.. _sphx_glr_api_gallery_structure_point_group:

Point-group determination
--------------------------

Genuine symmetry-element detection from 3D coordinates (rotation axes,
mirror planes, an inversion center), combined into a point-group
classification, and the corresponding character tables.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Arthur Schoenflies named every point group after the symmetry elements that generate it: a principal axis C_n, extra C_2 axes (D_n), mirror planes \sigma_v or \sigma_h, an inversion centre i, and the special cubic groups T_d and O_h. This example builds four molecules from their real 3D geometry, lets determine_point_group find the symmetry elements by testing the coordinates directly, and shows how the elements it finds spell out each Schoenflies symbol: C_{2v}, C_{3v}, T_d and D_{\infty h}.">

.. only:: html

  .. image:: /api/gallery/structure/point_group/images/thumb/sphx_glr_plot_01_point_groups_thumb.png
    :alt:

  :doc:`/api/gallery/structure/point_group/plot_01_point_groups`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Schoenflies point-group symbols for water, ammonia, methane, and carbon dioxide</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="In 1893 Alfred Werner proposed that a six-coordinate metal complex places its ligands at the corners of an octahedron. No experiment could see the shape directly, so he argued from isomer counts. A complex \mathrm{MA_4B_2} such as [\mathrm{Co(NH_3)_4Cl_2}]^+ has only two forms (green trans &quot;praseo&quot; and violet cis &quot;violeo&quot;). An octahedron gives exactly two ways to place the two B ligands; a flat hexagon or a trigonal prism gives three. For \mathrm{MA_3B_3} the octahedron again gives two (fac and mer).">

.. only:: html

  .. image:: /api/gallery/structure/point_group/images/thumb/sphx_glr_plot_02_octahedral_symmetry_thumb.png
    :alt:

  :doc:`/api/gallery/structure/point_group/plot_02_octahedral_symmetry`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Werner's coordination theory: counting isomers to prove the octahedron</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Robert Mulliken&#x27;s 1955 report fixed how chemists label the irreducible representations in a character table. The rules read the label straight off the characters:">

.. only:: html

  .. image:: /api/gallery/structure/point_group/images/thumb/sphx_glr_plot_03_mulliken_symbols_thumb.png
    :alt:

  :doc:`/api/gallery/structure/point_group/plot_03_mulliken_symbols`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Mulliken symbols: reading the labels of irreducible representations</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Hans Bethe asked in 1929 what happens to a free ion&#x27;s degenerate levels when it sits in a crystal, surrounded by charges with only the symmetry of a point group. Group theory alone gives the answer. Under a rotation by angle \alpha, the 2l+1 orbitals of angular momentum l have the character">

.. only:: html

  .. image:: /api/gallery/structure/point_group/images/thumb/sphx_glr_plot_04_crystal_field_splitting_thumb.png
    :alt:

  :doc:`/api/gallery/structure/point_group/plot_04_crystal_field_splitting`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Bethe's crystal-field splitting: d orbitals in an octahedral field</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/structure/point_group/plot_01_point_groups
   /api/gallery/structure/point_group/plot_02_octahedral_symmetry
   /api/gallery/structure/point_group/plot_03_mulliken_symbols
   /api/gallery/structure/point_group/plot_04_crystal_field_splitting