Exact computation of the 3-f and 6-f symbols - iucaa

350 Computer Physics Communications 61 (1990) 350—360
North-Holland
Exact computation of the 3-f and 6-f symbols
Shan-Tao Lai ~ and Ying-Nan Chiu 2
Center for Molecular Dynamics and Energy Transfer, Department of Chemistry, The Catholic University ofAmerica,
Washington, DC 20064, USA
Received 21 November 1989
A simple FORTRAN program for the exact computation of 3-f and 6-f symbols has been written for the VAX with VMS
version v5.1 in our university’s computing center. It goes beyond and contains all of the 3-f and 6-f symbols evaluated in
the book by M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten Jr. The 3-f symbols up to
“‘2 ~] and 6-f symbols up to ( ~ ~ } can be computed exactly by this program. Approximate values for larger f’s up to
(~~ ~) and (~~3~3~ } can also be computed by this program.
1. Introduction
Angular-momentum couplings play an important role in quantum mechanics. For quantum-mechanical
problems involving two or three particles, many Wigner 3-f and 6-f values are needed. A table of these
values was compiled by M. Rotenberg et a!. in 1959 [1]. But, their table was only convenient for some
manual theoretical calculations where people can look up the values. We have therefore rebuilt “the table”
in order to contain all the results in a computer program which can be used as a subroutine in machine
calculations. Our program can be incorporated into various computer calculations that make use of these
symbols. Our general results go beyond and cover the values of Rotenberg et a!. [1]. And our results are
exact up to j = 30 for 3-f and j = 20 for 6-f symbols in contrast to other approximate computation for
large angular momenta [2,3]. For larger j ‘s our program can also compute their approximate values which
agree with the 1976 values of Schulten and Gordon [21.
2. Program logic
We rewrite the algebraic expression [1] for the Wigner 3-f symbol as
11 12 13
m 1 m2 m3
— 1 ~~I2~’~3 (ft + m1)!(fi — m1)!(j2 + m2)!(j2 — m2)!(f3 + m3)!(j3 — 1/2
— ~ U~+12 f3)!(J1 ~12 +f3)!(31 ~12 +13)!(fl ~12+13 + 1)!
x ~ ( —1) L fi + f2—f3 1112 + 13 ~11 +12±13 (1)
L L j1m1L j2+m2—L
Permanent address: Department of Chemistry, Xiamen University, Xiamen, Fujian, 361005 P.R. China.
2 To whom reprint requests should be addressed.
0010-4655/90/$03.50 © 1990 — Elsevier Science Publishers B.V. (North-Holland)

S. -T. La,, Y -N. Chiu / Exact computation of the 3-f and 6-f symbols 351
and for the 6-f symbol as
{ } Ji J2 ~ = (_1)h+J2~4±J5([(f, —12 +f 3)!( f4 +J~ J3)!(J4 —15 +f3)!(—f4 ~f2 +j3)!
14 15 16
x (f ~f2 —f6)!(14 ~12 +16+ 1)!( —14 ~J2 +f6)!(jl +f~—j6)!(j1 —15 +16)!
x (—f’ ~15 +f~)!J/ [(f1+f~—f~)!(f1~f2 +13 + 1)!(_f~+12+13)!
x (14 ~15~J6 + 1)!(f4 12 +f6)!(fl +15+16 + 1)!] }1/2
L f~+12 —13 14 12 +16 —11 +J~+13 [ L + 1
x~(—1) L—f4—f5—f3 L—f1—f2—f3 L—f1—f5—f6 ~L—f4—j2—f6
where (~)‘s are the binomial coefficients. In eqs. (1) and (2), the summation is over all positive integral
values of L such that none of the factorials is negative. The main point is, how can we conveniently deal
with the large factorials exactly. We designed subroutines DECOM(L1,QK,R9) and PRIME(DIA,QK,R9)
which decompose each integer into a product of prime numbers with certain powers. In eqs. (1) and (2),
the value inside the square root is easy to compute, even if the jr’s (x = 1, 2, 3,.. . ,6) inside the square root
are very large. But there is a limitation on the f values that can be manipulated in the summation part.
Since each binomial coefficient is always an integer, the limitation of our computer requires that the
product of the three or four binomial coefficients cannot be beyond iO~~when we use Real * 1 6(H — floating)
on the Digital VAX computer. The value of H~floatingdata is in the approximate range 0.84 x 10 4932
to 0.59 x 104932. The precision of H_floating data is approximately one part in 2112, i.e. typically 33
decimal digits. According to our computation the 3-f symbols up to ~ ~ m3) and 6-f symbols up to
~ } can be handled on the VAX/VMS computer. 3g ~ That ~), ( means ~ _ ~) theand values ( ~ of~ 3-f }, and ( ~ 6-f ~} symbols and some are
exact approximate for thevalues abovefor values. larger We f’sgive in appendices examples of A ( and B.
(2)
3. Description of the program
The program contains five subroutines and one function which are named A3JSY (for computation of
3-f symbol), A6JSY (for computation of 6-f symbol), PRIM1(Q1) (for computation of selected prime
numbers below 602), DECOM(L1,QK,R9), PRIME(DIA,QK,R9) and FUNCTION DIFAC(N) (for
computation of factorials). For 3-f symbol calculation with integer f, we input angular momentum
numbers f~12’ 13’ and magnetic quantum numbers m
1, m2, m3. We give examples that are available in
Rotenberg’s book and can be checked against the latter. For example, for (1~1~~ we input 16.0, 16.0, 6.0,
0.0, 0.0, 0.0 from the WIG.DAT file, for half-integer 3-f symbol, for example, (~—9/2 —1/2)’ we input 8.0,
7.5, 6.5, 5.0, —4.5, —0.5. The procedures for computations of 6-f symbols are parallel to that for 3-f
symbols. The input WIG.DAT file format can be found in appendix C.
4. Output
The output shows the 3N-f symbols (N = 1, THRJ; N = 2, SIXJ) equal to a sign (+ 1.0 and —1.0)
multiplied by the prime numbers of given powers. The program output is as follows:

352 S. -T. Lai, Y. -N. Chiu / Exact computation of the 3-f and 6-f symbols
N~?FOR 35—SYMBOL N1, FOR 6]—STMBOr. N~2 N~?FOR 35—SYMBOL N~1, FOR 65—SYMBOL N~2
N~2
.71—PJ2~?33—?Mj~?M2—3M3=?
