ 2
+
t
"
22
v
acting
from the right on the unprimed variable of G Ä Sk ( ur $, ur)
D
the arrow on top of the gradient indicates the direction in
ª
which the derivative has to be taken . Using Eq. 21
$, we can
(
t v from the right on Eq. 17 $,
act with the operator hˆ D ( ur)
k
N
leading to

57
k
APPROXIMATE RELATIVISTIC OPTIMIZED . . .
2
r hˆ D r! k $" #&%
' ( r)$*,+ A M -/. r01
¯
k
xc
798 : ;
u
k
L
@u ) . This fact also motipotential ÿ A xc ( 0 A ROPM
xck
xc +vates the boundary
condition assumed above. In T x -only
t
v
D
theory, @u M xc k ( ur) is the local, orbital-dependent RHF exchange
t v
4
potential so that k ( r) is the first-order shift of the RKS
ª
wave function towards the RHF wave function. However,
0
one has to realize that the first-order
shift of the orbital?
(
@
quantity
u
dependent
has
been
neglected.
xck
i
D
Now we use Eq. 23 to further transform the ROPM
+
18 . As a first step, we solve Eq. 23 for
equations
¶ t v t
v
A Ä 0S ( r) k ( r) , leading to

k
5
u
r6
C
"
Ö ROPM ?
@ M A $
¯
r<>= ¯
A xck
u xck ,
B
23
ROPM M
where we have introduced Ā
xck t vas a shorthand notation H for
ROPM
D
the average value of DFE A M xcG ( r) with respect to the k th
0
orbital, i.e.,
ª
D
P
3 ¯ K
d r J k rL$MFN A M ROPM
rQSR
xcO

Ā M ROPM
:I
xck

k
T
rU
V
24W
¶
Ö
ROPM
3 è Y E xc
@ M
¯
u xck : X
d r Z[

f
"
\
k
r ]_^
`

k
a
u
c
(
Y
The differential equation e 23 has the structure of a RKS
equation with an additional inhomogeneity term. Together
ª
with the boundary condition
m r npo
hi
8
rjlk
k
q
)
\
¶ c
8
e k g rhikj+l k m rn#o ,c.c. p 0,
A Ä 0S rdf¯

] c 2 ^`_ k `a b <F
.
t D
and employ Eq. s 32 to obtain
ROPM
u
E M xc

| } r~+ A M ROPM
¯
r#
k
xc
wyx
v c 2 k z`{ F
k r
Eq. s 23t Zuniquely determines u kv ( r) . To prove this statement, we assume that
are two
z t there
| t v independent solutions of
v
the difference of
Eq. w 23x $, namely, y k ,1( r) and { k ,2 ( r) . Then
t v
t v t v
D

(
r)
these two solutions, } k~ ( r):
k ,1
k ,2( r) , satisfies the
(
homogeneous RKS equation
t u v

u
ª
k
u $
80,
r> hˆ D
k
r

"
u
k
¯ u # ¸ L ½ 2 O P Ä u #¡O¢ ¶ 0£ ¤
c
AS r
k r ic
k
¥O¦ ¶ 0 §¨© ª
,
8
k r«#¬ c.c. 0.

"
ur $,
r9 k

28
j
if the above boundary condition is fulfilled. However, this
3
solution
leads to a contradiction to the orthogonality relation
" 3
t u v
20
so
that
¡ k¢ ( r) can only be the trivial solution of Eq.
£
27¤ $,
¥
¦§ u
k
8
ª
r¨© 0,
½
29
L
E M xc
$,
¾¿
u xck r³ : ´ ¹
Â
À Á
*º » r¼
k rÃ
k
and defining
*
Ñ
ÌÎ Í
a M xck rÏ : Ð
Å
.
Ä
30
?
with Æ and Ç denoting spinor indices running from 1 to 4,
we can rewrite Eq. È 17É as
ª
ª
Ê
ËÌ u
k
ÍÎÏ Ð
r
¦ ÑÒ
Ó ¦ ÔÕ
l
u
rÖ
¥ l × k ØÙ ¦ l
Ú Û 3 è ÜÞÝ
d r

u
ß
k* à
¶ æFç
r á_â>ãåä 0
Ö
u xck í urî ïñð$ò·ó ¦ l ôÂõ ur _ö ÷ $,
ü
ª
û
¶ ÙÚÛ
0
Ü rÝ
Ó Ô Õ
k
Ø
Ö
×
r
k
ç è
¶ í
Þ¯ß k à rá#â ¸ic ã äåLæ ½c ¾ 2 O
AÄ S é rê#ëOì 0 k î
ïOð ¶ 0 ñòó ô u õ $
ö .36÷
k r ,
.
ù ú ûüþý u ÿ Ö ROPM u r A xc
r
é u ê_ë
A ROPM
r
xc è

