Comment on ‘‘Analytic value of the atomic three-electron correlation integral with Slater

wave functions’’

James S. Sims

National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8911, USA

Stanley A. Hagstrom

Department of Chemistry, Indiana University, Bloomington, Indiana 47405, USA

~Received 27 March 2003; published 11 July 2003!

Reference is made to the paper by Remiddi @Phys. Rev. A 44, 5492 ~1991!#. In this Comment we point out

two misprints in the paper. We provide further verification of the corrections with a short table comparing 30

significant digit results using Remiddi’s formula ~with our corrections! and the output of our recently developed triangle integral package.

DOI: 10.1103/PhysRevA.68.016501 PACS number~s!: 31.15.2p

Reference is made to the paper by Remiddi @1#, where

misprints occur in two important equations.

In Eq. ~33! of Ref. @1#, the sum of the ln@ ̄#ln@ ̄# type

terms should be

lnF

w11w2

w3

G lnF

w11w21w3

w11w2

G 1lnF

w11w3

w2

G lnF

w11w21w3

w11w3

G

1lnF

w21w3

w1

G lnF

w11w21w3

w21w3

G , ~1!

while in Eq. ~42! of Ref. @1#, line 8 should be (w12w2

1w3) instead of (w12w21w3)

2

. With these two changes,

both equations are now invariant under interchange of the

nonlinear parameters w1 and w3

, as is required by the symmetry of the electron interaction operator.

Harris @2# has shown that the formulas of Remiddi are

equivalent to the older, more general formulas of Fromm and

Hill @3# in their common region of applicability @Remiddi’s

Eq. ~33! only#. However, he failed to point out the misprints.

We provide further verification of the correctness of the

modified Eq. ~42! of Ref. @1# for several values of the nonlinear parameters with a short table comparing 30 significant

digit results using Remiddi’s formula ~with the above corrections! and the output of our recently developed triangle integral package ~details to be published!.

We calculate, for various values of w1

, w2, and w3,

Z~w1

,w2,w3

!5 E r12r23r13

21

e

2w1r12w2r22w3r3dr1dr2dr3

.

~2!

Table I shows the results of the usual expansion technique

@4# ~18 terms! followed by a Levin u transform (kmax517)

@5# and compared with Remiddi’s formula results. ~The 64p

3

factor in Remiddi’s Eq. ~42! is not included in these numbers.! The Remiddi results are accurate to 30 digits and

therefore indicate an accuracy of 20 to 23 places for our

Levin extrapolated values ~details to be published!.

@1# E. Remiddi, Phys. Rev. A 44, 5492 ~1991!.

@2# F.E. Harris, Phys. Rev. A 55, 1820 ~1997!.

@3# D.M. Fromm and R.N. Hill, Phys. Rev. A 36, 1013 ~1987!.

@4# J.S. Sims and S.A. Hagstrom, J. Chem. Phys. 55, 4699

~1971!.

@5# D. Levin, Int. J. Comput. Math. 3, 371 ~1973!.

TABLE I. Z(w1,w2,w3) results for various w1, w2, and w3.

w1 w2 w3 Direct1Levin u-transform Remiddi

1.50 1.00 1.00 8.7835058015481861493257 8.78350580154818614932496534961

1.50 1.00 1.50 2.53956460526041533302745 2.53956460526041533302749242495

1.50 1.00 2.00 1.05391036650781776777 1.05391036650781776783539343043

1.50 1.50 1.00 2.53956460526041533302745 2.53956460526041533302749242495

1.50 2.00 1.50 3.511260150977273542310 3.51126015097727354230916222419

2.00 1.00 1.00 2.98667298227213234381 2.98667298227213234375647088278

2.00 2.00 1.00 3.24350301286034626758316 3.24350301286034626758309854140

PHYSICAL REVIEW A 68, 016501 ~2003!

1050-2947/2003/68~1!/016501~1!/$20.00 68 016501-1 ©2003 The American Physical Society