Download EXPLICIT FORMULAS FOR CLEBSCH—GORDAN COEFFICIENTS

Computer Physics Communications 10 (1975) 245 —250
© North-Holland Publishing Company
EXPLICIT FORMULAS FOR CLEBSCH—GORDAN COEFFICIENTS
Georges
RUDNICKI~BUJNOWSKI*
Department of Physics, Technion-IsraelInstitute of Technology,
Haifa, Israel
Received 12 November 1974, revised manuscript received 17 July 1975
PROGRAM SUMMARY
Title of program: CLEBSCH—GORDAN EXPLICIT
Method of solution
FORMULAS
Algebraic method based on the Racah formula using the
FORMAC programming language.
Catalogue number: ACWL
Program obtainable from: CPC Program Library, Queen’s
Restriction on the complexityof the problem
University of Belfast, N. Ireland (see application form in this
issue)
Formac limitations. Numbers which are too large may be
generated during intermediate stages of the program for high
values of angular momentum.
Computer: IBM 370/168:Installation: Technion Computer
Center, Haifa, Israel
Operating system: OS-MVT-Rsl. 7
+
HASP, Formac system
Typical running time
Compilation and linkage: 7 seconds
(Minimac etc.), PL1/F compiler
Program language used: FORMAC (includes PL1)
Execution time
High speed storage required: 65,536 words
Depending on the values demanded: 11 seconds for /2
3/2; 6 minutes forj
2 = 2,5/2, 3,7/2,4, 9/2.
=
1/2,
No. of bits in a word: 32
Overlay structure: None
Unusual features of the program
No. of magnetic tapes required: None
The results are algebraic formulas in which numbers can be
substituted to get numerical results.
Other peripherals used: Card reader, line printer
No. of cards in combined program and test deck: 170
Card punching code: EBCDIC
Keywords: Atomic, nuclear, Clebsch—Gordan, spectroscopy,
algebraic manipulation, Formac.
Nature of the physical problem
To obtain explicit algebraic formulas for Clebsch—Gordan
coefficients for high values of angular momentum.
*
Mailing address: 6/8 Remez Street, Rishon-le-Zion, Israel.
References
[11 PL/I-FORMAC Symbolic Mathematic Interpreter IBM
Corporation, Program Information Department (PID),
40 Saw Mill River Road, Hawthorne, New York 10532,
USA.
[21 A. Messiah, Mécanique Quantique (Dunod, Paris, 1960)
Appendix C.
G. Rudnicki-Bu/nowski/Explicit formulas for aebsch—Gordan coefficients
246
LONG WRITE-UP
1. Introduction
A program capable of printing algebraic formulas,
giving the Clebsch—Gordan coefficients for a given
value of angular momentum (/2) has been developed.
The program uses the Formac algebraic language [1].
will generate the tables for!2 = 9/2.
The second integer must be larger than or equal to the
first one, otherwise an error print will result, and the
program will be stopped.
2.2. Output
The program prints the values of m2 and J, followed by the line
2. Method
The program is based on the Racah formula [2]
which allows us to write
K/1 j2m1 m2IJM~
~~‘~‘fl ‘~“~1’~2’
~
CG
=
“value of the Clebsch—Gordan coefficient”.
In the printing of this line, the symbol W stands for
(2/1)and the symbol ROOT.
( ) stands
for \/( ) (e.g.
X -../(j1+m1)(f1—m1)(j2+m2)(j2—m2)(J+M)(J—M)
2.3. Structure
X ~t(_)t[t!(J_f2+t+mi)!(J_/i+t_m2)!
X ~ +12 ~
t)!(j1 t— m1)!(j2 t +m2)!~—1
—
—
where
~O1’J2’~)
(/~+j2—J)!(f~+J—j~
)!(J+j1 —/2)!
=
; +~ ~
Vi ‘2
,
“
and E~.indicates the summation over all the integer
values of t for which the arguments of the factorials
are non-negative (0! = 1). The notation used is the
standard one in current textbooks on Quantum Mechanics. (/1’I2’~~)
represent respectively the angular momenta of the first particle, of the second particle and
of the two-particles system; (m1, m2 M) the corresponding projections on the quantization (z) axis,
,
2. The program
2.1. Input
The program must receive as input two integers,
printed anywhere on a c~rd(free format). They represent the lowest and highest values of 2/2 for which
a table will be generated. For instance:
5
7
will generate the tables for /2 = 5/2, 3,7/2, and
9
9
The program consists of the main procedure, and
one internal subroutine, SIMPLEX (see below). For
a given value of J2, all the possible values of J are generated, and for each value of (J2, J) the program generates all the values ofM2. For a given value of(J2,J,
M2) all the possible terms of the implicit summation
in the Racah formula are then generated.
2.4. The subroutine SIMPLEX
The subroutine SIMPLEX receives as input a number, called IM in the program (for instance 24), decomposes it into prime factors (24 becomes 2~,3) and
returns two integers defining the square root of the
original number. One integer is the square root of the
largest factor of the original number which is a perfect square (IRT in the program), the other (called
INRT) is the residue of the original number after division by this factor. For this example IRT = 2 and
INRT = 6 since ‘~J~
=
We have limited our search for factors to the first
168 prime numbers. In the case of veiy large values of
‘2~the user should decide whether to accept values of
IRT which could be factored further, or to add more
prime numbers. To add more prime numbers:
1) Add them in the list of prime numbers in the INIT
statement before the SIMPLEX subroutine.