@
Ò
ROPM
E M xc
we rewrite Eq. ø 34 as
ª
l k
ì
34
¶ Ä ÅÆ u #È É ,
.
Ë
Ç r
¯ k À urÁ Â+ÃÂ 0
¹
c.c. Ê 35
k
2 º c 2 »`¼ `
½k ¾ <F ¿
which
completes the proof.
@
As an interesting aside, we briefly consider the physical
t v
¬
meaning of the quantity k® ( ur) . Defining
µ·¶$¸
®. ¯
1
± ²´³¶µ u ·
r :¸ "
ª
±
@ M ¯$°²

33
Introducing the 4 ° 4 matrix
«
"
q. r
¯
@
8 Ā M ROPM

u xck < k r
xck
27
which has a unique solution

r
We then multiply the ROPM equations Z 18[ by¶ the zeroth
t v
,
component of the effective RKS four-potential, A Ä S0 ( r) , yieldj
ing
26r
0,
k
687 ¯ M 9 ¯
A xck @u M xc k :<; k= > ur?
@B¯A k C urDFE ¸ic G HJILK ½c ¾ 2 MON PAÄ S Q ur#R SOT ¶ 0U k V . W .32X
25
+
g
ROPM
& ' ur( )+* Ö A ROPM - ur#. /10 2 E3 xc
¯
k
xc,
4 u 5
Ö
d
"
rb .
v
L
u #! "%$
r
k
A Ä 0S ur
and
L
t
Ö
4
E ROPM
xc
34
¶ Ö
141
ø
.
ù
c 2 k 31
j
is implicitly understood.
where summation over ú and
t v
From this equation, it is obvious that ý kþ ( ur) is the usual
first-order shift in the wave function caused by a perturbing
<F
* r
@

¯
a M xck r j k r Ā M ROPM
u xck ,c.c.,
xck
.
37
H
with j k! ( r) being the four-current with respect to the k th
0
orbital:
ª
r
t u v

D
142
r
j k"# ur$ : %'¯
& k ( ur) *,+..

k
4
.
2
1 38
/ ur0 .
t
?
det
>@?BA u C DEGF r ¶ 0 H u I J
r
j r
2 r
j
ÿ
1
@
@ ¯* Ð ì
Ðk
¯
M lk Ā ¯ RKLI
u Ð ¯
u xck ,
xc
2 xck
Ä c 2 È k F
ö ý þ
Ā xcl¦ ü Ā ¯ Sxc¦ l
v
ROPM
5 ( r) , we first have to
In order to solve Eq. 3 37 for A M xc
t v
investigate
whether the 4 6 4 matrix 7 ( r) , defined
by Eq.
8. 9
1t u v
35 $, is nonsingular, i.e., whether the inverset :<
(
r
)
exists.
;
v
Y
We therefore calculate the determinant of = ( r) , yielding
KL u M r NO u P Q R 4 S u T
r j r n r 1U
57
44
øwhere we have defined
´ ¯ýS V ¾2 W
3 · 1 · d r jl r
r
A xcl : j rX
$,
c Rn 2 Y urZ [ . \
39
½ 2
1
º
2 ! Ä c 2 "# È $% k
F
ª
where the last equality follows from the decomposition of
the four-current into the scalar density and the vector com4
ponents according to
& Ì ROPM
E
Õ -/.0 · 23 å
Ð k 1 r c.c.
'( xcÐ ) · *,+ 0
k
r
D
r
]^
j
V
Rn a rb $, 1 V j c rd .
½
r_`
t
v
?
t
det
H 1± @
´ RKLI
@ *Ð ² J ú
Å2 ( ¯
The unknown coefficients G ¯
A xckÐ
u ¯ xcÐ k I ¯
u xck ) are then
K
determined by the linear equation
pÃ q
t
L
M Ä c 2 NO Èk PQ
41
v
and therefore that the matrix r ( r) is nonsingular.
.
t
ROPMt v
Solving then Eq. s 37 for A xc
u ( r) yields
w
ROPM
v
A M xc
rxy
1 1
r
2 z|{~}

