*DECK RC3JJ
SUBROUTINE RC3JJ (L2, L3, M2, M3, L1MIN, L1MAX, THRCOF, NDIM, IER)
C***BEGIN PROLOGUE RC3JJ
C***PURPOSE Evaluate the 3j symbol f(L1) = ( L1 L2 L3)
C (-M2-M3 M2 M3)
C for all allowed values of L1, the other parameters
C being held fixed.
C***LIBRARY SLATEC
C***CATEGORY C19
C***TYPE SINGLE PRECISION (RC3JJ-S, DRC3JJ-D)
C***KEYWORDS 3J COEFFICIENTS, 3J SYMBOLS, CLEBSCH-GORDAN COEFFICIENTS,
C RACAH COEFFICIENTS, VECTOR ADDITION COEFFICIENTS,
C WIGNER COEFFICIENTS
C***AUTHOR Gordon, R. G., Harvard University
C Schulten, K., Max Planck Institute
C***DESCRIPTION
C
C *Usage:
C
C REAL L2, L3, M2, M3, L1MIN, L1MAX, THRCOF(NDIM)
C INTEGER NDIM, IER
C
C CALL RC3JJ (L2, L3, M2, M3, L1MIN, L1MAX, THRCOF, NDIM, IER)
C
C *Arguments:
C
C L2 :IN Parameter in 3j symbol.
C
C L3 :IN Parameter in 3j symbol.
C
C M2 :IN Parameter in 3j symbol.
C
C M3 :IN Parameter in 3j symbol.
C
C L1MIN :OUT Smallest allowable L1 in 3j symbol.
C
C L1MAX :OUT Largest allowable L1 in 3j symbol.
C
C THRCOF :OUT Set of 3j coefficients generated by evaluating the
C 3j symbol for all allowed values of L1. THRCOF(I)
C will contain f(L1MIN+I-1), I=1,2,...,L1MAX+L1MIN+1.
C
C NDIM :IN Declared length of THRCOF in calling program.
C
C IER :OUT Error flag.
C IER=0 No errors.
C IER=1 Either L2.LT.ABS(M2) or L3.LT.ABS(M3).
C IER=2 Either L2+ABS(M2) or L3+ABS(M3) non-integer.
C IER=3 L1MAX-L1MIN not an integer.
C IER=4 L1MAX less than L1MIN.
C IER=5 NDIM less than L1MAX-L1MIN+1.
C
C *Description:
C
C Although conventionally the parameters of the vector addition
C coefficients satisfy certain restrictions, such as being integers
C or integers plus 1/2, the restrictions imposed on input to this
C subroutine are somewhat weaker. See, for example, Section 27.9 of
C Abramowitz and Stegun or Appendix C of Volume II of A. Messiah.
C The restrictions imposed by this subroutine are
C 1. L2 .GE. ABS(M2) and L3 .GE. ABS(M3);
C 2. L2+ABS(M2) and L3+ABS(M3) must be integers;
C 3. L1MAX-L1MIN must be a non-negative integer, where
C L1MAX=L2+L3 and L1MIN=MAX(ABS(L2-L3),ABS(M2+M3)).
C If the conventional restrictions are satisfied, then these
C restrictions are met.
C
C The user should be cautious in using input parameters that do
C not satisfy the conventional restrictions. For example, the
C the subroutine produces values of
C f(L1) = ( L1 2.5 5.8)
C (-0.3 1.5 -1.2)
C for L1=3.3,4.3,...,8.3 but none of the symmetry properties of the 3j
C symbol, set forth on page 1056 of Messiah, is satisfied.
C
C The subroutine generates f(L1MIN), f(L1MIN+1), ..., f(L1MAX)
C where L1MIN and L1MAX are defined above. The sequence f(L1) is
C generated by a three-term recurrence algorithm with scaling to
C control overflow. Both backward and forward recurrence are used to
C maintain numerical stability. The two recurrence sequences are
C matched at an interior point and are normalized from the unitary
C property of 3j coefficients and Wigner's phase convention.
C
C The algorithm is suited to applications in which large quantum
C numbers arise, such as in molecular dynamics.
C
C***REFERENCES 1. Abramowitz, M., and Stegun, I. A., Eds., Handbook
C of Mathematical Functions with Formulas, Graphs
C and Mathematical Tables, NBS Applied Mathematics
C Series 55, June 1964 and subsequent printings.
C 2. Messiah, Albert., Quantum Mechanics, Volume II,
C North-Holland Publishing Company, 1963.
