Function sh_c2r_mat

pub fn sh_c2r_mat(
    l: u32,
    csphase: bool,
    pureorder: &PureOrder,
) -> Array2<Complex<f64>>

Obtains the transformation matrix $\boldsymbol{\Upsilon}^{(l)}$ allowing complex spherical harmonics to be expressed as linear combinations of real spherical harmonics.

Let $Y_{lm}$ be a real spherical harmonic of degree $l$. Then, a complex spherical harmonic of degree $l$ and order $m$ is given by

Y_l^m =
    \begin{cases}
        \frac{\lambda_{\mathrm{cs}}}{\sqrt{2}}
        \left(Y_{l\lvert m \rvert}
              - \mathbb{i} Y_{l,-\lvert m \rvert}\right)
        & \mathrm{if}\ m < 0 \\
        Y_{l0} & \mathrm{if}\ m = 0 \\
        \frac{\lambda_{\mathrm{cs}}}{\sqrt{2}}
        \left(Y_{l\lvert m \rvert}
              + \mathbb{i} Y_{l,-\lvert m \rvert}\right)
        & \mathrm{if}\ m > 0 \\
    \end{cases}

where $\lambda_{\mathrm{cs}}$ is the Condon–Shortley phase as defined in complexc. The linear combination coefficients can then be gathered into a square matrix $\boldsymbol{\Upsilon}^{(l)}$ of dimensions $(2l+1)\times(2l+1)$ such that

    Y_l^m = \sum_{m'} Y_{lm'} \Upsilon^{(l)}_{m'm}.

Arguments

• l - The spherical harmonic degree.
• csphase - If true, $\lambda_{\mathrm{cs}}$ is as defined in complexc. If false, $\lambda_{\mathrm{cs}} = 1$.
• pureorder - A PureOrder struct giving the ordering of the components of the pure Gaussians.

Returns

The $\boldsymbol{\Upsilon}^{(l)}$ matrix.