r

2
c
k
ROPM @
j kk r Ā M xc k
Y
<F
[
*
a M xck r
¯
u xck ,c.c.
42
D
RKLI ¶
A xc
µ
¿
·r¸¹ º1 '
» ¼¾½ 1 À ·rÁ
2
Â
Ã Ä cÅ2 Æ Ç Èk ÉÊ
Ë Ì ROPM
E xc
Õ ÖØ×Ù · Û
Ðk Ú r
ÍÏÎ Ð Ñ · ÒÔÓ 0
F
k
r
Ü Ý Ð Þkß
@
Ð â ¯
j k ràá Ā ¯ RKLI
u xckÐ ã ä åc.c.
xck
è
æ 43ç
This equation establishes the generalization of the nonrelativistic KLI approximation to the realm of relativistic sysé
tems.
ì
í In î ï contrast
ð to the ROPM equation ê 15ë , the relativistic
KLI RKLI ñequation, although still being an integral equaé
can be solved explicitly in terms of the RKS spinors
ò~tion,
ó
ö ÷
Ý ± ²
:
Multiplication of Eq. õ 43 øwith j ¦ lù ( ·r) , summing over
iô
úall û ì, and integrating over space yields
é
F
W 1 @
RTS U
@ ¯* Ð Z
V Ā ¯ RKLI
Ð
¯
u Ð Y ¯
u xck
lk M lk
xck
2 X xck
´ ¯ýS \ 1 ¯
@ * _
A xcl º ] @u ¯ xc l ^ ¯
u xcl .
2
ö a
` 47
b
We emphasize that Eq. 42 is an ±exact transformation of the
ROPM equation 15 . In particular, Eq. 42 is still an inteq
gral equation. However, its advantage lies in the fact that it
I
naturally lends itself as a starting point for deriving systemÖ ROPMt u v
atic approximations of A xc
( r) : We only need approximate
.
¢ )
t v
¡
(
r)
in
Eq.
36
by
a
suitable
function of the set of RKS
k
0
orbitals £~¤ i¥ . The simplest possible approximation is obt v
D
neglecting the terms involving ¦ k§ ( r)
¨tained byª completely
«
in Eq. © 36 . Although this approximation may seem to be
¬rather crude, it was shown to produce highly accurate results
in the nonrelativistic case 29,31® .
ROPM± ²
° ( r) is then approximately deterThe xc potential A ¯ xc
³mined by the following equation:
´
E 46F
è
v
ikjBl u m no 8
r
0
3 78
>
Ý
d r j l r9:<;=1 ? r@ j Ð kACB rD .
M lk : 6
Since the current j( r) divided by the density Rn ( r) is the
+velocity field of the system, it follows from g t( urv) h ½c Dthat
455
úand
e 40f
c
4
Solving this equation and substituting the result into Eq. c 43d
finally
RKLI± ²leads to an expression for the xc four-potential
e ( r) that depends explicitly on the set of single-particle
A ¯ xc
fspinors gh . We thus have obtained a method of calculating
ii
ROPM± ²
é
j ( r) in an approximate way,
the xc four-potential A ¯ xc
øwhich is numerically much less involved compared to the
full solution of the ROPM integral equations.
V. ELECTROSTATIC LIMIT
The ROPM and RKLI methods, developed in the preceding sections, can be applied to systems subject to arbitrary
fstatic external four-potentials. In particular, the methods alk
low us to deal with external magnetic fields of arbitrary
fstrength. Yet, in electronic structure calculations of atoms,
molecules, and solids, we most commonly encounter
situa¨
é
where
no
magnetic
fields
are
present
in
a
suitable
tions,
l
m
Lorentz frame, typically the rest frame of the nuclein .
Thus in this section we consider four-potentials whose
fspatial components vanish, i.e., PAo ±( ·r²) p q0. r èThis also inåcludes a partial fixing of the gauge.ext
s In this situation, a simtplified Hohenberg-Kohn-Sham scheme can be developed,
fstating that the zeroth component un ±( ·r²) v Ý j Õ 0 ±( ·r²) of the groundfstate current density alone determines the external potential
u x ú
u | }
V
~ o ext w n and the ground-state wave function y{z n uniquely
for a discussion on this so-called ‘‘electrostatic case’’ cf.
Refs. ± 21,23
² . Consequently, only a scalarº effective potené
tialY V ý S ( ·r) is present in the RKS equation 2 .
When orbital-dependent functionals are used for the xc
ñenergy in this context, the corresponding scalar xc potential
± ²
V xc( r) can be calculated by repeating the steps of Sec. III.
One then finds the ROPM integral equation for the ‘‘electrofstatic case’’:
APPROXIMATE RELATIVISTIC OPTIMIZED
57

Ä cÅ2 Èk
3 Ð · ROPM ·
d r
r
k r V xc
F
ROPM
ý · ì · ¢¡ Ð £ · ¤¥ å ¦ q
xc

E
G Sk r , r k r c.c. 0.
Ð
k r
Y
§ö ¨
48
V RKLI
xc
Ç ·rÈÊÉ
V ROPM
xc
un Ð Ó ·rÔÊÕ×Ö
k
1
º u Ë · Ì Å ÏÍ Ð
2 n r Î Ä c 2 È ÑÒ
k
Þ å
@
Ð ØÚÙ V̄ ROPM
Ð Û ¯
u xckÐ ÜTÝ
xck
xck
F
c.c.,
øwhere
1
á ¯ Ð â rã : ä
un Ð å ræ
xck
k
ç
ß ª50à
ROPM
E ¯ xc
õ Ð ö
Ð k î rïð ñic òôóÊ¯
Ðû
èé Ð ê
k r÷ùøú k rüTý þ
í
ë
ì
r
k
ª ÿ
51
úand
Ð k ·r : Ì ROPM
3 Ð · ROPM · E xc

