Compact representation of helium wave functions in perimetric and hyperspherical coordinates

Robert�

C. ForreyBerks-Lehigh�

Valley College, Penn State University, Reading, Pennsylvania 19610-6009, USA�Received26 September2003;published9 February2004�

V�

ariationalcalculationsof the ground-stateenergy of helium areperformedusinga basissetrepresentationthat�

includesan explicit treatmentof the Fock expansionin hypersphericalcoordinates.The constructionofbasis�

functionsthat havethe correctcuspbehaviorat three-particlecoalescencepointsand the evaluationofintegralscontainingthesefunctionsis discussed.Thebasissetin hypersphericalcoordinatesis addedto a basisset� consistingof productsof Laguerrepolynomialsin perimetriccoordinates.It is demonstratedthat theuseofFock�

basisfunctionsprovidesa substantialimprovementin the convergencerateof the basissetexpansion.

DOI: 10.1103/PhysRevA.69.022504 PACS number s�� :� 31.15.Pf,31.10. z

I. INTRODUCTION

A�

two-electron atomic system is an excellent testingground� for new quantumtheoriessince it is the simplestsystem� with enoughcomplexityto containthe main featuresof� a many-body theory. This complexity arises from theelectron-electron� Coulombenergy which dependson the in-terelectronic�

distancer12��� r� 1 � r� 2 � . Furthermore,the scaledHamiltonianfor infinite nuclearmassandcharge Z,� given inatomic� units by

H ��� 1

2 "!

12#%$'&

2#2#)(+* 1

r1 ,1

r2

- 1

Zr12,� . 1/

does0

not dependon anyexperimentalconstantswhosevalueschange1 with improvements in measurementtechniques.Therefore,it providesa standardfor theoreticalcalibration.After�

the wave function belongingto this Hamiltonianhasbeen2

obtained,it is possibleto computeperturbativecorrec-tions�

to thenonrelativisticenergy. It hasbeenestablished3 14that�

relativistic andQED correctionsto the energy levelsofan� atomicor molecularsystemrequirehighly accuratenon-relativistic5 wave functions.Rayleigh-Ritzvariationalcalcu-lationsprovidea wave function with relativeerror approxi-matelyproportionalto thesquareroot of the relativeerror inthe�

energy. Therefore, if nonrelativistic variational wavefunctions6

areto be usedfor perturbativeapplications,andifthe�

energies are used to estimatethe quality of the wavefunctions,thenit is necessaryto calculatethe nonrelativisticener� gies to far greateraccuracythan would otherwisebeneeded.7

The successof variationalcalculationsdependsupon therateof convergenceof the basisset expansionusedto con-struct� the trial wave function. The convergencerate is con-trolled�

by the analytic structureof the exactwave function82–79 . For problems in atomic or molecular physics, the

mainconsiderationwhenconstructingtrial wavefunctionsisthe�

cuspbehaviorat two particlecoalescences.This behavioris:

describedby the Kato condition ; 2<>=?%@ ˆArB i j rC i j

DFE 0G%HJI i jq

KiqK

jL"MON rB i j P 0

QSR,� T 2<>U

whereV Wi j is the reducedmassof particlesi

Xand� j

Y,� qK i and� qK j

Lare� the chargesof the two particles,and the caretover thewaveV function denotesa sphericalaverage.A lesserbut stillveryZ importantconsiderationis the cuspbehaviorat three-particle[ coalescences.This behavioris believedto be prop-erly� describedby the Fock expansion\ 8–

]13

_O`r,�ba ,�dcfe+gih

kj+k

0Glnm

lo"p

0G

[ kj

/2]qsr

kj

,lout)v ,�dwyx rk

j{zln r | l

o,� } 3~>�

whereV useof the hypersphericalcoordinatesr,�d� ,� and ��� de-0

fined�

in Eqs. � 18� – � 20<��

below2 �

reveals5 the presenceof alogarithmic�

singularityasthehyperradiustendsto zero.Thisregionis only a tiny part of the full configurationspaceandis oftenneglectedwhenconstructingtrial wavefunctions.Insuch� cases,the convergencerate can be very slow, particu-larly�

if the basisfunctionsthat mustapproximatethe neigh-borhood2

of thesingularityareinflexible.A primaryobjectiveof� this work is to handlethe logarithmicsingularitydirectlyso� thattheremainderof thebasissetis freeto concentrateonregions5 of configurationspacethatarefurtherawayfrom thenucleus.If successful,this strategyshould provide an im-proved[ convergence rate for the basis set expansion,orequivalently� , a more compactrepresentationof the heliumwaveV function.

Many differentbasissetshavebeenappliedto theheliumHamiltonian � 1� withV varyingdegreesof success.It is not theintent:

of this work to provide a completeaccountof themany� contributionsthat havebeenreportedin the literaturefor this problem.However, we would like to benchmarkthepresent[ method,so it is useful to provide a brief review ofsome� of thecurrentstandards.Onevery successfulmethodisto�

use a Hylleraas-typebasis set � 14� whichV effectivelyhandlestheelectroncorrelationthroughuseof r12 as� a coor-dinate.0

A relatedmethodintroducedby Drake � 15,16� uses�‘‘doubled’’ basis sets in which two different exponentialscale� factorsareusedfor eachcombinationof rB 1

i rB 2jLrB 12

kj

. Thismethod,� originally designedfor statesof high angularmo-mentum� � 15� ,� has beensuccessfullyapplied to S

�-statesby

Drake � 16� and� alsoby Kleindienst,Luchow1 , andMerckens�17� . Thesebasissetsbuild in the two-particlecuspsdueto

the�

linear termsin rB 1 ,� rB 2,� andrB 12,� but generallyhavediffi-culty1 handling the three-particlecusp characterizedby thelogarithmictermsof theFockexpansion.Nevertheless,these

PHYSICAL�

REVIEW A 69�

, 022504 � 2004� �

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10¤ /$22.50¢

©2004TheAmericanPhysicalSociety69�

022504-1

basis2

setstend to be very flexible, particularly the doubledbasis2

sets,so that the three-particlecuspmay be adequatelyapproximated� if enoughfunctionsareused.Anothermethod,introduced:

by FrankowskiandPekeris¥ 18,19¦ ,� usesthebasisfunctions

§n¨ ,lo,m© , jL«ª s¬ ,� t­ ,� u®S¯±° s¬ n¨ t­ lo u® m©³² ln� s¬>´ j

Lexp�¶µ¸· s¬ /2¹»º±¼ cosh1 ½ ct¾À¿

sinh� Á ct¾�ÂÄÃ4Å

in:

the Hylleraascoordinates

s¬ÇÆ r1 È r2# ,� t­«É r2

#ËÊ r1 ,� u®ÍÌ r12,�subject� to the constraints

lÎ,� mÏÑÐ 0,

Qand�

ÒnÓÍÔ l

ΫÕmÏ×Ö±Ø 2

<jYSÙ

0 iQ

f nÓ³Ú 0,Q

ÛnÓÍÜ l

ΫÝmÏ×Þ±ß 2 j

YSà2 á 0 i

Qf nÓ³â 0.

