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#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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