d r
r
Ð · Gý Sk ·r ì, ·r
k r V xc
k r
ª 52
fsimilar to Eq. 17 . Equation ª50! represents the ‘‘electrofstatic case’’ analog of Eq. " ö42# úand can also be approximated
¨
in the same way, namely, by neglecting all terms involving
ª '
$ Ð% ± · ²
k ( r) in Eq. & 51 .
In the context of the ‘‘electrostatic case’’ considered here,
fsome more insight into the nature of this approximation can
(
be gained: It can be interpreted as a ‘‘mean-field’’-type aptproximation in the sense that the average of the neglected
é
Kterms with respect to the ground-state density vanishes. To
demonstrate this, we note that the neglected terms averaged
)over *n ±( r)² are given by
+
ñ
ic
Å
/
, Äc 2 . Èk 021 F
3 3547¯
d r 6 Ð k 8 ·r9;:< Ð k = ·r>@?;A åc.c.
B ª53C
Applying the divergence theorem, this integral can be transformed to a surface integral which vanishes for finite systems
G ·rHI º 1 · K
2 *n J r M Ä c Å2 N/L O È P/Q
k
F
W *n Ð X r7Y Z V̄ ¯RKLI
Ð [
k
xck
`
R Ì ROPM
E
ST xcÐ U · V
k
@
¯
u xckÐ \
r
] åc.c.
^ ª54_
From this form it is obvious that the RKLI potential closely
S
In
· fact,
K using this¹ approach, Shadwick, Talman, and Norman
¸
22
derived the x -only limit of the ROPM integral equation
º
48¼ » .
Compared to the four ROPM integral equations ½ 15¾ ì,
øwhich determine the xc four-potential ´ A ROPM ì, Eq. À ö48Á ¨is
xc¿
åconsiderably simpler. Still, its numerical solution
is a rather
K
demanding task which has been achieved so far only
for
fsystems of high symmetry, i.e., spherical atoms Â º21–24Ã .
Ä
Again, an approximate ROPM scheme can be derived: Following the arguments of Sec. IV, the ROPM integral equaé
Å
tion 48Æ åcan be exactly rewritten as
143
if the surface is taken to infinity. Hence, neglecting the terms
± ²
involving D Ð kE ( r) means replacing them by their average
Fvalue, which is zero.
è
ROPM± · ²
( The xc potential V xc ( r) can therefore approximately
ïbe determined by the following equation, leading to the
RKLI equation for the ‘‘electrostatic case’’:
We mention that the same result is obtained if one de³mands that V ý ±( r)² be the variationally best local effective
S
tpotential yielding
single-particle
spinors minimizing the
ª «
é
total-energy functional © 5 ì, i.e.,
¬ Ì ROPM
E
q
´ö µ
® tot
0.
49
V ý S ¯ ·r° V ± ² V± ROPM ³
S
¶
...
¬resembles the relativistic Dirac-Slater potential as well as the
nonrelativistic KLI potential. Whether the accuracy of the
åcorresponding nonrelativistic scheme is maintained in the
¬relativistic domain will be investigated in the following secé
tion.
¶
aVI. RESULTS
In this section, we test the accuracy of the approximate
ROPM scheme, derived within the framework of the ‘‘elecé
trostatic case’’ in the last section, for atomic systems. In
)order to assess the quality of this approximation, exact refsults either for the xc energy E )or for the xc potential
xc
±·²
where relativ¨V xc( r) would be useful. However, for systems
b
istic effects become important, e.g., high-Z úatoms, exact refsults are presently not available. Consequently, we have to
look
b for a different standard to compare with.
is available within the ¹ x -only
k Such a standard
º reference
d
limit of RDFT c 21,23,24 . As in the nonrelativistic case, the
¹ x -only limit of the xc energy functional is defined by the use
)of the exact exchange energy functional, i.e., by the relativfg
istic
h i Fockj term, Eq. e 6 ì,b in the case of only longitudinal
Coulomb interactions. k Since, in the present context, our
tprincipal goal is to the test the quality of the RKLI method,
øwe restrict ourselves to this longitudinal case and neglect
é
transverse contributions.l As explained in the preceding sec± ²
é
tion, the exact longitudinal exchange potential V Lx ( ·r) can
é
then
be obtained by solving the
p full ROPM integral equation
m
15n øwith E ¯ xc replaced by E o Lx ,exact . Simultaneous solution of
é
é
the ROPM integral equation and the RKS equation q 2r therefore represents the exact implementation of the longitudinal
¹ x -only limit of RDFT. This scheme will serve as a reference
fstandard in the following.
It is first compared — of course — to the RKLI method,
øwhich employs the same exact expression s f6t ufor the exåchange energy and only approximates the local exchange
tpotential V o L ±( ·r²) by means of Eq. v ª54w . Besides that, we list
x
é
the results from traditional RKS calculations obtained with
é
¹ RLDAx úand two recently
the longitudinal ¹ x -only RLDA (x
introduced
gradient approximation
yï
|
z u relativistic{ º generalized
RGGA
functionals
24
:
The
first
one is based on the
ö ï
}