J1”?J2~?J3..?34..?JS,.?J6~?
16.0 16.0 6.0 0.0 8.0 0.0
0.0 0.0 0.0 8.0 8.0 0.0
EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 6—3 SYMBOL
THR3~—1.0
SIX3=—1.0
* 2** 2.0 • 2’’ —1.0
• 3” —1.0 • 3** 1.0
* 5** J.0 * 5•’ —1.0
11’’ —0.5 * 53** ~
* 13” —0.5 * 17’~ —1.0
* 17’’ 0.5 * 19’’ —1.0
• 18** 0.5 • 23’’ —1.0
* 29” —0.5 MULTIPLY
* 31” —0.5 4073.00000000000000000000300000000
• 37’’ —0.5
MULTI PLY
1 .00000000000000000000000000000000
where ““ means multiplication and “s” means exponential. They correspond to the following
Rotenberg’s notation
16 16 6) = * 4220, 1111,0111
and
{
= * 24.3_2.52.70.11_1.13_1.171.191.230.29_1.31_1 .37~
8 8 8)2 *2220,om,2(4073)
2 (3)
Their “*“ means negative sign after the square root and underline means negative exponents. Each
number represents the power of the prime numbers in increasing sequence, respectively, as we illustrate in
eq. (3). In their notation the exceptionally large prime numbers beyond 31 are given explicitly with their
powers, e.g. (4073)2.
In our program, we store the prime numbers 2 to 601 into Q1(110) (from Q1(1) to Q1(110)) and we use
PP(110) to store the corresponding powers of each prime number (from PP(1) to PP(110)). For prime
numbers beyond 601 we use QK to stand for this number and the first power only. This QK represents the
left-over in the multiplication after the powers of prime numbers have been factored out. Therefore, our
values of 3N-f symbols are very convenient for other computational purposes when our program is
incorporated as a subroutine.
5. Conclusions
We are very fortunate to have in Rotenberg’s book more than 44500 values of the 3-f and 6-f symbols
which can be checked against our program. We pick up some small f ‘s and some of their biggest f ‘s and
some 3-f or 6-f symbols which look very unusual. After computing them with our program, our values
agree with theirs perfectly. Therefore we have confidence in our 3-f and 6-f symbols that are beyond those
in their book (e.g. general rn’s for f> 8 in 3-f symbols, and f> 8 for 6-f symbols). Also, the symmetry
properties and orthogonality properties of 3-f and 6-f symbols [1,3] can be used to check the results of our
computations. We give some exact 3-f and 6-f values that are not available in Rotenberg’s book (see
appendix A). They are exact values and not approximate values like those in refs. [2,3]. We have also

S. -T La~Y.-N. Chiu / Exact computation of the 3-f and 6-f symbols 353
compute some approximate values for larger f‘s (also see appendix A). And these also agree with those in
ref. [2].
Finally, we add appendix D where we list the complete FORTRAN source program for both exact and
approximate computation of 3-f and 6-f symbols. When we input angular momentum f‘s that are beyond
the limitation of exact computation of 3-f and 6-f symbols, the program will automatically execute
approximate computation.
Appendix A. Test run output
In the following, we give sample results of our computation of 3-f symbols for f 1 = 20, 12 = 20, f~= 20
and some special rn values which are not available in the literature.
N?FOR 3]—SYMBOL N—i, FOR 65—SYMBOL N—2 N—7FOR 35—SYMBOL N—i, FOR 65—SYMBOL N—S N—?FOR 35—SYMBOL N—i, FOR 65—SYMBOL N—2
N— 1 N— 1 N 1
3i—?32.?J3—?Mi~?M2—PM3—? J1—732—?33—Yfli—7M3?M3—7 3i?32? 33?M1OM2?M3?
20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 28.0
0.0 0.0 0.0 0.0 1.0 —1.0 0.0 2.0 —2.0
EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL
THRJ 1.0 THRJ—1.S THRJ~—i.0
• 2” 1.0 * 7** —0.5 * 7~ —5.5
• 7’’ —0.5 * 11” 1.0 • 13’ 1.5
* ii’’ 1.0 • 13” 1.5 • 11’’ 1.0
• 13’’ 1.5 * 17” 1.0 • 31’’ —0.5
* 17” 1.0 * 19” 1.0 • 37” —0.5
* 19’’ 1.0 * 31” —0.5 • 4i’ —0.5
* 3i” —0.5 • 37” —0.5 * 43” —0.5
• 37” —0.5 * 41” —0.5 • 47” —0.5
* 41” —0.5 * 43’ —0.5 * 53” —0.5
* 43” —0.5 • 47’’ —0.5 • 59” —0.5
• 47” —0.5 * 53” —0.5 * 61** —0.5
* 53” —0.5 * 59” —0.5 * 2ii” 1.0
* 59” —0.5 * 61’’ —0.5 MULTIPLY
• 61” —0.5 MULTIPLY i.00000000000000500000000000008500
MULTIPLY 1.00000000000000000060000000000000
1. 00000 000 00000000 0000 0000 000 00 000
N—?FOR 3)—SYMBOL N—i. FOR 65—SYMBOL N—2 N—?FOR 35—SYMBOL N—i, FOR 65—SYMBOL N2 Ol—OFOR 35—SYMBOL N1, FOR 65—SYMBOL N—S
N—i N—i Ri
31—?32—?33—?Mi?M2—?M3—? Ji—?32—?33—?M1?X2?M3—? 3i?32?33?M1?M2—?M3?
