forte2.symmetry.sph_harm_utils
==============================

.. py:module:: forte2.symmetry.sph_harm_utils




Module Contents
---------------

.. py:function:: sph_real_to_complex(l)

   
   Conversion matrix from real spherical harmonics to complex spherical harmonics for a given l.


   :Parameters:

       **l** : int
           The angular momentum quantum number.



   :Returns:

       NDArray
           The conversion matrix of shape (2*l+1, 2*l+1).








   .. rubric:: Notes

   See https://en.wikipedia.org/wiki/Spherical_harmonics#Real_form
   Y^m_l are the complex spherical harmonics, and Y_{lm} are the real spherical harmonics.



   ..
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.. py:function:: sph_complex_to_real(l)

   
   Conversion matrix from complex spherical harmonics to real spherical harmonics for a given l.


   :Parameters:

       **l** : int
           The angular momentum quantum number.



   :Returns:

       NDArray
           The conversion matrix of shape (2*l+1, 2*l+1).











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.. py:function:: clebsh_gordan_spin_half(l, msdouble, jdouble, mjdouble)

   
   Clebsch Gordon coefficient specialized for coupling orbital angular momentum l
   with spin 1/2 to form total angular momentum j = l + 1/2, |l - 1/2|

   .. math::
       C^{j,m_j}_{l,m_l;1/2,m_s} = \langle l,m_l;\frac{1}{2},m_s|j,m_j\rangle

   Since m_s + m_l must equal m_j, specifying m_j and m_s uniquely determines m_l.

   :Parameters:

       **l** : int
           The orbital angular momentum quantum number.

       **jdouble** : int
           The total angular momentum quantum number
           j is either l + 1/2 or l - 1/2.

       **mjdouble** : int
           The magnetic quantum number of the total angular momentum.
           m_j can take a value in {j, j-1, ..., -j}.

       **msdouble** : int
           The magnetic quantum number of the z-component of the total angular momentum.
           mz is either 1/2 or -1/2.











   .. rubric:: Notes

   1. Ch. 4.1.2, Atkins, P. W. (2011). Molecular Quantum Mechanics Fifth Ed.
   2. https://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients#Special_cases



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.. py:function:: real_sph_to_j_adapted_per_l(l)

   
   Transformation matrix that transforms real spherical harmonics 
   of a given angular momentum to (j-adapted) spinor basis.


   :Parameters:

       **l** : int
           The angular momentum quantum number.



   :Returns:

       tuple[NDArray]
           The transformation matrices for alpha and beta spinors.
           Each matrix has shape (2*l+1, 4*l+2).








   .. rubric:: Notes

   The transformation is based on the Clebsch-Gordan coefficients for coupling
   the orbital angular momentum l with spin 1/2 to form total angular momentum j = l + 1/2, l - 1/2.
   l = 0 is a special case where the transformation is trivial.

   .. math::
       |j, m_j> = \sum_{m_l=-l}^l\sum_{m_s=-1/2}^{1/2} |l, m_l; 1/2, m_s\rangle\langle l, m_l; 1/2, m_s|j, m_j\rangle\\
               = \sum_{m_l=-l}^l\sum_{m_s=-1/2}^{1/2} |l, m_l; 1/2, m_s\rangle * C^{j,m_j}_{l,m_l;1/2,m_s}.



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.. py:function:: real_sph_to_j_adapted(basis)

   
   Transformation matrix that transforms real-spherical GTOs to spinor
   GTOs for all basis functions
















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