Module qsym2::angmom::sh_conversion

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,