20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0
0.0 3.0 —3.0 0.0 4.0 —4.0 0.0 5.0 —5.0
EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL
TIORJ 1.0 YHR3~—i.S YHR3—1.0
* 7’ —0.5 * 2** 2.0 • 2’’ —1.0
* 13’’ 1.5 * 7’ —0.5 * 7” —0.0
* 17** 1.0 * 13’’ 1.5 * 13’’ 1.5
* 31’’ —0.5 * 31” —0.5 * 23’ 1.0
* 37’ —0.5 * 37” —0.5 * 31** —0.5
* 41** —0.5 * 41’’ —0.5 • 37’ —0.5
* 43” —0.5 * 43” —0.5 * 41” —0.5
* 47” —0.5 * 47” —0.5 * 43’’ —0.5
* 53** ~0.S • 53” —0.5 * 47” —0.5
• 59” —0.5 * 59” —0.5 * 53” —0.5
* 61” —0.5 • 61” —5.5 * 59~ —0.5
* 421” 1.0 MULTIPLY * 61” —0.5
MULTIPLY 053.000508000000000000000000000000 • 347” 1.0
1.00000000000000000000000000000000 MULTIPLY
1.00000000000005000000000000000000

354 5. -T Lai, 1’. -N. Chiu / Exact computation of the 3-f and 6-f symbols
N—UFOR 35—SYMBOL N1, FOR 65—SYMBOL N—S N7FOR 35—SYMBOL Ni, FOR 65—SYMBOL N—) R—?FOR 35—SYMBOL Ni, FOR 65—SYMBOL N2
N1 N—i N—i
31?32—?J 3—?Mi..7M2—7M3—? 3i—?32—?33—?Mi—?M2—?M3—? 31—332—333=?Mi—?M2—?M3~P
20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.3
0.0 6.0 —6.0 0.0 7.0 —7.0 0.0 B.0 —6.0
EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL
TNRJ iS TUR3—i.0 T14R3=—i.0
• 2” —i.0 * 2” —1.0 * 7” ~5~5
• 7” —0.5 • 7” —0.5 * 03’’ —0.5
* 13” 0.5 • i3’’ 0.5 * 3i’’ —0.5
* 31” —0.5 • 3i” —o.s * 37” —0.5
• 37” —0.5 * 37’’ —0.5 * 41’’ —0.5
• 4i’ —0.5 * 4i” —0.5 * 43” —0.5
* 43** —0.5 * 43” —0.5 * 47** —S.5
* 47~ —0.5 * 47’’ —0.5 * 53” —0.5
* 53’’ —0.5 * 53*’ —5.5 * 59** ~5~5
* 59** —0.5 * 59” —0.5 • 61” —0.5
• 6i” —0.5 * 61’’ —0.5 MULTIPLY
• 307’’ i.0 MULTIPLY - 05i379.000000000000000050050055005
MULTIPLY 67339.0000000000000000009000000000
617 .000000000000000000000000000000
N—PFOR 35—SYMBOL N1, FOR 65—SYMBOL N—S N?FOR 35—SYMBOL N—i, FOR 65—SYMBOL N—2 N7FOR 35—SYMBOL N—i, FOR 6)—SYMBOL N—S
Ni N—I N.j
3i.?32—?33.?Mi*?M2—?M3.? 3i—?32?33?M1?M2?M3? 31~?32—?33—?Mi—?M2—PM3.?
20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0
0.0 9.0 —9.0 0.0 10.0 —iS.0 0.0 u.S —ii.o
EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OP 3—3 SYMBOL
THR3 1.0 TNRJ—i.0 THR3.—i.0
* 2’’ —1.0 • 2” —1.0 * 2’’ —i.0
* 7” —0.5 • 7*’ —0.5 * 7” —0.5
* i3” —0.5 * i3” —0.5 * ii’’ 1.0
* 31” —0.5 * 3i” —0.5 * iS’’ —0.5
* 37” —0.5 * 37’’ —0.5 * 3i•• 0.5
* 4i” —0.5 * 4i” —0.5 * 37~ —0.5
* 43” —0.5 * 43~ —0.5 * 4i** —0.5
* 47•’ —0.5 • 47*’ -0.5 * 43” —0.5
* 53” —0.5 * 53” —0.5 * 47” —0.5
* 59** —0.5 • 59•• —0.5 * 53’’ —0.5
* 62’’ —0.5 * 6i** —0.5 • 59” —0.5
• 1i3” 1.0 MULTIPLY * 61’’ —0.5
MULTIPLY 5301.00000000000000005000000050005 MULTIPLY
2i34i.0000000000000000000060000000 75i7.00000000000000000000000000000
N—PFOB 35—SYMBOL N—i, FOR 65—SYMBOL N—S N?FOR 35—SYMBOL N—i. FOR 65—SYMBOL N2 N.?FOR 35—SYMBOL N.1, FOR 65—SYMBOL N—S
N—i N—i N—i
3i—?3S?33?Mi*7MS?M3? Ji—?3S—733—?Mi—?MS?M3? 3i—732—?33—?M1—?MS—?M3—?
20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0
0.0 12.0 —i2.0 0.0 13.0 —13.0 0.0 i4.0 —14.0
EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL
THRJ 1.0 TNR3 1.0 THRJ~i.O
* 2’’ 1.0 • 7” —0.5 * 7” —0.5
* 7” —0.5 * i3” i.5 • i3” —0.5
• 13” —0.5 * 31” 0.5 * 31” 0.5
* 3i’’ 0.5 * 37” —0.5 * 37~ —0.5
• 37” —0.5 * 4i* —0.5 * 41’’ —0.5
• 41” —0.5 * 43” —0.5 • 43 —0.5
* 43” —0.5 • 47” —0.5 * 47~ —0.5
* 47” —0.5 • 53’’ —0.5 • 53” —0.5
• 53’’ —0.5 • 59” —0.5 * 59’’ —0.5
• 59’• —0.5 • 61** ~0.S • 61” —0.5
* 6i” _0.S • 191” i.0 MULTIPLY
MULTIPLY MULTIPLY 39569.0000000000000000000000000000
12823 .0000000000000000000000000000 i .0000o000000000000000000000000000

S. -T Lai, Y. -N. C’hiu / Exact computation of the 3-f and 6-f symbols 355
N?FOR 3)—SYMBOL N—i, FOR 65—SYMBOL N—S N?FOB 35—SYMBOL Ni, FOB 65—SYMNOL N—S N—UPON 35—SYMNOL Ni, FOR 65—SYMBOL NS
N—i N—i N—i
3i—?32—233—?Mi—?MS—?M3—7 3i=232—?33—?Mi—?M2—?M3—? 3i?32?33=?Mi—?MS—?N3—?
