Function qsym2::angmom::sh_conversion::sh_cart2cl_mat
source · pub fn sh_cart2cl_mat(
l: u32,
lcart: u32,
cartorder: &CartOrder,
csphase: bool,
pureorder: &PureOrder,
) -> Array2<Complex<f64>>
Expand description
Obtains the matrix $\mathbf{V}^{(l, l_{\mathrm{cart}})}
$ containing linear combination
coefficients of complex solid harmonic Gaussians of a specific degree in the expansion of
Cartesian Gaussians, i.e., briefly,
\mathbf{g}^{\mathsf{T}}(l_{\mathrm{cart}})
= \tilde{\mathbf{g}}^{\mathsf{T}}(l)
\ \mathbf{V}^{(l, l_{\mathrm{cart}})}.
Let $\tilde{g}(\alpha, \lambda, l_{\mathrm{cart}}, \mathbf{r})
$ be a complex solid harmonic
Gaussian as defined in Equation 1 of Schlegel, H. B. & Frisch, M. J. Transformation between
Cartesian and pure spherical harmonic Gaussians. International Journal of Quantum Chemistry
54, 83–87 (1995), DOI with
$n = l_{\mathrm{cart}}
$, and let $g(\alpha, \lambda_{\mathrm{cart}}, \mathbf{r})
$ be a
Cartesian Gaussian as defined in Equation 2 of the above reference. Here, $\lambda
$ is a
single index labelling a complex solid harmonic Gaussian of spherical harmonic degree $l
$
and order $m_l
$, and $\lambda_{\mathrm{cart}}
$ a single index labelling a Cartesian
Gaussian of degrees $(l_x, l_y, l_z)
$ such that $l_x + l_y + l_z = l_{\mathrm{cart}}
$.
We can then write
g(\alpha, \lambda_{\mathrm{cart}}, \mathbf{r})
= \sum_{\substack{\lambda\\ l \leq l_{\mathrm{cart}}}}
\tilde{g}(\alpha, \lambda, l_{\mathrm{cart}}, \mathbf{r})
V^{(l_{\mathrm{cart}})}_{\lambda\lambda_{\mathrm{cart}}}
where $V^{(l_{\mathrm{cart}})}_{\lambda\lambda_{\mathrm{cart}}}
$ is given by the inverse
complex coefficients
V^{(l_{\mathrm{cart}})}_{\lambda\lambda_{\mathrm{cart}}} =
c^{-1}(l_x, l_y, l_z, l, m_l, l_{\mathrm{cart}})
defined in complexcinv
.
We can order the rows $\lambda
$ of $\mathbf{V}^{(l_{\mathrm{cart}})}
$ that have the same
$l
$ into rectangular blocks of dimensions
$(2l+1) \times \frac{1}{2}(l_{\mathrm{cart}}+1)(l_{\mathrm{cart}}+2)
$
which give contributions from complex solid harmonic Gaussians of a particular degree $l
$.
We denote these blocks $\mathbf{V}^{(l, l_{\mathrm{cart}})}
$.
They contain only zero elements if $l
$ and $l_{\mathrm{cart}}
$ have different parities.
§Arguments
l
- The degree of the complex spherical harmonic factor in the solid harmonic Gaussian.lcart
- The total Cartesian degree for the Cartesian Gaussians and also for the radial part of the solid harmonic Gaussian.cartorder
- ACartOrder
struct giving the ordering of the components of the Cartesian Gaussians.csphase
- Set totrue
to use the Condon–Shortley phase in the calculations of the $c^{-1}
$ coefficients. Seecomplexc
andcomplexcinv
for more details.pureorder
- APureOrder
struct giving the ordering of the components of the pure Gaussians.
§Returns
The $\mathbf{V}^{(l, l_{\mathrm{cart}})}
$ block.