Racah V-Coefficient

The Racah $V$-Coefficients are written

$$V(j_1j_2j;m_1m_2m)$$ (1)

and are sometimes expressed using the related Clebsch-Gordan Coefficients

$$C^j_{m_1m_2}=(j_1j_2m_1m_2\vert j_1j_2jm),$$ (2)

or Wigner 3j-Symbol. Connections among the three are

$$(j_1j_2m_1m_2\vert j_1j_2m)=(-1)^{-j_1+j_2-m} \sqrt{2j+1}\pmatrix{j_1 & j_2 & j\cr m_1 & m_2 & -m\cr}$$ (3)

$$(j_1j_2m_1m_2\vert j_1j_2jm)=(-1)^{j+m}\sqrt{2j+1}\,V(j_1j_2j; m_1m_2\, -m)$$ (4)

$$V(j_1j_2j;m_1m_2m)=(-1)^{-j_1+j_2+j}\pmatrix{j_1 & j_2 & j_1\cr m_2 & m_1 & m_2}.$$ (5)

See also Clebsch-Gordan Coefficient, Racah W-Coefficient, Wigner 3j-Symbol, Wigner 6j-Symbol, Wigner 9j-Symbol

References

Sobel'man, I. I. ``Angular Momenta.'' Ch. 4 in Atomic Spectra and Radiative Transitions, 2nd ed. Berlin: Springer-Verlag, 1992.