approximate relativistic optimized potential method · sity,3 provided the orbitals come from a...

Approximate relativistic optimized potential method

T.�

Kreibich andE. K. U. GrossInstitut�

fur� Theoretische Physik, Universitat Wurzburg,� Am Hubland, D-97074 Wurzburg,� Germany

E. EngelInstitut�

fur Theoretische Physik, Universitat� Frankfurt, Robert Mayer Straße 8-10, D-60054 Frankfurt-am-Main, Germany�Received28 July 1997�

Approximatesemianalyticalsolutionsof the integral equationfor the relativistic optimized potential areconstructed� by extendinga methodrecently proposedby Krieger, Li, and Iafrate � Phys.Lett. A 146,� 2561990�� to the relativistic regime.The quality of the approximationis testedin the longitudinal x -only limit

where� fully numericalsolutionsof the relativistic optimizedeffectivepotentialintegralequationareavailablefor sphericalatoms.The resultsobtainedturn out to be in excellentagreementwith the exactx -only values.The�

proposedmethodprovidessignificant improvementover the conventionalrelativistic local densityap-proximation� andgeneralizedgradientapproximationschemes.� S1050-2947� 97� 04912-3�PACSnumber� s� � :� 31.10.� z, 71.10.� w, 31.30.Jv,31.15.Ew

I.�

INTRODUCTION

Since�

the seminalwork of HohenbergandKohn � 1� and�Kohn

and Sham ! 2" # ,$ density functional theory % DFT& '

has(

become)

a powerful tool for ab* initio electronic+ structurecal-culations, of atoms,molecules,andsolids - 3–5

. /. The devel-

opment0 of more and more refined approximationsof theexchange-correlation+ 1 xc2 energy+ functional Exc has led tosignificant3 improvementover the standardlocal densityap-proximation4 5 LDA

6 7. In particular, the so-calledoptimized

potential4 method 8 OPM9 :<;

6,7= >

employing+ explicitly orbital-dependent?

functionals rather than the traditional density-dependent?

functionalshas achievedhighly accurateresults@7–2A

0B .MostC

of theadvancesin DFT havebeenmadein thecon-textD

of nonrelativisticphysics.For high-ZE

atoms,� however,relativisticF contributionshaveto beconsidered.For example,theD

ground-stateenergyof mercury G HgH decreases?

from itsnonrelativisticI value J 18408a.u.to K 19649a.u.if relativ-istic effects are taken into account.Furthermore,even forsystems3 with moderateZ

E,$ relativisticcontributionsto E

LxcM are�

largerN

thanthe differencesbetweenthe currentlybestxc en-ergy+ functionalsO 21P . Until now, thecalculationof suchrela-tivisticD

contributionshasmostly beenbasedon the relativis-ticD

local densityapproximationQ RLDAR S

. To go beyondtheRLDA, an xT -only version of the OPM was formulatedforrelativisticF systemsU 22,21

" V. As in thenonrelativisticcase,the

solution3 of theresultingequationsis a ratherdemandingtaskand� hasbeenachievedso far only for systemsof high sym-metry, e.g.,sphericalatoms W 21,23–25X .

The�

purposeof the presentpaperis to developa simpli-fiedY

versionof therelativisticOPM Z ROPMR [

scheme3 leadingtoD

a generalizationof the approximationof Krieger, Li, andIafrate \ KLI ]<^ 8_ –10,26–36 to

Dthe realm of relativistic sys-

tems.D

This will bedonefor systemssubjectto arbitrarystaticexternal+ four-potentials.The paperis organizedas follows.In Sec.II we give a brief review of the foundationsof rela-tivisticD

DFT a RDFTb . After that, in Sec.III, we developtheROPMR

generalizinga nonrelativisticderivation of GorlingFand� Levy c 37

. dtoD

therelativisticdomain.TherelativisticKLI

eRKLI f scheme3 is developedin Sec.IV beforesomelimiting

cases, arediscussedin Sec.V. In Sec.VI, numericalresultsof0 the ROPM and RKLI methodsare presentedand com-pared4 to otherRDFT methods.

II. THEORETICAL BACKGROUND

On9

its most generallevel, RDFT is basedon quantumelectrodynamics+ g QED

h iand� thuscontainsnot only relativis-

ticD

but also radiativeeffects.For a detailedderivation,alsoincludingj

questionsof renormalization,the readeris referredtoD

recentreviews k 23,38" l

. Thecentralstatementof RDFT —theD

relativisticversionof theHohenberg-Kohnm HK n theoremDo

39. p

— canbestatedin thefollowing way: Therenormalizedground-stateq four-current j

r s(tru )v of an interactingsystemof

Dirac particlesuniquelydetermines,up to gaugetransforma-tions,D

the external four-potential Aextw xzy jr {}| as� well as theground-stateq wave function ~�� jr �}� . As a consequence,anyobservable0 of the relativistic many-bodysystemundercon-sideration3 is a functionalof its ground-statefour-current.Asinj

thenonrelativisticcase,theexactground-statefour-currentincluding all quantumelectrodynamicaleffectscan in prin-ciple, be obtainedfrom an auxiliary noninteractingsystem—theD

relativistic Kohn-Sham� RKSR �

system3 � 21,38,40,41" �

:

jr ��

r�� c� 2� ��

k� ��

F

�k� � r�� k

� ¡ r¢ £ jr

V¤¦¥ r§ ,$ ¨ 1©whereª j

rV« (t r)

vdenotesthe vacuumcontribution to the four-

current., The four-componentspinors ¬ k� (t r)

vare solutionsof

an� effective single-particleDirac equation in atomic units®°¯e± ² m³ ´ 1)

µ0¶ · ic

¸ ¹ º¼»c½ 2¾ ¿ÁÀÃÂ

ASÄ Å�Æ jr Ç}È�É rÊ Ë k

� Ì rÍ�ÎÐÏ k� Ñ

k� Ò rÓ ,$ Ô 2Õ

withª AÖ

SÄ ×�Ø jr Ù}Ú (t ru )v beingthe effectivefour-potential,which can

be)

decomposedaccordingto

AÖ

SÄ Û�Ü jr Ý}Þ�ß ru à�á A

Öextw âzã ru ä�å d

æ 3çrè é jr êìë

r íïîðru ñ ru òôó õ A

Öxcöø÷ jr ùûú�ü ru ý . þ 3. ÿ

PHYSICAL REVIEW A JANUARY�

1998VOLUME 57, NUMBER 1

57�

1050-2947/98/57� 1� /138� �

11� /$15.00�

138 © 1998The AmericanPhysicalSociety

Here,�

AÖ

extw � (tru )v is a staticbut otherwisearbitraryexternalfour-

potential,4 thesecondtermon theright-handsiderepresentsaHartree-like�

potential,andthe last term,definedby

AxcM � jr �� r� : ��

EL

xcM � jr ��jr ��

r��4�

denotes?

the xc four-potentialcontainingby constructionallnontrivial many-bodyeffects.

