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Gaunt Coefficient

Gaunt coefficients are the coefficients defined by integrals of products of three spherical harmonics,

G(l_1,m_1;l_2,m_2;l_3,m_3)=int_0^(2pi)int_0^piY_(l_1)^(m_1)(theta,phi)Y_(l_2)^(m_2)(theta,phi)Y_(l_3)^(m_3)(theta,phi)sinthetadthetadphi.

In terms of the Wigner 3j-symbols,

G(l_1,m_1;l_2,m_2;l_3,m_3)=sqrt(((2l_1+1)(2l_2+1)(2l_3+1))/(4pi))(l_1 l_2 l_3; 0 0 0)(l_1 l_2 l_3; m_1 m_2 m_3).

The coefficient is zero unless the corresponding Wigner 3j-symbol selection rules are satisfied; in particular, m_1+m_2+m_3=0 and the angular momenta obey the triangle condition.

See also

Clebsch-Gordan Coefficient, Spherical Harmonic, Wigner 3j-Symbol

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References

Arfken, G. "Spherical Harmonics" and "Integrals of the Products of Three Spherical Harmonics." §12.6 and 12.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 680-685 and 698-700, 1985.Lehtola, S. "libwignernj: A Reusable C/C++/Fortran/Python Library for Exact Wigner Symbols and Related Coefficients." 2 Jul 2026. https://arxiv.org/abs/2605.06634.

Cite this as:

Weisstein, Eric W. "Gaunt Coefficient." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GauntCoefficient.html

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