C 3. Schulten, Klaus and Gordon, Roy G., Exact recursive
C evaluation of 3j and 6j coefficients for quantum-
C mechanical coupling of angular momenta, J Math
C Phys, v 16, no. 10, October 1975, pp. 1961-1970.
C 4. Schulten, Klaus and Gordon, Roy G., Semiclassical
C approximations to 3j and 6j coefficients for
C quantum-mechanical coupling of angular momenta,
C J Math Phys, v 16, no. 10, October 1975,
C pp. 1971-1988.
C 5. Schulten, Klaus and Gordon, Roy G., Recursive
C evaluation of 3j and 6j coefficients, Computer
C Phys Comm, v 11, 1976, pp. 269-278.
C***ROUTINES CALLED R1MACH, XERMSG
C***REVISION HISTORY (YYMMDD)
C 750101 DATE WRITTEN
C 880515 SLATEC prologue added by G. C. Nielson, NBS; parameters
C HUGE and TINY revised to depend on R1MACH.
C 891229 Prologue description rewritten; other prologue sections
C revised; LMATCH (location of match point for recurrences)
C removed from argument list; argument IER changed to serve
C only as an error flag (previously, in cases without error,
C it returned the number of scalings); number of error codes
C increased to provide more precise error information;
C program comments revised; SLATEC error handler calls
C introduced to enable printing of error messages to meet
C SLATEC standards. These changes were done by D. W. Lozier,
C M. A. McClain and J. M. Smith of the National Institute
C of Standards and Technology, formerly NBS.
C 910415 Mixed type expressions eliminated; variable C1 initialized;
C description of THRCOF expanded. These changes were done by
C D. W. Lozier.
C***END PROLOGUE RC3JJ
C
INTEGER NDIM, IER
REAL L2, L3, M2, M3, L1MIN, L1MAX, THRCOF(NDIM)
C
INTEGER I, INDEX, LSTEP, N, NFIN, NFINP1, NFINP2, NFINP3, NLIM,
+ NSTEP2
REAL A1, A1S, A2, A2S, C1, C1OLD, C2, CNORM, R1MACH,
+ DENOM, DV, EPS, HUGE, L1, M1, NEWFAC, OLDFAC,
+ ONE, RATIO, SIGN1, SIGN2, SRHUGE, SRTINY, SUM1,
+ SUM2, SUMBAC, SUMFOR, SUMUNI, THREE, THRESH,
+ TINY, TWO, X, X1, X2, X3, Y, Y1, Y2, Y3, ZERO
C
DATA ZERO,EPS,ONE,TWO,THREE /0.0,0.01,1.0,2.0,3.0/
C
C***FIRST EXECUTABLE STATEMENT RC3JJ
IER=0
C HUGE is the square root of one twentieth of the largest floating
C point number, approximately.
HUGE = SQRT(R1MACH(2)/20.0)
SRHUGE = SQRT(HUGE)
TINY = 1.0/HUGE
SRTINY = 1.0/SRHUGE
C
C LMATCH = ZERO
M1 = - M2 - M3
C
C Check error conditions 1 and 2.
IF((L2-ABS(M2)+EPS.LT.ZERO).OR.
+ (L3-ABS(M3)+EPS.LT.ZERO))THEN
IER=1
CALL XERMSG('SLATEC','RC3JJ','L2-ABS(M2) or L3-ABS(M3) '//
+ 'less than zero.',IER,1)
RETURN
ELSEIF((MOD(L2+ABS(M2)+EPS,ONE).GE.EPS+EPS).OR.
+ (MOD(L3+ABS(M3)+EPS,ONE).GE.EPS+EPS))THEN
IER=2
CALL XERMSG('SLATEC','RC3JJ','L2+ABS(M2) or L3+ABS(M3) '//
+ 'not integer.',IER,1)
RETURN
ENDIF
C
C
C
C Limits for L1
C
L1MIN = MAX(ABS(L2-L3),ABS(M1))
L1MAX = L2 + L3
C
C Check error condition 3.
IF(MOD(L1MAX-L1MIN+EPS,ONE).GE.EPS+EPS)THEN
IER=3
CALL XERMSG('SLATEC','RC3JJ','L1MAX-L1MIN not integer.',IER,1)
RETURN
ENDIF
IF(L1MIN.LT.L1MAX-EPS) GO TO 20
IF(L1MIN.LT.L1MAX+EPS) GO TO 10
C
C Check error condition 4.
