forte2.symmetry.mo_sym_detect
=============================

.. py:module:: forte2.symmetry.mo_sym_detect






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

.. py:function:: local_sign(l, m, op)

   
   Return phase describing how the spherical harmonic Y_lm transforms under Abelian symmetry
   operations `op`, which is one of [E, C2z, C2x, C2y, σ_xy, σ_xz, σ_yz].
















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.. py:function:: get_symmetry_ops(point_group)

   
   Compute 3x3 matrix representations for the symmetry operators in `point_group`.
   These representation perform reflections/rotations in the molecular principal frame.
















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.. py:class:: MOSymmetryDetector(system, info, S, C, eps, tol=1e-06)

   
   Class to detect the irreducible representation (irrep) labels of molecular orbitals.


   :Parameters:

       **system** : forte2.System
           Forte2 System object with symmetry information

       **info** : forte2.BasisInfo
           Forte2 BasisInfo object with basis set information

       **S** : ndarray
           AO overlap matrix

       **C** : ndarray
           MO coefficient matrix (columns are MOs)

       **eps** : ndarray
           MO energies

       **tol** : float, optional, default=1e-6
           tolerance for matching atomic positions under symmetry operations

   :Attributes:

       **irrep_indices** : list of int
           List of irrep indices for each MO according to COTTON_LABELS

       **labels** : list of str
           List of irrep labels for each MO (e.g. 'a1', 'b2', etc.)










   .. rubric:: Notes

   We compute symmetry irreps for MOs a posteriori by computing the character of each MO
   under each symmetry operation of the point group. In a nutshell, what we want is the character

   .. math::

       \chi(g)_{p} = \langle p | \hat{R}(g) | p \rangle = \sum_{uv} c_{pu}^* c_{pv} \langle u | \hat{R}(g) | v \rangle

   Each AO function :math:`|u\rangle \sim R_{nl}(r) Y_{lm}(\theta,\phi)`.
   For Abelian point groups, we only need to consider
   C2 rotations and mirror planes. None of these affect the radial part, but they transform the
   angular part to a symmetric partner on the same or a different atom with a phase.

   If :math:`\hat{R}(g)|v\rangle = \sum_{w} U_{vw} |w\rangle`, then :math:`\langle u | \hat{R}(g) | v \rangle = \sum_{w} U_{vw} \langle u | w \rangle`,
   and :math:`\chi(g)_{p} = \sum_{uvw} c_{pu}^* c_{pv} U_{vw} S_{uw}.`

   The above procedure works if the symmetry operations do not mix MOs (i.e., the true molecular point group is Abelian).
   If some MOs are mixed by the symmetry operations, then we need diagonalize subsets of the MO space
   that are mixed together to obtain MOs that purely transform as irreps of the Abelian subgroup.



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   .. py:attribute:: system


   .. py:attribute:: info


   .. py:attribute:: S


   .. py:attribute:: C


   .. py:attribute:: eps


   .. py:attribute:: tol
      :value: 1e-06



   .. py:attribute:: two_component


   .. py:method:: run()