QThisã

basissetprovidesthe correctcuspstructureat all two-particle[ coalescencesdue to linear terms in r1 ,� r2

# ,� and r12and� thecorrectsingularitystructureat the three-particlecoa-lescence�

sincethe logarithmic termsof the Fock expansionare� expandablein powers of s¬ ,� ln(s¬ ),ä and (t­ /¹ s¬ )ä . For theground� state,the nonlinearvariationalparameterc¾ is

:zero,

and� it is requiredthat lÎ

be2

evenin order to ensurethat thebasis2

set has the proper symmetryunder exchangeof par-ticles�

rB 1 å rB 2,� or equivalently, t­«æèç t­ . For excitedstates,c¾ is:

used� to build in thecorrelationthatoccurswhenoneelectronis closeto the nucleuswith the otherelectronfar away. Thechoice1 of cosh(ct¾ )

äor sinh(ct¾ )

äis madeto give the basisfunc-

tion�

thepropersymmetryunderexchangeof particles.Hencefor6

singlet statescosh(ct¾ )ä

is usedwhen the index lÎ

is:

evenand� sinh(ct¾ )

äis usedwith oddvaluesof l

Î. For triplet statesthe

situation� is reversed.The Frankowski-Pekerismethod de-scribed� above has been used by Baker, Freund,Hill, andMorgan é 20ê to

�calculatevery accuratewave functions for

the�

low-lying S�-statesof helium usingonly severalhundred

basis2

functions.The basisset was very successfulin appli-cation1 to high-energy double photoionization ë 21

<OìwhereV a

proper[ descriptionof electroncorrelationnearthe nucleusisessential.� It can,however, be difficult to usein somecasesbecause2

of possiblecomputationallinear dependenceprob-lems�

asthe sizeof the basisset increases.Anothersuccessfulmethod í 22–25î is to usea Laguerre

basis2

in perimetriccoordinates.This basisset is analyticallyequivalent� to Hylleraas-typebasissets.Therefore,it buildsinthe�

two-particle cusps but fails to effectively model thethree-particle�

cuspcharacterizedby the logarithmictermsofthe�

Fock expansion.The method,however, is very efficientsince� the matrix elementintegralscanbe evaluatedanalyti-cally1 . Also, since Laguerre polynomials are numericallystable� andsincetherelevantmatricesaresparse,manyfunc-tions�

canbe includedin the basisset.In fact, Burgersetï al.

ð25ñ have usedbetter than ten thousandsuch functions in

order� to obtaina benchmarknonrelativisticground-stateen-er� gy.

Larò

genumbersof basisfunctionsarenow routinelybeingused� togetherwith advancesin computingpowerto produceremarkablenew benchmarksfor the ground-stateenergy ofheliumó ô

26<

–30õ . Schwartzhaslikenedthis effort to thecom-petition[ betweenmathematiciansto computeevermoredig-its:

of the number öø÷ 30~Où

. Someof the recentcalculations,most� notably those of Korobov ú 27

<Oû,� have achievedtheir

success� usingsimpleexpansionsthatseemto ignoretheana-lytic structureof the wavefunction.Like the Laguerrebasisin:

perimetric coordinatesü 25<�ý

,� the latest benchmarksþ 26<

–30~Oÿ

require5 several thousandsof basis functions. In thepresent[ work, we desireto utilize the insightsprovidedbyour� current understandingof the analytic propertiesof thewaveV function, andfind a basiswhich will cut down on thenumber7 of functionsneededto getgoodconvergence.In do-ing so,we removethe flexibility providedby othermethodsand� confirm the importanceof the logarithmicsingularity.

Theã

presentapproachusesthe Laguerrebasisin perimet-ric5 coordinatesas a primary basis in order to build in thecorrect1 two-particle cusps.A small number of functionswhichV exactly incorporatethe first few terms of the Fockexpansion� arethenaddedto theprimaryLaguerrebasis.Thecomputational1 linear dependenceproblemthat plaguessomemethodsis not anissueheresincetheLaguerrefunctionsareclose1 to orthogonaland the number of Fock functions issmall.� Furthermore,we expectthat the numberof Laguerrefunctions6

will be considerablyreducedasa consequenceofdealing0

with the three-particlecuspdirectly. In the followingsections,� we will show how these‘‘Fock functions’’ can bederived0

andalsohow thematrix elementintegralscontainingthe�

Fockfunctionscanbeevaluatedto 30 digits via Gaussianquadrature� in threedimensions.We usethe new basisset tovariationallyZ solve the Schrodinger

0equationfor the Hamil-

tonian� �

1� . Theoptimizednonlinearparametersaregiven forthe�

newbasissets,andthevariationalenergiesarecomparedwithV well-known valuesobtainedusing the standardbasissets� describedabove.

II. COORDINATES AND NOTATION

The Hamiltonian � 1� contains1 six independentcoordi-nates.7 The most commonset of six coordinatesconsistsofthe�

electronicsphericalpolar coordinatesr1 ,��� 1 ,��� 1,� andrB 2 ,� 2 ,�� 2. Thekinetic-energy operatorin thesecoordinatesis given by

� 1

2 ��

12#����

22#������ 1

2

�n¨�� 1

2 � 2

�rn¨2#��

2<rn¨

rn¨�!"

n¨rn¨2# ,� # 5$&%

whereV'

n¨�( 1

sin�*)n¨++-,

n¨ sin�*.n¨0//-1 n¨ 2

1

sin� 2 3n¨4 2#

576n¨2# . 8 69&:

In order to take accountof the effects of electron-electroncorrelation,1 it is convenientto use coordinatesystemsthat

ROBERT C. FORREY PHYSICAL REVIEW A 69,; 022504 < 2004=

022504-2

explicitly� containthe interelectronicdistancerB 12. The mostcommon1 setis r1 ,� r2 ,� r12,�?> ,�A@ ,� and B ,� where C ,�ED ,� and Fare� thethreeEulerangleswhich specifytherotationfrom thespace-fixed� axes to the body-fixed axes.For S

�states,� the

EulerG

anglesareignorableandthe kinetic-energy operatorisgiven� by

H 1

2<JILK 1

2#�M�N

22#�O�P�Q 1

2< 2

<rB 1

RR

rB 1 S2<rB 2#TT

rB 2#�U 4

VrB 12

WW

rB 12 XY 2#Z

r12#

[]\ 2

^r2#2 _ 2 ` 2

ar12

2 b r12 c r2

#2 d r122

r1r12

e 2fr1 g r12

h rB 22 i rB 1

2 j rB 122

rB 2rB 12

k 2#

lrB 2 m rB 12

,� n 7o&p

withV the volumeelement

dqsrut

8]wv 2 x rB 1rB 2

# rB 12y drq

1drq

2# drq

12. z 8]&{Theã

coordinatesrB 1 ,� rB 2# ,� andrB 12 are� easilytransformedto the

perimetric[ coordinates| 22<~}

qK 1 ��� rB 1 � rB 2#�� rB 12,� � 9�&�

qK 2#�� rB 1 � rB 2

#�� rB 12,� � 10�qK 12� r1 � r2

#�� r12,� � 11�withV eachcoordinatehavingdomain � 0,

Q����as� a consequence

of� the triangle condition for rB 1 ,� rB 2 ,� and rB 12. The kineticener� gy in perimetriccoordinatesis