IER=4
CALL XERMSG('SLATEC','RC3JJ','L1MIN greater than L1MAX.',IER,1)
RETURN
C
C This is reached in case that L1 can take only one value,
C i.e. L1MIN = L1MAX
C
10 CONTINUE
C LSCALE = 0
THRCOF(1) = (-ONE) ** INT(ABS(L2+M2-L3+M3)+EPS) /
1 SQRT(L1MIN + L2 + L3 + ONE)
RETURN
C
C This is reached in case that L1 takes more than one value,
C i.e. L1MIN < L1MAX.
C
20 CONTINUE
C LSCALE = 0
NFIN = INT(L1MAX-L1MIN+ONE+EPS)
IF(NDIM-NFIN) 21, 23, 23
C
C Check error condition 5.
21 IER = 5
CALL XERMSG('SLATEC','RC3JJ','Dimension of result array for 3j '//
+ 'coefficients too small.',IER,1)
RETURN
C
C
C Starting forward recursion from L1MIN taking NSTEP1 steps
C
23 L1 = L1MIN
NEWFAC = 0.0
C1 = 0.0
THRCOF(1) = SRTINY
SUM1 = (L1+L1+ONE) * TINY
C
C
LSTEP = 1
30 LSTEP = LSTEP + 1
L1 = L1 + ONE
C
C
OLDFAC = NEWFAC
A1 = (L1+L2+L3+ONE) * (L1-L2+L3) * (L1+L2-L3) * (-L1+L2+L3+ONE)
A2 = (L1+M1) * (L1-M1)
NEWFAC = SQRT(A1*A2)
IF(L1.LT.ONE+EPS) GO TO 40
C
C
DV = - L2*(L2+ONE) * M1 + L3*(L3+ONE) * M1 + L1*(L1-ONE) * (M3-M2)
DENOM = (L1-ONE) * NEWFAC
C
IF(LSTEP-2) 32, 32, 31
C
31 C1OLD = ABS(C1)
32 C1 = - (L1+L1-ONE) * DV / DENOM
GO TO 50
C
C If L1 = 1, (L1-1) has to be factored out of DV, hence
C
40 C1 = - (L1+L1-ONE) * L1 * (M3-M2) / NEWFAC
C
50 IF(LSTEP.GT.2) GO TO 60
C
C
C If L1 = L1MIN + 1, the third term in the recursion equation vanishes,
C hence
X = SRTINY * C1
THRCOF(2) = X
SUM1 = SUM1 + TINY * (L1+L1+ONE) * C1*C1
IF(LSTEP.EQ.NFIN) GO TO 220
GO TO 30
C
C
60 C2 = - L1 * OLDFAC / DENOM
C
C Recursion to the next 3j coefficient X
C
X = C1 * THRCOF(LSTEP-1) + C2 * THRCOF(LSTEP-2)
THRCOF(LSTEP) = X
SUMFOR = SUM1
SUM1 = SUM1 + (L1+L1+ONE) * X*X
IF(LSTEP.EQ.NFIN) GO TO 100
C
C See if last unnormalized 3j coefficient exceeds SRHUGE
C
IF(ABS(X).LT.SRHUGE) GO TO 80
C
C This is reached if last 3j coefficient larger than SRHUGE,
C so that the recursion series THRCOF(1), ... , THRCOF(LSTEP)
C has to be rescaled to prevent overflow
C
C LSCALE = LSCALE + 1
DO 70 I=1,LSTEP
IF(ABS(THRCOF(I)).LT.SRTINY) THRCOF(I) = ZERO
70 THRCOF(I) = THRCOF(I) / SRHUGE
SUM1 = SUM1 / HUGE
SUMFOR = SUMFOR / HUGE
X = X / SRHUGE
C
C As long as ABS(C1) is decreasing, the recursion proceeds towards
C increasing 3j values and, hence, is numerically stable. Once
C an increase of ABS(C1) is detected, the recursion direction is
C reversed.
C
80 IF(C1OLD-ABS(C1)) 100, 100, 30
C
C
C Keep three 3j coefficients around LMATCH for comparison with
C backward recursion.
C
100 CONTINUE
C LMATCH = L1 - 1
X1 = X
X2 = THRCOF(LSTEP-1)
X3 = THRCOF(LSTEP-2)
NSTEP2 = NFIN - LSTEP + 3
C
C
C
C
C Starting backward recursion from L1MAX taking NSTEP2 steps, so
C that forward and backward recursion overlap at three points
C L1 = LMATCH+1, LMATCH, LMATCH-1.
