Singular and nonsingular three-body integrals for exponential wave functions

Singular and nonsingular three-body integrals for exponential wavefunctionsFrank E. Harris, Alexei M. Frolov, and Vedene H. Smith Citation: J. Chem. Phys. 121, 6323 (2004); doi: 10.1063/1.1786912 View online: http://dx.doi.org/10.1063/1.1786912 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v121/i13 Published by the American Institute of Physics. Related ArticlesDirect perturbation theory in terms of energy derivatives: Scalar-relativistic treatment up to sixth order J. Chem. Phys. 135, 194114 (2011) Accurate non-Born-Oppenheimer calculations of the complete pure vibrational spectrum of D2 with includingrelativistic corrections J. Chem. Phys. 135, 074110 (2011) Nuclear magnetic resonance shielding constants and chemical shifts in linear 199Hg compounds: A comparisonof three relativistic computational methods J. Chem. Phys. 135, 044306 (2011) Cavity quantum electrodynamics for photon mediated transfer of quantum states J. Appl. Phys. 109, 113110 (2011) Electronic spectra of GdF reanalyzed by decomposing state functions according to f-shell angular momentum J. Chem. Phys. 134, 164310 (2011) Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Singular and nonsingular three-body integralsfor exponential wave functions

Frank E. Harrisa)

Department of Physics, University of Utah, Salt Lake City, Utah 84112 and Quantum Theory Project,University of Florida, Gainesville, Florida 32611

Alexei M. Frolov and Vedene H. Smith, Jr.b)

Department of Chemistry, Queen’s University, Kingston, Ontario K7L 3N6, Canada

~Received 17 May 2004; accepted 7 July 2004!

Integrals which are individually singular, but which may be combined to yield convergentexpressions, are needed for computations of relativistic effects and various properties of atomic andquasiatomic systems. As computations become more detailed and precise, more such integrals arerequired. This paper presents general formulas for the radial parts of the singular and nonsingular~regular! integrals that occur when three-body systems are described using wave functions thatinclude exponentials in all three interparticle coordinates. Our results are compared with those foundin the literature for some of the integrals, and are also shown to be consistent with previouslyreported results for Hylleraas functions~a limiting case in which one of the exponential parametersis set to zero!. © 2004 American Institute of Physics.@DOI: 10.1063/1.1786912#

I. INTRODUCTION

Calculations of relativistic and quantum electrodynamiceffects in few-body systems are becoming of increasing im-portance as experimental precision improves, and such cal-culations need ever-higher accuracy to extract better infor-mation about quantities such as the fine-structure constant. Ageneral approach to this topic was long ago sketched in theimportant monograph by Bethe and Salpeter;1 we refer alsoto the recent review by Sapirstein.2

A considerable body of work in this area has been car-ried out on the He atom, using Hylleraas wave functions,i.e., those containing exponentials in the interparticle co-ordinatesr 1 and r 2 ~distances relative to the nucleus!, towhich are appended powers of the interelectron distancer 125ur12r2u. Important recent contributions include thoseof Drake3 and Pachucki and Sapirstein.4

The above studies, while impressive, do not indicate anoptimum starting point for ‘‘nonadiabatic’’ systems in whichall the particles are of comparable mass. In such systems itwould be preferable to use wave functions which treat all theparticles symmetrically. It is for this reason that we are ex-amining here the integrals that arise from the use of wavefunctions containing exponentials in all three interparticleseparationsr 1 , r 2 , andr 12 ~hereafter sometimes referred togenerically as ‘‘r i j ’’ !. Such wave functions were introducedfor nonrelativistic calculations by Delves and Kalotas5 andby Thakkar and Smith,6 and their use in that context has beendiscussed recently by the present authors.7–10

An important aspect of the calculation of relativistic ef-fects is that the matrix elements involved contain powers ofthe interparticle distances more negative than22, andhigher-order relativistic contributions include increasingly

negative powers ofr i j . Since an r p dependence withp<23 produces a divergence, these matrix elements are ofFrullanian character11,12 ~i.e., they are normally written as asum of integrals, each of which is individually divergent, butwhich combine to yield a convergent result!. A detailed dis-cussion of the singularity cancellation for Hylleraas func-tions has been provided by Yan and Drake.13

The present communication analyzes the integrals aris-ing in relativistic and nonrelativistic calculations on three-body systems, giving formulas applicable to general rangesof powers of the variousr i j . Attention has been given to theexplicit identification of the divergent parts of singular inte-grals so as to facilitate verification that the divergences can-cel. By taking the limit as one of the exponential parametersgoes to zero, one can recover expressions for integrals overHylleraas functions, which can in some cases be checkedagainst formulas for those functions given by Yan andDrake.13,14 A few of the integrals discussed here have beenpreviously reported by investigators other than Yan andDrake, but with some errors and without identification oftheir divergent parts. These discrepancies are noted at appro-priate places below.

II. DEFINITIONS

This paper deals with integrals based on three-particlespatial wave functions constructed from a basis of the ge-neric form

C5YLM exp~2ar 12br 22gr 12!, ~1!

where YLM is an angular-momentum eigenfunction of thethree-body system, anda, b, andg are parameters. The con-dition thatC be normalizable requires that the real parts ofa1b, a1g, andb1g be positive; there is no such require-ment individually ona, b, or g. Matrix elements of the op-erators of interest here can be evaluated by separating them

a!Electronic mail: [email protected]!Electronic mail: [email protected]

JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 13 1 OCTOBER 2004

63230021-9606/2004/121(13)/6323/11/$22.00 © 2004 American Institute of Physics


http://dx.doi.org/10.1063/1.1786912

into a radial part~dependent only uponr 1 , r 2 , andr 12), andan angular part~dependent also upon a set of Euler anglesdescribing the orientation of the triangle defined byr i j ). Theangular integrations can be carried out by well-known pro-cedures for angular-momentum eigenfunctions, and in gen-eral lead to polynomials inr i j that must be included in theradial integrations@for details applicable to wave functionsof the form in Eq.~1!, see Refs. 8 and 9#. The exponentialAnsatzof Eq. ~1! implies that flexibility in the wave functionwill be achieved by the use of a number of parameter sets~a,b,g! rather than by appending powers ofr i j to the formgiven above forC. However, a few powers ofr i j ~both posi-tive and negative! will be needed to describe operators ofinterest and~for functions that are not spherically symmetric!as a result of the angular integrations.