� 1

2<J�L� 1

2#����

22#������

rB 1rB 2# rB 12��  1 ¡

¡ qK 12

¢Q£

1 ¤ Q£

2#�¥§¦¦ qK 12

¨ ©© qK 2

P1

ªª

qK 2 «¬¬

qK 1P2

­­

qK 1 ®¯¯

qK 12Q£

1

°°

qK 2

± ²² qK 2# Q£

1

³³

qK 12 ´µµ

qK 12Q£

2#0¶¶ qK 1

· ¸¸ qK 1

Q£

2

¹¹

qK 12,� º 12»

whereVP1 ¼ 2Q

£1 ½ Q£

12¾ 2qK 1qK 2qK 12,� ¿ 13ÀP2 Á 2Q

£2 Â Q£

12Ã 2qK 1qK 2qK 12,� Ä 14Åand�

Q£

1 Æ qK 122 qK 2#�Ç qK 12q

K22 ,� È 15É

Q£

2#�Ê qK 12

2#

qK 1 Ë qK 12qK

12#

,� Ì 16ÍQ£

12Î qK 12qK 2 Ï qK 1qK 2

2 . Ð 17Ñ

Theã

conventionalhypersphericalcoordinatesrB ,�ÓÒ ,� and Ô are�obtained� from r1 ,� r2,� andr12 viaZ the definitions

r Õ�Ö r12#�×

r2#2# ,� Ø 18Ù

Ú?Û 2 tanÜ 1rB 2#

r1,� Ý 19Þ

ßáàcos1ãâ 1

r12#�ä

r2#2#�å r12

2#

2r1r2# . æ 20ç

Theã

kinetic-energy operatorin hypersphericalcoordinatesisgiven� by

è 1

2 é�ê 12#�ë�ì

2#2#�í�î�ï 1

2

ð 2

ñr2 ò 5

$r

óó

r ô2<r2õ ,� ö 21÷

whereVøJùûú 2

ü-ý 2#Eþ 2 cotÿ�� ���� � � csc1 2

#���� � 2 �� 2#�� cot1��������� � . �

22�Another�

set of hypersphericalanglesthat turned out to beveryZ useful in the presentcontext was first introducedbyPluvinage� 31

~��. Theseanglesaredefinedby

���tan�! 1

"rB # 1 $ rB % 2 &('*) rB 1 + rB 2 ,.- rB 12/ rB 1 0 rB 2 12r3 1 4 r5 2#�6(7 r1 8 r2

#�9.: r12; r1 < r2#�= ,� > 23

<�?

@�A tan�!B 1 C rD 1 E rF 2 G(H r1 I r2 J.K r12L r1 M r2 NO

rB P 1 Q rB R 2#�S(T rB 1 U rB 2#�VXW rB 12Y*Z rB 1 [ rB 2

#�\ . ] 24<�^

Theangularkinetic-energy operatorin Pluvinagecoordinatesis

_�`wacbed ,��f�g.h 1 iikj sin�ml 2<on�prqq n s

tt�u sin�wv 2<yx�z�{{ x ,�*|25}

whereVwac~e� ,�����.��� sin�w� 2 ���.� sin�w� 2 ����� . � 26�

Using�

the Jacobian���r1 ,� r2

# ,� r12����r,��� ,������� r2 

2cos1 ¡�¢¤£

2,� ¥ 27¦

together�

with Eqs. § 8]�¨ and� © 26ª ,� givesthe volumeelement

d«­¬�®°¯ 2

±r5²w³µ´·¶ ,��¸�¹ drd

« ºd«¼»

. ½ 28¾III. FOCK EXPANSION

In order to constructbasisfunctionsthat incorporatetheFock¿

expansion,it is necessaryto know the angularcoeffi-cients1 À

kj

,lo in Eq. Á 3~� . The standardmethod for obtaining

these�

coefficients is to put the Fock expansioninto the

COMPACT REPRESENTATION OF HELIUM WAVE . . . PHYSICAL REVIEW A 69, 022504 Ã 2004� Ä

022504-3

SchroÅ

¨dinger0

equationto get the recursionrelation

ÆrÇÉÈ 1

4V kÊ�Ë

kÊ�Ì

4V�Í ÎÐÏ

kj

, Ñ�Ò*Ó�ÔeÕ�Ö 1 ×XØ kÊ�Ù 2<oÚXÛÐÜ

kj

, Ý.Þ 1 ß*à�áeâ�ã 1 äåçæeè�é

2 êXëÐì kj

, í.î 2 ïð 1

2

r

r1

ñ r

r2

ò róZrô

12

õÐökjX÷

1,ø�ùú 1

2< EûýüÐþ

kj�ÿ

2,±���� .�29<��

Theinhomogeneousproblem � 29� wasV solvednumericallybyFeagin,¿

Macek, and Starace 32~�

. It has also been solvedindependently:

by Hill � 33~��

using hyperanglesexpressedinPluvinage coordinates � 23� and� � 24� . The operator � inPluvinagecoordinates� 25� is self-adjointfor � and� � in theinterval� �

0,���

/2¹��

with� thevolumeelement 28!�"

. Theeigenval-ues of # are� given by $ n% (

&n%(' 2) with n% a� non-negative

integer�

, andthe eigenvectorsaregiven by ) 33,34* +

,n¨ ,m©.-0/ ,1�24365 2m© m7 !

82m7:9 1 ;=< n%(> 1 ?=@ n%.A m7CB !DFE n%(G m7:H 1 I !

JcosKMLON

P:QSRUT mV PmV cosKMWOX.Y:ZS[cosKM\O].^:_S` Cnacb mVmVed 1 f singihcj.k:lSmUn ,1

o30*�p

where� Pq

mV and� CnamV are� theLegendreandGegenbauerpolyno-mials,r respectively. Theoperatoron the left-handsideof Eq.s29t is singular if k

Êvu2n% . Therefore,the right-handside of

Eq. w 29x must be orthogonalto all of the eigenvectorsbe-longingy

to this n% if�

solutionsareto exist.If solutionsdo exist,theyz

will containlinear combinationsof homogeneoussolu-tionsz

for eachvalue of { . Apart from the |(} 0�

case,theconstantsK in theselinearcombinationscanbeuniquelydeter-mined if the right-handsideof Eq. ~ 29� is madeorthogonaltoz

theeigenvectorsfor �.� 1. Thefirst few termsof theFockexpansion� havebeenworkedout analytically � 8,12,33

� �. The

result� in Pluvinagecoordinatesis � 00��� 1 and

�10�

�2

2Zsing �.�:�

2 ��

2 cos �.��

2,1 � 31

*��

�20��� 1

6� 2 � E � 1

2Z2 �2!

3*

Z   sing¢¡4£ sing¥¤4¦6§ singi¨c©«ª­¬¯®

° ±F²:³v´¶µ3*¸·

Zô ¹ 1

2! cosKMºO».¼:½S¾ 1 ¿ 1

3*

Zô À 1

3*�Á

Zô singiÂOÃ

Ä:ůÆ6Ç F2dÈOÉ0ʯË6Ì F2d

ÈOÍcÎ4ÏUÐOÑ cosKMÒOÓ.Ô:ÕSÖ C × 1

2Zln 2

Ø 1

3*

Zô lny

cosK Ù2!�ÚÜÛ

4Ý cosK Þ

2!�ßáà

4Ý ,1 â 32

*�ã

ä21å 1

3*

Z

2æ­ç 1 cosKMèOé.ê:ëSì ,1 í 33*�î

where� C is the coefficient of the homogeneoussolution atsecondg order. The function F

ï2�

dÈ (ðOñ )

òand its derivative are

definedó

by ô 33*�õ

Fï

2�

dÈOöø÷úù=û i

ü2!þý Liÿ

2��� 1 � exp��� 2! i

����� �Liÿ

2��� 1 � exp��� i����� 2

!i��������

���ln � 1 exp��! i"�#$�% & i

')( 2

8� ,1 * 34

*,+dF-

2dÈ .�/10

d243 576 2

!48:9<;>=1 ? cosKA@�BC�D

singFE 2!4G1H ,1 I 35*,J

where� Li2� is the dilogarithm function K 35

*,L. The first-order

polesM in Eq. N 35*,O

indicatethat the functionF2dÈ (P Q )

Rhasloga-

rithmic� branch-pointsingularitiesat SUT n%1V /2W

for n%YX 0�

andn%YZ 1. Although thesesingularitiesare outsidethe physicalrange� 0 []\:[<^ /2,

_they will neverthelesshavean important

impact on the convergencerate of the basisset expansionused to obtainthe higher-ordertermsof the Fock expansionand� ultimatelyon theeffectivenessof thenumericalmethodsused to computethe matrix elementintegrals.To obtain thehigher-order terms of the Fock expansion,the inhomoge-neousproblem ` 29a was� solvednumerically b 33