C
NFINP1 = NFIN + 1
NFINP2 = NFIN + 2
NFINP3 = NFIN + 3
L1 = L1MAX
THRCOF(NFIN) = SRTINY
SUM2 = TINY * (L1+L1+ONE)
C
L1 = L1 + TWO
LSTEP = 1
110 LSTEP = LSTEP + 1
L1 = L1 - ONE
C
OLDFAC = NEWFAC
A1S = (L1+L2+L3)*(L1-L2+L3-ONE)*(L1+L2-L3-ONE)*(-L1+L2+L3+TWO)
A2S = (L1+M1-ONE) * (L1-M1-ONE)
NEWFAC = SQRT(A1S*A2S)
C
DV = - L2*(L2+ONE) * M1 + L3*(L3+ONE) * M1 + L1*(L1-ONE) * (M3-M2)
C
DENOM = L1 * NEWFAC
C1 = - (L1+L1-ONE) * DV / DENOM
IF(LSTEP.GT.2) GO TO 120
C
C If L1 = L1MAX + 1, the third term in the recursion formula vanishes
C
Y = SRTINY * C1
THRCOF(NFIN-1) = Y
SUMBAC = SUM2
SUM2 = SUM2 + TINY * (L1+L1-THREE) * C1*C1
C
GO TO 110
C
C
120 C2 = - (L1 - ONE) * OLDFAC / DENOM
C
C Recursion to the next 3j coefficient Y
C
Y = C1 * THRCOF(NFINP2-LSTEP) + C2 * THRCOF(NFINP3-LSTEP)
C
IF(LSTEP.EQ.NSTEP2) GO TO 200
C
THRCOF(NFINP1-LSTEP) = Y
SUMBAC = SUM2
SUM2 = SUM2 + (L1+L1-THREE) * Y*Y
C
C See if last unnormalized 3j coefficient exceeds SRHUGE
C
IF(ABS(Y).LT.SRHUGE) GO TO 110
C
C This is reached if last 3j coefficient larger than SRHUGE,
C so that the recursion series THRCOF(NFIN), ... ,THRCOF(NFIN-LSTEP+1)
C has to be rescaled to prevent overflow
C
C LSCALE = LSCALE + 1
DO 130 I=1,LSTEP
INDEX = NFIN - I + 1
IF(ABS(THRCOF(INDEX)).LT.SRTINY) THRCOF(INDEX) = ZERO
130 THRCOF(INDEX) = THRCOF(INDEX) / SRHUGE
SUM2 = SUM2 / HUGE
SUMBAC = SUMBAC / HUGE
C
C
GO TO 110
C
C
C The forward recursion 3j coefficients X1, X2, X3 are to be matched
C with the corresponding backward recursion values Y1, Y2, Y3.
C
200 Y3 = Y
Y2 = THRCOF(NFINP2-LSTEP)
Y1 = THRCOF(NFINP3-LSTEP)
C
C
C Determine now RATIO such that YI = RATIO * XI (I=1,2,3) holds
C with minimal error.
C
RATIO = ( X1*Y1 + X2*Y2 + X3*Y3 ) / ( X1*X1 + X2*X2 + X3*X3 )
NLIM = NFIN - NSTEP2 + 1
C
IF(ABS(RATIO).LT.ONE) GO TO 211
C
DO 210 N=1,NLIM
210 THRCOF(N) = RATIO * THRCOF(N)
SUMUNI = RATIO * RATIO * SUMFOR + SUMBAC
GO TO 230
C
211 NLIM = NLIM + 1
RATIO = ONE / RATIO
DO 212 N=NLIM,NFIN
212 THRCOF(N) = RATIO * THRCOF(N)
SUMUNI = SUMFOR + RATIO*RATIO*SUMBAC
GO TO 230
C
220 SUMUNI = SUM1
C
C
C Normalize 3j coefficients
C
230 CNORM = ONE / SQRT(SUMUNI)
C
C Sign convention for last 3j coefficient determines overall phase
C
SIGN1 = SIGN(ONE,THRCOF(NFIN))
SIGN2 = (-ONE) ** INT(ABS(L2+M2-L3+M3)+EPS)
IF(SIGN1*SIGN2) 235,235,236
235 CNORM = - CNORM
C
236 IF(ABS(CNORM).LT.ONE) GO TO 250
C
DO 240 N=1,NFIN
240 THRCOF(N) = CNORM * THRCOF(N)
RETURN
C
250 THRESH = TINY / ABS(CNORM)
DO 251 N=1,NFIN
IF(ABS(THRCOF(N)).LT.THRESH) THRCOF(N) = ZERO
251 THRCOF(N) = CNORM * THRCOF(N)
C
RETURN
END