Equations � 1 – ! 4" representthe relativistic KS schemethatD

hasto besolvedself-consistently.However,thecalcula-tionD

of the vacuum contribution jr

V# (t ru )v to the four-currentrequirestheknowledgeof aninfinite numberof $ positive4 andnegativeI energy% states,3 so that onewould haveto dealwithan� infinite systemof coupledequations.Sincesucha proce-dure?

is highly impractical,we will, in the following, ignoreall� vacuumcontributionsto the variousenergycomponentsand� to the four-current,which is thengiven just by the firsttermD

on the right-handside of Eq. & 1' . This meansthat werestrictourselvesto the calculationof relativistic effectsandneglectradiative corrections.Since we are aiming at elec-tronicD

structurecalculationsfor atoms,molecules,andsolids,weª expectthe neglectedtermsto be small. If onewereafterall� interestedin the radiativecontributions,an a* posterioriperturbative4 treatmentshouldbe sufficientandrepresentsinfact(

the standardapproach.

III. RELATIVISTIC OPTIMIZED POTENTIAL METHOD

In)

orderto derivea relativisticgeneralizationof theOPMintegralj

equation,we start out from the total-energyfunc-tionalD

of a systemof interactingDirac particles * neglectingvacuum+ contributions, subject3 to a static external four-potential4 Aextw - (

tr):v

E tot.ROPM/ 0

jr 1�243 5

6 c� 2 798k� :�;

F< d

= 3>r ? k� @ rACBED ic

¸ F GIHc½ 2 JLK

k� M rN

Od= 3>r jè P�Q ru R AÖ extw SUT ru V

W 1

2d= 3>r d

= 3>r X jr Y�Z

r[ jr \�]

r ^`_ar b r ced

fExcMROPMgihkj

i lnm .o5p q

In)

contrastto the ordinary RDFT approach,the xc energyfunctionalis given hereasan explicit functionalof the RKSfour-spinorsrts i u . Still, ExcMROPM

/ vxwtyi zn{ representsa functional

of0 thedensity:Via theHK theoremappliedto noninteractingsystems,3 j

r |(tr)v

uniquely determinesthe effective potentialASÄ }�~ jr �� . With this very potential, the Dirac equation � 2� is

solved3 to obtain the set of single-particleorbitals �t� i � jr ��whichª are then used to calculate the quantityEL

xcROPM/ �t�

i � jr ��e� . Thereforeeveryfunctional,dependingex-±plicitly� on0 RKS spinors,is an implicit

¸functionalof the den-

sity,3 providedthe orbitalscomefrom a local potential.Thisallows� us to use the exact expressionfor the longitudinalexchange+ energy,i.e., the relativistic Fock term

Ex�L� ,exact� jr �� 1

2

�� c� 2 ��

l� , � k� e¡ F

< d= 3>r d

= 3>r ¢

£¥¤ l¦ §©¨ rªL« k

� ¬ rL® k� ¯±° r ²`³L´ l

¦ µ r ¶`·¸ru ¹ ru º�» . ¼ 6= ½

One9

major advantageof suchan exacttreatmentof the ex-change, energy lies in the fact that the spurious self-interactionsj

containedin the Hartreeenergyare fully can-celed., Theprice to bepaid for theorbital dependenceof E

LxcM

is that thecalculationof thexc four-potentialAxcM ¾ (tr)v, defined

by)

Eq. ¿ 4À ,$ is somewhatmorecomplicated.It hasto be de-terminedD

by an integral equation,as will be shown in thefollowing.(

Starting�

from the definition of AxcÁ (tru )v , Eq. Â 4Ã Ä , w$ e can

calculate, the xc four-potentialcorrespondingto an orbital-dependent?

xc energyfunctional by applying the chain rulefor(

functionalderviatives:

AxcM ÅROPMÆ rÇ�È ÉÊ c� 2� Ë�Ì

k� Í9Î

F

d= 3>r Ï d

= 3>r Ð

Ñ Ò EL

xcROPMÓxÔtÕ

i Ön×Ø±Ùk� Ú ru Û`Ü

Ý±Þk� ß r à`áâ

AÖ

SÄ ãEä ru åEæ©ç c.c., è AS

Ä éEê rëEìíjr î�ï

ru ð .

ñ7A ò

The�

last term on the right-handside of Eq. ó 7A ô isj

readilyidentifiedj

with theinverseof thestaticresponsefunctionof asystem3 of noninteractingDirac particles

õSÄ ök÷Eø r,$ r ù`ú : û

üjr ý�þ

ru ÿ�ASÄ �� r �� ,$ � 8_ �

so3 that Eq. 9 � can, be rewrittenas

AxcM �ROPM/

r�� c� 2 ��k� ��

F

d= 3>r � d

= 3>r �

� � EL

xcROPM��

i !#"$&%k� ' ru (�)

*&+k� , r -�./

AÖ

SÄ 0�1 ru 2�354 c.c., 6

SÄ 798: 1 ; r< ,$ r= .>

9 ?

Acting@

with the responseoperator A 8_ B on0 Eq. C 9 D and� usingtheD

identity

d= 3>r E SÄ FHGI 1 J rK ,$ rL�M S

Ä NPORQ r,$ r S�T�UWVYX[Z\V^] r_a` r b�c d 10eweª obtain f after� rearrangingthe indicesg

d= 3>r h AxcM iROPM

/ jr k�l�m S

Ä nHoqp r r ,$ rst uv c� 2

� wqxk� y�z

F

d= 3>r { | E

LxcROPM}�~ �

i �#��&�k� � ru ��

�&�k� � r ��

AÖ

SÄ �q� ru �� c.c., � 11�

To furtherevaluatethis equation,we notethat thefirst func-tionalD

derivativeon the right-handsideof Eq. � 11� isj

readily

57 139APPROXIMATE�

RELATIVISTIC OPTIMIZED . . .

computed, oncean expressionfor EL

xcROPM in

jtermsof single-

particle4 spinorsis given.Theremainingfunctionalderivativeisj

calculated by using standard first-order perturbationtheory,D

yielding

�&�k� � ru ��

AÖ

SÄ �q� ru 5¡£¢ ¤

l¥

l¦ § ¨

k�©

l¦ ª ru «�¬k� ®°¯

l¦ ±¯ l¦ ² ru ³a´¶µ^· k

� ¸ ru ¹ . º 12»This�

equationalsoenablesus to give an explicit expressionfor(

the responsefunction

¼SÄ ½P¾À¿ r,$ r Á�Â : Ã ÄÅ

ASÄ Æ�Ç r È�É ÊË c� 2 Ì�Í

k� Î�Ï

F< Ð¯k

� Ñ rÒ�Ó^Ô^Õ k� Ö r× Ø

13Ùin termsof the RKS spinors:

ÚSÄ ÛPÜÀÝ ru Þ ,$ ru ß�à áâ c� 2

� ãqäk� å�æ

F<ç è

l¥

l¦ é ê

k�ë

k� ì r í�îaï¶ð^ñ l

¦ ò r ó�ôPõ l¦ ö r÷aø^ù�ú k

� û rüýk� þ°ÿ

l¦

�c.c., � 14�

Finally, puttingEqs. � 11� ,$ � 12� ,$ and � 14� togetherD

leadsto theROPMR

integral equationsfor the local xc four-potentialAÖ

xcROPM

(tru ):v�

� c� 2� ��

k� ��

F

d� 3�r � � k

� � r �� AxcM �ROPM� r �!

