Function qsym2::angmom::sh_conversion::sh_r2c_mat
source · pub fn sh_r2c_mat(
l: u32,
csphase: bool,
pureorder: &PureOrder,
) -> Array2<Complex<f64>>
Expand description
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
- 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)\dagger}
$ matrix.