In keeping with the discussion of the preceding para-graph, we define the generic radial integral

G l ,m,n~a,b,g!

5E0

`

dr1E0

`

dr2Eur 12r 2u

r 11r 2dr12r 1

l r 2mr 12

n e2ar 12br 22gr 12. ~2!

This integral is invariant under the simultaneous permutationof l ,m,n anda,b,g, and its range of integration correspondsto all values ofr 1 , r 2 , andr 12 that can form a triangle. Forcomparison with other ways of writing the same integrals,we note that

G l ,m,n~a,b,g!

51

8p2 E dr1 dr2r 1l 21r 2

m21r 12n21e2ar 12br 22gr 12, ~3!

where the integration in Eq.~3! is over the full three-dimensional space ofr1 andr2 . It can then be seen that our

G l ,m,n is 1/8p2 times the corresponding definition of Sack,Roothaan, and Kolos,15 and we also see thatG l ,m,n occurswhen describing expectation values ofr 1

l 21r 2m21r 12

n21 . Thisobservation has as a consequence that integrals such asG21,0,0 are regular, while those such asG22,0,0 are singular.We will also find it useful to relate ourG l ,m,n to a radialintegral defined forg50 by Drake,14

G l ,m,n~a,b,0!5I 0~ l 21,m21,n21;a,b!. ~4!

We treat singular integrals by first permutingr 1 , r 2 , andr 12 as necessary to cause the singularity to arise from a di-vergence in the neighborhood ofr 150 and we change thelower limit of r 1 integration toe, a small positive quantity.We then retain thee dependence that does not vanish in thelimit e→01. Processed in this way, nonsingular integralswill exhibit no e dependence, while those which are singularwill contain terms of type lne and possibly also negativepowers ofe.

III. INTEGRALS WITH l ,m ,nÐ0

These integrals are nonsingular. The simplest integral ofthis type isG0,0,0, which by direct integration of Eq.~2! isfound16 to have the value

G0,0,0~a,b,g!52

~a1b!~a1g!~b1g!. ~5!

Repeatedly applying the relation

G l 11,m,n~a,b,g!52]

]aG l ,m,n~a,b,g! ~6!

and its analogs containing derivatives with respect tob andg, one can obtain forl ,m,n.0 the general formula

G l ,m,n~a,b,g!52l !m!n! (l 850

l

(m850

m

(n850

n S m2m81 l 8l 8 D S l 2 l 81n8

n8 D S n2n81m8m8 D

~a1b!m2m81 l 811~a1g! l 2 l 81n811~b1g!n2n81m811. ~7!

IV. INTEGRALS GÀp ,0,0

The formulas presented in this section are derived inAppendices A and B. We give and discuss here the results.

The integral G21,0,0 is nonsingular, but the integralsG2p,0,0 are singular for integersp.1. A formulation validfor all integersp.0 can be written in terms of an auxiliaryfunction Lp of definition

Lp~x!5Ee

`

t2pe2xtdt. ~8!

For p.0, Lp has evaluation~dropping terms that vanish ase→0)

Lp~x!5~2x!p21Fc~p!2 ln~xe!

~p21!!

1 (j 51

p21~2xe!2 j

j ~p2 j 21!! G ~p.0!. ~9!

Here c is the digamma function; for positive integerp,c(p)52gE1(m51

p21 m21, and gE is Euler’s constant,0.577 21 . . . ~usually just writteng but to do so here wouldcause a notational conflict!. For p<0, Lp is essentially agamma function; it has value

Lp~x!5~2p!!

x12p ~p<0!. ~10!

As shown in Appendix A, the formula forG2p,0,0 is

6324 J. Chem. Phys., Vol. 121, No. 13, 1 October 2004 Harris, Frolov, and Smith


G2p,0,0~a,b,g!52

b22g2 @Lp~a1g!2Lp~a1b!#.

~11!

Using Eqs.~9! and ~11!, we may work out the explicitforms for the first few nonzero values ofp. We find

G21,0,0~a,b,g!52

b22g2 @ ln~a1b!2 ln~a1g!#, ~12!

G22,0,0~a,b,g!

52

b1g@c~2!2 ln e#1

2

b22g2

3@~a1g!ln~a1g!2~a1b!ln~a1b!#, ~13!

G23,0,0~a,b,g!

51

b1g H 2

e2@2a1b1g#@c~3!2 ln e#J

11

b22g2 @~a1b!2 ln~a1b!

2~a1g!2 ln~a1g!#, ~14!

G24,0,0~a,b,g!

51

b1g H 1

e2 22a1b1g

e

1Fa21ab1ag1b21g21bg

3 G@c~4!2 ln e#J1

1

3~b22g2!@~a1g!3 ln~a1g!

2~a1b!3 ln~a1b!#, ~15!

G25,0,0~a,b,g!51

b1g H 2

3e3 22a1b1g

2e2 11

e S a21ab1ag1b21g21bg

3 D21

12~2a1b1g!

3~2a21b21g212ab12ag!@c~5!2 ln e#J 11

12

~a1b!4 ln~a1b!2~a1g!4 ln~a1g!

b22g2 . ~16!

Note that althoughG21,0,0 involves two instances ofL1 ,each of which is singular, the singularities cancel and thefinal result exhibits noe dependence. This observation isconsistent with the fact thatG21,0,0 is nonsingular. However,for p.1, the singularity cancellations of the two instances ofLp are incomplete, leaving a nete dependence, indicative ofthe fact that the integralsG2p,0,0 for p.1 are singular.