"$# EL

xcROPM

%'&k� ( ru )�* G

+SkÄ , ru - ,$ ru .0/ 0

¶ 1�2�3k� 4 ru 506 c.c., 7 0,

8

9;: 0,1,2,38 <

15=whereª G

+SkÄ (tru > ,$ ru )v is definedas

G+

SkÄ ? ru @ ,$ ru A : B$C D

l¥

l¦ E F

k�G

l¦ H ru I�JLK l

¦ M'N ru OPk� QSR

l¦ . T 16U

NowV

the ROPM schemeis complete:For a given approxi-mationof the xc energy,the ROPM integralequationshavetoD

be solved for AxcM WROPM(tr)v

simultaneouslywith the RKSequation+ X 2" Y untilZ self-consistencyis achieved.Note thatEq.[15\ determines

?the xc four-potentialA

Öxc] (tru )v only up to an

arbitrary� constant, which can be specified by requiringAxcM ^ROPM(

tr)v

to vanishasymptotically_ for finite systems` .To�

concludethis section,we notethat exchangeandcor-relationF contributionscan be treatedseparatelywithin theROPMscheme.This is mosteasilyseenby startingout withonly0 the exchange four-potential, defined asAxa b (

tr)v ced

Exa /f g

jr h

(tr)v, insteadof Eq. i 4j and� repeatingthe

steps3 which leadto Eq. k 15l and� likewise for the correlationpotential.4

IV. TRANSFORMATION OF THE ROPMINTEGRAL EQUATIONS

ANDm

THE RELATIVISTIC KLI APPROXIMATION

In order to usethe ROPM equationsderivedin the pre-ceding, section,we have to solve Eq. n 15o for the xc four-potential.4 Unfortunately,thereis no known analyticsolutionfor(

AÖ

xcpROPM

(tru )v dependingexplicitly on the set of single-

particle4 spinorsqsr i t . We thereforehaveto dealwith Eq. u 15vnumerically,I which is a ratherdemandingtask.Thusa sim-plified4 schemefor the calculation of A

ÖxcwROPM(

tru )v appears

highly desirable.To�

this end we shall perform a transformationof theROPMR

integral equationssimilar to the one recently intro-duced?

by KLI in thenonrelativisticdomain x 29,31y . This willleadto an alternativebut still exactform of the ROPM inte-gralq equationwhich naturally lendsitself asa startingpointfor(

systematicapproximations.We startout by defining

zk� {}| ru ~ : � d

� 3�rè � �

k� � ru �� AÖ xc�ROPM� ru ��

�$� EL

xcROPM

�'�k� � ru �� G

+SkÄ � ru � ,$ ru � ,$ � 17�

such3 that the ROPM integralequationscanbe rewrittenas

�� c� 2 ��

k� ��

F< �¯k

� � ru �� ¢¡¤£ k� ¥ ru ¦0§ c.c., ¨ 0,

8 ©18ª

whereª the adjoint spinor « k� (t ru )v is definedin the usualway,

i.e.,j

¬k� ru ® : ¯±° k

� ²'³ ru ´�µ 0¶. ¶ 19·

Since�

the RKS spinors ¸s¹ i º span3 an orthonormalset, onereadily provesthe orthogonalityrelation

d� 3�rè » k� ¼}½ ru ¾L¿ k

� À ru Á0Â 0.8 Ã

20" Ä

WeÅ

now usethe fact that G+

SkÄ (tru Æ ,$ ru )v is the Greenfunction of

theD

RKS equationprojectedonto the subspaceorthogonaltoÇk� (t ru )v , i.e., it satisfiesthe equation

G+

SkÄ È ru É ,$ ru ÊÌË hÍˆ DÎ}Ï ru Ð0ÑSÒ k

� Ó�ÔÖÕÖ×ÙØ�Ú ru ÛÙÜ ru Ý0Þàßk� á ru â�ãLä k

� å'æ ru çéè .ê21ë

The operatorhìˆ

Dí î (tr)v

denotesthe Hermitian conjugateof theRKSR

Hamiltonian,

hìˆ

Dï}ð ru ñ : ò±ó 0¶ ô

ic¸ õ öø÷úù

c½ 2 û±ü�ý AÖ SÄ þ ÿ ru �� ,$ � 22

" �acting� from the right on the unprimedvariableof G

+SkÄ (tru � ,$ ru )v�

theD

arrow on top of the gradientindicatesthe direction inwhichª thederivativehasto betaken� . UsingEq. 21 , w$ e canact� with theoperator� h�ˆ D (

tru )v �� k

� � from(

theright on Eq. � 17� ,$leadingN

to

140 57T. KREIBICH, E. K. U. GROSS,AND E. ENGEL

�k� �� r�� h�ˆ D

� �� r�� ! k� "$#&% '

k� ( r)$*,+ AxcM -/. r0�1

2EL

xcROPM3�4k� 5 ru 6

798k� :�; ru <>= AÖ¯xck

�ROPM? u@ xcM k� A ,$ B 23

" Cwhereª we haveintroducedA xcM k

�ROPM as� a shorthandnotationfortheD

averagevalue of DFE AxcM GROPM

(tr)v

with respectto the kHthD

orbital,0 i.e.,

A xcM k�ROPM

: I d

� 3�r J k� K rL$MFN AxcM OROPM

PrQSR k

� T rU V 24Wand�

u@ xcM k� : X d

� 3�rè Y EL

xcROPM

Z�[k� \ r ]_^ ` k

� a ru b . c 25" d

The�

differential equatione 23" f

has(

the structureof a RKSequation+ with an additional inhomogeneityterm. Togetherwithª the boundarycondition

gk� h�i rjlkrm npo 0,

8 q26r

Eq. s 23t uniquelyZ determinesu k� v (tru )v . To prove this state-

ment,we assumethat therearetwo independentsolutionsofEq. w 23x ,$ namely, y k

�,1

z(tr)v

and { k�

,2

|(tr)v. Thenthedifferenceof

theseD

two solutions,} k� ~ (tr):v ��

k�

,1

�(tr)v ��

k�

,2

�(tr)v, satisfiesthe

homogeneous(

RKS equation

�k� �� ru �>� h�ˆ D�� ru �� k

� �$� 0,8 �

27" �

whichª hasa uniquesolution

�k� �� ru ��9�

k� �� ru � ,$ � 28

" �ifj

the aboveboundarycondition is fulfilled. However, thissolution3 leadsto a contradictionto theorthogonalityrelation�20"

so3 that ¡ k� ¢ (tru )v can only be the trivial solution of Eq.£

27¤ ,$ ¥k� ¦�§ ru ¨�© 0,

8 ª29" «

whichª completesthe proof.As@

an interestingaside,we briefly considerthe physicalmeaning¬ of the quantity k

� ® (tru )v . Defining

u@ xcM k�¯$°²± r³ : ´

µ·¶$¸¹

k� º* » r¼

½EL

xcM¾�¿k� ÀÂÁ rÃ ,$ Ä 30

. Å

withª Æ and� Ç denoting?

spinor indicesrunning from 1 to 4,weª canrewrite Eq. È 17É as�

Êk� Ë�Ì ru Í�Î�Ï Ð

l¥

l¦ Ñ Ò

k�Ó

l¦ Ô�Õ ru Ö×k� Ø�Ù

l¦ d

Ú 3Ûrè ÜÞÝ k

� ß* à ru á_â>ãåä 0¶ æFç

AÖ

xcèROPMé ru ê_ëì u@ xck

� í ru î�ïñð$ò·ó l¦ ôÂõ ru ö_÷ ,$ ø 31

. ùwhereª summationover ú and� û is

jimplicitly understood.

