Does anybody know about an accurate quadrature rule over three Euler angles $\theta, \phi, \chi$? I am trying to calculate the average value of an arbitrary function $f(\theta, \phi, \chi)$ for a given probability distribution $\rho(\theta, \phi, \chi)$:
$$ \langle f\rangle_{\rho} = \int_0^{2\pi} d\chi \int_0^{2\pi} d\phi\int_0^{\pi} \sin\theta d\theta f(\theta, \phi, \chi)\rho(\theta, \phi, \chi) \approx \sum_{l} w_l g(\theta_l, \phi_l, \chi_l) $$ where $f(\theta, \phi, \chi)$ is not sum-of-products of single angle-coordinate functions. $w_l$ are quadrature weights and $(\theta_l, \phi_l, \chi_l)$ represents a single grid point. $g(\theta, \phi, \chi) = \sin\theta f(\theta, \phi, \chi)\rho(\theta, \phi, \chi)$ and an appropriate quadrature weight-function is implicitly included in $f$.
Such an integration is sometimes needed in orientation averaged calculations of material/molecule's properties.
What we can assume about $\rho(\theta, \phi, \chi)$ is that it is a linear combination of the products of the Wigner D-matrices: $\rho(\theta, \phi, \chi) = \sum_{K,K'} c_{KK'} D^{(J)}_{KM}(\theta, \phi, \chi)D^{(J')}_{K'M'}(\theta, \phi, \chi)$.
So to sum up. It seems there is a need for a simple quadrature scheme for integration over three Euler angles: $\theta, \phi, \chi$.
- Update:
Ways around the need for explicit three-Euler angle $\theta, \phi, \chi$ quadrature:
Expanding $f(\theta, \phi, \chi)$ in the Wigner matrices basis is one way of proceeding here. The appropriate integrals can be calculated analytically. But for some functions $f(\theta, \phi, \chi)$ this method is very inefficient, as the Wigner expansion has many terms.
Another possibility, if we are dealing with a quantum system, is to solve the Schroedinger equation for wavefunctions in three separate system-fixed embeddings of the coordinate frame. Then one can choose the system-fixed z-axis to be the axis in the system that we want to quantify (average over). In such a case only $\theta, \phi$ are needed. The down-side to this approach is that one needs to repeat calculations for three independent embeddings, which for some embeddings and some external potentials can be very unnatural. In the case of rotational dynamics problems the complete symmetric-top basis sets guarantee accuracy of the solutions regardless of the embedding chosen. Rotational wavefunction representations among different embeddings differ only by the coefficients. In good embeddings the rotational wavefunction can be reprsented compactly in the basis, but if the embedding is poor, often many basis functions are needed. In the case of problems which add coupling of rotational degrees of freedom to some other internal or external degrees of freedom, the choice of embedding is often critical for quick convergence of the variational procedure.