A limiting case of the integral G21,0,0, namely,G21,0,0(0,a,b), agrees with the corresponding integral givenby Drake,14 I 0(21,21,22;a,b). This integral is an in-stance of Formula 5 of his Table 11.1. The limiting caseG22,0,0(0,a,b) is also in agreement with Drake’s value ofI 0(21,21,23;a,b) ~Formula 7 of Drake’s Table 11.1!. Val-ues ofG21,0,0 and the nondivergent parts ofG22,0,0, G23,0,0

and G24,0,0 for generala, b, and g have been reported byKorobov,17 but his results disagree with ours by a factor of 2and also contain more serious errors.

The formulas given in Eqs.~12!–~16! are ill-conditionedfor small values ofb2g, and forb5g become indetermi-nate forms. It is interesting to note that the indeterminacyaffects only the convergent parts of the integrals; the diver-gent parts remain well defined atb5g. However, numeri-cally stable expressions in the regimeb'g can be obtainedby expandingLp(a1g) aboutg5b. When such an expan-sion is inserted into Eq.~11!, the bracketed expression in thatequation is seen to contain a factorb2g ~with the remainderof the expression nonzero atb5g). Cancellation of that fac-

tor against a similar factor ofb22g2 removes the numericalinstability.

The details of the process outlined above are given inAppendix B. Following Thakkar and Smith,6 it is convenientto express its results in terms of a dimensionless variablet5(b2g)/(a1b). As shown in Appendix B, we obtain

G2p,0,052~a1b!p22

b1g H (j 51

p21t j 21

j !@~a1b! j 2p11

3Lp2 j~a1b!#1tp21

p! 2F1~1,1;p11;t!J . ~17!

The quantity2F1 is a hypergeometric function~for defini-tion, see theHandbook of Mathematical Functions18!, and isto be evaluated using numerical methods that are stable forsmallt. Note that forp51, the summation overj in Eq. ~17!is to be omitted.

Special cases of Eq.~17! include those fort50 (g5b). In particular,

G21,0,0~a,b,b!51

b~a1b!, ~18!

G2p,0,0~a,b,b!51

bLp21~a1b!, ~p.1!. ~19!

6325J. Chem. Phys., Vol. 121, No. 13, 1 October 2004 Three-body integrals for exponential wave functions


In obtaining Eq. ~18! we used the fact that2F1(1,1;p11;0)51. Further specializing Eq.~18! to G21,0,0(0,b,b),we retrieve Drake’sI 0(21,21,22;b,b), in agreement witha case of Formula 6 of his Table 11.1.

Alternatively, specializing Eq.~17! by settinga50 butwith nonzerot ~here equal to 12g/b), we reach forp52

G22,0,0~0,b,g!52

b1g FL1~b!1t

2 2F1~1,1;3;t!G . ~20!

Inserting an expression forL1 , we reach agreement withI 0(21,21,23;b,g) as given by Formula 8 of Drake’s Table11.1 ~Ref. 14! after correcting a minor misprint in that equa-tion: there replacec~1! by 2 c~2!. The agreement is mosteasily verified by substituting closed explicit expressions forthe hypergeometric functions~see Appendix B!. We remindthe reader that these closed expressions are not optimum fornumerical evaluation at smallt; series expansions of the hy-pergeometric functions are then preferable.

V. INTEGRALS GÀp ,m ,n FOR POSITIVE m ANDÕOR n

For the first fewm and n, a simple approach to thecomputation ofG2p,m,n is simply to start from an analyticalformula forG2p,0,0 and use a symbolic algebra system, suchas MAPLE,19 to obtainG2p,m,n by differentiation, using Eq.~6! or its analogs. Some formulas obtained in this way in-clude the following:

G21,1,1~a,b,g!516bg@ ln~a1g!2 ln~a1b!#

~b22g2!3

14

~b22g2!2 F b

a1g1

g

a1bG , ~21!

G22,1,1~a,b,g!524ln e2c~1!

~b1g!3 14

~b22g2!3

3@~4abg13b2g1g3!ln~a1b!

2~4abg13bg21b3!ln~a1g!#

216bg

~b1g!3~b2g!2 , ~22!

G23,1,1~a,b,g!

54a@ ln e2c~2!#

~b1g!3 14

e~b1g!3 14bg~2a1b1g!

~b1g!3~b2g!2

14 ln~a1g!@ab~3g21b2!1bg~2a21b21g2!#

~b22g2!3

24 ln~a1b!@ag~3b21g2!1bg~2a21b21g2!#

~b22g2!3 .

~23!

However, if formulas for general values ofm andn aredesired, it is convenient to use the recursive process of Sack,Roothaan, and Kolos15 to obtain values ofG2p,m,n for posi-tive m and/orn from those withm5n50. The relation to beused is

G2p,m,n~a,b,g!5S 1

b1g D @mG2p,m21,n~a,b,g!

1nG2p,m,n21~a,b,g!

1G2p,m,n~a,b,g!#, ~24!

where, as shown in Appendix C,

G2p,m,n~a,b,g!

52(j 50

n S nj D ~21!n2 j~m1n2 j !!

~b2g!m1n2 j 11 Lp2 j~a1g!

22(j 50

m S mj D ~21!n~m1n2 j !!

~b2g!m1n2 j 11 Lp2 j~a1b!. ~25!

The summations may have extents causingp2 j to becomezero or negative, in which caseLp2 j can be evaluated usingEq. ~10!. For positivep2 j , Lp2 j is given by Eq.~9!.

Equation ~25! is suitable for computation whenb2gdoes not too closely approach zero. For smallb2g, onemay proceed, as was done forG2p,0,0, by expandingLp(a1g) about g5b. The resulting formulas forG2p,m,n fallinto two cases which are distinguished according to whetheror not it is singular. The first case, corresponding to nonsin-gular G2p,m,n , is characterized by the conditionp<m1n11. In this case, as discussed in detail in Appendix C, wehave the compact result

G2p,m,n~a,b,g!52m!n! ~m1n2p11!!