20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0
0.0 i5.0 —15.0 0.0 16.0 —16.0 0.0 17.0 —i7.0
EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OP 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL
-
TNR3.—i.0 YNRJ 1.0 THR3 iS
• 7’’ —0.5 * 2’’ 1.0 • 7” —0.5
* 13’’ —0.5 * 7’’ —0.5 i3” —0.5
* 31’’ 0.5 • i3’’ —0.5 • 17’’ 1.0
• 37” —0.5 * 31’’ 0.5 * 31’’ 0.5
* 4i’ —0.5 * 37” —0.5 • 37” 0.5
* 43•* —0.5 * 41’’ —0.5 • 41” —0.5
• 47’’ —0.5 * 43’’ —0.5 * 43” —0.5
* 53” —0.5 * 47’’ —0.5 * 47~ —0.5
* 59” —0.5 • 53” —0.5 • 53’’ —0.5
• 61” —0.5 * 50 0.5 * 59’’ —0.5
MULTIPLY * 61” —0.5 * 61” —0.5
3i307.0500000000000000000050000050 • 347” 1.0 * 101’’ 1.0
MULTIPLY
MULTIPLY
1.00500500000000000000000000000060 i.00000000000000000000000000000005
N—UPON 35—STMNOL N1, FOR 65—SYMBOL N—S N?FOR 35—SYMBOL N—i, FOR 65—SYMBOL N2 N?FON 35—SYMBOL NI, FOR 65—SYMBOL N—2
N1 N—i N2
3iP32—?33?M1—UMS.?M30 3i.?32.?33—?M1.UMS—2M3—? 3i—?35P33?Mi—7M2—7M3?
20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0
0.0 iS.0 —iO.0 0.0 i9.0 —19.0 0.0 20.0 —20.0
EXACT VALUE OP 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL EXACT VALUE OF 3—3 SYMBOL
YERJ— 1.0 THRJ 1.0 TUB). 1.0
* 7” —0.5 • 7’’ —0.5 • 2’’ 1.0
* 13’’ —0.5 * 13’’ 0.5 * 7’’ —0.5
* 23** 1.0 * 19’’ i.0 • 13’’ 0.5
• 3i* 0.5 • 3i” 0.5 * 31’’ 0.5
* 37’’ 0.5 * 37** 0.5 • 37” 0.5
* 41’’ 0.5 * 41” —0.5 * 41’’ —0.5
• 43” —0.5 • 43*• —0.5 * 43~ —0.5
• 47** —0.5 • 47’’ —0.5 • 47” —0.5
* 53*’ —0.5 * 53” —0.5 • 53’’ —0.5
• 59” —0.5 * 59’’ —0.5 * 59’’ —0.5
* 61’’ —0.5 * 61’’ —0.5 * 6i** —0.5
MULTIPLY MULTIPLY MULTIPLY
i.00000000000050000000000000000000 1 .00000000000000005000000000000000 1.00000000000000000000000000000000
N—UPON 35—SYMBOL N—i. FOR 65—SYMBOL N—2 N?POR 35—SYMBOL Ni, FOR 6)—SYIINOL N—S
NS
N—S
3i.332—?33.?34P35—?36..?
3i=?3S—?33—U34~?35?36—7
20.0 20.0 20.0 20.0 20.0 20.0
20.0 20.0 20.0 10.0 10.0 10.0
EXACT VALUE OF 6—i SYMBOL EXACT VALUE OF 6—3 SYMBOL
SIXJ—1.0 SIX)— 1.0
• 7** —2.0 * 2’’ —2.0
* 3’’ 2.0 * 3’’ —1.0
• 7’’ —1.0 * 5’’ —0.5
* 13’’ —1.0 * 11*’ ~0.S
* 31’’ —1.0 * 13’’ —1.0
* 37~ —i.0 • 23” —0.5
* 41’’ —1.0 * 29’’ —0.5
• 43’’ —1.0 * 31’’ —1.0
• 47” —1.0 • 37’’ —i.0
* 53” —1.0 • 41” —1.0
* 59” —1.0 * 43” 0.5
* 61” —1.0 • 47” 0.5
MULTIPLY * 53” 0.5
3607626304291.00000000000000000000 • 59” 0.5
* 61’’ 0.5
MULTIPLY
1 .00000000000005000000000000000000
The above are exact values not available in the literature. Below are some approximate values using the
same program. They may be compared with the results of Schulten and Gordon [2]. The major difference
is that we have computed up to 33 decimal places (vs. their maxima of 16) and our computation is stable
up to 32 decimal places regardless of the permutation order of the f’s.

356 5. -T Lai, Y. -N. Chiu / Exact computation of the 3-f and 6-f symbols
N?FOR 35—SYMBOL N1, FOR 65—SYMBOL N—2 N—UFOR 35—SYMBOL N.i, FOR 65—SYMBOL N.2 N?FOR 15—SYMBOL Ni. FOR 65—SYMBOL N2
N 1 N 1 N. 1
Ji—PJS.?33.?M1.?MS.?M3? 3i?32.?33?Mi.?M2?M3.? ii.?3S—?33.?Mi—?MS—?M3—?