*,cusing the

Schwingerd

-Levine variationalprinciple

e�fmaxg h

˜ ij k�l,mon prq 2sut v Awyxoz { . | 36

*~}

The�

maximumis achievedwhen

�o� ��� c�A

� �~�,1 � 37

*,�where� c� is an arbitrary constant.In the presentcase,A�7����� 1

4� kÊ(�kÊ��

4Ý

) and � is�

equalto the right-handside ofEq. � 29� . For eachcombinationof k

Êand� � ,1 a trial functionof

the�

form

� ��� ,1)���¡ £¢mV4¤ 0�

N¥ ¦

na�§ 0�

N¥

c� mV ,na TmVY¨ x©«ª Tna ¬ y­¯® ° 38*~±

was� used ² 33*,³

,1 whereT´

mV are� Chebyshevpolynomialsof thefirstµ

kind with x¶ and� y­ related� to the Pluvinageanglesby

·¹¸£º4 » x¶½¼ 1 ¾ ,1 ¿ 39

*,À

ÁãÄ4ÅÇÆ yÈÉ 1 Ê . Ë 40

Ì,ÍRapidconvergenceof the trial function Î 38

*~Ïis necessaryin

orderÐ to devise an efficient quadratureschemefor matrixelementÑ evaluation Ò seeg Sec.V Ó . The convergencerate forChebyshevÔ

expansionsmay be analyzedÕ 36,37* Ö

by×

consid-eringÑ the expansionof an arbitraryanalytic function

ROBERT C. FORREY PHYSICAL REVIEW A 69,Ø 022504 Ù 2004Ú

022504-4

fÛÃÜ

xÝ«Þ¡ßáàna�â 0�ã

c� na Tä na å xÝçæuè 1 é xÝ 2 êuë 1/4,1 ì 41í,î

where�

c� naðï 2ñ ò1

1

fÛÃó

xÝ«ô Tä na�õ xÝ«ö)÷ 1 ø xÝ 2��ù¡ú

1/4dxû ü

42í,ý

forþ

n%Yÿ 0�

. Thecoefficientsc� na canK beevaluatedby usingcon-tour�

integrationin the complexz� plane.M The result is�34*��

c� na�� 2!�� nai�

Cf��

z�� �� 1 � z� 2���

1/4� z��� 1 � 2� � na�� 1

�Fï

1 n%�� 1,n%�� 1

2! ;2n%�� 1;

2!

1 � z� dzû

. 43í�!

Thecontouris shownin Fig. 1. Thelarge-n% behavior×

of c� na isdominatedó

by R " na ,1 where R is the largest value such that(1#%$

z� 2)& 1/4fÛ

(#z� )& is analyticwithin the ellipse ' 36,37

* (Re) z��*�+ 1

2 , R - R . 1 / cosK10�243 ,1 5 446Im 7 z�98�: 1

2 ; R < R = 1 > sing@?�A4B ,1 C 45Dwith� 0 EGFIH 2 J . For R K 1, theseriesL 41M converK gesto f

Û(#z� )&

for all z� inside the ellipse N seeg Fig. 1O in analogywith thecircularK region of convergenceof Taylor-seriesexpansions.AsP

shownabove,thenearestsingularitiesof theFockexpan-siong appearto occur in the Q 20

� term�

at RTS /2U

and V . Theinverse�

transformationsof Eqs. W 39*�X

and� Y 40í[Z

mapr thesesin-gularities\ to ] 3

*. Therefore,thecoefficientsc� na decrease

ólike

R^`_ na with� R acb z��dce z� 2

��f1 gih 3

*�j2 k 2 l 5.828.

mThis estimate

is confirmedby the observedconvergencebehaviorand isuseful whenanalyzingmatrix elementevaluationin Sec.V.

IV. BASIS SET

Fromtheconsiderationsof theprecedingsection,we haveat� our disposalall the termsof the Fock expansionthroughany� desiredorder. We note,however, that we cannotsimply

add� the termsof theFockexpansiondirectly to our basissetsinceg they do not falloff properlyat large distances.We canget\ aroundtheproblemby building in theexponentialdecayat� large distancesandthenremovingthe exponentialeffectsat� small distances.The resultingfunction is

nK o rp ,1rq ,1sIt�uwv

kx�y

0�

Kz {

l|r}

0�

[ kx

/2]~

g� K � kx� rp ,1r� ,1�I�9� k

x,l|���� ,1�I� rp k

x�ln�

rp�� l|,1 �46�

where�

g� kx�� rp ,1r� ,1�I�����

mV�� 0�

kx

1

m7 ! � r fpw��� ,1���¡  mV exp¢¤£¦¥ r fpw§�¨ ,©ªI«¡¬ . ­ 47í[®

The¯

contributionfrom the function fÛ

(#¦°

,©�± )&, which is needed

to²

providethe properfalloff at large distance,is removedatsmall³ rp so³ that ´ K

z reproducesµ the Fock expansionthroughorder¶ K

·. We note that the unknown energy E

¸and¹ the un-

known coefficients cº mV of¶ the homogeneoussolutions arecontained» within the single function ¼ K . We could in prin-ciple» treattheseunknownparametersasnonlinearvariationalparameters½ andattemptto optimizethem.A betterapproachis¾

to usethe expansion

¿kx

,l|�À�Á ,©Â�Ã�ÄÆÅ

mV�Ç 0�

M Èna�É 0�

NÊ

cº mV E¸ na�Ë

kx

,l|,mV ,na9ÌÎÍ ,©ÏIÐ ,© Ñ 48

í�Òin¾

order to break up Ó Kz into¾

separatebasisfunctions,andthen²

usethe diagonalizationto determinethe valuesof theunknownÔ parameters.This procedureuniquely definestheFock functionswhich we denote Õ mÖ ,na(

×K)Ø

. For example,for KÙ 7Ú

, which is the largestvalueof K considered» in this work,weÛ obtainthe basisfunctionsÜ

0,0�(7)×ÞÝ

cº 0�9ß g� 7à4á

0000�¤â g� 6

ã�ä1000r å g� 5

²4æ2000� r2 ç g� 5

²�è2100� r2ln ré

g� 4ê4ë

3000ì rp 3

ì�íg� 4ê�î

3100ì rp 3

ìln�

rpðï g� 3ì�ñ

4000ê rp 4ò

g� 3ì4ó

4100ê r4ln r ô g� 3

ì4õ4200ê r4ln2r ö g� 2

��÷5000² r5

²ø

g� 2 ù 5100² r5

²ln r ú g� 2 û 5200

² r5²ln2�r ü g� 1 ý 6000

ã r6ã

þg� 1 ÿ 6100

ã rp 6ãln�

rp�� g� 1�

6200ã rp 6

ãln� 2�rp�� g� 1 � 6300

ã rp 6ãln� 3ìrp�

g� 0���

7000à rp 7

àg� 0��

7100à rp 7

àln�

rp�� g� 0�

7200à rp 7

àln� 2rp�

g� 0���

7300à r7

àln3ìr � ,© � 49�

�1,0(7)��

cº 1 � g� 5²��

2010� rp 2 � g� 4

ê�3010ì r� 3

�g� 3�

4010ê r� 4 � g� 3

ì�4110ê r� 4ln

�r�

g� 2��!