~m1n11!! ~a1b!n1m2p12

32F1~n11,m1n2p12;m1n12;t!.

~26!

Here t has the same definition as in the preceding section:t5(b2g)/(a1b). Equation ~26! will be numericallystable if the hypergeometric function is evaluated in a suit-able fashion.

The other smallb2g case, for whichG2p,m,n is singu-lar, is characterized by the conditionp.m1n11. For thiscase, also discussed in Appendix C, we find

G2p,m,n~a,b,g!

52 (m50

p2m2n22m! ~m1n!! tm

~m1m1n11!!m!

3~a1b!mLp2m2n2m21~a1b!

12m! ~p2m21!! ~a1b!p2m2n22 tp2m2n21

p! ~p2m2n21!!

33F2~1,1,p2m;p11,p2m2n;t!. ~27!

Here3F2 is a generalized hypergeometric function~for defi-nition, see Ref. 20, Sec. 9.14!. It should be evaluated by amethod that is numerically stable for smallt.



VI. INTEGRALS GÀp ,À1,0

Again, we give and discuss here the results, with detailsdeferred to Appendix D. As in previous sections, the formu-las describe limiting behavior at smalle; contributions whichvanish in the limite50 are omitted.

The integralG21,21,0 is nonsingular, but the integralsG2p,21,0 are singular for integersp.1. All these integralscan be evaluated in terms of the auxiliary functionLp intro-duced at Eqs.~8! and ~9!, and an additional auxiliary func-tion I p with definition

I p~x,y!5Ee

`

t2pe2xtE1~yt!dt. ~28!

HereE1 is an exponential integral~defined in Formula 5.1.1of Ref. 18!. We shall need its expansion,

E1~x!52gE2 ln x2 (n51

`~2x!n

n~n! !. ~29!

As shown in Appendix D,I p can be generated using therecurrence formula

I p~x,y!5e2xeE1~ye!

~p21!ep21 2x

p21I p21~x,y!2

1

p21Lp~x1y!

~30!

starting with given values ofI 1 , obtainable using Eqs.~32!and~33! ~vide infra!. When applying Eq.~30!, we expand itsfirst term using Eq.~29! and the Maclaurin series forexp(2xe), keeping only contributions that do not vanish inthe limit e50.

The formula forG2p,21,0, derived in Appendix D, is

G2p,21,0~a,b,g!

51

glnS b1g

b2g DLp~a1g!

21

gI p~a1g,b2g!1

1

gI p~a2g,b1g!. ~31!

We now see that for the recursive generation ofI p wewill need the following specific values ofI 1 , also derived inAppendix D:

I 1~a1g,b2g!51

2~gE

21 ln2 e!1gE ln e2p2

12

21

2ln2~a1g!1gE ln~b2g!

1 ln~b2g!ln e1 ln~a1g!ln~b2g!

2dilogS a1b

a1g D , ~32!

I 1~a2g,b1g!51

2~gE


12

1gE ln~b1g!1 ln~b1g!ln e

11

2ln2~b1g!1dilogS a1b

b1g D . ~33!

We see thatI 1 involves a relatively unfamiliar function,the dilogarithm, here denoted ‘‘dilog.’’ We use the definitiongiven in Ref. 18; it is the function denotedf (x) in Formula27.7.1. Properties of the dilogarithm needed here are summa-rized in Appendix E.

Using Eq.~31! and the formulas for the auxiliary func-tions, we obtain for the first fewp the explicit forms

G21,21,0~a,b,g!51

g F1

2ln2S a1g

b1g D1dilogS a1b

a1g D1dilogS a1b

b1g D1p2

6 G , ~34!

G22,21,0~a,b,g!

5@ ln~@b1g#e!2c~2!#2112S a2g

g DdilogS a1b

b1g D2

p2a

6g2S a1g

g D F1

2ln2S a1g

b1g D1dilogS a1b

a1g D G ,~35!

G23,21,0~a,b,g!522

e@ ln~@b1g#e!2c~1!#2a@ ln~@b1g#e!2c~2!#22b@ ln~@b1g#e!2c~3!21#1

~a21g2!p2

12g

1~a2g!2

2gdilogS a1b

b1g D1~a1g!2

2g F1

2ln2S a1g

b1g D1dilogS a1b

a1g D G2~a1b!lnS a1b

b1g D , ~36!

G24,21,0~a,b,g!522 ln~@b1g#e!22c~1!21

2e2 1b

e1

2a

e@ ln~@b1g#e!2c~1!#

1S 3a21g2

6 D H @ ln~@b1g#e!2c~4!#215

4J1S 5a216ab1b2

6 D F ln~@a1b#e!2c~2!25

3G11

18S b21g2

3 D2S a2

31g2D ap2

12g2

~a1g!3

6g F1

2ln2S a1g

b1g D1dilogS a1b

a1g D G2~a2g!3

6gdilogS a1b

b1g D . ~37!



It should be noted that although Eq.~31! contains ln(b2g),this quantity cancels in the final expressions forG2p,21,0 as aconsequence of routine simplification. As a result, the formu-las in Eqs.~34!–~37! do not exhibit instabilities at or nearb5g. We also note that thee dependence of the individualterms comprisingG21,21,0 cancel, reflecting the fact that thisintegral is nonsingular. The integralsG2p,21,0 for p.1 aresingular and are seen to have a nete dependence.

A limiting case of Eq.~34!, G21,21,0(a,b,0), can beshown to be equivalent to the integralI 0(22,22,21;a,b)reported by Drake as Formula 1 of his Table 11.1~Ref. 14!.The demonstration is complicated by the fact thatG21,21,0

becomes an indeterminate form atg50 and that the diloga-rithm is multiple valued. Details are provided in Appendix F.Korobov17 reports expressions forG21,21,0 ~which, thoughregular, he labels as singular!, G22,21,0, and G23,21,0, butall in error by a factor of 2 and, for the latter two integrals,with the omission of some of the regular and all of the sin-gular terms.

