Function qsym2::angmom::sh_conversion::sh_c2r_mat
source · pub fn sh_c2r_mat(
l: u32,
csphase: bool,
pureorder: &PureOrder,
) -> Array2<Complex<f64>>
Expand description
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
- Iftrue
, $\lambda_{\mathrm{cs}}
$ is as defined incomplexc
. Iffalse
, $\lambda_{\mathrm{cs}} = 1
$.pureorder
- APureOrder
struct giving the ordering of the components of the pure Gaussians.
§Returns
The $\boldsymbol{\Upsilon}^{(l)}
$ matrix.