*DECK RC3JM
SUBROUTINE RC3JM (L1, L2, L3, M1, M2MIN, M2MAX, THRCOF, NDIM, IER)
C***BEGIN PROLOGUE RC3JM
C***PURPOSE Evaluate the 3j symbol g(M2) = (L1 L2 L3 )
C (M1 M2 -M1-M2)
C for all allowed values of M2, the other parameters
C being held fixed.
C***LIBRARY SLATEC
C***CATEGORY C19
C***TYPE SINGLE PRECISION (RC3JM-S, DRC3JM-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 L1, L2, L3, M1, M2MIN, M2MAX, THRCOF(NDIM)
C INTEGER NDIM, IER
C
C CALL RC3JM (L1, L2, L3, M1, M2MIN, M2MAX, THRCOF, NDIM, IER)
C
C *Arguments:
C
C L1 :IN Parameter in 3j symbol.
C
C L2 :IN Parameter in 3j symbol.
C
C L3 :IN Parameter in 3j symbol.
C
C M1 :IN Parameter in 3j symbol.
C
C M2MIN :OUT Smallest allowable M2 in 3j symbol.
C
C M2MAX :OUT Largest allowable M2 in 3j symbol.
C
C THRCOF :OUT Set of 3j coefficients generated by evaluating the
C 3j symbol for all allowed values of M2. THRCOF(I)
C will contain g(M2MIN+I-1), I=1,2,...,M2MAX-M2MIN+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 L1.LT.ABS(M1) or L1+ABS(M1) non-integer.
C IER=2 ABS(L1-L2).LE.L3.LE.L1+L2 not satisfied.
C IER=3 L1+L2+L3 not an integer.
C IER=4 M2MAX-M2MIN not an integer.
C IER=5 M2MAX less than M2MIN.
C IER=6 NDIM less than M2MAX-M2MIN+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. L1.GE.ABS(M1) and L1+ABS(M1) must be an integer;
C 2. ABS(L1-L2).LE.L3.LE.L1+L2;
C 3. L1+L2+L3 must be an integer;
C 4. M2MAX-M2MIN must be an integer, where
C M2MAX=MIN(L2,L3-M1) and M2MIN=MAX(-L2,-L3-M1).
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 g(M2) = (0.75 1.50 1.75 )
C (0.25 M2 -0.25-M2)
C for M2=-1.5,-0.5,0.5,1.5 but none of the symmetry properties of the
C 3j symbol, set forth on page 1056 of Messiah, is satisfied.
C
C The subroutine generates g(M2MIN), g(M2MIN+1), ..., g(M2MAX)
C where M2MIN and M2MAX are defined above. The sequence g(M2) 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; MMATCH (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 RC3JM
C
INTEGER NDIM, IER
REAL L1, L2, L3, M1, M2MIN, M2MAX, THRCOF(NDIM)
C
INTEGER I, INDEX, LSTEP, N, NFIN, NFINP1, NFINP2, NFINP3, NLIM,
+ NSTEP2
REAL A1, A1S, C1, C1OLD, C2, CNORM, R1MACH, DV, EPS,
+ HUGE, M2, M3, NEWFAC, OLDFAC, ONE, RATIO, SIGN1,
+ SIGN2, SRHUGE, SRTINY, SUM1, SUM2, SUMBAC,
+ SUMFOR, SUMUNI, THRESH, TINY, TWO, X, X1, X2, X3,
+ Y, Y1, Y2, Y3, ZERO
C
DATA ZERO,EPS,ONE,TWO /0.0,0.01,1.0,2.0/
C
C***FIRST EXECUTABLE STATEMENT RC3JM
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 MMATCH = ZERO
C
C
C Check error conditions 1, 2, and 3.
IF((L1-ABS(M1)+EPS.LT.ZERO).OR.
+ (MOD(L1+ABS(M1)+EPS,ONE).GE.EPS+EPS))THEN
IER=1
CALL XERMSG('SLATEC','RC3JM','L1-ABS(M1) less than zero or '//
+ 'L1+ABS(M1) not integer.',IER,1)
RETURN
ELSEIF((L1+L2-L3.LT.-EPS).OR.(L1-L2+L3.LT.-EPS).OR.
+ (-L1+L2+L3.LT.-EPS))THEN
IER=2
CALL XERMSG('SLATEC','RC3JM','L1, L2, L3 do not satisfy '//
+ 'triangular condition.',IER,1)
RETURN
ELSEIF(MOD(L1+L2+L3+EPS,ONE).GE.EPS+EPS)THEN
IER=3
CALL XERMSG('SLATEC','RC3JM','L1+L2+L3 not integer.',IER,1)
RETURN
ENDIF
C
C
C Limits for M2
M2MIN = MAX(-L2,-L3-M1)
M2MAX = MIN(L2,L3-M1)
C
C Check error condition 4.
IF(MOD(M2MAX-M2MIN+EPS,ONE).GE.EPS+EPS)THEN
IER=4
CALL XERMSG('SLATEC','RC3JM','M2MAX-M2MIN not integer.',IER,1)
RETURN
ENDIF
IF(M2MIN.LT.M2MAX-EPS) GO TO 20
IF(M2MIN.LT.M2MAX+EPS) GO TO 10
C
C Check error condition 5.
