862
The Journal of Physical Chemistry, Vol. 83, No. 7, 1979
P.
L. Corio
Exact and Approximate Formulas for Vector Coupling Coefficients P. L. Corio Department of Chemistry, University of Kentucky, Lexington, Kentucky 40506 (Received August 14, 1978) Publication costs assisted by the Graduate School, Universlty of Kentucky
The vector coupling coefficients characterizing the coupling of two angular momenta J1,J2to a resultant J3 are obtained by a method different from those used previously. The present method exhibits the mathematical structure of the coefficients clearly, and permits facile derivation of four approximate formulas. The relation of the coupling coefficients to the rotation matrices gives three approximate formulas for D~~,lO,O,O).
1. Formulas for the Coupling Coefficients The coefficients ~ l m j , m 2 1 j j ~ 3 mcharacterizing 3) the coupling of two angular momenta J1,J2to a resultant J3 satisfy the partial difference equation'
ml + 1)G2- mJG2 - m2 + l)I1'*G1ml + 1j2m2- l l j j j 3 m J + lilG1 + 1) + j2G2 + 1)- j 3 G 3 + 1) + 2mlmzlGlmj2mzljj2/3m3) - [GI+ ml)Gl - ml + 1)G2- m2)G2+ m2 + 1)]14.jlml - lj2mz+ 11jjj3m3)= 0 (1)
ynomial system (Qn(x))satisfies the orthogonality relations Cw(x)Qn(X)Qnl(X)= p ( n ) a n n f C [ ~ ( n ) l - ~ Q f i ( x ) Q n=( [w(x)l-'axx( ~? n
[G1- m1)G1+
The sum of the variable arguments in each coupling coefficient is ml m2 = m3, so this condition may be used to reduce (1)to an ordinary difference equation in ml. For this purpose, let
+
O'lmdZmZbji3m3) =
(8)
X
(9)
Equation 8 may be established by substituting (7) for Qn(x) and performing n' partial summations, obtaining the orthogonality condition when n' # n, and formula (4) when n' = n. Equation 9 follows from (8), for the latter expresses row orthogonality of the (real) matrix 0 with matrix elements On, = [ w ( ~ ) / p ( n ) ] ~ / ~ Q , and ( x ) ,row orthogonality implies column orthogonality. A symmetrical expression for Qn(x) may be obtained from (7) by writing3 An = ( E - l)n, where E is the displacement operator: E f ( x ) = f ( x + 1). A straightforward calculation yields eq 10. Introducing (3), (4),and (10) into
[w(~;r,s,N)/p(n;r,s,N)ll'~Q,(~;r,s,N) (2) where
x ! ( r - x ) ! ( N- x)!(x t s - N ) ! $(-1)"
(3) rts-2n p ( n ; r , s , ~ ) = ( L > ( A > (N - n
> (r t s n- n t l > X
V=O
u ! ( n- u ) ! ( r- x - v ) ! ( N- x -t u ) ! ( xt s - N - n t v)! (10)
(2), one obtains Racah's f ~ r m u l aeq , ~ 11, where the second
[ ( 2s)yrP)l-' ( 4 ) r = 2j, s = 2j2 x = jl + ml (5) n = j , + j, - j 3 N = j, + j, + m3 The definitions2of w ( x ) and p ( n ) show that x = 0, 1, 2, ..., r or N,whichever is smaller, n = 0, 1, 2, .,., r , s, or N , whichever is smaller, and N = 0, 1, 2, ..., r + s, p ( n ) is a
normalization factor whose derivation will be indicated subsequently. The substitution (2) transforms (1) into
(N- x)(r - x)Qn(x + 1) - [ ( N- x)(r n(n - r
-
s - l)]Qn(x) + x(x
2x) + sx + + s - N)Q,(x 1) = 0 -
-
(6)