Becke88 GGA ~ 42 RB88 ì, the second one on a GGA funcé
tional
due to Engel, Chevary, MacDonald, and Vosko 11

.
RECMV92
è
These
various
approaches are analyzed for spherical
å
closed-shell úatoms. To this end, the spin-angular part of the
144
57
ªTABLE I. Longitudinal ground-state energy « E ¬L from various self-consistent ® x -only and RHF calcula¯tions. ° ¯ denotes the mean absolute deviation and ¯± totthe average relative deviation ² in 0.1 percent³ from the
´exact ROPM values µ ¶all energies are in hartree units· ¸.
® x RLDA
RHF
ROPM
RKLI
RB88
RECMV92
¹He
ºBe
»Ne
¼Mg
Ar
½Ca
Zn
¾Kr
¿Sr
Pd
½Cd
ÀXe
Ba
Yb
¹Hg
Rn
Ra
»No
ï
Á¯
Â
¯
2.862
14.576
128.692
199.935
528.684
679.710
1794.613
2788.861
3178.080
5044.400
5593.319
7446.895
8135.644
14067.669
19648.865
23602.005
25028.061
36740.682
2.862
14.575
128.690
199.932
528.678
679.704
1794.598
2788.848
3178.067
5044.384
5593.299
7446.876
8135.625
14067.621
19648.826
23601.969
25028.027
36740.625
2.862
14.575
128.690
199.931
528.677
679.702
1794.595
2788.845
3178.063
5044.380
5593.292
7446.869
8135.618
14067.609
19648.815
23601.959
25028.017
36740.609
2.864
14.569
128.735
199.952
528.666
679.704
1794.892
2788.907
3178.111
5044.494
5593.375
7446.838
8135.612
14068.569
19649.141
23602.038
25028.105
36741.900
2.864
14.577
128.747
199.970
528.678
679.719
1794.880
2788.876
3178.079
5044.442
5593.319
7446.761
8135.532
14068.452
19649.004
23601.892
25027.962
36741.783
2.724
14.226
127.628
198.556
526.337
677.047
1790.458
2783.282
3172.071
5036.677
5585.086
7437.076
8125.336
14054.349
19631.622
23582.293
25007.568
36714.839
0.006
0.189
0.168
8.668
0.002
0.103
0.108
6.20
tproaches and the variationally best local tpotential
ö
L,ROPM± · ²
RKS wave function is treated analytically and the remaining
¬radial Dirac equation is solved numerically on a logarithmic
V

R nucl 1.0793 A 1/3 0.735 87 fm,
åclosely: For the mean absolute deviation from the exact
ï
ª
55
ROPM data of the 18 neutral atoms listed in Table I, one
only 5 mhartree. Thus the RKLI method impresfsively improves on the RLDA results, for which we find a
³mean absolute deviation of 6092 mhartree. The accuracy of
é
the RKLI scheme becomes even more obvious when comtpared to the RGGAs. Both RGGAs improve significantly
)over the RLDA method. Still, their deviations from the exact
ï
ROPM data are more than one order of magnitude larger
åcompared to the RKLI results.
The trends found in the above discussion are almost idené
tical to the ones found in the nonrelativistic case. In order to
úanalyze the relativistic effects more directly, we additionally
åconsider the relativistic contribution to ÌE L ì, defined by
tot
)obtains
øwhere A is the atomic mass taken from 43 . We mention in
tpassing that employing finite nuclei is not necessary to en-
fsure convergent results as, for example, in the relativistic
è
Thomas-Fermi model. We incorporate finite nuclei because
they represent the physically correct approach.
In Table I, we show the longitudinal ground-state energy
E Ltot )obtained from the various self-consistent ¹ x -only RDFT
úapproaches and, in addition, from relativistic Hartree-Fock

RHF åcalculations. Comparing the first two columns, we
¬recognize that the RHF and the ROPM data are very close.
è
The largest deviation is found for Be with 41 ppm. With
increasing atomic number, the inner orbitals, contributing
most to the total energy, become more and more localized
fsuch that the difference between the nonlocal RHF potential
úand the local ROPM decreases. In fact, for No, the difference
Kis down to 2 ppm. We emphasize that these differences are
ïdue to the different nature of the two approaches. While the
RHF method, by construction, yields the variationally best
ñenergy, the ROPM scheme additionally constrains the exåchange potential to be local. Consequently, we expect the
ROPM results to always be somewhat higher, which is confirmed in Table I. In the third column, the total energies
)obtained from the RKLI approximation are presented. They
úalways lie above the corresponding ROPM values since the
fsame exchange energy functional is employed in both ap-
( r) is approximated by Eq. 43 in the RKLI ap-
xc
tproach.
However, the results are clearly seen to agree very
mesh 21 . In all our calculations we use finite nuclei modñeled by a homogeneously charged sphere with radius
é
¨