Fromü

this equation,it is obvious that ý k� þ (tru )v is the usual

first-ordershift in the wavefunction causedby a perturbing

potential4 ÿ AÖ

xc � (t � 0¶ ��

AÖ

xc�ROPM� u@ xck� )v . This fact also moti-

vates+ the boundary condition assumedabove. In xT -onlytheory,D

u@ xcM k� (t ru )v is the local, orbital-dependentRHF exchange

potential4 so that � k� � (tr)v

is the first-order shift of the RKSwaveª function towards the RHF wave function. However,one0 has to realize that the first-order shift of the orbital-dependent?

quantityu@ xck� � �

i �� has(

beenneglected.Now�

we use Eq. � 23� toD

further transform the ROPMequations+ � 18� . As a first step, we solve Eq. � 23� forASÄ0¶ (t r)

v �k� � (tr)v, leadingto

AÖ

SÄ0¶ � ru �� k

� � � ru !#"%$ &k� ' ru (�)+* AÖ xc,ROPM- ru .#/10 E

LxcROPM2 3k� 4 ru 5

687AÖ¯

xcM k� 9 u@ xcM k

� :<;k� = > ru ?

@BAk� C ru DFE iç G HJILK

c½ 2¾ MON

AP

SÄ Q ru R#SOT 0

¶ Uk� V . W 32

. XWeY

then multiply the ROPM equationsZ 18[ by)

the zerothcomponent, of theeffectiveRKS four-potential,AS

Ä0¶ (t r)v, yield-

ingj

\] c� 2 ^`_

k� a`b

F< AS

Ä0¶ c rdfe k� g rh�ikj+l k

� m rn#o c.c., p 0,8 q

33. r

and� employEq. s 32. t

toD

obtain

uv c� 2� wyx

k� z`{

F

|k� } r~��+� AxcM �ROPM� r�#�

�ExcMROPM� �

k� � r�

�8� A xcM k�ROPM� u@ xck

� �<�k� � � r�

��k� � ru �#� iç � ��L�

c½ 2 �O� AP

SÄ � ru #¡O¢ 0

¶ £k� ¤

¥O¦ 0¶ §�¨�©

k� ª r«#¬ c.c., 0.

8 ®34. ¯

Introducingthe 4 ° 4 matrix

± ²´³¶µru · : ¸ 1

2" ¹º c� 2 »`¼

k� ½`¾

F< ¿¯k

� À ru Á�Â+Ã�Â 0¶ Ä�Å�Æ

k� Ç ru È#É c.c., Ê 35

. Ë

and� defining

a* xcM k�ÌÎÍ

rÏ : ÐÑ

ExcMROPMÒ

Ó Ôk� Õ rÖØ× 0

¶ Ù�Ú�Ûk� Ü rÝ

Þ�ßk� à rá#â iç ã ä�åLæ

c½ 2¾ çOè

ASÄ é rê#ëOì 0

¶ ík� î

ïOð 0¶ ñ�ò�ó

k� ô ru õ ,$ ö 36

. ÷weª rewrite Eq. ø 34

. ùas�

ú û üþýru ÿ AÖ xc�ROPM

� �ru �

� �� c� 2 ��k� ��

F< a* xcM k

��r�� j

rk� �� r�� A xcM k

�ROPM� �

u@ xck� �� c.c.,,

�37.

withª jr

k� ! (t ru )v being the four-current with respectto the k

HthD

orbital:0



jr

k� "# ru $ : %'& k

� ( ru )�*,+.- k� / ru 0 . 1 38

. 2In order to solveEq. 3 37

. 4for AxcM 5ROPM

�(tr)v, we first haveto

investigatewhether the 4 6 4 matrix 7 (tr)v, definedby Eq.8

35. 9

,$ is nonsingular,i.e., whetherthe inverse :<; 1(tru )v exists.

WeY

thereforecalculatethe determinantof = (tr)v, yielding

det? >@?BA

ru C�D�EGF jr 0¶ H

ru I�J 2 jr KL

ru M jr NO ru P�Q nR 4 S ru T 1 U jV 2¾ W

rXc½ 2nR 2 Y ru Z ,$[

39. \

whereª the last equality follows from the decompositionoftheD

four-currentinto the scalardensityand the vector com-ponents4 accordingto

jr ]^

r_�` nR a rb ,$ 1c½ jV c

rd . e 40fSince�

the current jV(tr)v

divided by the density nR (tr)v

is thevelocity+ field of the system,it follows from g (

tru )v h c½ that

Ddet? ikjBl

ru m�n�o 08 p

41Ã q

and� thereforethat the matrix r (tr)v

is nonsingular.Solving�

thenEq. s 37. t

for AxcuROPM�

(tr)v

yields

AxcM vROPM� w

rx�y 1

2 z|{~}� 1 � r� �� c� 2 ��k� ��

F<�a* xcM k

��r�

�jr

k� �k� r�� A xcM k

�ROPM� �

u@ xck� �� c.c., � 42�

WeY

emphasizethatEq. � 42� is anexact± transformationD

of theROPM equation � 15� . In particular,Eq. � 42� is still an inte-gralq equation.However,its advantagelies in the fact that itnaturallyI lendsitself asa startingpoint for deriving system-atic� approximationsof A

Öxc�ROPM(

tru )v : We only needapproximate�

k� (tr)v

in Eq. ¡ 36. ¢

by)

a suitablefunction of the set of RKSorbitals0 £~¤

i ¥ . The simplest possibleapproximationis ob-tainedD

by completelyneglectingthe termsinvolving ¦ k� § (tr)v

in¨

Eq. © 36ª «

. Although this approximationmay seemto berather¬ crude,it wasshownto producehighly accurateresultsin the nonrelativisticcase 29,31® .

The xc potential Axc¯ °ROPM(±r)²

is then approximatelydeter-mined³ by the following equation:

A´

xcµRKLI ¶ r· ¸�¹ 1

2º »'¼¾½¿ 1 À r· Á ÂÃ cÄ 2

Å ÆÇkÈ É�Ê

F

ËEÌ

xcROPMÍÏÎkÐ Ñ r· ÒÔÓ 0

Õ ÖØ×�ÙkÐ Ú r· Û

ÜjÝ

kÐ Þkß rà�á A xc¯ k

ÐRKLI â u@ xckÐ ã ä c.c.å æ 43ç

Thisè

equationestablishesthe generalizationof the nonrela-tivisticé

KLI approximationto the realm of relativistic sys-tems.é

In contrastto the ROPM equation ê 15ë ,ì the relativisticKLIí î

RKLIï ð

equation,ñ althoughstill beingan integralequa-tion,é

can be solved explicitly in terms of the RKS spinorsò~ói ô : Multiplication of Eq. õ 43

ö ÷withø j

Ýl¦ ù (±r· )² , summingover

allú û ,ì andintegratingover spaceyields

A xcl¦ ü A xc¯ l

¦Sý þ ÿ� cÄ 2 ��

kÈ ��

F� M lk

� A xc¯ kÐRKLI

� 1

2u@ xckÐ � u@ xc¯ k

Ð* � ,ì 44ö �

whereø we havedefined

A´¯

xcl�Sý

: � d� 3�r j� l� �� r· �� 1 � r· �

� 1

2º ! cÄ 2 "�#

kÈ $�%

F�

&EÌ

xcROPM�

'�(kÐ ) r· *,+ 0

Õ -/.�0kÐ 1 r· 2�3 c.c.å

4455

andúM lk� : 6 d

� 3�r j l� 7�8 r9�:<;�=> 1 ? r@ j

ÝkÐ ACB rD . E 46F

Theè

unknowncoefficients G A´¯xckÐRKLI

� H12Å (± u@ xc¯ k

Ð I u@ xckÐ* )² J

areú thendeterminedK

by the linear equation

LM cÄ 2 N�O

kÈ P�Q

F

RTSlk� U M lk

� V A xc¯ kÐRKLI

� W 1

2 X u@ xckÐ Y u@ xc¯ k

Ð* Z[ A

´¯xcl�Sý \ 1

2º ] u@ xc¯ l

� ^ u@ xcl�* _ . ` 47

ö aSolvingb

this equationandsubstitutingtheresultinto Eq. c 43dfinally leads to an expressionfor the xc four-potentialAxc¯ eRKLI�

(±r)²

that dependsexplicitly on the setof single-particlespinorsf g�h

i i . We thushaveobtaineda methodof calculatingtheé

xc four-potential Axc¯ jROPM�

(±r)²

in an approximateway,whichø is numerically much less involved comparedto thefull solutionof the ROPM integralequations.