VII. INTEGRALS GÀp ,À1,n FOR POSITIVE n

For the first fewn, it is practical to obtain the integralsG2p,21,n using a symbolic algebra program to differentiateG2p,21,0. In this way one may obtain, for example,

G21,21,1~a,b,g!51

g2 F1

2ln2S a1g

b1g D1dilogS a1b

a1g D1dilogS a1b

b1g D1p2

6 G12

g F b

b22g2 lnS a1g

a1b D1a

a22g2 lnS b1g

a1b D G ,~38!

G22,21,1~a,b,g!52@c~2!2 ln e2 ln~b1g!#

b1g1

2b

g~b1g!lnS a1b

b1g D2

a

g2 F1

2ln2S a1g

b1g D1dilogS a1b

a1g D1dilogS a1b

b1g D1p2

6 G12b~a1g!

g~b22g2!lnS a1b

a1g D , ~39!

G23,21,1~a,b,g!52

~b1g!e1

2a@ ln e1 ln~a1b!2c~3!#

b1g2

a

glnS a1b

b1g D1S a22g2

2g2 D F1

2ln2S a1g

b1g D1dilogS a1b

a1g D1dilogS a1b

b1g D1p2

6 G2b~a1g!2

g~b22g2!lnS a1b

a1g D , ~40!

G24,21,1~a,b,g!51

e2~b1g!2

2a

e~b1g!2

g

3@ ln e1 ln~b1g!2c~4!#22

3a21g2

3~b1g!@ ln e1 ln~b1g!2c~4!#

249g

1081

a~3g22a2!p2

3bg2 2~a1g!2~a22g!

6g2 F1

2ln2S a1g

b1g D1dilogS a1b

a1g D G2

~a2g!2~a12g!

6g2 dilogS a1b

b1g D1g2b1a2b22a2g

3g~b1g!lnS a1b

b1g D2b~a1g!3

3g~b22g2!lnS a1g

a1b D . ~41!

Equation~38! is numerically stable for all nonzero parametervalues, under the condition that the individual terms in itsfinal square brackets be evaluated appropriately in the re-spective regionsb'g anda'g. For the first of these terms,

b

b22g2 lnS a1g

a1b D5b

b22g2 lnS 12b2g

a1b D5

2b

~a1b!~b1g! (n51

`1

n S b2g

a1b D n21

.

~42!

The second can be treated analogously.In each of the Eqs.~39!–~41! only the final term is po-

tentially unstable asb→g. These terms can be treated by ananalysis similar to that presented in Eq.~42!.

VIII. CONCLUSIONS

Formulas have been presented for a broad set of three-body integrals involving exponentials in all the interparticledistances. Included are many integrals that, although singu-lar, arise~in combinations without net singularity! in relativ-istic and quantum electrodynamic corrections to the nonrel-ativistic energy levels of various three-body systems. Inorder to facilitate an understanding of the singularity cancel-lation, the singular portion of each such integral has beenexplicitly identified and included in the tabulation.

Many of the formulas have been derived or checked us-ing symbolic manipulation programs, and limiting cases~corresponding to the use of Hylleraas wave functions! havebeen checked against the work of others. Except for minormisprints, our expressions in the Hylleraas limit agree withthe definitive work by Drake.14 However, the expressionsgiven here differ significantly from those given for some ofthe same integrals by Korobov,17 and we have therefore pro-



vided, in appendices, the detail needed to validate our work.We believe the present study provides material prerequisiteto careful and comprehensive investigations of relativisticand quantum electrodynamic effects in three-body systemssuch as the He atom, the Ps2 ion, and the H2

1 ion.

ACKNOWLEDGMENTS

It is a pleasure to thank the Natural Sciences and Engi-neering Research Council of Canada for financial support.One of the authors~F.E.H.! also acknowledges support fromthe U.S. National Science Foundation, Grant No. PHY-0303412.

APPENDIX A: DERIVATION OF FORMULASFOR GÀp ,0,0

Starting from Eq.~2! with l set to2p andm andn set tozero, we break ther 2 integration into regions dependent onthe sign ofr 12r 2 ,

G2p,0,0~a,b,g!5E0

` dr1

r 1p E

0

r 1dr2E

r 12r 2

r 11r 2dr12

3e2ar 12br 22gr 12

1E0

` dr1

r 1p E

r 1

`

dr2Er 22r 1

r 11r 2dr12

3e2ar 12br 22gr 12. ~A1!

The r 12 and r 2 integrations are elementary, leading aftersome simplification to

G2p,0,0~a,b,g!51

g S 1

b2g2

1

b1g D E0

` dr1

r 1p

3@e2(a1g)r 12e2(a1b)r 1#. ~A2!

Now, as indicated in the main text, we change the lower limitof the r 1 integration to a small positive quantitye and con-sider the limiting behavior ase→01, retaining only termsthat do not vanish in this limit. Nonsingular integrals willthen exhibit noe dependence, while those that are singularwill have a residual dependence one. Identifying each termin the integral appearing in Eq.~A2! as an instance of theauxiliary functionLp defined in Eq.~8!, we confirm Eq.~11!of the main text.

For positive integersp, function Lp is a scaled versionof the generalized exponential integralEp ,

Lp~x!51

ep21 Ep~xe!; ~A3!

the formula for its evaluation, Eq.~9!, is an instance of For-mula 5.1.12 of Ref. 18. For integersp.0, Lp is singular ate50 whether or not the relatedG2p,0,0 is singular. For inte-gers p<0, Lp is a convergent integral proportional to(2p)!, with the value given in Eq.~10!.