¡;¢
L * R
n
E tot : E tot
£ ¤ £ ¥
* NR .
E NR
tot n
ª §
¦ 56
Via this decomposition, we are able to test the quality of the
RKLI scheme independently of the accuracy of its nonrelaé
tivistic equivalent. Yet, at first, we want to point out that the
¬relativistic treatment leads to drastic corrections especially
for high-Z úatoms. For example, Table II shows that the relaé
tivistic correction of Hg amounts for about 6.7% of the total
ñenergy thus demonstrating the need for a fully relativistic
é
treatment. Furthermore, by comparing the second and third
åcolumns of Table II, we realize that the ROPM and the RKLI
³method yield almost identical results for the relativistic coné
©
tribution E tot . In other words, almost no additional deviaé
tions are introduced by the relativistic treatment of the KLI
APPROXIMATE RELATIVISTIC OPTIMIZED
57
...
145
ªTABLE II. Relativistic contribution Ì5Í ÎE ¬L from various self-consistent ® x -only and RHF calculations. Ï ¯
tot

Ðdenotes the mean absolute deviation and ¯Ñ the
Ó
ÕROPM values Ö all energies are in hartree units× . average relative deviation Ò in 0.1 percentÔ from the exact
® x RLDA
RHF
ROPM
RKLI
RB88
RECMV92
¹He
»Be
Ne
¼Mg
Ar
½Ca
ØZn
¾Kr
¿Sr
½Pd
Cd
Xe
ÙBa
Yb
¹Hg
ÕRn
Ra
»No
Ú¯
Û
¯
0.000
0.003
0.145
0.320
1.867
2.953
16.771
36.821
46.554
106.527
128.245
214.860
252.223
676.559
1240.521
1736.153
1934.777
3953.172
0.000
0.003
0.145
0.320
1.867
2.953
16.770
36.820
46.553
106.526
128.243
214.858
252.222
676.551
1240.513
1736.144
1934.770
3953.155
0.000
0.003
0.145
0.320
1.867
2.953
16.770
36.820
46.553
106.526
128.243
214.858
252.221
676.549
1240.511
1736.142
1934.768
3953.151
0.000
0.003
0.145
0.321
1.867
2.952
16.779
36.822
46.552
106.526
128.243
214.825
252.176
676.590
1240.543
1736.151
1934.781
3953.979
0.000
0.003
0.145
0.321
1.867
2.953
16.779
36.821
46.551
106.525
128.241
214.822
252.173
676.588
1240.538
1736.151
1934.783
3954.015
0.000
0.002
0.138
0.308
1.821
2.888
16.555
36.432
46.092
105.715
127.323
213.522
250.725
673.785
1236.349
1730.890
1929.116
3944.569
0.001
0.056
0.058
1.788
0.009
1.14
1.35
33.7
fscheme. Considering now the relativistic corrections calcumethods, the conclusions
Klated with the conventional RDFT
L å
drawn in the discussion of E tot can be repeated: Compared to
the RKLI method, the RGGA results are worse by more than
)one order of magnitude whereas the RLDA is the by far least
úaccurate approximation.
These trends also remain valid when other quantities of
Ãinterest are considered. For example, in Tables III and IV we
have listed the relativistic contributions to the longitudinal
ñexchange energy, defined analogously to Eq. Ä ª56Å . Again, the
RKLI method almost exactly reproduces the ROPM
results.
It is worthwhile noting that the exchange energy E Æ xL is influñenced quite substantially by relativistic effects, too. Taking
úagain Hg as an example, we realize that the 5.8% contribu
é
tion to E Æ xL is of the same order as for the total energy. Furé
thermore, even for lighter atoms such as Mg, the relativistic
åcorrections to ÌE Æ L úare comparable or even larger than the
x
K
differences between the currently best nonrelativistic exåchange functionals. As a consequence, a relativistic treat³ment is indispensable for the ultimate comparison with extperiments Ç 21È . Next, we turn our attention to the calculation
)of É ÌEÆ ufrom the RGGA functionals listed in the third and
x
u
fourth columns of Table III. With the exception of No, the
results are also in excellent agreement with the exact ones.
However, when we turn our attention to other systems like
fsome ions of the neon-isoelectronic series shown in Table
¶
IV, we recognize that the RGGAs cannot reproduce the exact
results to the same level of accuracy as obtained for the
neutral atoms above Ê 24Ë . This can be explained by the fact
é
that the RGGAs are optimized for the neutral atoms. In coné
trast, the agreement of the RKLI results with the exact ones
for these ions is as good as for the neutral atoms.
é
ªTABLE
Î
III. Relativistic contribution Ü5Ý E Þ x to the exchange
energy from various self-consistent ® x -only calculations. ß ¯ denotes
the mean absolute deviation and ¯
à ¯the average relative deviation á Óin
percentâ from the exact ROPM values ã all energies are in hartree
unitsä ¸.
He
Be
Ne
Mg
Ar
Ca
Zn
Kr
Sr
Pd
Cd
Xe
Ba
Yb
Hg
Rn
Ra
No
å¯
æ
¯
ROPM
RKLI
RB88
RECMV92
® x ÕRLDA
0.000
0.001
0.015
0.029
0.118
0.172
0.627
1.215
