:orphan:

Examples
========

This gallery walks through every public feature of
``chemistrykit.structure``: VSEPR geometry prediction with real 3D
coordinate generation; point-group determination from 3D coordinates
(genuine symmetry-element detection, not a formula lookup) and character
tables; bond order from the Pauling length correlation and from
Huckel-theory MO coefficients; formal-charge/oxidation-state
assignment from a Lewis structure and Pauling electronegativities;
Kekule-structure enumeration; Baeyer ring angle strain; and dipole
moments.

Each script in this gallery is self-contained and can be run directly with
``python examples/structure/<section>/<script>.py``. Every script also
carries an RST module docstring as its title/description and uses ``# %%``
markers to split narrative text from code, which is exactly what
Sphinx-Gallery renders into the pages below -- the script *is* the source of
truth for what you see, not a copy of it.

Sections
--------

- **vsepr** -- van't Hoff and Le Bel's tetrahedral carbon and chirality,
  steric number to idealized molecular shape, and hypervalent molecules.
- **point_group** -- Schoenflies symbols from detected symmetry elements,
  Werner's octahedral isomer counting, Mulliken's irrep labels, and
  Bethe's crystal-field splitting of the d orbitals.
- **bonding** -- the Coulson pi bond order from Huckel molecular-orbital
  theory, and the Pauling bond-order/bond-length correlation.
- **lewis** -- formal charges of competing Lewis structures, oxidation
  states in Kossel's ionic limit, and Pauling electronegativities from
  bond energies.
- **kekule** -- Kekule structures of benzene and larger aromatics.
- **ring_strain** -- Baeyer's angle strain in planar cycloalkanes.
- **dipole** -- molecular dipole moments from bond dipoles and charges.


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Bond order
----------

The Coulson pi bond order from Huckel molecular orbitals, and Pauling's
bond-order/bond-length correlation.


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    <div class="sphx-glr-thumbcontainer" tooltip="Charles Coulson (1939) defined the pi bond order between neighbouring atoms from the Huckel molecular orbitals themselves,">

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  .. image:: /api/gallery/structure/bonding/images/thumb/sphx_glr_plot_01_coulson_bond_order_thumb.png
    :alt:

  :doc:`/api/gallery/structure/bonding/plot_01_coulson_bond_order`

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      <div class="sphx-glr-thumbnail-title">Coulson's molecular-orbital bond order in butadiene, hexatriene, and benzene</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Linus Pauling (1947) related the length of a bond to its order,">

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  .. image:: /api/gallery/structure/bonding/images/thumb/sphx_glr_plot_02_pauling_bond_length_thumb.png
    :alt:

  :doc:`/api/gallery/structure/bonding/plot_02_pauling_bond_length`

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      <div class="sphx-glr-thumbnail-title">Pauling's bond-order and bond-length correlation for carbon-carbon bonds</div>
    </div>


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Dipole moments
--------------

Molecular dipole moments from partial charges and from vector sums of
bond dipoles.


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    <div class="sphx-glr-thumbcontainer" tooltip="Peter Debye (1912) explained why some gases have a dielectric constant that falls with temperature: their molecules carry a permanent electric dipole moment \boldsymbol\mu = \sum_i q_i \mathbf r_i, which thermal motion partly disorders. The measured moment tests a proposed shape. Water&#x27;s 1.85 D rules out a linear H-O-H, while carbon dioxide&#x27;s zero moment requires a linear O=C=O.">

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  .. image:: /api/gallery/structure/dipole/images/thumb/sphx_glr_plot_01_debye_dipole_moments_thumb.png
    :alt:

  :doc:`/api/gallery/structure/dipole/plot_01_debye_dipole_moments`

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      <div class="sphx-glr-thumbnail-title">Debye's permanent dipole moments: polar and nonpolar molecules from geometry</div>
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Kekulé structures
-----------------

Enumerating the alternating single/double-bond structures of benzene and
other conjugated hydrocarbons.


