Gaunt's formula is a particular instance of this theorem, which is most often proven using the definition of a spherical basis.
In crystal field theory, this statement can be used to write the crystal field Hamiltonian in terms of both crystal field coefficients and Stevens coefficients. One term on one side of this statement appears within a double bar and has a normalization factor of the square root of quantity 2j plus 1. A particular instance of this result can be used to simplify Slater integrals, which contain the product of three spherical harmonics. Applying this theorem to a vector operator is used to find a statement used, in turn, to derive Lande's formula for the g-factor; that corollary of this theorem is the (*) projection theorem. Gaunt's formula is a particular instance of this theorem, which is most often proven using the definition of a spherical basis. One way of writing this theorem relies on a mathematical device that is invariant under even permutations of its three columns; that is its first namesake's 3j symbols. For 10 points, name this theorem that decomposes matrix elements of certain tensor operators into an orientation-independent component and a Clebsch-Gordon coefficient.
In crystal field theory, this statement can be used to write the crystal field Hamiltonian in terms of both crystal field coefficients and Stevens coefficients.
Wigner-Eckart theorem [accept projection theorem and Gaunt’s formula before mention]