2) Change the number 168 into the new number of
G. Rudnicki-Bu/nowski/Explicit formulas for Clebsch—Gordan coefficients
primes in PRIME(168) and in all the occurrences
of 168 in the SIMPLEX subroutine,
2.5. Restriction on use
The two restrictions most likely to occur in practice are:
1) The argument of a FAC function (factorial) is
too large. One possible modification to overcome this
problem would be to add the statement: OFFSET
(NOTRANS); after the card with the statement
FORMAC-OPTIONS; the result will be that the fac.
tonal functions will be retained in a symbolic form
(e.g. FAC (3) = 3! instead of 6). In that case the
statement: CALL SIMPLEX; should be replaced by
the statement:
LET (INRT
IM; IRT
=
1);
2) A number generated is too large to be handled
by Formac (more than 2000 digits approximately).
A possible solution would be to print the factors of
the Clebsch—Gordan coefficient separately.
2.6. Recommended form ofuse
a) To calculate numerical values of the Clebsch—
Gordan coefficient from the algebraic formulas.
b) To calculate the values of the numerator and
of the denommator separately (or their squares), using
integer arithmetic. In this case all the calculations
247
should be done inside the Formac system, and the
numerator and the denominator should be divided
by the largest common denominator in case it is decided to pass them into PL1, (using for instance the
INTEGER function). The reason is that the numbers
generated may surpass the PLI handling facilities.
c) To produce complete algebraic formulas for use
in a PL1 (or Fortran) program as this will reduce the
execution time of the program which calculates the
numerical results. In this case, too, the user should
consider the necessity of calculating various parts of
the formulas separately, and then performing simplifications in order to avoid numerical overflow in the
integer arithmetic of the host language.
Acknowledgements
I would like to express my gratitude to Professor
Asher Peres for his support, helpful discussions and
continuous encouragement.
References
[1] PL/l FORMAC Symbolic Mathematic Interpreter, IBM
Corporation,
Program
Information
Dept. 10532,
(PID), U.S.A.
40 Saw
Mill River Road,
Hawthorne,
New York
[21 A. Messiah, Mécanique Quantique (Dunod, Paris, 1960)
Appendix C.
G. Rudnicki-Bujnowski/Explicit formulas for Clebsch—Gordan coefficients
248
TEST RUN OUTPUT
CAlCULATION OF CLF8SCH GORCAN COEFFICIENTS CC
(JL.J7.M1.M2.IILL.,J.M)
W=2*Ji
4 0.5
OUTPUT FOR J2=
-‘
-
_-4-_’_
4,
_‘-_
-I-,
7P’
.-J.-.7~_’_
~‘;
M
2=
0..5
C~. =
M
J=J1+L
1/2
+
.11
+
=
M2=
(
—
M
+
.11
1/2
1/2 )
M2= —0.5
cc.
0.51
/ ( w
,J=J1+(
1/2
1/2 )
.
C.5
1 1
+
C.~
1/2
/
(
W
+
1
1
J=J1+L —C.51
1/2
-Cc,
1/2
—
I
—
P’ + ii
M2= —0.5
0M+.111i-l/?)
CC.=
--
+
1/2
1
/
J~J1_~L____~_0.5i
1/22 /in+-~1j
(
W +
1/7
1
G. Rudnicki-Bufnowski/Explicitformulas for aebsch—Gordan coefficients
249
t1IJTPIJT FOR J2= 1.0
M2=
S.C
C(,.
J=J1+L1.O)~
1/2
(M+J1)
-
4M+.i1+1)
t~~2= 0.0
=
I
—
,i=J1+t
1/2
M + .11 + 1
1
-
--
—
1/?
1/2
1/2
/g(v~.?J
(W+L)
I
1.0)
1/2
1 ?
+ ,J1
l I
+
[/2
ROOT..(
2 1 /
I
I
1/2
W + 2 I
I
—-———-—————-——————-—-————
J=J1+L_L.0I
N2= —1.0
Cc=(
1.0
I
J=J14-t
—M+JlI
—(
P47=
—14+J1•1
0..0
I
en
=
I
— P4 + JI
I
/1
(W+2)
1/2
(~+LJ
I
1/2
ROOT.(2)/(
(Ws21
1/2
U
[/2
I
0.0)
-
-
1/2
-
d M
-
1/2
O01
JJ1+I
1/2
-
1/2
iM+J1)
JJI+(
1/2
IW+2)
P42= —l.C
—
00)
1/2
cr.=
--
1/2
—M+j1+1)
P42=
_
1/2
-
U • 1 1
+
.11
+
1
1
1/2
RL~OT.( 2 ) / I
( b~ + 2 1
1/2
U
I
G. Rudnicki-Bujnowski/Explicit formulas for Clebsch—Gordan coefficients
250
-
42=
1.0
-
.1=J1+t
—1.01
--
1/2
1/2
—M+JI+1)
CC,=I
M?=
=
0.0
—
I
—
~
+
M2= —1.0
cr.=
(M+JJ+I)
I
—M’i-Jl)
J=J1+i~ —1.01
1/?
.11 1
( P4 + Ji
J=J1+( —1.01
1/2
IM+J1)
1/2
/1
(W4-II
-
1/2
W
-
1/2
I
1/2
RGOT.(
1/2
2
1
/
I
1/?
/1(w+l)
I ~
+ 1
I
1/2
I
1/2