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    <div class="sphx-glr-thumbcontainer" tooltip="August Kekulé (1865) proposed that benzene&#x27;s six carbons form a ring with alternating single and double bonds, and soon after that the two alternating arrangements interconvert, which explained why benzene has only one ortho-disubstituted isomer. A Kekulé structure is a choice of double bonds that gives every carbon exactly one; in graph language, a perfect matching of the carbon skeleton.">

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  .. image:: /api/gallery/structure/kekule/images/thumb/sphx_glr_plot_01_kekule_structures_thumb.png
    :alt:

  :doc:`/api/gallery/structure/kekule/plot_01_kekule_structures`

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      <div class="sphx-glr-thumbnail-title">Kekulé's benzene: enumerating the Kekulé structures of aromatic hydrocarbons</div>
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Lewis structures, oxidation states, and electronegativity
----------------------------------------------------------

Lewis's shared electron pair and formal charge, Kossel's ionic limit and
oxidation states, and Pauling's electronegativity scale from bond
energies.


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    <div class="sphx-glr-thumbcontainer" tooltip="G. N. Lewis (1916) described a covalent bond as a pair of electrons shared by two atoms, and each atom as seeking an octet of eight valence electrons. Often several octet-satisfying structures can be drawn for one molecule. Formal charge ranks them: split every bonding pair evenly, count the electrons each atom then owns, and compare with the free atom,">

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  .. image:: /api/gallery/structure/lewis/images/thumb/sphx_glr_plot_01_lewis_formal_charge_thumb.png
    :alt:

  :doc:`/api/gallery/structure/lewis/plot_01_lewis_formal_charge`

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      <div class="sphx-glr-thumbnail-title">Lewis's shared electron pair: choosing between Lewis structures by formal charge</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Walther Kossel (1916) described bonding as the complete transfer of electrons from one atom to another, leaving ions with noble-gas configurations held together by electrostatic attraction. Oxidation states are the bookkeeping version of that picture: give every bonding pair entirely to the more electronegative atom (sharing only between identical atoms) and compare the electron count with the free atom.">

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  .. image:: /api/gallery/structure/lewis/images/thumb/sphx_glr_plot_02_kossel_oxidation_states_thumb.png
    :alt:

  :doc:`/api/gallery/structure/lewis/plot_02_kossel_oxidation_states`

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      <div class="sphx-glr-thumbnail-title">Kossel's ionic bond: oxidation states as the fully ionic limit</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Linus Pauling (1932) noticed that a bond A-B is usually stronger than the average of the A-A and B-B bonds. He called the excess the extra ionic energy,">

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  .. image:: /api/gallery/structure/lewis/images/thumb/sphx_glr_plot_03_pauling_electronegativity_thumb.png
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  :doc:`/api/gallery/structure/lewis/plot_03_pauling_electronegativity`

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      <div class="sphx-glr-thumbnail-title">Pauling's electronegativity scale from bond energies</div>
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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.


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    <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}.">

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  .. image:: /api/gallery/structure/point_group/images/thumb/sphx_glr_plot_01_point_groups_thumb.png
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  :doc:`/api/gallery/structure/point_group/plot_01_point_groups`

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

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  .. image:: /api/gallery/structure/point_group/images/thumb/sphx_glr_plot_02_octahedral_symmetry_thumb.png
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  :doc:`/api/gallery/structure/point_group/plot_02_octahedral_symmetry`

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

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  .. image:: /api/gallery/structure/point_group/images/thumb/sphx_glr_plot_03_mulliken_symbols_thumb.png
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  :doc:`/api/gallery/structure/point_group/plot_03_mulliken_symbols`

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

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  .. image:: /api/gallery/structure/point_group/images/thumb/sphx_glr_plot_04_crystal_field_splitting_thumb.png
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  :doc:`/api/gallery/structure/point_group/plot_04_crystal_field_splitting`

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

Baeyer's angle-strain theory of cycloalkanes, and why puckered rings
escape it.