155.0 60.0 143.0 143.0 100.0 60.0 200.0 200.0 200.0
60.0 —50.0 —iO.0 —15.0 60.0 —50.0 —15.0 60.0 —50.0
APPROXIMATE VALUE OF 3—3 SYMBOL APPROXIMATE VALUE OF 3—0 SYMBOL APPROXIMATE VALUE OF 3—3 SYMBOL
TUB)— 0 .5500050054i67i07920654939550458ii2 Y053= 0. 00000050041671O7928654939555450112 TNRJ.—Q . 5044047i79909i076538i3288365070586
N?FOR 35—SYMBOL N.j FOR 65—SYMBOL N—2 N.?FOR 35—SYMBOL Ni, FOR 65—SYMBOL N.2 N?FOR 35—SYMBOL N—i, FOR 65—SYMBOL N2
N— 2 N. S N. S
3i732?33734—?35?36? 31.?32.733?)4—235—?36..? Ji?32.733*’34—?35—?36—’
200.0 205.0 200.5 112.0 00.0 120.0 128.0 72.0 120.0
200.0 200.0 200.0 128.0 72.0 40.0 112.0 80.0 40.0
APPROXIMATE VALUE OF 6—3 SYMBOL APPROXIMATE VALUE OF 6—3 SYMBOL APPROXIMATE VALUE OF 6—3 SYMBOL
SIX)— S. 005i5589244242i229442065508224691i SIX)— 0.000000330050S50011937752289ui5522i SIX)— 3. 0050555000500050119377S22891155213
N?FOR 3)—SYMBOL N—i, FOR 65—SYMBOL N—S
N. 2
31..732.3 33—? 34? 35?36?
i2B.0 iSO.S 72.0
112.0 40.0 00.0
APPROXIMATE VALUE OF 6—3 SYMBOL
SIX)—
0 .00055500000050001i937702209i155224
Appendix B. Explanation of detailed intermediate steps not given in the test run output
We will give several detailed intermediate computational steps leading to the final 3-f and 6-f symbol
values. In principle, these can be checked by using manual calculation. We give these in the same way as
one gives detailed steps in the derivation of a final mathematical formula. They serve to prove the validity
of the final values. We explain one of the examples of the computation as follows.
For example (see hereafter), for the computation of (3~~ ~), the fourth and fifth line from the top
represent the f 1’ f~,f~,and rn1, rn2, rn3, respectively. On the sixth line, 0 and 30 indicate the summation
over the three binomial coefficients of eq. (1) which ranges from L = 0 to 30. Therefore, the output for
subtotals has 31 terms, the figure on line 31 is the grand total of the summation which will be sent to
subroutine PRIME(DIA,QK,R9) where this figure is decomposed into a product of prime numbers with
appropriate powers. Our computation result of this 3-f symbol corresponds to
(30 30 30\ 24x32x5xl7x19x23x29Vi~
~, 0 0 o) 7%/13x47x53x59x61x67x71x73x79x83x89

S. -T Liii, Y.-N. C’hiu / Exact computation of the 3-f and 6-f symbols 357
The output of the above example, and for two other 3-f and 6-f symbols is as follows:
N?POR 35—SYMBOL N—i, FOR 65—SYMBOL N.2 N—UPON 35—SYMBOL N1, FOR 65—SYMBOL N—S N.?FOR 35—SYMBOL N—i. FOR 65—SYMBOL N.2
tB_i N—i N—S
Ji—?32—?33.?Mi—?M5?M3—P 31?32.?33?Mi.?M2.?M3? 3i?JS.?33?34.?35?36?
30.0 30.0 30.0 30.0 30.0 30.0 20.0 20.0 20.0
0.0 0.0 0.0 0.0 15.0 —15.0 10.0 10.0 iS.0
5 35 15 30 60 60
i .os055500000soooooosoossoossooooo —2406i4450i0950405.0000000500000500 6S36646703759395.00005550050005005
—26099.0000500000000000005000500500 6i037ii87ii37i8355.S05555S53550500 EXACT VALUE OF 6—) SYMBOL
82580876.0500000000000S00500050000 —5628582S402i6S69i55.05055S00555000
—6684u130i24 .os0000000000000s005000 247445943S164046B3S0.0005000000000 SIX). 1.0
2S5i524632500i.5550055500505505555 —570350i459283i981650.0055000000000 * 2” —2.0
—2873490956565215.00000505005005005 7160476B629633i82255 .0000000000000 * 3’’ —1.0
2064730i8995794160.0000500055500SO —49937732525370255300.5000000000005 * 5” —0.5
—8235062703716205840.50500550553055 i9B02i20557662ui9700.0000000509000 * ii~ —0.5
i9227i2i40i0027747205.O0S005505005 —4455219645134BB0300.00050500005000 * 13’’ —1.0
—2736324589u10748i277j5.55500555550 58903435536950$455.000005550000055 * 23’’ —0.5
243853986736256S7S50660 .0000000000 —2i1i860053u432090 .0000000000000000 * 29’’ —0.5
—138630216429222759749340 .000000000 199083524624504i0.0500000000005000 * 31” —1.0
50643234375S4)B45i01.6285.550550005 i84B26327388404i0.0000000000050005 * 37” —i.0
—12092139S43S77353456007i5.S5500550 18500294296456660.0000000000000000 • 41** —i.0
i066i49091367606758j88i60 0050S500 18505163416049160.0000000000000000 • 43’’ 0.5
—i866S0S586347392i3SSi984S .00000000 10505i635711666B0 .0000000000000000
EXACT VALUE OF 3—) SYMBOL *
47”
53”
0.5
o~5
i209160429347949970977535. 00000000
—509485838730213S25647965.000500005
THR3—1.0
*
•
59~
61’’
0.5
0.5
i3957672i449437385ii7665 .000000000
—244388936534ii03iBBS340.OS00000000 2” —1.5 MULTIPLY
* 3” —1.0 i.00000000005050005050550000000005
2682829309325373496035.00000000500 * 7’’ —i.0
—245766194695402378965.000000000000 * 13” —0.5
—45S64i170Bi658455840. 0000000000000
—5370i452804370425840.0500000000550
• 19”
47’’
—0.5
—0.5
*
—53492106204460066460.0005000000000 • 53’’ —0.5
—53495000300620956681 .0000000000000
• 59” —0.5
—534949797i853350i556.0000000000505 * 61’’ —0.5
—534949797854569i7556.5000055505050 * 67’’ —0.5
—534949797B537460468i.0000005000005 — 71’’ —0.5
—534949797B537463i68i.0000000505000 * 73” —0.5
—5349497978537463i680.0050000090009 * 79~• —0.5
YNRJ.—i.S ‘ 93” —0.5
• 2’’ 4.3 * 09’’ —0.5
* 3” 2.0 * 101’’ 1.0
* 5’’ iS MULTIPLY
• 7” —.1.0 i3465205i.00005S0550000000505005SO
• i3’’ —0.5
* 17” 1.5
• 19’’ 1.5
• S3~ i.0
* 29” 1.0
• 47’’ —0.5
* 53” —0.3
* 59” —0.5
• 61” —0.5
• 67’’ —0.5
71’’ —0.5
* 73’’ —0.5
* 79” —0.5
• 63’’ —0.5
• 89*~ .55
MULTXPLY
i.0000900BOB0000000000
0550088QOO0Ø
Obviously, as f increases it is tedious to evaluate one 3-f symbol manually or by calculator. But, using
our program to compute 3-f and 6-f with such large f ‘s, the CPU time was less than 1 second.