APPENDIX B: EVALUATION OF GÀp ,0,0„a,b,g…

FOR SMALL bÀg

The formulas in Eqs.~12!–~16! become ill-conditionedwhenb2g is small. Starting from the general formula~forp>1),

G2p,0,0~a,b,g!52

b22g2 @Lp~a1g!2Lp~a1b!#,

~B1!

a more stable formulation in this regime may be obtained byexpandingLp(a1g) about a1b. Noting that the deriva-tives of Lp satisfy

S d

dxDj

Lp~x!5~21! jLp2 j~x! ~B2!

we have

Lp~a1g!5(j 50

`

~21! jLp2 j~a1b!~g2b! j

j !. ~B3!

Inserting this expansion into Eq.~B1!, canceling its first termagainstLp(a1b), and separating the remaining terms ac-cording to the sign of the index ofL, we have

G2p,0,052

b22g2 F (j 51

p21

~21! jLp2 j~a1b!~g2b! j

j !

1(j 5p

`

~21! jLp2 j~a1b!~g2b! j

j ! G . ~B4!

Now, using Eq.~10!, we insert in the second summationLp2 j (a1b)5( j 2p)!/(a1b) j 2p11, and rewrite in termsof a dimensionless variablet5(b2g)/(a1b). We alsochange the summation index of the second sum ton5 j2p. We then get after some rearrangement

G2p,0,052~a1b!p22

~b1g!t F (j 51

p21t j

j !@~a1b! j 2p11

3Lp2 j~a1b!#1tp(n50

`n! tn

~n1p!! G . ~B5!

The second summation can now be recognized as a hyper-geometric function, and Eq.~B5! thereby becomes

G2p,0,052~a1b!p22

b1g F (j 51

p21t j 21

j !@~a1b! j 2p11

3Lp2 j~a1b!#1tp21

p! 2F1~1,1;p11;t!G ,

~B6!

written also as Eq.~17!. The formulation given in Eq.~B6! isexact and exhibits explicit stability att50. However, thehypergeometric series will be the preferred computationalform only whent is small.

For checking the present formulas against those given byDrake,14 the following hypergeometric function formulas areuseful:



2F1~1,1;2;x!52ln~12x!

x, ~B7!

2F1~1,1;3;x!52~12x!ln~12x!

x2 12

x. ~B8!

APPENDIX C: DERIVATION OF FORMULASFOR GÀp ,m ,n

The recurrence relation of Sacket al.15 applies to any setof functions whose members can be expressed in the generalform

Fmn~b,g!5S 2]

]b D mS 2]

]g D n f ~b,g!

b1g, ~C1!

with the recurrence relation assuming the form

Fmn51

b1g FmFm21,n1nFm,n21

1S 2]

]b D mS 2]

]g D n

f ~b,g!G . ~C2!

When applied in the present context, we have

G2p,m,n~a,b,g!5S 2]

]b D mS 2]

]g D nS 1

b1g D 2

b2g

3@Lp~a1g!2Lp~a1b!#. ~C3!

Comparing with Eq.~24! of the main text, we see thatG2p,m,n has the formal definition

G2p,m,n~a,b,g!5S 2]

]b D mS 2]

]g D n 2

b2g

3@Lp~a1g!2Lp~a1b!#. ~C4!

Repeated differentiation ofLp(a1g)/(b2g), calling uponEq. ~B2!, yields the first summation of the formula forG2p,m,n given in Eq. ~25!. The second summation of thatformula comes from the differentiation ofLp(a1b)/(b2g).

If b2g is small, an alternative approach to the compu-tation of G2p,m,n is to start from the expansion ofLp(a1g) given in Eq. ~B3!. Inserting that expansion into Eq.~C4!,

G2p,m,n~a,b,g!52S 2]

]b D mS 2]

]g D n

3(j 51

`~b2g! j 21

j !Lp2 j~a1b!. ~C5!

Taking theg derivative first, we have


]b D m

(j 5n11

`~b2g! j 2n21

j ~ j 2n21!!

3Lp2 j~a1b!, ~C6!

which we rewrite so the summation starts from zero,


]b D m

(j 50

`~b2g! j

~ j 1n11! j !

3Lp2 j 2n21~a1b!. ~C7!

Now we take theb derivative,

G2p,m,n~a,b,g!52(j 50

`

(k50

min( j ,m) S mk D ~21!k~b2g! j 2k

~ j 1n11!~ j 2k!!

3Lp2m2n1k2 j 21~a1b!. ~C8!

We next interchange the order of the summations, causingthe range ofk to be from 0 tom and that of j from k toinfinity, after which we change the summation index fromjto m5 j 2k, reaching

G2p,m,n~a,b,g!52(k50

m

(m50

` S mk D ~21!k

m1k1n11~b2g!m

3Lp2m2n2m21~a1b!. ~C9!

To continue we notice@cf. Formula 4.2.2~45! of Ref. 21# that

(k50

m S mk D ~21!k

m1k1n115

m! ~m1n!!

~m1m1n11!!, ~C10!

so

G2p,m,n~a,b,g!52m! (m50

`~m1n!!

~m1m1n11!!m!~b2g!m

3Lp2m2n2m21~a1b!. ~C11!

Further steps depend upon whether theLp2m2n2m21 oc-cur with positive index values. No such values occur ifp2m2n21<0, i.e., if p<m1n11. In that case,Lp2m2n2m215(m1m1n2p11)!/(a1b)m1m1n2p12,and, again using the definitiont5(b2g)/(a1b), Eq.~C11! becomes

G2p,m,n~a,b,g!52m!

~a1b!m1n2p12

3 (m50

`~n1m!! ~m1n2p1m11!! tm

~m1n1m11!!m!.

~C12!

Identifying the summation as a hypergeometric function, weobtain Eq.~26!.

The above case, in whichG2p,m,n can be seen to benonsingular, is the only one required ifp51, but for largerpone must also consider the casep.m1n11. It is now ad-visable to break them summation into two regions distin-guished by the sign of the index ofLp2m2n2m21 . Doing so,and inserting the values of theL functions with negativeindices, we have



G2p,m,n~a,b,g!52 (m50

p2m2n22m! ~m1n!!

