compact representation of helium wave functions in
TRANSCRIPT
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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022504-10