Blog Entry (c) Wednesday, November 6, 2024, by James Pate Williams, Jr. Small Angular Momentum Quantum Numbers Gaunt Coefficients

// GauntCoefficients.cpp (c) Monday, November 4, 2024
// by James Pate Williams, Jr., BA, BS, MSWE, PhD
// Computes the Gaunt angular momentum coefficients
// Also the Wigner-3j symbols are calculated 
// https://en.wikipedia.org/wiki/3-j_symbol
// https://doc.sagemath.org/html/en/reference/functions/sage/functions/wigner.html#
// https://www.geeksforgeeks.org/factorial-large-number/
#include <iostream>
using namespace std;
typedef long double real;
real pi;
// iterative n-factorial function
real Factorial(int n)
{
    real factorial = 1;

    for (int i = 2; i <= n; i++)
        factorial *= i;
    if (n < 0)
        factorial = 0;
    return factorial;
}
real Delta(int lt, int rt)
{
    return lt == rt ? 1.0 : 0.0;
}
real Wigner3j(
    int j1, int j2, int j3,
    int m1, int m2, int m3)
{
    real delta = Delta(m1 + m2 + m3, 0) * 
        powl(-1.0, j1 - j2 - m3);
    real fact1 = Factorial(j1 + j2 - j3);
    real fact2 = Factorial(j1 - j2 + j3);
    real fact3 = Factorial(-j1 + j2 + j3);
    real denom = Factorial(j1 + j2 + j3 + 1);
    real numer = delta * sqrt(
        fact1 * fact2 * fact3 / denom);
    real fact4 = Factorial(j1 - m1);
    real fact5 = Factorial(j1 + m1);
    real fact6 = Factorial(j2 - m2);
    real fact7 = Factorial(j2 + m2);
    real fact8 = Factorial(j3 - m3);
    real fact9 = Factorial(j3 + m3);
    real sqrt1 = sqrtl(
        fact4 * fact5 * fact6 * fact7 * fact8 * fact9);
    real sumK = 0;
    int K = (int)fmaxl(0, fmaxl((real)j2 - j3 - m1,
        (real)j1 - j3 + m2));
    int N = (int)fminl((real)j1 + j2 - j3, 
        fminl((real)j1 - m1, (real)j2 + m2));
    for (int k = K; k <= N; k++)
    {
        real f0 = Factorial(k);
        real f1 = Factorial(j1 + j2 - j3 - k);
        real f2 = Factorial(j1 - m1 - k);
        real f3 = Factorial(j2 + m2 - k);
        real f4 = Factorial(j3 - j2 + m1 + k);
        real f5 = Factorial(j3 - j1 - m2 + k);
        sumK += powl(-1.0, k) / (f0 * f1 * f2 * f3 * f4 * f5);
    }
    return numer * sqrt1 * sumK;
}
real GauntCoefficient(
    int l1, int l2, int l3, int m1, int m2, int m3)
{
    real factor = sqrtl(
        (2.0 * l1 + 1.0) *
        (2.0 * l2 + 1.0) *
        (2.0 * l3 + 1.0) /
        (4.0 * pi));
    real wigner1 = Wigner3j(l1, l2, l3, 0, 0, 0);
    real wigner2 = Wigner3j(l1, l2, l3, m1, m2, m3);
    return factor * wigner1 * wigner2;
}
int main()
{
    pi = 4.0 * atanl(1.0);
    cout << "Gaunt(1, 0, 1, 1, 0, 0)  = ";
    cout << GauntCoefficient(1, 0, 1, 1, 0, 0);
    cout << endl;
    cout << "Gaunt(1, 0, 1, 1, 0, -1) = ";
    cout << GauntCoefficient(1, 0, 1, 1, 0, -1);
    cout << endl;
    real number = -1.0 / 2.0 / sqrtl(pi);
    cout << "-1.0 / 2.0 / sqrt(pi)    = ";
    cout << number << endl;
    return 0;
}