V. ELECTROSTATIC LIMIT

The ROPM andRKLI methods,developedin the preced-ing sections,can be applied to systemssubjectto arbitrarystaticf externalfour-potentials.In particular,the methodsal-lowk

us to deal with external magneticfields of arbitrarystrength.f Yet, in electronicstructurecalculationsof atoms,molecules,and solids,we most commonlyencountersitua-tions,é

where no magneticfields are present l in¨ a suitableLorentzm

frame,typically the rest frameof the nuclein .Thus in this section we considerfour-potentialswhose

spatialf componentsvanish, i.e., AP

exto (±r· )² p 0.

q rThisè

also in-cludeså a partial fixing of the gauge.s In this situation,a sim-plifiedt Hohenberg-Kohn-Shamschemecan be developed,statingf that thezerothcomponentnu (

±r· )² v j

Ý 0Õ(±r· )² of theground-

statef currentdensityalonedeterminesthe externalpotentialVexto w nu x andú the ground-statewave function y{z nu | uniquely}~for a discussionon this so-called‘‘electrostatic case’’ cf.

Refs. � 21,23�� . Consequently,only a scalareffective poten-tialé

VSý (± r· )² is presentin the RKS equation� 2º � .

WhenY

orbital-dependentfunctionalsare usedfor the xcenergyñ in this context,the correspondingscalarxc potentialVxc(

±r)²

can be calculatedby repeatingthe stepsof Sec.III.One�

thenfinds theROPMintegralequationfor the ‘‘electro-staticf case’’:


�� cÄ 2Å ��

kÈ ��

F

d� 3�r� � �

kÐ �� r· �� Vxc

ROPM� r· �� Exc

ROPM�

��kÐ � r �� G

�Ský � r· � ,ì r· ¢¡ k

Ð £ r· ¤�¥ c.c.å ¦ 0.q §

48ö ¨

WeY

mention that the sameresult is obtainedif one de-mands³ that VS

ý (± r)²

be the variationally best local effectivepotentialt yielding single-particle spinors minimizing thetotal-energyé

functional © 5ª « ,ì i.e.,¬EÌ

totROPM®

VSý ¯ r· °

VS± ² V

S±ROPM³ 0.

q ´49ö µ

In¶

fact, usingthis approach,Shadwick,Talman,andNorman·22 derived

Kthe x¹ -only limit of the ROPM integralequationº

48» .Compared¼

to the four ROPM integral equations ½ 15¾ ,ìwhichø determinethe xc four-potentialA

´xc¿ROPM�

,ì Eq. À 48ö Á

is¨

considerablyå simpler.Still, its numericalsolution is a ratherdemandingK

task which has been achievedso far only forsystemsf of high symmetry, i.e., sphericalatoms Â 2º 1–24Ã .Again,Ä

an approximateROPM schemecanbe derived:Fol-lowing the argumentsof Sec.IV, the ROPM integral equa-tioné Å

48Æ canå be exactlyrewrittenas

VxcROPMÇ r· ÈÊÉ 1

2º

nu Ë r· Ì ÍÎ cÄ 2Å Ï�Ð

kÈ Ñ�Ò

F

nu kÐ Ó r· ÔÊÕ×Ö xck

Ð ØÚÙ V xckÐROPMÛ u@ xck

Ð ÜTÝÞ

c.c.,å ß 50ª à

whereøá

xc¯ kÐ â rã : ä 1

nu kÐ å ræ

çExc¯ROPM�

è�ékÐ ê rëíì k

Ð î rï�ð icñ òôóÊõ

kÐ ö r÷ùø�ú k

Ð û rüTý þ51ª ÿ

andú�

kÐ �� r· � : � d

� 3�r� � �

kÐ �� r· �� Vxc

ROPM r· �� EÌ

xcROPM

��kÐ � r· �� G

�Ský � r· � ,ì r· ��

52ª �

similarf to Eq. � 17� . Equation 50ª !

representsthe ‘‘electro-staticf case’’analogof Eq. " 42

ö #andú canalsobeapproximated

in¨

the sameway, namely,by neglectingall termsinvolving$kÐ % (±r· )² in Eq. & 51

ª '.

In thecontextof the‘‘electrostaticcase’’consideredhere,somef moreinsight into the natureof this approximationcanbe(

gained:It canbe interpretedasa ‘‘mean-field’’-type ap-proximationt in the sensethat the averageof the neglectedtermsé

with respectto the ground-statedensityvanishes.TodemonstrateK

this, we note that the neglectedtermsaveragedover) n* (

±r)²

aregiven by

+, cÄ 2Å -/.

kÈ 021

F� icñ

d� 3�r� 35476 k

Ð 8 r· 9;:�< kÐ = r· >@?;A c.c.å B 53

ª C

Applying the divergencetheorem,this integralcanbe trans-formedto a surfaceintegralwhich vanishesfor finite systems

if thesurfaceis takento infinity. Hence,neglectingthetermsinvolving D k

Ð E (±r)²

meansreplacing them by their averagevalue,F which is zero.

Theè

xc potential VxcROPM(

±r· )² can thereforeapproximately

be(

determinedby the following equation, leading to theRKLIï

equationfor the ‘‘electrostaticcase’’:

VxcRKLI G r· H�I 1

2º

n* J r· K LM cÄ 2Å N/O

kÈ P/Q

F

REÌ

xcROPMS�TkÐ U r· V

Wn* kÐ X rY7Z V xc¯ k

ÐRKLI [ u@ xckÐ \ ] c.c.å ^ 54

ª _

From`

this form it is obviousthat the RKLI potentialcloselyresembles¬ therelativisticDirac-Slaterpotentialaswell asthenonrelativisticKLI potential. Whether the accuracyof thecorrespondingå nonrelativistic schemeis maintainedin therelativistic¬ domainwill be investigatedin the following sec-tion.é

VI.a

RESULTS

In¶

this section,we test the accuracyof the approximateROPM scheme,derivedwithin the frameworkof the ‘‘elec-trostaticé

case’’ in the last section, for atomic systems.Inorder) to assessthe quality of this approximation,exact re-sultsf either for the xc energy Exc or) for the xc potentialVxc(

±r· )² would beuseful.However,for systemswhererelativ-

istic¨

effectsbecomeimportant,e.g.,high-Zb

atoms,ú exactre-sultsf are presentlynot available.Consequently,we havetolook for a different standardto comparewith.