Appendix C. Input data file format
In our program, we use WIG.DAT as input data file. Before running this program, one has to compile
and link the program first and create WIG.DAT file. The format for creating the WIG.DAT file is as
follows:
For example, for the computation of the 3-f symbol (143 100 60) the format is
10 60 —50
5 10 20 30 40 50 60
1
143.0 100.0 60.0 —10.0 60.0 —50.0

358 S.-T Liii, Y. -N. Chiu / Exact computation of the 3-f and 6-f symbols
For example, for the computation of the 6-f symbol (~‘~ ~ }, the format is
5 10 20 30 40 50 60
2
128.0 120.0 72.0 112.0 48.0 80.0
Appendix D. Source program for the exact and approximate computation of the values of 3-f and 6-f
symbols
IMPLICIT NNALO (A—N,O—Z) DIAi—OIFACtINTtJi+J2—33))
NEAL*B 3i,32,33.Mi,MS.M3,34,35,36 D1A2.DIPAC(INT(Si—J2+33))
REAL16 DIAi,DIA2,DIA3,DIA4,DIBi,DIBS,DIB3,DIB4 DIA3—DIFAC(INY(—JL+32+33))
NEAL’i6 DONS .DIR6 ,DXB7,DIBB ,DIPAC.DY3 , DId • DICS DIN1DIFAC(L)
REAL’i6 DIC3,DIC4,01CC,DIDS,51A,KQ,QIi,QP,Q33 DSB2—DIFAC(INT(3isJS—33—PL))
INYEOER2 Qi(uio) DIB3—DI?ACEIN’r(Ji—Mi—pL))
DIMENSION PP(iiO) DIB4—DIFAC(INT(33—32s.Mi+PL))
COMMON/PRI/Q1,PP
DIBS—DIFAC(INThT2+MS—PL))
OPEN(UNITSi,FILE’WSJ.OUY’,STATIJS’NEW.)
OIB6—DIPAC(XNT(33—3i—M2+PL))
OPEN(UNIT—22,FX5.E—’WIO.DAY’ .STATUS’OLD’ ) DICiDIA1/(DIB1’DIB2)
CALL PRIMi(Qi) DIC2.DIAS/(DINS’DIB4)
WRITR(21,5)
DIC3—OIA3/(DIB5’DIN6)
5 FORMAY(iX, ‘N—?’,’FOR 35—SYMBOL Ni, FOR 65—SYMBOL N—2’ DICC—DICi~DIC2’DIC3
READ(2S,$)N
D105—DIDS+(—i)’’L’DICC
B PORMATtiX,I5) 250 CONTINUE
WRIYE(2i,10)N IF(010S.EQ,0,0Q9) 0010 500
10 FORMAT(iX,’N~’,IS) IF(DIDS.LT.0.OQO) CC——i.O’MSION
IP(N.EQ.1) 001015 IP(DIDS.OT.0.OQO) CC~i.0’MSI0N
IF(N.EQ.2) 0OTO 20 DIA.QABS(DIDS)
iS CALL A33BY IF(Ji.GT.30.S) 0010 255
IF(N.EQ.i)OOTOSS
IF(35,OT.35.5)O0TO555
20 CALL A6JSY IF(33.OT.30.0) OOTO 255
25 CLOSE(UNIT.55) R9.1.0
CLOSE(UNITS1) CALL PRIME(DIA,QE,R9)
STOP 255 Li.INT~i+32—33)
END
R9.—.5
C CALL DECOM(L1,QE,R9)
SUBROUTINE A335Y Li.INT~1i÷Mi)
IMPLICIT REAL9(A—19,O—01 R9—.5
REALi6 DIAI.DIA2,DIA3,DIA4,DIBL,DIB5,DIB3,01B4 CALL DECOM(Li,QK,R9)
BEAL’16 DIBS,DIB6,DIB7,DIBO,01PAC,DY3,DIC1,DIC5 Li.INThli—35+33)
REAL’i6 DIC3,01C4,DICC,DXDS,DIA,EQ,QK,Qp,Q33 R9.—.5
REAL’B )i,3S,33,Mi,M2,M3 CALL DECOM(Li,QE,R9)
INTE0ER*S Qitii0) Li.INT(3i—Mi)
DIMENSION PPt 110) R9.5
COMMON/PPI/51,PP CALL DECOM(Li,QK,R9)
5 0010 IU1,iio Li.INT)—3i,3S*)3)
PP(IU)~0.0
R9——.5
iO CONTINUE CALL DECOM)Li,QK,R9)
QK.i.OQO
Li—INT~S2+M2)
iS WRITE(2i,i5) R9.5
15 FORMAY(iX, ‘Si—?’ • ‘32—?’ ‘33.?’, ‘Mi?’, ‘MS—U’, ‘M3—?’/) CALL DECOM(Li,QK.R9)
READ(S2.i9) 3i..S2,33,Mi,542,M3 Li—INY(3i+3S+33+i.)