~m1m1n11!!m!~b2g!mLp2m2n2m21~a1b!

12 (m5p2m2n21

`m! ~m1n!! ~n1m2p1m11!! ~b2g!m

~m1m1n11!!m! ~a1b!m1n2p1m12 . ~C13!

We now change the summation index of the second sum toj 5m1m1n2p11, with range from 0 to , after whichthat summation can be identified as a generalized hypergeo-metric function. The result is

G2p,m,n~a,b,g!

52 (m50

p2m2n22m! ~m1n!! tm

~m1m1n11!!m!

3~a1b!mLp2m2n2m21~a1b!

12m! ~p2m21!! ~a1b!p2m2n22tp2m2n21

p! ~p2m2n21!!

33F2~1,1,p2m;p11,p2m2n;t!. ~C14!

This expression, which contains both regular and singularterms, is the result presented at Eq.~27!.

APPENDIX D: DERIVATION OF FORMULASFOR GÀp ,À1,0

Starting from the case of Eq.~2! appropriate toG2p,21,0, we perform ther 12 integration, and then partitionthe resulting two-dimensional integral to treat distinctly theregionsr 1.r 2 andr 1,r 2 , reaching after some manipulation

G2p,21,0~a,b,g!

51

g E0

` dr1

r 1p e2(a1g)r 1E

0

` dr2

r 2@e2(b2g)r 22e2(b1g)r 2#

21

g E0

` dr1

r 1p e2(a1g)r 1E

r 1

` dr2

r 2e2(b2g)r 2

11

g E0

` dr1

r 1p e2(a2g)r 1E

r 1

` dr2

r 2e2(b1g)r 2. ~D1!

The firstr 2 integral is an instance of Formula 5.1.32 of Ref.18, and the other twor 2 integrals can be identified as casesof the exponential integralE1 , so Eq.~D1! reduces to

G2p,21,0~a,b,g!51

glnS b1g

b2g D E0

` dr1

r 1p e2(a1g)r 1

21

g E0

` dr1

r 1p e2(a1g)r 1E1~b2g!

11

g E0

` dr1

r 1p e2(a2g)r 1E1~b1g!.

~D2!

Now, introducing the auxiliary functionsLp andI p , defined,respectively, in Eqs.~8! and~28!, we can expressG2p,21,0 inthe compact symbolic form

G2p,21,0~a,b,g!

51

glnS b1g

b2g DLp~a1g!

21

gI p~a1g,b2g!1

1

gI p~a2g,b1g!. ~D3!

To complete the evaluation we now require a formula forI p . Applying an integration by parts to its definition, Eq.~28!, we obtain~valid for p.1) the recurrence relation givenas Eq. ~30! of the main text. As noted there, use of thisrecurrence formula to increasep requires a starting value ofI 1 , namely

I 1~x,y!5Ee

`

t21e2xtE1~yt!dt. ~D4!

Inserting the expansion forE1(yt), using Eq. ~29!, it isstraightforward to obtain

I 1~x,y!5~2gE2 ln y!E1~xe!2I log~x!

2Ee

`

(j 51

`~2y! j

j ~ j ! !t j 21e2xtdt, ~D5!

where

I log~x!5Ee

`

t21e2xt ln tdt. ~D6!

After exchanging the order of the summation and integrationin the last term of Eq.~D5!, its t integration~if extended tot50) can be evaluated to yield (j 21)!/xj , so, withinO(e),that term becomes~cf. Appendix E!

2(j 51

`1

j 2 S 2y

xD j

52dilogS x1y

x D . ~D7!

Returning to I log , we integrate by parts, noting thatln t/t5(d/dt)(ln2 t)/2, thereby reaching

I log~x!521

2ln2 e1

x

2 E0

`

e2xt ln2 tdt. ~D8!

Note that because the integral in the above equation is con-vergent we changed its lower limit to zero, affecting the re-sult only by an amountO(e). Evaluating the definite integral@Formula 4.335~1! of Ref. 20#, we find

I log~x!51

2 S p2

61gE

21 ln2 x2 ln2 e D1gE ln x. ~D9!



Inserting into Eq.~D5! the results derived in Eqs.~D7! and~D9!, and using Eq.~29! to write the small-e limit of E1(xe),we obtain

I 1~x,y!51

2 FgE22

p2

61 ln2 e2 ln2 xG1gE@ ln y1 ln e#

1 ln y@ ln x1 ln e#2dilogS x1y

x D . ~D10!

The specific instances ofI 1 which are needed are

I 1~a1g,b2g!51

2~gE


12

21

2ln2~a1g!1gEln~b2g!


2dilogS a1b

a1g D , ~D11!

I 1~a2g,b1g!51

2~gE


12

21

2ln2~a2g!1gE ln~b1g!


2dilogS a1b

a2g D . ~D12!

The first of these two equations is in a form convenient forfurther operations, but the second can be manipulated to amore useful form by application of Eq.~E9!, leading to

I 1~a2g,b1g!51

2~gE


12

1gE ln~b1g!1 ln~b1g!ln e

11

2ln2~b1g!1dilogS a1b

b1g D . ~D13!

Equations~D11! and~D13! are the forms written as Eqs.~32!and ~33! of the main text.

APPENDIX E: PROPERTIES OF THE DILOGARITHM

The definition of the dilogarithm used here is

dilog~x!5E1

x ln t

12tdt. ~E1!

Its properties are discussed in detail in a monograph byLewin,22 who denotes it Li2 and defines it differently thanEq. ~E1!,

Li 2~x!5dilog~12x!. ~E2!

Properties of dilog(x) needed in the current work in-clude its power series expansion aboutx51 ~convergent forux21u<1),

dilog~x!5 (n51

`~12x!n

n2 ~E3!

the formula for its derivative,

d dilog~x!

dx5

ln x

12x, ~E4!

and the functional equations

dilog~x!1dilog~12x!52 ln x ln~12x!1p2

6, ~E5!

dilog~x!1dilog~x21!521

2ln2~x!. ~E6!