Suchb

a standardreferenceis availablewithin the x¹ -onlylimitk

of RDFT c 21,23,24º d

. As in the nonrelativisticcase,thex¹ -only limit of thexc energyfunctionalis definedby theuseof) the exactexchangeenergyfunctional, i.e., by the relativ-istic Fock term, Eq. e 6f g ,ì in the caseof only longitudinalhCoulombi j

interactions. k Since,b

in the presentcontext,ourprincipalt goal is to the test the quality of the RKLI method,weø restrict ourselvesto this longitudinal caseand neglecttransverseé

contributions.l As explainedin theprecedingsec-tion,é

the exact longitudinal exchangepotential VxL(±r· )² can

thené

beobtainedby solving the full ROPMintegralequationm15n withø Exc¯ replacedby ExoLp ,exact. Simultaneoussolutionof

theé

ROPMintegralequationandtheRKS equationq 2r there-é

fore representsthe exactimplementationof the longitudinalx¹ -only limit of RDFT. This schemewill serveasa referencestandardf in the following.

It is first compared— of course— to the RKLI method,whichø employs the sameexact expressions 6f t for

uthe ex-

changeå energy and only approximatesthe local exchangepotentialt VxoL(

±r· )² by meansof Eq. v 54

ª w. Besidesthat, we list

theé

resultsfrom traditional RKS calculationsobtainedwiththeé

longitudinal x¹ -only RLDA (x¹ RLDA x andú two recentlyintroduced relativistic generalizedgradient approximationyRGGAï z

functionalsu {

24º |

: The first one is basedon theBecke88}

GGA ~ 42ö ��

RB88ï �

,ì thesecondoneon a GGA func-tionalé

due to Engel, Chevary,MacDonald,and Vosko � 11��RECMV92� .

Theseè

various approachesare analyzed for spherical�closed-shellå � atoms.ú To this end,thespin-angularpartof the



RKSï

wavefunction is treatedanalyticallyandthe remainingradial¬ Dirac equationis solvednumericallyon a logarithmicmesh � 21� . In all our calculationswe usefinite nuclei mod-eledñ by a homogeneouslychargedspherewith radius

Rnucl� � 1.0793A1/3� 0.735�

87 fm, � 55ª �

whereø A is the atomicmasstakenfrom � 43� . We mentioninpassingt that employingfinite nuclei is not necessaryto en-suref convergentresultsas, for example,in the relativisticThomas-Fermiè

model.We incorporatefinite nuclei becausetheyé

representthe physicallycorrectapproach.In TableI, we showthe longitudinalground-stateenergy

E tot�L� obtained) from the variousself-consistentx¹ -only RDFT

approachesú and, in addition, from relativistic Hartree-Fock�RHF� calculations.å Comparingthe first two columns,we

recognize¬ that the RHF and the ROPM dataarevery close.Theè

largestdeviation is found for Be with 41 ppm. Withincreasingatomic number, the inner orbitals, contributingmost to the total energy,becomemore and more localizedsuchf that the differencebetweenthe nonlocalRHF potentialandú thelocal ROPMdecreases.In fact, for No, thedifferenceis down to 2 ppm. We emphasizethat thesedifferencesaredueK

to the different natureof the two approaches.While theRHFï

method,by construction,yields the variationally bestenergy,ñ the ROPM schemeadditionally constrainsthe ex-changeå potential to be local. Consequently,we expect theROPM resultsto alwaysbe somewhathigher,which is con-firmed�

in Table I. In the third column, the total energiesobtained) from the RKLI approximationarepresented.Theyalwaysú lie abovethe correspondingROPM valuessincethesamef exchangeenergyfunctional is employedin both ap-

proachest and the variationally best local�

potentialtVxc

L,ROPM(±r· )² is approximatedby Eq. � 43

ö �in�

the RKLI ap-proach.t However,the resultsare clearly seento agreeveryclosely:å For the mean absolutedeviation from the exactROPMï

data of the 18 neutral atomslisted in Table I, oneobtains) only 5 mhartree.Thus the RKLI method impres-sivelyf improveson the RLDA results,for which we find amean³ absolutedeviationof 6092mhartree.The accuracyoftheé

RKLI schemebecomesevenmore obviouswhen com-paredt to the RGGAs. Both RGGAs improve significantlyover) theRLDA method.Still, their deviationsfrom theexactROPMï

data are more than one order of magnitudelargercomparedå to the RKLI results.

Thetrendsfound in theabovediscussionarealmostiden-ticalé

to theonesfound in thenonrelativisticcase.In ordertoanalyzeú the relativisticeffectsmoredirectly, we additionallyconsiderå the relativistic contributionto E

Ìtot�L� ,ì definedby

�E tot� : � E tot

�L � n* R ¡;¢

E tot�NR£ ¤

n* NR£ ¥

. ¦ 56ª §

Via¨

this decomposition,we areableto testthequality of theRKLI schemeindependentlyof the accuracyof its nonrela-tivisticé

equivalent.Yet, at first, we want to point out that therelativistic¬ treatmentleadsto drastic correctionsespeciallyfor high-Z atoms.ú For example,TableII showsthat the rela-tivisticé

correctionof Hg amountsfor about6.7%of the totalenergyñ thus demonstratingthe needfor a fully relativistictreatment.é

Furthermore,by comparingthe secondand thirdcolumnså of TableII, we realizethattheROPMandtheRKLImethod³ yield almostidenticalresultsfor the relativistic con-tributioné ©

E tot� . In otherwords,almostno additionaldevia-

tionsé

are introducedby the relativistic treatmentof the KLI

TABLEª

I. Longitudinalground-stateenergy « E tot¬L from

variousself-consistentx® -only andRHF calcula-

tions.¯ °¯denotesthe meanabsolutedeviationand ± the averagerelativedeviation ² in 0.1 percent³ from theexact´ ROPM values µ all¶ energiesare in hartreeunits· .¸

RHF ROPM RKLI RB88 RECMV92 x® RLDA

He¹

2.862 2.862 2.862 2.864 2.864 2.724Beº

14.576 14.575 14.575 14.569 14.577 14.226Ne»

128.692 128.690 128.690 128.735 128.747 127.628Mg¼

199.935 199.932 199.931 199.952 199.970 198.556Ar�

528.684 528.678 528.677 528.666 528.678 526.337Ca½

679.710 679.704 679.702 679.704 679.719 677.047Zn 1794.613 1794.598 1794.595 1794.892 1794.880 1790.458Kr¾

2788.861 2788.848 2788.845 2788.907 2788.876 2783.282Sr¿

3178.080 3178.067 3178.063 3178.111 3178.079 3172.071Pd 5044.400 5044.384 5044.380 5044.494 5044.442 5036.677Cd½

5593.319 5593.299 5593.292 5593.375 5593.319 5585.086XeÀ

7446.895 7446.876 7446.869 7446.838 7446.761 7437.076Ba 8135.644 8135.625 8135.618 8135.612 8135.532 8125.336Yb 14067.669 14067.621 14067.609 14068.569 14068.452 14054.349Hg¹

19648.865 19648.826 19648.815 19649.141 19649.004 19631.622Rn 23602.005 23601.969 23601.959 23602.038 23601.892 23582.293Ra 25028.061 25028.027 25028.017 25028.105 25027.962 25007.568No»

36740.682 36740.625 36740.609 36741.900 36741.783 36714.839Á¯ 0.006 0.189 0.168 8.668Â 0.002 0.103 0.108 6.20


scheme.f Consideringnow the relativistic correctionscalcu-latedwith the conventionalRDFT methods,the conclusionsdrawnK

in thediscussionof E tot�L canå berepeated:Comparedto

theé

RKLI method,theRGGA resultsareworseby morethanone) orderof magnitudewhereastheRLDA is theby far leastaccurateú approximation.