which defines a system of homogeneous linear equations for the quantities Q,(O), Q,(l), Qn(2), ... . The direct solution of this system can be avoided by observing that (6) is satisfied by Q n ( x ) = (-1)"[w(x)
(r
S>]-lAn
;( - 1 ) V u
=11
[ x ! n ! ( r- x ) ! ( r- n ) ! ( s - q)!(iV- n ) ! ( N - x)! X s - N ) ! ( rt s - N - n ) ! / ____ ( rt s - n t 1 ) ! ] " 2 v ! ( n - u ) ! ( r- x - .)!(Ar - x - u ) ! X ( r - n t u ) ! ( x ts-iV- n t u ) ! (11)
(x
member shows the relation to Wigner's 3 - j symb01.~ The symmetries6 of the 3 - j symbol are, of course, neatly exhibited by formula 11. The group of 72 symmetries of the 3 - j symbol consists of these linear transformations of x,r, n, and N that leave the quantity within the square brackets in eq 11 invariant and transform the interval of orthogonality into itself. The polynomial Qn(x)may be expressed as a generalized hypergeometric function of the type 3F2with unit argument:
' i"11
{w(~)(:)(~
(7)
where the phase factor has been chosen to secure agreement with the customary conventions for the coupling coefficients, and A is the finite difference operator3 for unit increment in x: Af(x) = f ( x + 1)- f ( x ) . It follows that Qn(x) is a polynomial of degree n in x, and that the pol0022-3654/79/2083-0862$01 .OO/O
This relation may be used to deduce eight independent recurrence relations that subsist among the coupling coefficients7 and the numerous transformations to which 0 1979 American
Chemical Society
The Journal of Physical Chemistry, Vol. 83, No. 7, 1979 883
Formulas for Vector Coupling Coefficients
TABLE I: -
a
Numerical Comparisons of Eq 11, 20, and 28 ( j , m , j z m zl j l j 2 j 3 m 3 )
eq 11
eq 20a
N / ( r f s)
(10,- 10,312,- 312 110,3/2,23/2,-2312) (10,- 9,312,- 312 110,3/2,23/2,-21/2) (10,-8,312,- 3/2 110,3/2,23/2,-1912) (10,- 7,312,- 3/2110,3/2,23/2,-1712) (10,- 6,312,- 3/2110,3/2,23/2,-1512) (10,-5,3/2,- 3/2110,3/2,23/2,- 1312) ( 10,0,3/2,- 3/2110,3/2,23/2,- 312) (3,- 3,2,- 113,2,4,- 4) (3,- 2,2,- 2 13,2,4,- 4) (112,- 1/2,20,- 1011/2,20,41/2,- 2112) (1/2,1/2,20,- 1111/2,20,41/2,-2112) ( 1,- 1,40 ,- 3 0 I1,40,41,- 31) (1,0,40,- 31 11,40,41,- 31) (1,1,40,- 32 11,40,41,- 31)
1.00000 0.9 325 0 0.86660 0.80231 0.73969 0.67879 0.40186 - 0.77460 0.7 07 11 0.86954 0.49386 0.87729 0.46562 0.11640
1.00000 0.93250 0.86957 0.81087 0.75614 0.705 11 0.49718 -0.77460 0.63245 0.88519 0.437 16 0.88519 0.437 16 0.15266
0.00 0.04 0.09 0.13 0.17 0.22 0.43 0.10 0.10 0.24 0.24 0.12 0.12 0.12
The first nine entries in this column computed were with eq 20; the remaining entries were computed with eq 29.
formula 11 may be !subjected.8 Two such transformations yield
Q n ( x ) =[ n ! ( r : s ) ] - l x
These formu1.a~yield less symmetrical expressions for the 3 - j symbol as compared with Racah‘s formula; the latter exhibits explicitly the maximum symmetry with respect to all argumentsag
2. Approximate Formulas T h e representation ( 2 ) is particularly useful for developing approximate formulas. Case 1. Let N 1 is required for the rotation matrices. Case 2. When N >> 1 and p
