# Simplifying Wigner sums

Rotations of second rank tensors are common in quantum mechanics and Nuclear Magnetic Resonance theory. The sums can often be simplified into simplified real components, but when many sums exist, simplification can be challenging to do by hand. For example,

$$V_{20}' = \sum_m V_{2m} D_{m0}^{(2)} (\phi, \theta, 0)$$

has 5 terms. The simplified result should be:

$$V_{20}' = V_{20} \cdot \tfrac{1}{2} (3\cos^2\theta - 1) + \sqrt{\tfrac{3}{8}} (V_{22} + V_{2-2}) \sin^2\theta cos2\phi$$

The tensor is symmetric and $$V_{2\pm1}=0$$.

I've used Mathematica's built-in Wigner matrices as well as my own, and I can't simplify the equation to yield a compact expression. For example,

Sum[Subscript[V, 2, m]*
WignerD[{2, m, 0}, \[Phi], \[Theta], 0], {m, -2, 2, 2}] // FullSimplify


yields

$$\frac{1}{4} \left(V_{2,0} (3 \cos (2 \theta )+1)+\sqrt{6} e^{-2 i \phi } \sin ^2(\theta ) \left(V_{2,-2}+e^{4 i \phi } V_{2,2}\right)\right)$$

This expression is correct, but it is not compact with trig functions. The expression has complex exponentials with components that cancel, and the 2nd Legendre polynomial is eliminated in the $$m=0$$ term. This example can be solved by hand, but rotations with 25 or 125 components are not uncommon, and it would be immensely helpful to simplify these expressions. Are there techniques or tools in Mathematica to more easily work with sums of Wigner matrices?

Thank you,

Justin