5010² r� 5

²"g� 2�#

5110² r� 5

²ln�

r��$ g% 1 & 6010ã r� 6

ã'

g% 1 ( 6110ã r� 6

ãln�

r�*) g% 1 + 6210ã r� 6

ãln, 2r��- g% 0

�.7010à r� 7

à/

g% 0��0

7110à r7

àln r 1 g% 0

��27210à r7

àln2r 3 ,© 4 50

5768

2,0(7)�9

cº 2�;: g% 3ì�<

4020ê r� 4 = g% 2

�>5020² r� 5

²�?g% 1 @ 6020

ã r� 6ãA

g% 1 B 6120ã r� 6

ãln,

r�Cg% 0��D

7020à r� 7

àEg% 0�F

7120G r� 7

Gln,

r�IH ,J K 5157L

FIG.M

1. The integrationcontourC andthe ellipseof analyticityfor evaluatingthecoefficientsof a ChebyshevexpansionN 37

OQPof the

Fock expansion.The singularitiesthat limit the size of the ellipseoccurat R 3 andyield coefficientscnS that decreaselike R T nS withRUWV

3 X 2 Y 2.

COMPACT REPRESENTATION OF HELIUM WAVE . . . PHYSICAL REVIEW A 69, 022504 Z 2004[ \

022504-5

]3,0^(7)_�`

ca 3^;b g% 3

^�c4030r

� 4êd

g% 2 e 5030² r� 5

²�fg% 1 g 6030

h r� 6hi

g% 1 j 6130h r� 6

hln,

r�kg% 0l�m

7030G r7

Gng% 0lo

7130G r7

Gln r p ,J q 52

5srt

4,0ê(7)_�u

ca 4 v g% 1 w 6040h r6

h�xg% 0l�y

7040G r7

G{z,J | 53

5s}~

5,0²(7)_��

ca 5²�� g% 1 � 6050

h r� 6h��

g% 0l��

7050G r� 7

G{�,J � 54

5s��

0,1l(7)_��

ca 0l E � g% 5

²�2001r

2 � g% 4 � 3001^ r3

^�g% 3^�

4001r4

�g% 3^�

4101ê r4ln r � g% 2

��5001² r5

²��g% 2���

5101² r5

²ln r�

g% 1 � 6001h r6

h��g% 1 � 6101

h r6hln r � g% 1 � 6201

h r6hln2�r�

g% 0l�

7001G r� 7

G��g% 0l��

7101G r� 7

Gln,

r��  g% 0l�¡

7201G r� 7

Gln, 2�r�I¢ ,J�£

555s¤

¥1,1(7)_�¦

ca 1E§©¨

g% 3^�ª

4011ê r� 4 « g% 2

�¬5011² r� 5

²�­g% 1 ® 6011

h r� 6h

¯g% 1 ° 6111

h r� 6hln,

r��± g% 0l�²

7011G r� 7

G³g% 0l´

7111G r� 7

Gln,

r�Iµ ,J·¶565s¸

¹2,1�(7)_�º

ca 2E » g% 1 ¼ 6021h r6

h½g% 0l¾

7021G r7

G{¿,J À 57

5sÁÂ

3,1^(7)_�Ã

ca 3^ E§©Ä g% 1 Å 6031

h r� 6hÆ

g% 0lÇ

7031G r� 7

G{È,J É 58

5sÊË

0,2l(7)_�Ì

ca 0l E2�;Í

g% 3^Î

4002r4ê�Ï

g% 3^�Ð

4102r4êln r Ñ g% 2 Ò 5002

² r5²

Óg% 2��Ô

5102² r5

²ln r Õ g% 1 Ö 6002

h r6h×

g% 1 Ø 6102h r6

hln rÙ

g% 0l�Ú

7002G r7

GÛg% 0lÜ

7102G r7

Gln r Ý ,J Þ 59

5sßà

1,2(7)_�á

ca 1E2 â g% 1 ã 6012h r6

h�äg% 1 å 6112

h r6hln r æ g% 0

l�ç7012G r7

Gè

g% 0l�é

7112G r7

Gln r ê ,J ë 60

ìsíî

0,3l(7)_�ï

ca 0l E§ 3^;ð

g% 1 ñ 6003h r� 6

h�òg% 1 ó 6103

h r� 6hln,

r��ô g% 0l�õ

7003G r� 7

Gö

g% 0l�÷

7103G r� 7

Gln,

r�Iø . ù 61ìsú

The functional dependencieshave been ignored in Eqs.û49ísü

– ý 61ìsþ

inÿ

orderto savespace.The � k�

,l�,m� ,n� are� determined

by�

the solutionto Eq. � 29� . Thereare13 basisfunctionsforK�

6ì

andK��

7

. Similar analysisyields7 basisfunctionsforK � 4 and K � 5

5, and 3 basis functions for K � 2 and K� 3

�. The basissetmay be compactedfurther by combining�

m� ,n�(_K)�

with� the samevalue of m� . For K��

7

, this procedureyields� the basisset

�0l(7)_����

0,0l(7)_ �

E ! 0,1l(7)_ "

E2 #0,2l(7)_�$

E3^&%

0,3l(7)_

,J ' 62ì)(

*1(7)_�+-,

1,0(7)_�.

E§0/

1,1(7)_�1

E§ 2 2

1,2(7)_

,J 3 63ì54

62(7)_�7�8

2,0(7)_�9

E§0:

2,1(7)_

,J ; 64ì5<

=3^(7)_�>�?

3,0^(7)_�@

E A 3,1^(7)_

,J B 65ì5C

D4ê(7)_�E�F

4,0ê(7)_

,J G 66ì5H

I5J(7)_�K�L

5,0M(7)_

. N 67ì5O

In this procedure,the energy E may be input asan approxi-mation or treatedas a nonlinearvariationalparameter. TheonlyP remaining nonlinear variational parametersare con-tainedQ

in fÛ

(RTS

,JVU )W which we chooseto obeythe condition

expXZY\[ b]

1r1 ^ b_

2r2 ` b_

12r12acb expXZd\e r f fhg ,JVikjml ,J n 68ì)o

sop that

fÛrqts

,JVukvcw 1x2y cosz {

2y cosz |

2y } b~ 1 � b

~2 ����� 2y b

�12� b�

1

�b�

2 � tanQ �

2 ��� 2b�

12� b�

1 � b�

2 � tanQ �

2

���b�

1 � b�

2�T� tanQ �

2y tanQ �

2y . � 69

ì5�W�

e now addtheFock functionsin hypersphericalcoordi-natesto the primary basisset consistingof productsof La-guerre� polynomialsin perimetriccoordinatesin orderto ob-tainQ

the compactrepresentationof the full wave function.Using�

a mixed coordinatenotation,we have

�����l�,m� ,n� ca l

�,m� ,n� U l

�,m� ,n�¡  q¢ 1 ,J q¢ 2

� ,J q¢ 12£c¤�¥m� ,n� ca m� ,n�&¦ m� ,n�(

_K)��§

r� ,J©¨ ,JVªk« ,J¬70 )­

where�U l�,m� ,n�¡® q¯ 1 ,J q¯ 2

� ,J q¯ 12°c± u² l�´³ aµ 1 ;q¶ 1 · u² m�¹¸ aµ 2

� ;q¶ 2�Tº u² n�¡» aµ 12;q¶ 12¼ ,J ½

71 )¾

with� the unnormalizedLaguerrebasisfunctionsdefinedby

u² n�À¿ aÁ ;xÝÃÂ�Ä expXÆÅTÇ axÁ /2ÈÊÉ

LË

n�¡Ì axÁÎÍ . Ï 72 5Ð

TheÑ

LaguerrelengthscalesaÒ 1 ,J aÒ 2� ,J andaÒ 12 combinez with the

Fock length scalesbÓ

1 ,J bÓ

2,J and bÓ

12 toQ

total six nonlinearvariationalÔ parameters.However, the dimensionalityis re-ducedÕ

from six to four by recognizingthat aÒ 1 must be thesamep asaÒ 2

� ,J andbÓ

1 mustÖ be the sameasbÓ

2� sincep the elec-

tronsQ

are indistinguishable.Further simplifications,such asassuming� a hydrogenicexponentialfalloff for eachelectron,are� helpful whenfine tuning the optimizations.