1.478
2.785
3.264
5.021
5.739
12.043
19.963
26.637
29.241
52.403
0.000
0.001
0.015
0.029
0.118
0.172
0.626
1.214
1.477
2.787
3.264
5.020
5.736
12.024
19.956
26.620
29.218
52.402
0.000
0.001
0.015
0.029
0.117
0.171
0.632
1.212
1.473
2.782
3.255
4.977
5.684
12.027
19.965
26.612
29.225
53.168
0.000
0.001
0.015
0.029
0.118
0.171
0.632
1.211
1.472
2.780
3.252
4.974
5.680
12.024
19.957
26.610
29.224
53.205
0.000
0.000
0.007
0.015
0.069
0.104
0.402
0.814
1.005
1.958
2.322
3.657
4.215
9.194
15.734
21.307
23.513
43.683
0.004
0.053
0.056
1.819
0.079
1.03
0.857
35.9
146
57
TABLE IV. Relativistic contribution E Þ x to the exchange
energy from various self-consistent ® x -only calculations for some
Ð
ions of the neon-isoelectronic series. ¯ denotes the mean absolute
the average relative deviation in percent from the
deviation and ¯
exact ROPM values all energies are in hartree units .
Ca 10 Zr30 Nd 50
Hg70
! "
Fm90
ROPM
RKLI
RB88
RECMV92
® x RLDA
0.159
1.528
5.780
15.599
#
36.475
0.160
1.528
5.783
15.606
36.494
0.158
1.507
5.667
15.293
36.258
0.158
1.509
5.675
15.325
36.366
0.093
0.983
3.970
11.365
28.173
0.006
0.132
0.102
2.992
0.041
1.34
1.17
31.8
$¯
%
¯
è
To further proceed with our investigation, we next raise
the question of how± well
local properties such as the exåchange potential V Æ L ( r)² are reproduced within the different
x
ï
uRDFT methods. In Fig. 1, the exchange potential is plotted
for the case of Hg. As expected, the RKLI result follows the
ñexact curve most closely. Yet, the pronounced shell structure
úapparent in the ROPM curve is not fully reproduced, alé
å
ì
éthough clearly visible ç cf. the more detailed plot in Fig. 2è , in
the RKLI approximation. However, it again improves sigénificantly over the conventional RDFT results, where the intershell peaks are strongly smeared out or even absent. In
Fig. 3, the asymptotic region, which is of particular imporé
ètance for excitation properties, is plotted in greater detail.
There we recover the known deficiencies of the conventional
RDFT functionals: The RLDA as well as both RGGAs fall
)off much too rapidly. In contrast, the RKLI and ROPM
åcurves become indistinguishable in the asymptotic region,
(
both decaying as 1/r úas r éëê úand thus reflecting the proper
åcancellation of self-interaction effects. Since all these obserFvations are already known for the corresponding nonrelativistic functionals, we again consider the relativistic contribué
tion separately. This relativistic contribution to the exchange
tpotential, given by
é
& ' + (
FIG. 2. Longitudinal exchange potential V Þ Lx ( r) for Hg showing
Ó
the shell structure of Fig. 1 in more detail , in hartree units- .
ì
ð
VÆx í r î : ï
£ ö £ ÷ø
ú û ñ £ NRüý
V Æ OPM
n
rþ
x
ñn NR r ù
V Æ Lx ñn R òó r ôõ V Æ NR
x
& '
(
ì,
ÿ ª57
FIG. 1. Longitudinal exchange potential V Þ Lx (r) for Hg from
various self-consistent ® x -only calculations ) in hartree units* .
is plotted in Fig. 4. We first observe strong oscillations between 0.1 a.u. and 5 a.u. These oscillations are introduced by
é
the displacement of the density due to relativistic effects and
é
thus represent a direct consequence of the atomic shell strucé
ture. As the shell structure of the exchange potential is not
fully reproduced within the RKLI approach, the amplitudes
)of the oscillations are somewhat smaller compared to
ú OPM± ²
V Æ x ( r ) . While these deviations are clearly visible, the
RKLI curve is still closest to the exact one, especially in the
kregion near the nucleus and in the asymptotic region, where
large deviations occur for the conventional RDFT methods.
The properties of the ¹ x tpotential in the large-r úasymptotic
¬region also strongly influence the eigenvalue of the highest
)occupied orbital shown in Table V: Due to the correct asymptotics, the energies calculated within the RKLI scheme
úare almost identical to the exact ROPM results. On the con(
é
trary, the lack of the correct 1/r behavior of the RGGAs and
é
the RLDA shows up in rather poorly reproduced eigenvalues
)of the highest occupied state. Since in the exact nonrelativ
istic theory, the highest occupied energy level coincides with
é
the
potential 44 ì, we also list this quantity, given
( ionization
Ì
Ì
by I E tot N 1 E tot N ì, in Table VI. Again, the RKLI
úand ROPM methods provide almost identical results. In ad-
é
FIG. 3. Asymptotic region of the longitudinal exchange potential of Fig. 1 . in hartree units/ .