19 FORMATC6(F10.1)) R9—.5
WRITE(Si 20)31,32 ~ CALL DECOM(L1,QE.R9)
SO PORJIAT)iX,3(FB.i)/) Li—INT)JS—MS)
WBIYE(2i,22)Mj,MS,M3
R9..5
22 PORMAT(iX,3(F0.i)/) CALL DECOM(Li,Qx,R9)
IF(ABSCM1).GT.Ji) 0010 509 Li—INT(33~M3)
IF)ABSCM2).G1.3S) 0010 500 R9—.5
IF(ABS(M3).GT.33) GOTO 500 CALL DECOM(Li,QK,R9)
IP)ABS),Si+32—33).LT.9.0) 0010 500 Li.INT()3—M3)
IF(kIS(33+32—JL).L?.0.0) OOTO 500 R9..5
IF)ANN(33+33.—32).LT.0.0) 00T0 ~ CALL DECOM(L1,QK.R9)
IF(M1+M2+113.NE.O.0) 0010 500 IF(3i.GT.30.S GOTO 400
IMXN—INT(23—Ji—M2) IF(3S.OY.30.S) GOTO 400
NMINi—ENZN IP(33.OT.30.0) GOTO 400
EMZN2—INT(J3—32+M1)
WRITE(Si.4i0)
IP(KMIN2.LT.RMIN) EMIN—KMIN2 410 PORMAI(II. ‘EXACT VALUE OF 3—) SYMBOL’/)
EMIN—)—1)EMIN WRITE)S1,420) CC
EMAX—INT(Si+32—33) 420 PORMAT(iX,’TUR)’,F4.i)
XMAX1.XXAX DO 425 II—i,iio
EMAX2—INT)J1—M1) IF(ABS(PP)II)).LT.i.0E—5) 0010 425
KMAX3INT(3S+M2) WRITE(21,422) Qi)II),PP)II)
IP)NMAX2.LT.EMAX) RMAXIMAX2 422 FORMAT)iOX,’”,14,’”’,P7.i)
IF(KMAX3.LT.NMAX) IMAXEMAX3 425 CONTINUE
IF(KMXN.LT.0) EMIN~0 WRITE)Si,42B)
IF)KMIN.GY.RMAX) 0010 500 429 FORMAT(1X,’MULTIPLY’)
MSI0N)—i)”)IBT)Ji—32—M3))
KO—QINT(QK+.SQO)
DIDS.0.000
IP(KQ.LT.60i.000)KQ.i.SQO
DO 230 L—IMIN.EMAX WRITE(2i, * EQ
PL.QFLOAT)L) 0010 600

S. -T Lai~Y.-N. C’hiu / Exact computation of the 3-f and 6-f symbols 359
455 QP.i.OQO L1—INT)Si+32+33+1)
WRITE)2i,429)
N9—.5
429 FORMAT(11, ‘APPROXIMATE VALUE OF 3—) SYMBOL’/) CALL DECOM(Li.QR.R9)
00430 II.i,ui0 Li—INT)34535—33)
IF)ABS)PP)II)).LT.i.OE—5) PP)II).0.S RB—S
IF)ABS(PP)II)).LY.i.SE—5) GOYO 430 CALL DECOM)Li,QK,R9)
QPQP’Qi(II)’’PP(II)
Li—IIOT)Ji+32—33)
430 CONTINUE N9—.
Q3)—CC’QP’QABS(DIDS) CALL DECOM)L1,QK,R9)
WRIYE)2i,450) Q3) Li.INY(34—35*33)
450 ?ORMAT(iI, ‘TUB)—’ ,F37.34) R9—.5
0010 600 CALL DECOM)L1.Qk.B9)
550 TBR)—S.0 Li.INT)—Ji+32+33)
IP)TNR).EQ.S.0) WRITE(21,550) R9—.5
550 PORMAT(iI.’YHR)—O.S’) CALL DECOM)Li.QE.R9)
600 CONTINUE Li.INT)—34.35+33)
RETURN
R9.S
END CALL DECOM)Li,QK,R9)
C
L1ENY(24+35+33+i)
SUBROUTINE A6)SY R9—.S
IMPLICIT REAL’9(A—H,O—Z) CALL DECOM)Li,QK,R9)
REAL’S 31,32.33,34,35,36,) Li.INT)34s.JS—36)
REAL’i6 DIAi,DIA2,01A3,DIA4,DIB1,DIB2,DIB3,DIB4 R9.S
REAL’i6 DIBS,DIB6 ,DIB7,DIBB ,DEFAC.0Y3 ,DIC1 ,DIC2 CALL DECOM(Li,QE.R9)
REAL*i6 DIC3,DIC4,DICC,DIDS,01A,KQ,QE,QP,Q3) Li—INT)34+)2+36+i)
INTEOER’2 Qi(uio) RB.5
DIMENSION PP(ii0) CALL DECOM)Li,QK,R9)
COMMON/PRI/Q1,PP
LiINT))4—32+36)
5 DO 10 lu—illS
PP)IU)0.O CALL DECOM)L1.QE,R9)
10 CONTINUE Li—INT)—34+)S’36)
QK.i.OQO R9.5
12 WRITE)2i,iS) CALL DECOM)Li,QK,R9)
15 FORMAT(iX.’)i—?’,’32.?’,’33—P’.’)4.?’.’)5.?,’36—?’/) L5—INT)Ji+3S—36)
REAO)S2,i9) Ji,32,33,34,3S,36 R9..S
19 FORMAT(6CPiO.i)) CALL DECOM)L1,QE,R9)
WRIYE)2i .20)31 .32,33
L1INT)ii+)S+36+i)
SO FORMAT)iX,3)F9.i)/) R9—.S
BORIYE)2i.2S)34,JS,36 CALL DECOM)Li.QK,R9)
22 FORMAT)SX,3)FB.i)/) L1INT)J1—)5S)6)
C
R9—.5
iMi.Ji+JS+33 CALL DECOM)Li,QK,R9)
3MS—)i*35+36
Li.INT(—31,3S+)6)
3N3.i3*34÷)6 RB— .5
3M4)3+)4.35 CALL DECOM)L1.QE,R9)
)Xi)i+3S+)4+35 IP)Ji.GY.S0.0) OOTO 400
3X2—)5+33+)5+36 IF)32.GT.20.0) 0010 400
3X3.)i+)3+34+.76 IP))3.GT.20.0) 0010 400
C IF)34.GT.20.0) OOTO 400
)M)Mi IF)35.OT.S0.0) 0010 400
IP)3M2.G1.JM)JM—3M2 IF)36.GY.S0.S) OOYO 400
IF))M3.GT.JM)JM.3M3
WBITE)2i,4i9)
IP(7M4.GT.3M)JM.)M4 419 PORMAT)1X, ‘EXACT VALUE OP 6—) SYMBOL’/)
)XJXi VRITE(Si,4S0) CC
IP))XS.LT.3X))X—3X2 420 FORMAT)iX,’SIXJ.’,F4.i)
IP))X3.LT.)X))X.)X3 0042511—1,110
KM.IMT))M) IF)ABS(PP)II)).LY.i.OE—5) 8010 425
KX.INT))X) WRITE)Si,422) Qi(II),PP)II)
IF(EM.GY.RX) GOYO 500 422 FORMAT)iOX,’’,I4,’’’’,F7.i)
DIDS—0.008 425 CONTINUE
DO 250 LKM,KX WRITE)21,420)
PL*QFLOAT)L) 428 FORMAT(1X,’MULTIPLY)
DIA1—DIPAC)L.1)
KQ..QINY)QK+.500)
DIA2.DIFAC(INT)Ji,J2—33)) IF)KQ.LT.601.OQO) tCQ—i.OQO
DEA3—DIPAC)INT)34—)S+36))
WRITE)2i,’)EQ
DIA4—DIFAC)INT)—Ji+JS+JS)) GOTO 650
DEN1DIPAC)INT)PL—.21—)2—33)) 400 QP1.OQO
0182.DEFAC)INT))4*.32+36+i,))
WRITE)21,429)
DIB3—DIFAC)INT)PL—J4—35—33)) 429 FORMAT)1X,’APPROXIMATE VALUE OF 6—) SYMBOL’!)