V. MATRIX ELEMENT EVALUATION

TheÑ

matrix elementscontainingonly Laguerrefunctionsinÿ

perimetric coordinateshave been reported previously×23,24Ø . The matrix elementscontainingthe Fock functions

canz beevaluatedusingGaussianquadratureÙ 38�5Ú

in the threecoordinatesz r� ,JÜÛ ,J and Ý . We useGauss-Legendrequadraturefor the Pluvinageangularintegrations

Þ 1

1

fÛrß

z�cà dzûÎáãâ

k��ä

1

Nå

wæ k� fÛrç z� k�mècé E

§Nêìë fÛÜí ,J î 73

)ï

where� the nodesz� k� are� the N

ðzerosñ of the Legendrepolyno-

mial PNê and� the weightswò k

� are� given by

ROBERT C. FORREY PHYSICAL REVIEW A 69,ó 022504 ô 2004õ

022504-6

wò k�Vö 2 ÷ 1 ø�ù z� k

�mú 2ûüNðþý

1 ÿ 2��� PNê��

1 � z� k����� 2� ,J � 74

with� theerrorE

§Nê dependentÕ

on thesingularitiesof fÛ. We use

Gauss-Laguerre�

quadraturefor the hypersphericalradial in-tegrationsQ

0l�

z�� expX���� z��� fÛ�� z��� dzû����

k���

1

Nê

wò k� fÛ�� z� k�� "! EN

ê$# fÛ&% ,J ' 75 )(

where� the nodesz� k� are� the N

ðzerosñ of the Laguerrepolyno-

mialÖ L*

Nê(_,+ )�

and� the weightswò k� are� given by

wò k�"- .0/ Nð214365 1 7 z� k

�Nð

! 8:9 Nð2; 1 < LNê�=

1(_,>

)�@?

z� k��A�B 2

,J C 76 D

with� theerrorEE

Nê again� dependenton thesingularitiesof f

Û. It

was� shownin Sec.III that the termsof the Chebyshevex-pansionF decreaselike (3 G 2 H 2) I n� where� nJ is the order oftheQ

Chebyshevpolynomial.Therefore,a twenty termCheby-shevp expansionwill providedoubleprecision( K 10L 15)

Mand

a� forty term Chebyshevexpansionwill provide quadrupleprecisionF ( N 10O 30

^)M. Sincethe Gaussianquadraturerule ex-

actly� integratesa polynomial of order 2Nð2P

1, we canachieve� double precisionfor the angularintegralsusing N

ðQ 10 andquadrupleprecisionusingNðSR

20.T

Theseestimatesfor Nð

shouldp be doubledwhencomputinginner productsofbasis�

functions.The singularitiesintroducedinto the angularintegrationsby theexponentialfactor f

Û(U�V

,J�W )M

in Eq. X 69ìY

maybe�

movedin thecomplexplane Z seep Fig. 1[ beyond�

thenear-estX singularitiesof theFockexpansionby choosingappropri-ate� valuesfor the variationalparametersb

\1 ,J b\

2,J andb\

12.TheGauss-Laguerrequadraturesfor the integrandswhich

containz logarithmsneedto be modified in order to achievetheQ

desiredaccuracy. The modificationis obtainedby differ-entiatingX the standardGauss-Laguerrequadraturerule withrespect] to ^ onP both sidesof the equalsign. The modifiedGauss-Laguerre�

rule is

0l_�`

ln,ba

z��c�d m� z��e expX�f�g z��h fÛ�i z��j dzû

kmln��n 0l

m� ok�"p

1

Nê

wò m� ,n�(_k�

)�bq n�r

z� k�n� fÛ�s

z� k��t�uwv m�xzy m� E

ENê${ fÛ&| ,J } 77

~�where�

wò m� ,n�(_k�

)���

wò m��� 1,n��� 1(_k�

)� � z� k

��&��� dû

dû�� wò m��� 1,n�(

_k�

)�

,J � 78~�

with�wò m� ,n�(_k�

)��� wò k

� for m��� nJ�� 0,�

0�

for nJ�� 0 o�

r m��� nJ . � 79~�

Since�

theweight functionsall dependon � and� z� k� ,J we must

use� the total derivativeoperator

dû

dû����w��&�

 m¡ z� k�¢z£ ¢¢

z� k� ,J ¤ 80

¥¦when� operatingon any of the weight functions.The deriva-tivesQ

of z� k� with� respectto § canz be obtainedby differentiat-

ingÿ

the nodecondition

L*

n�(_,¨ )�ª©

z� k��«�¬ 0

� ­81¥)®

toQ

get

¯z� k�°&±�²´³ µ

Ln�(_·¶ )�ª¸

z��¹ /º¼»&½»Ln�(_,¾ )�ª¿

z��À /º¼Á z�zÂ�à z kÄ . Å 82

¥)Æ

By construction,theFockbasisfunctionsaretruncatedatorderP K

. In the presentwork, K

ÈÇ7~

was the highestvalueconsidered.z Sincethe highestpowerof the logarithmof thehypersphericalÉ

radius is equal to K

/2º

for even K

,J and (KÊ 1)/2 for oddK,J theFockfunctionswill containup to three

powersF of thelogarithm,andthecorrespondingintegralswillcontainz up to six powersof thelogarithm.Becausethemodi-fied Gauss-LaguerrequadratureË 77

~ÌrequiresmÍ�Î 1 modified

weights,Ï the total numberof modifiedweightsfor K Ð 7~

willbeÑ

27.A very convenientway to computetheseweightswastoÒ

symbolically differentiatethe standardweights Ó 76~)Ô

withÏrespectto Õ andÖ z� andÖ defineeachderivativeasa new func-tion.Ò

The new derivative functionswere then differentiatedwithÏ respectto × andÖ z� to

Òdefinefurther new functions.We

proceedØ in this way until new functionsof all the requiredcombinationsÙ of the six derivativeswith respectto Ú andÖ z�havebeendefined.Onceall thedifferentiationhasbeencom-pleted,Ø thenodesz� k

Û andÖ thevalueof Ü isÝ

substitutedinto thenewly definedfunctionsmakingthemnumericalarrays.

In order to use the quadraturerules outlined above,thematrixÞ elementintegrandsneedto be separatedinto pieceswhichÏ contain no logarithms, pieces which contain onepowerØ of the logarithm,pieceswhich containtwo powersoftheÒ

logarithm,andso on. For the Grammatrix elementsthisseparationß procedureis no problem,but for the HamiltonianmatrixÞ elementstheseparationbecomesmoredifficult duetotheÒ

variousderivativesinvolved.The difficulties with differ-entiatingà andseparatingtheFockfunctionintegrandsmaybereducedá if a symbolic manipulationprogram is used.TheHamiltonianâ

derivativeoperationsandlogarithmicseparationproceduresØ wereperformedby consideringthe function

UK ã r,J,ä ,J�åçæ�è expà�é�ê r f ë¼ì ,J�í&î�ï�ð UKñ(0)_óò

r,J,ô ,J�õ&ö�÷ UKñ(1)_óø

r,J,ù ,J�úçûüUK

(2)_óý

r� ,J,þ ,J�ÿ���� UK(3)_��

r� ,J�� ,J�� � ,J � 83¥��

whereÏUKñ(0)_��

r,J�� ,J�������k��

0l

K

fÛ

kÛ

,0��� ,J��� rkÛ,J � 84

¥�

UKñ(1)_"!

r,J�# ,J$�%�&�'kÛ�(

2

K

fÛ

kÛ

,1)+* ,J,�- rkÛln r,J . 85

¥�/

COMPACT REPRESENTATION OF HELIUM WAVE . . . PHYSICAL REVIEW A 69, 022504 0 20041 2

022504-7

UKñ(2)_�3

r,J�4 ,J5�6�798kÛ:

4êK

fÛ

kÛ

,2;�< ,J=�> rkÛln2r,J ? 86

¥A@

UKñ(3)_�B

r,J�C ,JD�E�F9GkÛH

6h

K

fÛ

kÛ

,3I�J ,JK�L rkÛln3^r. M 87

¥ANTheO

angular fÛ

coefficientsÙ in Eqs. P 84¥AQ

– R 87¥AS

areÖ arbitraryfunctionsthat are to be determinedby the Fock basisfunc-tionÒ

of interest.The function UK isÝ

separatedin the abovemannerin orderto allow theGramandpotential-energy ma-trixÒ

elementsto be correctly separatedat the outset.Thekinetic-enerT

gy operatorwill mix up the logarithmic depen-dence,U

soa secondseparationprocedurewill beneededaftertheÒ

necessaryderivativeshavebeencompleted.UsingPluvi-nagehypersphericalangles,we breakthe probleminto fourpieces:Ø UK ,J VUK ,J RU

VK ,J and LU

WK ,J where V is

Ýthe

potential-enerØ gy operator, R is the radial kinetic-energy op-eratorà , andL is the angularkinetic-energy operator. The de-rivativesá were performedby MAPLE

X andÖ the separationwasenactedà by collecting the coefficientsof the logarithms.WemayÞ now treatVUK

ñ ,J RUV

Kñ ,J andLU

WKñ simplyß asfunctionsof

r,JZY ,J and [ thatÒ

havea form similar to Eq. \ 83¥A]

. After thedifU

ferentiationswere completedthe exponentialfactor wasremovedá since it will be put in by the Gauss-Laguerrequadrature^ rule. The angularf

ÛcoefficientsÙ and their deriva-

tivesÒ

weregiven scalarvariablenamesandsavedfor useintheÒ

next step.With the Hamiltonianoperationsalreadyper-formed,_

the next step is to perform the differentiationsre-quired^ for the modifiedquadrature77

~�a. If we let

z�cb rh�ed+f ,Jg�h ,J i 88¥Aj

thenÒ

we can perform the necessarydifferentiationswith re-spectß to z� . In analogywith the modified weight procedure,weÏ defineeachderivativewith respectto z� to

Òbea newfunc-

tion.Ò

After the differentiationshave been completed,thenodesaresubstitutedinto the new functionsto producenu-mericalÞ arrays.The final step is to assemblethe full inte-grands.k The requiredintegrandsareof the type

Il

1 m z� ,J�n ,Jo�p�q wòsr z� ,J�t ,Ju�v U lw,mx ,ny{z z� ,J�| ,J}�~ UK

ñ�� z� ,J�� ,J��� ,J�� 89�A�

I2�{� z� ,J�� ,J����� wòs� z� ,J�� ,J��� U l

w,mx ,ny{� z� ,J�� ,J��� HUK

ñ�� z� ,J�� ,J��� ,J�� 90�A�

I3^�� z� ,J�� ,J�� �¡ wò£¢ z� ,J�¤ ,J¥�¦ UK § z� ,J�¨ ,Jª©�« UK ¬ z� ,J�­ ,J®�¯ ,J ° 91

�A±I4 ² z� ,J�³ ,J´�µ�¶ wò£· z� ,J�¸ ,J¹�º UK » z� ,J�¼ ,J½�¾ HUK ¿ z� ,J�À ,JÁ� . à 92

�AÄInÅ

Eqs. Æ 89�AÇ

andÖ È 90�AÉ

itÝ

is assumedthat the perimetriccoor-dinatesU

areexpressedin termsof the hypersphericalcoordi-nates.The UK

ñ andÖ HUKñ piecesØ in Eqs. Ê 89

��Ë– Ì 92��Í

areÎ avail-ableÎ throughorderK Ï 7

Ðusingthemethodsdescribedabove.

In order to complete the requirementsof the modifiedquadratureÑ Ò 77

Ð�ÓitÔ

is necessaryto also have available thederivativesÕ

of the physical weight function wò andÎ the La-guerreÖ function U l

w,mx ,ny . For the integrandsI1 andÎ I2

� ,J thesefunctions×

needto bedifferentiatedwith respectto z� aÎ total ofthreeØ

timesdueto the threepowersof the logarithm.For theintegrandsÔ

Il

3^ andÎ I

l4ê ,J thephysicalweightfunctionwò needsÙ to

beÚ

differentiatedwith respectto z� aÎ total of six timesduetotheØ

six powers of the logarithm. We again use MAPLE toØ

separateÛ the logarithmic termsand perform the differentia-tionsØ

with respectto z� . We also usedMAPLEX to

Øwrite out a

FORTRAN codeÜ Ý 34Þ�ß

.

VI. RESULTS

Thesingletandtriplet eigenvaluesmaybecomputedwithaÎ singleLaguerrebasisset representationof the form giveninÔ

Eq. à 71ÐAá

. In this case,the numberof functionsis

nâ Lãåä 1

2

ælw�ç

0l

lwsumèêé

lë

sumìîí lë ï

1 ðñ lë sumìóò lëõô

2 ö÷ 1

6ø lë

sumì3^îùlë

sumì2 ú 11

6ø łû

sumìóü 1, ý 93�Aþ

whereÿ lë

sumì equals� the maximumsumof the threeLaguerreindices.In the presentwork, we aremainly concernedwiththeØ

singletgroundstate.Therefore,it is convenientto parti-tionØ

the different symmetries.In this case,the numberofLaguerre�

basisfunctionsis

nâ Lã�� 1

12lë

sumì3^�� 558� lë

sumì2��� 17

12lë

sumì�� 7Ð8� ,J 94

��

whereÿ the squarebracketsdenotethe nearestinteger. TheFock functionswereassembledandsystematicallyaddedtotheØ

Laguerrebasisin order to seethe effect on the conver-genceÖ rate. The total numberof basisfunctions is nâ L � nâ Fwhereÿ nâ F

equals� the numberof Fock functions.TablesI andIIÅ

showthe error in the ground-stateenergy asa function oftheØ

optimized nonlinearvariational parameters.Table I in-cludesÜ entriesfor K

���0,2,4,6�

correspondingto nâ F �� 0,3,7,13�

withÿ lë

sumì in therange10–15. In TableII, theFock functionswithÿ thesamehomogeneoussolutionwereaddedtogetherinorder� to speedup the calculation � seeÛ Eqs. � 62

ø��– � 67ø����

. ThiscausedÜ a significantdecreasein the accuracyof the energy,presumably� dueto thedecreasedflexibility of theFockfunc-tions.Ø

It is worth noting that the optimizationswere per-formed×

by trial anderror andarenot perfect.The tablesareincludedto providea startingpoint for future refinementsintheØ

optimization.Figure�

2 showsthesignificantimprovementin the rateofconverÜ gencethat canbe obtainedby addingFock functionstoØ

the basisset.The K � 0�

curvecorrespondsto an unmodi-fied�

Laguerrebasis set. It is easy to see that many moreLaguerre�

functionsarerequiredfor this basissetto matchtheaccuracyÎ obtainedin theothercurves,which areobtainedbyusing� Fock functionsthroughK

�thØ

order. Eventually, the im-provement� patternwe seeby going to higher ordersin theFock�

expansionbeginsto diminish. In fact, the K��

7Ð

curvewhichÿ is not shown, shows an insignificant improvementover� the K

�"!6ø

curve.The reasonthat the improvementdi-minishes# is that the cuspat the three-particlecoalescenceishandledwell enoughthat otherphysicaleffects,suchas ra-dialÕ

‘‘in-out’’ correlation,becomemore important.Figure 3showsÛ the optimizedvalueof the nonlinearparametera$ 12 asÎ

ROBERT C. FORREY PHYSICAL REVIEW A 69,ó 022504 % 2004&

022504-8

aÎ function of lë

sumì for×

different valuesof K�

. Becausethisparameter� is approximatelyequalto twice thevalueof a$ 1, iJ tdeterminesÕ

the lengthscaleof the Laguerrebasisfunctions.When'

Fock functionsareincludedin the basisset,the valueof� a$ 12 decreases

ÕandtheLaguerrefunctionsarebetterableto

represent( electronicbehaviorthat is further away from thenucleus.

