U111 Wigner 3-j, 6-j, 9-j Symbols; Clebsch-Gordan, Racah W-, Jahn U-Coefficients

Function subprograms RWIG3J, DWIG3J; RWIG6J, DWIG6J; RWIG9J, DWIG9J; RCLEBG, DCLEBG; RRACAW, DRACAW and RJAHNU, DJAHNU calculate the Wigner 3-j, 6-j and 9-j symbols, the Clebsch-Gordan coefficients, the Racah W-coefficients and the Jahn U-coefficients, respectively.

On CDC and Cray computers, the double-precision versions DWIG3J etc. are not available.

Structure:

FUNCTION subprograms
User Entry Names:

RWIG3J,	RWIG6J,	RWIG9J,	RCLEBG,	RRACAW,	RJAHNU
DWIG3J,	DWIG6J,	DWIG9J,	DCLEBG,	DRACAW,	DJAHNU

Usage:

In any arithmetic expression, for $t\; =\; R$ (type REAL), or $t\; =\; D$ (type DOUBLE PRECISION),

All the arguments must have integral or half-integral values (see Notes). They have the same type as the function name. For definitions and notations see References.
The following relations hold (see Refs. 1 and 3):
Clebsch-Gordan coefficient (in terms of the Wigner 3-j symbol):

Racah W-coefficient (in terms of the Wigner 6-j symbol):

Jahn U-coefficient (in terms of the Wigner 6-j symbol and the Racah W-coefficient):

Method:

The Wigner 3-j symbol and the Clebsch-Gordan coefficient are calculated from formulas (5.1) and (5.10) of Ref. 1, respectively. The Wigner 6-j symbol, the Racah W- and the Jahn U-coefficient are calculated from formulas (5.23) and (5.24) of Ref. 1. In both cases, the factorials are replaced by their logarithms during the calculation. The Wigner 9-j symbol is calculated from formula (5.37) of Ref. 1 in terms of Wigner 6-j symbols.

Notes:

A Wigner-3j symbol $($

$j$1	$j$2	$j$3
$m$1	$m$2	$m$3

$)$

is considered to be zero unless simultaneously

(i)	$j$i and $m$i have both either integral or half-integralvalues (each $i$ ),
(ii)	$j$i≥|mi| ≥0 (each $i$ ),
(iii)	$m$1+m2+m3=0 ,
(iv)	$j$1-j2-m3 is an integer, and
(v)	$j$1+j2+j3 is an integer and $j$1+j2≥j3, &quad;j2+j3≥j1, &quad;j3+j1≥j2 .

The conditions (v) are often denoted by $\delta (j$₁ j₂ j₃) and are called the triangle relations.

For a Clebsch-Gordan coefficient $(j$₁ j₂ m₁ m₂ | j₁ j₂ j₃ m₃) , condition (iii) reads $m$₁+m₂=m₃ and condition (iv) disappears.

A Wigner-6j symbol $$

$j$1	$j$2	$j$3
$l$1	$l$2	$l$3

$$

is considered to be zero unless simultaneously

(i)	all $j$i and $l$i have non-negative integral orhalf-integral values,
(ii)	the four triangle relations&quad;$\delta (j$1 j2 j3), &quad;δ(j1 l2 l3), &quad;δ(l1 j2 l3), &quad;δ(l1 l2 j3) &quad;hold.

A Wigner-9j symbol $$

$j$11	$j$12	$j$13
$j$21	$j$22	$j$23
$j$31	$j$32	$j$33

$$

is considered to be zero unless simultaneously

(i)	all $j$ik have non-negative integral or half-integralvalues,
(ii)	the arguments in each row and in each column satisfy thetriangle relations.

Restrictions:

The sum of arguments in any triangle relation must not exceed 100. No test is made.

References:

1. R.D. Cowan, The theory of atomic structure and spectra, (Univ. of California Press, Berkeley CA 1981).
2. A.F. Nikiforov, V.B. Uvarov and Yu.L. Levitan, Tables of Racah coefficients (Pergamon Press, Oxford 1965).
3. M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten, Jr., The 3-j and 6-j symbols (Crosby Lockwood, London 1959).
4. D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskii, Quantum theory of angular momentum (World Scientific, Singapore 1988).

U112