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Gaunt.c

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include "openmx_common.h"

#define S3J_0           1e-10
#define S3J_MAX_FACT    40

#define S3J_EQUAL(a,b)        (fabs((a)-(b))<S3J_0)
#define S3J_MAX(a,b,c,ris)    (((a)>(b)?(ris=(a)):(ris=(b)))>(c)?ris:(ris=(c)))
#define S3J_MIN(a,b,c,ris)    (((a)<(b)?(ris=(a)):(ris=(b)))<(c)?ris:(ris=(c)))

static double Clebsch_Gordan(int j1, int m1,
                             int j2, int m2,
                             int j, int m);
static double s3j(double j1, double j2, double j3,
                double m1, double m2, double m3);

double Gaunt(int l,  int m,
             int l1, int m1,
             int l2, int m2)
{

  /************************************************************
   Ref. Eq.(3.7.73) in Modern Quantum Mechanics by J.J.Sakurai 
  ************************************************************/

  int Ls;
  double tmp0,tmp1,tmp2,tmp3;
  double result,cleb1,cleb2;

  cleb1 = Clebsch_Gordan(l1,0,l2,0,l,0);
  cleb2 = Clebsch_Gordan(l1,m1,l2,m2,l,m);

  tmp0 = 2.0*(double)l1 + 1.0;
  tmp1 = 2.0*(double)l2 + 1.0;
  tmp2 = 4.0*PI*(2.0*(double)l + 1.0);
  tmp3 = sqrt(tmp0*tmp1/tmp2);
  result = tmp3*cleb1*cleb2; 

  return result;
}

double Clebsch_Gordan(int j1, int m1,
                      int j2, int m2,
                      int j, int m)
{
  int esp;
  double cgris;

  esp=(int)(j1-j2+m);
  if (!S3J_EQUAL(esp,j1-j2+m)) return 0;

  if (esp%2==0) cgris=1.0;
  else          cgris=-1.0;

  cgris*=sqrt(2*j+1)*s3j((double)j1,(double)j2,(double)j,
                         (double)m1,(double)m2,(double)(-m));
  return cgris;
}




double s3j(double j1, double j2, double j3,
         double m1, double m2, double m3)
{

  /******************************************************************************

   ************************************************************** 
     This program was adopted from the following webpage by
     T.Ozaki at Sep. 13 2002. T.Ozaki greatly thanks the original
     author (Dr. Gusmeroli).
   ************************************************************** 

   http://www.ph.surrey.ac.uk/~phs3ps/cleb.html

   The below is the original message.

   Name: s3j
        Evaluates 3j symbol

    Author: Riccardo Gusmeroli (web.address@libero.it)

    Notes:
       - defining S3J_TEST enables the compilation of a very small test suite.
       - the maximum allowed factorial is S3J_MAX_FACT (currently 25!). 

     This program is free software; you can redistribute it and/or 
     modify it under the terms of the GNU General Public License 
     as published by the Free Software Foundation; either version 2 
     of the License, or (at your option) any later version.
     This program is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the 
     GNU General Public License for more details.
     You should have received a copy of the GNU General Public License 
     along with this program; if not, write to the Free Software 
     Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA.

  ( j1 j2 j3 )
  (          ) = delta(m1+m2+m3,0) * (-1)^(j1-j2-m3) * 
  ( m1 m2 m3 )

    +-
    |  (j1+j2-j3)! (j1-j2+j3)! (-j1+j2+j3)!
  * | -------------------------------------- ...
    |
    +-
                                                               -+ 1/2
         (j1-m1)! (j1+m1)! (j2-m2)! (j2+m2)! (j3-m3)! (j3+m3)!  |
    ... ------------------------------------------------------- |     *
                            (j1+j2+j3+1)!                       |
                                                               -+
     +---
      \                       (-1)^k
  *    |   ---------------------------------------------------------------------
      /      k! (j1+j2-j3-k)! (j1-m1-k)! (j2+m2-k)! (j3-j2+m1+k)! (j3-j1-m2+k)!
     +---
      k

  Where factorials must have non-negative integral values:

   j1+j2-j3   >= 0  j1-j2+j3   >= 0  -j1+j2+j3 >= 0 j1+j2+j3+1 >= 0
   k          >= 0  j1+j2-j3-k >= 0   j1-m1-k  >= 0 j2+m2-k    >= 0
   j3-j2+m1+k >= 0  j3-j1-m2+k >= 0