The most accurateresult obtained in this work was) 2.903*

7243770341166 a.u.for theground-stateenergy ofhelium.+

This energy was computedusing lë

sumì�, 15 and K�

- 6ø

, correspondingto a total of 457 basis functions (nâ L. 444í

and nâ F / 13). This compares well to0 2.9037243770341184 a.u. calculated by Baker et1 al.220*�3

using� a Frankowski-Pekerisbasis set containing 476functions. The best standardis 4 2.9037243770341195a.u.Î calculatedby severalgroups 5 16,25–306 using� substan-tiallyØ

larger basissets.

VII.7

CONCLUSIONS

Theresultsof this work demonstratethat theconvergencerate( of a nearlyorthogonalbasisset,suchasthe triple prod-uct� of Laguerrepolynomialsin perimetriccoordinates,maybeÚ

substantiallyimprovedthroughtheadditionof a few spe-ciallyÜ designedbasisfunctions.Theseso-calledFock func-tionsØ

provide the exact analytic structureof the true wavefunction×

when two electronscoalescenearthe nucleus.Be-causeÜ the termsthat are logarithmic in the hyperradiushavebeenÚ

explicitly treatedby the methodsdescribedin this pa-per� , the remaining portion of configurationspacemay beeasily� represented.

Several8

usefulnumericalmethodshavebeendevelopedor

TABLE I. Optimized nonlinearparametersfor basissets thatinclude 9 m: ,; n<(

=K)>

.

lsum? K nF a@ 1 a@ 12 b1 b12 Error

10 0 0 1.30 2.35 1.02A 10B 8C

10 2 3 1.20 2.40 1.00 0.00 8.14D 10E 10

10 4 7 1.17 2.34 1.00 0.00 3.44F 10G 11

10 6 13 1.10 2.20 1.01 0.00 8.52H 10I 12

11 0 0 1.35 2.55 5.29J 10K 9L

11 2 3 1.20 2.40 1.00 0.00 4.46M 10N 10

11 4 7 1.20 2.40 1.00 0.00 7.82O 10P 12

11 6 13 1.10 2.20 1.00 0.00 1.82Q 10R 12

12 0 0 1.35 2.60 2.32S 10T 9L

12 2 3 1.25 2.50 1.00 0.00 1.74U 10V 10

12 4 7 1.25 2.50 1.00 0.00 2.72W 10X 12

12 6 13 1.10 2.20 1.00 0.00 4.75Y 10Z 13

13 0 0 1.40 2.80 1.18[ 10\ 9L

13 2 3 1.30 2.60 1.00 0.00 6.31] 10 11

13 4 7 1.30 2.60 1.00 0.00 8.99_ 10 13

13 6 13 1.19 2.39 1.07 0.04 9.13a 10b 14

14 0 0 1.45 2.90 5.01c 10d 10

14 2 3 1.30 2.60 1.00 0.00 3.64e 10f 11

14 4 7 1.30 2.60 1.00 0.00 2.69g 10h 13

14 6 13 1.20 2.40 1.00 0.01 9.50i 10j 15

15 0 0 1.50 3.00 2.75k 10l 10

15 4 7 1.30 2.60 1.00 0.00 8.88m 10n 14

15 6 13 1.25 2.50 1.00 0.02 2.80o 10p 15

TABLE II. Optimizednonlinearparametersfor basissetsthat

include q m:(= Kr )>tsvu

n< En<xwm: ,; n<(=Kr

)>

.ylsum? K nF

z a@ 1 a@ 12 b1 b12 Error

7 6 6 1.05 2.10 1.00 0.00 1.01{ 10| 7}

8 6 6 1.05 2.10 1.00 0.00 8.72~ 10� 9L

9 6 6 1.04 2.08 1.00 0.00 5.61� 10� 10

10 6 6 1.13 2.13 1.00 0.00 1.11� 10� 10

FIG. 2. Error in the ground-stateenergy asa function of lsum? .yThe k

���0�

curve correspondsto an unmodifiedLaguerrebasisset.The k

���0�

curvesinclude basisfunctions that reproducethe Fockexpansionthroughk

�th�

orderwhenthe hyperradiustendsto zero.

FIG. 3. Optimized value of the nonlinearparametera12 as afunctionof l

�sum? . As moreFockfunctionsareaddedto thebasisset,

the valueof a12 decreasesand the Laguerrefunctionsareconcen-tratedfurther awayfrom the nucleus.

COMPACT REPRESENTATION OF HELIUM WAVE . . . PHYSICAL REVIEW A 69, 022504 � 20041 �

022504-9

employed� in this work. The modified Gauss-LaguerrequadratureÑ allows an efficient andaccuratenumericalevalu-ationÎ of integralscontaininglogarithmsof the hyperradius.The�

useof Pluvinagehyperanglesprovidesa convenientco-ordinate� systemfor thenumericalevaluationof integralsandanÎ efficient method � 33

���for determiningthe angularcoeffi-

cientsÜ of the Fock expansion.The�

convergencerate of the basisset describedhere issimilarÛ to the Frankowski-Pekerisbasisset usedby Bakeret1 al. � 20� . The principle advantageof the presentrepresen-tationØ

is that it doesnot strugglewith numericallinear de-pendence� problemsdue to the small numberof Fock func-tionsØ

and the fact that the Laguerre basis is close toorthogonal.� The major disadvantageof the presentapproachisÔ

theneedto computematrix elementintegralsnumerically.Although�

thequadratureschemesdescribedhereareefficientandÎ accurate,they are still quite slow comparedto otherbasisÚ

set approachesthat allow integralsof one or more oftheØ

coordinatesto be evaluatedanalytically.Finally, thepresentwork allowstheanalyticbehaviornear

theØ

three-particlecoalescencepoint to be treatedwithout theneedÙ for invoking ‘‘flexibility’ ’ arguments.As Schwartzhasrecently( pointedout � 30

���,J flexibility is a vagueconceptthat

lacksmathematicalfoundation.While flexible basissetshavecertainlyÜ producedimpressivebenchmarks,they have cre-atedÎ a situationwhere theoreticalunderstandingof conver-genceÖ rates‘‘lags well behindthe power of availablecom-puting� machinery’’ � 30

���. It is hoped that the explicit

treatmentØ

of singularities,suchaspresentedherein regardtotheØ

Fock expansion,may provide a useful step toward thedevelopmentÕ

of a rigorous theory of convergenceratesfortwo-electronØ

systems.

ACKNOWLEDGMENTS

I would like to acknowledgeand thankProfessorRobertNyden�

Hill for introducingme to this problem � 34���

andÎ forproviding� manyhelpful commentsand insightsaswell asacomputerÜ code to generatethe angular coefficientsof theFock�

expansion.This work was initially supportedby theNational�

ScienceFoundationthrougha grantfor theInstitutefor TheoreticalAtomic, Molecular, and Optical PhysicsatHarvardUniversityandSmithsonianAstrophysicalObserva-toryØ

. The final stagesof the researchweresupportedby theNational�

ScienceFoundationGrantNo. PHY-0244066.

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ROBERT C. FORREY PHYSICAL REVIEW A 69, 022504 T 2004� U

022504-10