IER=5
CALL XERMSG('SLATEC','RC3JM','M2MIN greater than M2MAX.',IER,1)
RETURN
C
C
C This is reached in case that M2 and M3 can take only one value.
10 CONTINUE
C MSCALE = 0
THRCOF(1) = (-ONE) ** INT(ABS(L2-L3-M1)+EPS) /
1 SQRT(L1+L2+L3+ONE)
RETURN
C
C This is reached in case that M1 and M2 take more than one value.
20 CONTINUE
C MSCALE = 0
NFIN = INT(M2MAX-M2MIN+ONE+EPS)
IF(NDIM-NFIN) 21, 23, 23
C
C Check error condition 6.
21 IER = 6
CALL XERMSG('SLATEC','RC3JM','Dimension of result array for 3j '//
+ 'coefficients too small.',IER,1)
RETURN
C
C
C
C Start of forward recursion from M2 = M2MIN
C
23 M2 = M2MIN
THRCOF(1) = SRTINY
NEWFAC = 0.0
C1 = 0.0
SUM1 = TINY
C
C
LSTEP = 1
30 LSTEP = LSTEP + 1
M2 = M2 + ONE
M3 = - M1 - M2
C
C
OLDFAC = NEWFAC
A1 = (L2-M2+ONE) * (L2+M2) * (L3+M3+ONE) * (L3-M3)
NEWFAC = SQRT(A1)
C
C
DV = (L1+L2+L3+ONE)*(L2+L3-L1) - (L2-M2+ONE)*(L3+M3+ONE)
1 - (L2+M2-ONE)*(L3-M3-ONE)
C
IF(LSTEP-2) 32, 32, 31
C
31 C1OLD = ABS(C1)
32 C1 = - DV / NEWFAC
C
IF(LSTEP.GT.2) GO TO 60
C
C
C If M2 = M2MIN + 1, the third term in the recursion equation vanishes,
C hence
C
X = SRTINY * C1
THRCOF(2) = X
SUM1 = SUM1 + TINY * C1*C1
IF(LSTEP.EQ.NFIN) GO TO 220
GO TO 30
C
C
60 C2 = - OLDFAC / NEWFAC
C
C Recursion to the next 3j coefficient
X = C1 * THRCOF(LSTEP-1) + C2 * THRCOF(LSTEP-2)
THRCOF(LSTEP) = X
SUMFOR = SUM1
SUM1 = SUM1 + 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 MSCALE = MSCALE + 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
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 MMATCH for comparison later
C with backward recursion values.
C
100 CONTINUE
C MMATCH = M2 - 1
NSTEP2 = NFIN - LSTEP + 3
X1 = X
X2 = THRCOF(LSTEP-1)
X3 = THRCOF(LSTEP-2)
C
C Starting backward recursion from M2MAX taking NSTEP2 steps, so
C that forwards and backwards recursion overlap at the three points
C M2 = MMATCH+1, MMATCH, MMATCH-1.
C
NFINP1 = NFIN + 1
NFINP2 = NFIN + 2
NFINP3 = NFIN + 3
THRCOF(NFIN) = SRTINY
SUM2 = TINY
C
C
C
M2 = M2MAX + TWO
LSTEP = 1
110 LSTEP = LSTEP + 1
M2 = M2 - ONE
M3 = - M1 - M2
OLDFAC = NEWFAC
A1S = (L2-M2+TWO) * (L2+M2-ONE) * (L3+M3+TWO) * (L3-M3-ONE)
NEWFAC = SQRT(A1S)
DV = (L1+L2+L3+ONE)*(L2+L3-L1) - (L2-M2+ONE)*(L3+M3+ONE)
1 - (L2+M2-ONE)*(L3-M3-ONE)
C1 = - DV / NEWFAC
IF(LSTEP.GT.2) GO TO 120
C
C If M2 = M2MAX + 1 the third term in the recursion equation vanishes
C
Y = SRTINY * C1
THRCOF(NFIN-1) = Y
IF(LSTEP.EQ.NSTEP2) GO TO 200
SUMBAC = SUM2
SUM2 = SUM2 + Y*Y
GO TO 110
C
120 C2 = - OLDFAC / NEWFAC
C
C Recursion to the next 3j coefficient
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 + Y*Y
C
C
C See if last 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 MSCALE = MSCALE + 1
DO 111 I=1,LSTEP
INDEX = NFIN - I + 1
IF(ABS(THRCOF(INDEX)).LT.SRTINY)
1 THRCOF(INDEX) = ZERO
111 THRCOF(INDEX) = THRCOF(INDEX) / SRHUGE
SUM2 = SUM2 / HUGE
SUMBAC = SUMBAC / HUGE
C
GO TO 110
C
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((L1+L1+ONE) * SUMUNI)
C
C Sign convention for last 3j coefficient determines overall phase
C
SIGN1 = SIGN(ONE,THRCOF(NFIN))
SIGN2 = (-ONE) ** INT(ABS(L2-L3-M1)+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
C
C
RETURN
END