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Conversions between spherical/solid harmonics and Cartesian functions.
Functions§
- complexc
- Obtain the complex coefficients $
c(l, m_l, n, l_x, l_y, l_z)
$ based on Equation 15 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, but more generalised for $l \leq l_{\mathrm{cart}} = l_x + l_y + l_z
$. - complexcinv
- Computes the inverse complex coefficients $
c^{-1}(l_x, l_y, l_z, l, m_l, l_{\mathrm{cart}})
$ based on Equation 18 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, but more generalised for $l \leq l_{\mathrm{cart}} = l_x + l_y + l_z
$. - sh_
c2r_ mat - Obtains the transformation matrix $
\boldsymbol{\Upsilon}^{(l)}
$ allowing complex spherical harmonics to be expressed as linear combinations of real spherical harmonics. - sh_
cart2cl_ mat - 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, - sh_
cart2r - Returns a list of $
\mathbf{X}^{(l, l_{\mathrm{cart}})}
$ for $l_{\mathrm{cart}} \ge l \ge 0
$ and $l \equiv l_{\mathrm{cart}} \mod 2
$. - sh_
cart2rl_ mat - Obtains the real matrix $
\mathbf{X}^{(l, l_{\mathrm{cart}})}
$ containing linear combination coefficients of real solid harmonic Gaussians of a specific degree in the expansion of Cartesian Gaussians, i.e., briefly, - sh_
cl2cart_ mat - Obtains the matrix $
\mathbf{U}^{(l_{\mathrm{cart}}, l)}
$ containing linear combination coefficients of Cartesian Gaussians in the expansion of a complex solid harmonic Gaussian, i.e., briefly, - sh_
r2c_ mat - Obtains the matrix $
\boldsymbol{\Upsilon}^{(l)\dagger}
$ allowing real spherical harmonics to be expressed as linear combinations of complex spherical harmonics. - sh_
r2cart - Returns a list of $
\mathbf{W}^{(l_{\mathrm{cart}}, l)}
$ for $l_{\mathrm{cart}} \ge l \ge 0
$ and $l \equiv l_{\mathrm{cart}} \mod 2
$. - sh_
rl2cart_ mat - Obtain the matrix $
\mathbf{W}^{(l_{\mathrm{cart}}, l)}
$ containing linear combination coefficients of Cartesian Gaussians in the expansion of a real solid harmonic Gaussian, i.e., briefly,