DIB4—DIPAC)INT)JS+3i,)4+35—PL)) DO 438 1I1,iiO
DIBS.DIFAC)INT)PL—)4—32—)6)) IF)ABS(PP)II)).LT.i.SE—5) PP)II)—S.S
DIB6.DIFAC)INT)33+)6+)1+34—pL)) IF)ABS(PP)II)).LT.i.OE—5) 0010 430
DIB7—DIPAC)INT)PL—31—)5—)6))
QPQP’Q1)II)”PP(II)
DIBO—DIFAC)INT)35+334)S+.76—pL)) 430 CONTINUE
DIC1DIA1/)DIBSDIB2)
Q3).CCQP*QABS(DIOS)
DICSDIAS/)DIB3’DIB4) WRITE)2i,450) Q33
DIC3.DIA3/)DIBiDIB6) 450 FORMAT)iX,’SII)*’,F37.34)
DIC4.DIA4/)DLB7’DIB8) GOTO 650
DICC—OICi’DIC2DIC3’01C4 500 SIX)—S.0
DIDSDIDS+)—i)”L’DICC IF(SII).EQ.0 .0) WRITE(2i,600)
250 CONTINUE 600 PORMAT(iI,’SIX)=0.3)
IF)DIDS.EQ.0.500) 0010 580 650 CONTINUE
IF(DIDS.LT.S.OQO) CC.—i.0 RETURN
IF(DIDS.GT.0.SQO) CC.i.0 END
DIAQABS)D105)
C
IF))i.GT.2S.0) GOTO 255 C ‘CALCULATION OF Nt
IF))S.OT.S0.0) OOTO 255 C
IF)33.GT.20.O) GOTO 2SS FUNCTION DIPAC(N)
IP)34.0Y.20.0) OOTO 255 REAL16 DIFAC,Pi
IF)35.GT.2B.0) GOYO 255 INTEOER’2 K,N
IF)36.GY.SO,O) 0010 255 DIFAC.i.OQQ
NO.1.0 SF (N.LE.i) GOTO 20
CALL PRIME(DIA,OE,R9) 00 10 K1,N
255 L1.INT)31—32+33) Pi.QFLOAT(K)
R9.5 OIFACOIFACP1
CALL D!COM)Li.QX,R9) 10 CONTINUE

360 S.-T Lai, Y -N. Chiu / Exact computation of the 3-f and 6-f symbols
20 RETURN 15 CONTINUE
END
RETURN
C
END
SUBROUTINE PBIMi(Qit C
IMPLICIT REALB)A—U,O—I) SUBROUTINE PRIME)DIA,QK,BB)
INTEOER*2 Qi(11S) IMPLICIT REALO(A—N,O—Z)
LOGICAL KR INTEGER’S Qi)liS),LY
Qi(i)—2 REALi6 DIA.DY3,QE
NI2 LOGICAL LL,LQ
00101—3,601 DIMENSION PP)ii0)
DO 25 3—2,1—i COMMON/PRI/Q1,PP
KK—ABS(FL0AT(I/))—FLOAT(I)/FLOAT~fl).LT.i.0E—5
LY—0
IF(EE) GOTO iS LM1
20 CONTINUE DIA—QINT)DEA+.5Q5)
Qi(NI)—I 15 IF)LM.OT.LY) LYLM
NI—N1+i IF)DIA.EQ.i.558.OR.DIA.EQ.0.SQO) 001040
iO CONTINUE IF)01)LM).EQ.0) QKDIA
RETURN IF)Q1(LM).EQ.0) 001040
END SO DT3DEA/QFLOAT)Qi)LM))
C
LQ.QABS(DT3—i.000).LT.i.OQ—5
SUBROUTINE DECOM)Li,QE.R9) IF)LQ) PP)LM)—PP)LM)+R9
IMPLICIT REALB)A—U.O—Z) IF)LQ) 001040
REAL’16 DIA,DT3.QE LL.QABS(D13—QINT)DY3*.SQO)).LT. 1.SQ—iO
INTEOERS Qi)iio) IF)LL) PP)LM)—PP)LM)+R9
DIMENSION PP)uio) IF)LL) DIA:DY3
COMMON/PRI/Ql,PP IF)LL) 0010 50
IF(Li.EQ.0.OR.Li .EQ.i) RETURN LMLM+i
DO 10 LEi.Li 0010 10
DIA—QFLOAT)LK) 40 RETURN
CALL PRIME)DIA.QK,R9) END
References
[11 M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten Jr.. The 3-f and 6-f Symbols (The Technology Press. MIT, Cambridge,
MA, ~959).
[2] K. Schulten and R.G. Gordon, J. Math. Phys. 16 (1975) 1961; Comput. Phys. Commun. 11(1976) 269.
[3] RN. Zare, Angular Momentum (Wiley, New York, 1988).