Function sh_r2c_mat

pub fn sh_r2c_mat(
    l: u32,
    csphase: bool,
    pureorder: &PureOrder,
) -> Array2<Complex<f64>>

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

Let $Y_l^m$ be a complex spherical harmonic of degree $l$ and order $m$. Then, a real degree-$l$ spherical harmonic $Y_{lm}$ can be defined as

Y_{lm} =
    \begin{cases}
        \frac{\mathbb{i}}{\sqrt{2}}
        \left(Y_l^{-\lvert m \rvert}
              - \lambda'_{\mathrm{cs}} Y_l^{\lvert m \rvert}\right)
        & \mathrm{if}\ m < 0 \\
        Y_l^0 & \mathrm{if}\ m = 0 \\
        \frac{1}{\sqrt{2}}
        \left(Y_l^{-\lvert m \rvert}
              + \lambda'_{\mathrm{cs}} Y_l^{\lvert m \rvert}\right)
        & \mathrm{if}\ m > 0 \\
    \end{cases}

where $\lambda'_{\mathrm{cs}} = (-1)^{\lvert m \rvert}$ if the Condon–Shortley phase as defined in complexc is employed for the complex spherical harmonics, and $\lambda'_{\mathrm{cs}} = 1$ otherwise. The linear combination coefficients turn out to be given by the elements of matrix $\boldsymbol{\Upsilon}^{(l)\dagger}$ of dimensions $(2l+1)\times(2l+1)$ such that

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

It is obvious from the orthonormality of $Y_{lm}$ and $Y_l^m$ that $\boldsymbol{\Upsilon}^{(l)\dagger} = [\boldsymbol{\Upsilon}^{(l)}]^{-1}$ where $\boldsymbol{\Upsilon}^{(l)}$ is defined in sh_c2r_mat.

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