Thesetrendsalso remainvalid when other quantitiesofinterest�

areconsidered.For example,in TablesIII andIV wehaveÃ

listed the relativistic contributionsto the longitudinalexchangeñ energy,definedanalogouslyto Eq. Ä 56

ª Å. Again, the

RKLI methodalmostexactly reproducesthe ROPM results.It is worthwhilenoting that theexchangeenergyExÆL� is influ-encedñ quite substantiallyby relativistic effects,too. Takingagainú Hg asan example,we realizethat the 5.8% contribu-tioné

to ExÆL� is of the sameorderas for the total energy.Fur-thermore,é

evenfor lighter atomssuchasMg, the relativisticcorrectionså to E

ÌxÆL areú comparableor even larger than the

differencesK

betweenthe currently best nonrelativistic ex-changeå functionals.As a consequence,a relativistic treat-ment³ is indispensablefor the ultimate comparisonwith ex-perimentst Ç 21È . Next, we turn our attentionto thecalculationof) É E

ÌxÆ fromu

the RGGA functionalslisted in the third andfourthu

columnsof Table III. With the exceptionof No, theresultsare also in excellentagreementwith the exactones.However,when we turn our attentionto other systemslikesomef ions of the neon-isoelectronicseriesshown in TableIV,¶

we recognizethattheRGGAscannotreproducetheexactresults to the samelevel of accuracyas obtainedfor theneutralatomsabove Ê 24Ë . This canbe explainedby the factthaté

theRGGAsareoptimizedfor theneutralatoms.In con-trast,é

the agreementof the RKLI resultswith the exactonesfor theseions is asgoodasfor the neutralatoms.

TABLEª

II. Relativisticcontribution Ì5Í EÎ

tot¬L from

variousself-consistentx® -only andRHF calculations.Ï¯denotesÐ

the meanabsolutedeviationand Ñ the averagerelative deviation Ò inÓ 0.1 percentÔ from

the exactROPMÕ

values Ö all energiesare in hartreeunits× .RHF ROPM RKLI RB88 RECMV92 x® RLDA

He¹

0.000 0.000 0.000 0.000 0.000 0.000Be 0.003 0.003 0.003 0.003 0.003 0.002Ne»

0.145 0.145 0.145 0.145 0.145 0.138Mg¼

0.320 0.320 0.320 0.321 0.321 0.308Ar�

1.867 1.867 1.867 1.867 1.867 1.821Ca½

2.953 2.953 2.953 2.952 2.953 2.888ZnØ

16.771 16.770 16.770 16.779 16.779 16.555Kr¾

36.821 36.820 36.820 36.822 36.821 36.432Sr¿

46.554 46.553 46.553 46.552 46.551 46.092Pd 106.527 106.526 106.526 106.526 106.525 105.715Cd½

128.245 128.243 128.243 128.243 128.241 127.323Xe 214.860 214.858 214.858 214.825 214.822 213.522Ba 252.223 252.222 252.221 252.176 252.173 250.725YbÙ

676.559 676.551 676.549 676.590 676.588 673.785Hg¹

1240.521 1240.513 1240.511 1240.543 1240.538 1236.349Rn 1736.153 1736.144 1736.142 1736.151 1736.151 1730.890RaÕ

1934.777 1934.770 1934.768 1934.781 1934.783 1929.116No»

3953.172 3953.155 3953.151 3953.979 3954.015 3944.569Ú¯ 0.001 0.056 0.058 1.788Û 0.009 1.14 1.35 33.7

TABLEª

III. Relativistic contribution Ü5Ý EÎ

xÞ to the exchange

energyfrom variousself-consistentx® -only calculations.ß¯denotes

themeanabsolutedeviationand à the¯

averagerelativedeviation á inÓpercentâ from the exactROPM values ã all energiesare in hartreeunitsä .¸

ROPM RKLI RB88 RECMV92 x® RLDAÕ

He 0.000 0.000 0.000 0.000 0.000Be 0.001 0.001 0.001 0.001 0.000Ne 0.015 0.015 0.015 0.015 0.007Mg 0.029 0.029 0.029 0.029 0.015Ar 0.118 0.118 0.117 0.118 0.069Ca 0.172 0.172 0.171 0.171 0.104Zn 0.627 0.626 0.632 0.632 0.402Kr 1.215 1.214 1.212 1.211 0.814Sr 1.478 1.477 1.473 1.472 1.005Pd 2.785 2.787 2.782 2.780 1.958Cd 3.264 3.264 3.255 3.252 2.322Xe 5.021 5.020 4.977 4.974 3.657Ba 5.739 5.736 5.684 5.680 4.215Yb 12.043 12.024 12.027 12.024 9.194Hg 19.963 19.956 19.965 19.957 15.734Rn 26.637 26.620 26.612 26.610 21.307Ra 29.241 29.218 29.225 29.224 23.513No 52.403 52.402 53.168 53.205 43.683å¯ 0.004 0.053 0.056 1.819æ 0.079 1.03 0.857 35.9



Toè

further proceedwith our investigation,we next raisetheé

questionof how well local propertiessuch as the ex-changeå potential VxÆL(

±r)²

are reproducedwithin the differentRDFTï

methods.In Fig. 1, the exchangepotential is plottedforu

thecaseof Hg. As expected,theRKLI resultfollows theexactñ curvemostclosely.Yet, thepronouncedshellstructureapparentú in the ROPM curve is not fully reproduced,al-thoughé

clearlyvisible ç cf.å themoredetailedplot in Fig. 2è , iì ntheé

RKLI approximation.However, it again improvessig-nificantly over the conventionalRDFT results,wherethe in-tershellé

peaksare strongly smearedout or even absent.InFig. 3, the asymptoticregion,which is of particularimpor-tanceé

for excitation properties,is plotted in greaterdetail.Thereè

we recovertheknowndeficienciesof theconventionalRDFT functionals:The RLDA as well as both RGGAs falloff) much too rapidly. In contrast, the RKLI and ROPMcurveså becomeindistinguishablein the asymptoticregion,both(

decayingas1/r asú r éëê andú thusreflectingthe propercancellationå of self-interactioneffects.Sinceall theseobser-vationsF arealreadyknown for the correspondingnonrelativ-istic functionals,we againconsiderthe relativistic contribu-tioné

separately.This relativisticcontributionto theexchangepotential,t given by

ìVxÆ í r î : ï VxÆL� ð nñ R ò�ó r ô�õ VxÆNR

£ önñ NR£ ÷�ø

r ùVxÆOPMú û

nñ NR£ ü�ý

r þ ,ì ÿ 57ª �

is plotted in Fig. 4. We first observestrongoscillationsbe-tweené

0.1a.u.and5 a.u.Theseoscillationsareintroducedbytheé

displacementof thedensitydueto relativisticeffectsandthusé

representa directconsequenceof theatomicshellstruc-ture.é

As the shell structureof the exchangepotential is notfully reproducedwithin the RKLI approach,the amplitudesof) the oscillations are somewhat smaller compared to�

VxÆOPMú

(±r� )² . While thesedeviationsare clearly visible, the

RKLI curveis still closestto theexactone,especiallyin theregionnearthe nucleusandin the asymptoticregion,wherelargek

deviationsoccur for the conventionalRDFT methods.Thepropertiesof thex¹ potentialt in the large-r asymptoticú

region¬ also strongly influencethe eigenvalueof the highestoccupied) orbital shown in Table V: Due to the correctas-ymptotics,� the energiescalculatedwithin the RKLI schemeareú almostidentical to the exactROPM results.On the con-trary,é

the lack of thecorrect1/r behavior(

of theRGGAsandtheé

RLDA showsup in ratherpoorly reproducedeigenvaluesof) the highestoccupiedstate.Sincein the exactnonrelativ-istic�

theory,thehighestoccupiedenergylevel coincideswiththeé

ionizationpotential � 44� ,ì we alsolist this quantity,givenby(

I� �

EÌ

tot� � N� 1 �� E

Ìtot� N� � ,ì in Table VI. Again, the RKLI

andú ROPM methodsprovidealmostidentical results.In ad-

TABLE IV. Relativistic contribution �� ExÞ to the exchangeenergy from various self-consistentx® -only calculationsfor some

ions of the neon-isoelectronicseries.�¯denotesÐ

the meanabsolute

deviationand � theaveragerelativedeviation � in percent� from theexactROPM values � all energiesare in hartreeunits� .�

ROPM�

RKLI RB88 RECMV92 x® RLDA

Ca10� 0.159 0.160 0.158 0.158 0.093Zr30� �

1.528 1.528 1.507 1.509 0.983Nd50� �

5.780 5.783 5.667 5.675 3.970Hg70�

15.599 15.606 15.293 15.325 11.365Fm90! "

36.475#

36.494 36.258 36.366 28.173

$¯ 0.006 0.132 0.102 2.992% 0.041 1.34 1.17 31.8

FIG. 1. Longitudinal exchangepotential V&

xÞL' (r)(

for Hg fromvariousself-consistentx® -only calculations) in hartreeunits* .�

FIG. 2. LongitudinalexchangepotentialV&

xÞL' (+ r)(

for Hg showingthe shell structureof Fig. 1 in moredetail , inÓ hartreeunits- .

FIG. 3. Asymptotic regionof the longitudinalexchangepoten-tial of Fig. 1 . in hartreeunits/ .�


dition,0

whencomparingthe ionizationpotentialto the ener-gies1 of the highestoccupiedlevel, we seethat the relationI 243 N

£ — which we of courseexpectto hold only approxi-mately,5 since correlationcontributionsare neglected— isfulfilled6

within a few percentfor the ROPM andRKLI data,whereas7 the resultsof theconventionalx¹ functional

6differ to

a8 muchlargerextent.To conclude,we notethattheinclusionof9 relativistic effectsleadsto largecorrectionsfor heavyat-oms9 even for the outermostorbitals: Taking Hg as an ex-ample,8 we find a 20% shift for the ionizationpotential.

VII.:

CONCLUSIONS

In¶

this work, we derivedan approximateROPM schemegeneralizing1 the argumentsof Krieger,Li, andIafrateto therelativistic domain.As for the full ROPM, the advantageofthe;

RKLI methodlies in the fact that xc functionalsdepend-ing�

explicitly on a setof RKS single-particlespinorscanbetreated;

within the frameworkof RDFT. This, in particular,allows8 us to employ the exactexpressionfor the exchangeenergy< functional, i.e., the relativistic Fock term in the lon-gitudinal1 case.ThereforetheRKLI approachsatisfiesa num-ber=

of importantproperties,most notably the freedomfromself-interactions> implying the correctasymptoticdecay.

In¶

numericaltestson sphericalatomswithin the longitu-dinal0

x¹ -only limit of RDFT the RKLI methodwasfound tobe=

clearlysuperiorto theknownrelativistic x¹ -only function-als.8 The resultsobtainedare seento be in closeagreementwith7 theexactROPMvalues,thusreducingthedeviationof,for6

example,the widely usedRLDA by morethanthreeor-ders0

of magnitude.On the other hand,the numericaleffortinvolved is considerablylesscomparedto thesolutionof thefull ROPM scheme.We therefore expect that the RKLIscheme> will be successfullyusedfor larger ? nonspherical@ Asystems,> e.g.,molecules,includingalsocorrelationcontribu-tions.;

ACKNOWLEDGMENTSB

ManyC

valuablediscussionswith K. Capelle,T. Grabo,andM. Luders

0are greatly appreciated.One of us D T.K.E ac-8

knowledgesfinancial support from the Studienstiftungdesdeutschen0

Volkes. This work was supportedin part by theDeutscheF

Forschungsgemeinschaft.

G1H P. HohenbergandW. Kohn, Phys.Rev. 136, B864 I 1964J .K2L W.M

Kohn andL.J. Sham,Phys.Rev. 140,N A1133 O 1965P .�Q3R R.M.�

Dreizler and E.K.U. Gross,DensityS

Functional TheoryTSpringer,U

Berlin, 1990V .�W4X R.G.�

ParrandW. Yang, Density-FunctionalS

Theory of Atoms

andY Molecules Z Oxford University Press,New York, 1989[ .\5] DensityS

Functional Theory,N Vol. 337of NATO^

Advanced StudyInstitute, Series B: Physics, editedby E.K.U. GrossandR.M.Dreizler _ PlenumPress,New York, 1995 .�a

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TABLEf

V. Eigenenergygih Nj of the highestoccupiedorbital

from various self-consistentx® -only calculations. k¯ denotesthe

meanabsolutedeviationand l the¯

averagerelative deviation m inpercentn from

the exactROPM values o all energiesare in hartree

unitsp .�ROPM OPM RKLI RB88 RECMV92 x® RLDA

Be 0.309 0.309 0.309 0.181 0.182 0.170Mg 0.253 0.253 0.253 0.149 0.149 0.142Ca 0.196 0.196 0.196 0.116 0.116 0.112Zn 0.299 0.293 0.298 0.195 0.194 0.191Sr 0.181 0.179 0.181 0.108 0.108 0.104Cd 0.282 0.266 0.282 0.183 0.182 0.181Ba 0.163 0.158 0.163 0.097 0.097 0.095Hg 0.329 0.262 0.332 0.223 0.222 0.222

q¯ 0.001r

0.095 0.095 0.099s 0.156r

38.3 38.4 40.0

TABLE VI. Ionization potential calulatedfrom various self-

consistentx® -only calculations.t¯denotesthe meanabsolutedevia-

tion and u theaveragerelativedeviation v inÓ percentw from

theexactROPM values x all¶ energiesare in hartreeunitsy .�

ROPM�

OPM RKLI RB88 RECMV92 x® RLDA

Be 0.296 0.295 0.296 0.326 0.327 0.312Mg 0.243 0.243 0.243 0.269 0.269 0.261Caz

0.189 0.188 0.189 0.210 0.210 0.205Zn{

0.284 0.278 0.283 0.339 0.338 0.334SrU

0.175 0.172 0.175 0.195 0.196 0.192Cdz

0.269 0.253 0.268 0.315 0.315 0.314Baº

0.157 0.152 0.157 0.176 0.176 0.174Hg 0.313 0.250 0.312 0.364 0.363 0.364

|¯ 0.000 0.034 0.034 0.029} 0.130 13.5 13.6 11.6FIG.~

4. Relativisticcontribution � V�

xÞ (r)�, Eq. � 57

� �, for Hg from

variousself-consistentx® -only calculations� in hartreeunits� .�



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approximate relativistic optimized potential method · sity,3 provided the orbitals come from a...

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