  The 3j symbol is therefore non-null if

      j1+j2 >= j3 (1)
      j1+j3 >= j2 (2)
      j2+j3 >= j1 (3)

  and k values in the sum must be such that

      k <= j1+j2-j3     (4) k >= 0      (7)
      k <= j1-m1  (5) k >= -j3+j2-m1  (8)
      k <= j2+m2  (6) k >= -j3+j1+m2  (9)

  If no values of k satisfy the (4) to (9), the result is null
  because the sum is null, otherwise one can find   kmin < kmax
  such that
             kmin <= k <= kmax

   (4) to (6) => kmin=MAX(j1+j2-j3,  j1-m1,      j2+m2    )
   (7) to (9) => kmax=MIN(0,        -j3+j2-m1,  -j3+j1+m2 )

  The condition kmin < kmax includes (1) to (3) because

      (4) and (7)    =>    (1)
      (5) and (8)    =>    (2)
      (6) and (9)    =>    (3)

  Once the values of kmin and kmax are found,
  the only "selection rule" is kmin<kmax.

  ******************************************************************************/

  int k, kmin, kmax;
  int jpm1, jmm1, jpm2, jmm2, jpm3, jmm3;
  int j1pj2mj3, j3mj2pm1, j3mj1mm2;
  double ris, mult, f[S3J_MAX_FACT];
      
  f[0]=1.0;
  mult=1.0;
  for (k=1; k<S3J_MAX_FACT; ++k) {
    f[k]=f[k-1]*mult;
    mult+=1.0;
  }

  jpm1=(int)(j1+m1);
  if (!S3J_EQUAL(jpm1,j1+m1)) return 0.0;

  jpm2=(int)(j2+m2);
  if (!S3J_EQUAL(jpm2,j2+m2)) return 0.0;

  jpm3=(int)(j3+m3);
  if (!S3J_EQUAL(jpm3,j3+m3)) return 0.0;

  jmm1=(int)(j1-m1);
  if (!S3J_EQUAL(jmm1,j1-m1)) return 0.0;

  jmm2=(int)(j2-m2);
  if (!S3J_EQUAL(jmm2,j2-m2)) return 0.0;

  jmm3=(int)(j3-m3);
  if (!S3J_EQUAL(jmm3,j3-m3)) return 0.0;

  /* delta(m1+m2+m3,0) */
  if ((jpm1-jmm1+jpm2-jmm2+jpm3-jmm3)!=0) return 0.0;

  /* j1+j2-j3 = (j1+j2-j3) + (m1+m2+m3) = jpm1+jpm2-jmm3 */
  j1pj2mj3=jpm1+jpm2-jmm3;

  /* j3-j2+m1 = (j3-j2+m1) - (m1+m2+m3) = jmm3-jpm2 */
  j3mj2pm1=jmm3-jpm2;

  /* j3-j1-m2 = (j3-j1-m2) + (m1+m2+m3) = jpm3-jmm1 */
  j3mj1mm2=jpm3-jmm1;

  S3J_MAX(-j3mj2pm1, -j3mj1mm2,   0,         kmin);
  S3J_MIN(j1pj2mj3,  jmm1,       jpm2,       kmax);
  if (kmin>kmax) return 0.0;  

  ris=0.0;
  if (kmin%2==0) mult=1.0;
  else mult=-1.0;
  for (k=kmin; k<=kmax; ++k) {
    ris+=mult/(f[k]*f[j1pj2mj3-k]*f[jmm1-k]*f[jpm2-k]*f[j3mj2pm1+k]*f[j3mj1mm2+k]);
    mult=-mult;
  }

  /* (-1)^(j1-j2-m3)=(-1)^(j1-j2-m3+m1+m2+m3)=(-1)^(jpm1-jmm2) */
  if ((jpm1-jmm2)%2!=0) ris=-ris;

  ris*=sqrt(f[j1pj2mj3]*f[jpm1-jmm2+jpm3]*f[-jmm1+jpm2+jpm3]*
          f[jpm1]*f[jpm2]*f[jpm3]*f[jmm1]*f[jmm2]*f[jmm3]/
          f[jpm1+jpm2+jpm3+1]);

  return ris;
}

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