57
APPROXIMATE RELATIVISTIC OPTIMIZED . . .
147
TABLE VI. Ionization potential calulated from various selfconsistent ® x -only calculations. t ¯ denotes the mean absolute devia
Ó
tion and ¯
u the average relative deviation v in percentw from the exact
ROPM values x ¶all energies are in hartree unitsy .
Be
Mg
z
Ca
{
Zn
U
Sr
z
ºCd
Ba
Hg
~

FIG. 4. Relativistic contribution V Þ x (r) , Eq. 57 , for Hg from
various self-consistent ® x -only calculations in hartree units .
ROPM
OPM
RKLI
RB88
RECMV92
® x RLDA
0.296
0.243
0.189
0.284
0.175
0.269
0.157
0.313
0.295
0.243
0.188
0.278
0.172
0.253
0.152
0.250
0.296
0.243
0.189
0.283
0.175
0.268
0.157
0.312
0.326
0.269
0.210
0.339
0.195
0.315
0.176
0.364
0.327
0.269
0.210
0.338
0.196
0.315
0.176
0.363
0.312
0.261
0.205
0.334
0.192
0.314
0.174
0.364
0.000
0.034
0.034
0.029
0.130
13.5
13.6
11.6
|¯
}
¯
0
dition, when comparing the ionization potential to the ener1gies of the highest occupied level, we see that the relation
I 243 £ N — which we of course expect to hold only approxi5mately, since correlation contributions are neglected — is
6
fulfilled within a few percent for the ROPM and RKLI data,
7whereas the results of the conventional ¹ x 6functional differ to
8a much larger extent. To conclude, we note that the inclusion
9of relativistic effects leads to large corrections for heavy at9oms even for the outermost orbitals: Taking Hg as an ex8ample, we find a 20% shift for the ionization potential.
f
TABLE V. Eigenenergy gih jN of the highest occupied orbital
from various self-consistent ® x -only calculations. k ¯ denotes the
¯
l the average relative deviation m in
mean absolute deviation and ¯

percentn from the exact ROPM values o all energies are in hartree
unitsp .
Be
Mg
Ca
Zn
Sr
Cd
Ba
Hg
OPM
RKLI
RB88
RECMV92
® x RLDA
0.309
0.253
0.196
0.299
0.181
0.282
0.163
0.329
0.309
0.253
0.196
0.293
0.179
0.266
0.158
0.262
0.309
0.253
0.196
0.298
0.181
0.282
0.163
0.332
0.181
0.149
0.116
0.195
0.108
0.183
0.097
0.223
0.182
0.149
0.116
0.194
0.108
0.182
0.097
0.222
0.170
0.142
0.112
0.191
0.104
0.181
0.095
0.222
0.001
0.095
0.095
0.099
0.156
38.3
38.4
40.0
r
q¯
s
¯
r
G
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 I 1964J .
W. Kohn and L.J. Sham, Phys. Rev. 140,N A1133 O 1965P .
S
R.M. Dreizler and E.K.U. Gross, Density Functional Theory
TU
Springer, Berlin, 1990V .
W S
4X R.G. Parr and W. Yang, Density-Functional Theory of Atoms
K
1H
2L
Q
3R
ROPM
M
:
¶
VII. CONCLUSIONS
In this work, we derived an approximate ROPM scheme
1generalizing the arguments of Krieger, Li, and Iafrate to the
relativistic domain. As for the full ROPM, the advantage of
;
the RKLI method lies in the fact that xc functionals depending explicitly on a set of RKS single-particle spinors can be
;
treated within the framework of RDFT. This, in particular,
8allows us to employ the exact expression for the exchange
<energy functional, i.e., the relativistic Fock term in the lon1gitudinal case. Therefore the RKLI approach satisfies a num=
ber of important properties, most notably the freedom from
>self-interactions implying the correct asymptotic decay.
¶
In numerical tests on spherical atoms within the longitu0
¹ x -only limit of RDFT the RKLI method was found to
dinal
=
be clearly superior to the known relativistic ¹ x -only function8als. The results obtained are seen to be in close agreement
7with the exact ROPM values, thus reducing the deviation of,
6
0for example, the widely used RLDA by more than three orders of magnitude. On the other hand, the numerical effort
involved is considerably less compared to the solution of the
full ROPM scheme. We therefore expect that the RKLI
>scheme will be successfully used for larger ? @nonsphericalA
>systems, e.g., molecules, including also correlation contribu;
tions.
B
ACKNOWLEDGMENTS
C
Many
0 valuable discussions with K. Capelle, T. Grabo, and
M. Lüders are greatly appreciated. One of us D T.K.E 8ac0knowledges financial support from the Studienstiftung des
deutschen Volkes. This work was supported in part by the
F
Deutsche Forschungsgemeinschaft.
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