Both the logarithm and the dilogarithm are multiple valued,with a branch point at the origin. The above relations areconsistent with their definition on a complex plane with abranch cut along the negative real axis, and with the use ofthe principal values of both functions.

An identity needed in this work starts from a case of Eq.~E6!,

dilogS a1b

a2g D52dilogS a2g

a1b D21

2ln2S a1b

a2g D . ~E7!

If the dilogarithm on the right-hand side of this equation isfirst converted into dilog(@b1g#/@a1b#) using Eq. ~E5!and then to dilog(@a1b#/@b1g#) via Eq. ~E6!, we reach

dilogS a1b

a2g D52dilogS a1b

b1g D21

2ln2S b1g

a1b D1 lnS a2g

a1b D lnS b1g

a1b D2

1

2ln2S a1b

a2g D2p2

6. ~E8!

The logarithms on the right-hand side of Eq.~E8! can nowbe combined, leading to the final result

dilogS a1b

a2g D52dilogS a1b

b1g D2p2

62

1

2ln2S b1g

a2g D ,

~E9!

A detailed discussion of the numerical evaluation of thedilogarithm can be found in a number of places, includingearlier work by one of the authors.23 We note that an evalu-ation formula recently given~without attribution! byKorobov17 is an incorrect version of Formula 27.1.1 ofRef. 18.

APPENDIX F: EVALUATION OF GÀ1,À1,0IN HYLLERAAS LIMIT

We consider here the evaluation ofG21,21,0, shown be-low, for the special caseg50, a case encountered when us-ing Hylleraas wave functions



G21,21,0~a,b,g!51

g F1

2ln2S a1g

b1g D1dilogS a1b

a1g D1dilogS a1b

b1g D1p2

6 G . ~F1!

BecauseG21,21,0 should be finite atg50, the bracketedquantity in Eq. ~F1! ~its ‘‘numerator’’! must then becomezero, making the right-hand side of that equation an indeter-minate form of type 0/0. Forg50, the numerator is

1

2ln2S a

b D1dilogS 11b

a D1dilogS 11a

b D1p2

6.

Using a relation given in Appendix E, we transform thedilogarithms as follows:

dilogS 11b

a D52dilogS 2b

a D2 lnS 2b

a D lnS 11b

a D1p2

6,

~F2!

dilogS 11a

b D52dilogS 2a

b D2 lnS 2a

b D lnS 11a

b D1p2

6.

~F3!

We now wish to combine dilog(2b/a) with dilog(2a/b), but to do so it is necessary that the second of thesedilog arguments be the inverse of the first, keeping in mindthe fact that the dilogarithm has a branch cut on the negativereal axis. We note thata and b are both positive in thecurrent application, so if2b/a is represented as (b/a) eip,then we must identify2a/b as (a/b) e2 ip. Then we mayuse a dilogarithm identity from Appendix E, which for thepresent purposes we write

dilogS 2b

a D1dilogS 2a

b D521

2ln2S 2

b

a D , ~F4!

obtaining for the numerator the following:

1

2ln2S a

b D11

2ln2S b

aeipD2 lnS b

aeipD lnS 11

b

a D2 lnS a

be2 ipD lnS 11

a

b D1p2

2.

Finally, we expand the logarithms, reaching

1

2@ ln a2 ln b#21

1

2@ ln b2 ln a1 ip#2

2@ ln b2 ln a1 ip#@ ln~a1b!2 ln a#

2@ ln a2 ln b2 ip#@ ln~a1b!2 ln b#1p2

2.

This expression simplifies to zero, confirming that in thelimit g50, G21,21,0 is indeed an indeterminate form of thetype 0/0.

It is now straightforward to apply l’Hospital’s Rule; todifferentiate the dilogarithms we require the relation~alsofrom Appendix E!

d dilog~x!

dx5

ln x

12x. ~F5!

The result is

G21,21,0~a,b,0!52

alnS a1b

b D12

blnS a1b

a D ~F6!

in agreement with Drake’sI 0(22,22,21;a,b).

1H. A. Bethe and E. E. Salpeter,Quantum Mechanics of One- and Two-Electron Atoms~Springer, Berlin, 1957!.

2J. Sapirstein, Rev. Mod. Phys.70, 55 ~1998!.3G. W. F. Drake, Can. J. Phys.80, 1195~2002!.4K. Pachucki and J. Sapirstein, J. Phys. B33, 5297~2000!.5L. M. Delves and T. Kalotas, Aust. J. Phys.21, 1 ~1968!.6A. J. Thakkar and V. H. Smith, Jr., Phys. Rev. A15, 1 ~1977!.7A. M. Frolov and V. H. Smith, Jr., Phys. Rev. A53, 3853~1996!.8F. E. Harris, Adv. Quantum Chem.47, 128 ~2004!.9F. E. Harris, inFundamental World of Quantum Chemistry, A Tribute tothe Memory of Per-Olov Lo¨wdin, Vol. 3, edited by E. J. Bra¨ndas and E. S.Kryachko ~Kluwer, Dordrecht, 2004!, pp. 115–128.

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ited by G. W. F. Drake,~AIP, Woodbury, NY, 1996!, pp. 154–171.15R. A. Sack, C. C. J. Roothaan, and W. Kolos, J. Math. Phys.8, 1093

~1967!.16J.-L. Calais and P.-O. Lo¨wdin, J. Mol. Spectrosc.8, 203 ~1962!.17V. I. Korobov, J. Phys. B35, 1959~2002!.18Handbook of Mathematical Functions, edited by M. Abramowitz and I. A.

Stegun~Dover, New York, 1972!.19A product of Waterloo Maple Inc., Waterloo, Ontario, Canada~see: http://

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Singular and nonsingular three-body integrals for exponential wave functions