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    <div class="sphx-glr-thumbcontainer" tooltip="Adolf von Baeyer (1885) combined van&#x27;t Hoff&#x27;s tetrahedral carbon with the assumption that rings are flat. A planar ring of n carbons has interior angles 180^\circ(n-2)/n, so each bond must bend away from the tetrahedral 109.47 degrees by">

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  .. image:: /api/gallery/structure/ring_strain/images/thumb/sphx_glr_plot_01_baeyer_angle_strain_thumb.png
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  :doc:`/api/gallery/structure/ring_strain/plot_01_baeyer_angle_strain`

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      <div class="sphx-glr-thumbnail-title">Baeyer's strain theory: angle strain in planar cycloalkane rings</div>
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VSEPR geometry prediction
--------------------------

Steric number and lone-pair count to idealized 3D molecular geometry,
via real electron-domain coordinate generation -- not a shape-name
lookup table.


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    <div class="sphx-glr-thumbcontainer" tooltip="Builds four molecules spanning the five idealized electron-domain polyhedra -- methane (AX4E0, tetrahedral), water (AX2E2, bent from a tetrahedral parent), sulfur tetrafluoride (AX4E1, seesaw from a trigonal bipyramidal parent), and xenon tetrafluoride (AX4E2, square planar from an octahedral parent) -- via build_vsepr_molecule, and verifies the exact idealized bond angles each shape predicts.">

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  .. image:: /api/gallery/structure/vsepr/images/thumb/sphx_glr_plot_01_vsepr_geometries_thumb.png
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  :doc:`/api/gallery/structure/vsepr/plot_01_vsepr_geometries`

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      <div class="sphx-glr-thumbnail-title">VSEPR geometry prediction: from steric number to real 3D coordinates</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Jacobus van&#x27;t Hoff and Joseph Le Bel independently proposed in 1874 that a carbon atom&#x27;s four bonds point toward the vertices of a regular tetrahedron -- and that a carbon bonded to four different substituents therefore has no internal mirror symmetry: its mirror image cannot be superimposed on itself by any rotation, giving two distinct, non- interconvertible molecules (enantiomers). This builds a genuinely tetrahedral AX4 center -- bromochlorofluoromethane, CHFClBr, the textbook minimal chiral carbon -- from domain_positions&#x27;s real 3D vertex coordinates, confirms the exact 109.47-degree bond angles van&#x27;t Hoff and Le Bel&#x27;s model predicts, and then demonstrates the chirality itself numerically: a scalar-triple-product handedness invariant flips sign under reflection but is exactly preserved under any proper rotation (built with rotation_matrix), so no rotation can ever turn the molecule into its own mirror image.">

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  .. image:: /api/gallery/structure/vsepr/images/thumb/sphx_glr_plot_02_tetrahedral_chirality_thumb.png
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  :doc:`/api/gallery/structure/vsepr/plot_02_tetrahedral_chirality`

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      <div class="sphx-glr-thumbnail-title">Van't Hoff and Le Bel's tetrahedral carbon: chirality from real 3D coordinates</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="James Musher (1969) grouped molecules such as \mathrm{PCl_5}, \mathrm{SF_6}, \mathrm{ClF_3} and the xenon fluorides into one class: hypervalent molecules, whose central main-group atom appears to hold more than the eight electrons of Lewis&#x27;s octet. Counting the Lewis electrons around the central atom (two per bond plus two per lone pair) gives 10 or 12 for all of them.">

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  .. image:: /api/gallery/structure/vsepr/images/thumb/sphx_glr_plot_03_hypervalent_molecules_thumb.png
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  :doc:`/api/gallery/structure/vsepr/plot_03_hypervalent_molecules`

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      <div class="sphx-glr-thumbnail-title">Musher's hypervalent molecules: main-group centres beyond the octet</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/structure/bonding/index.rst
   /api/gallery/structure/dipole/index.rst
   /api/gallery/structure/kekule/index.rst
   /api/gallery/structure/lewis/index.rst
   /api/gallery/structure/point_group/index.rst
   /api/gallery/structure/ring_strain/index.rst
   /api/gallery/structure/vsepr/index.rst



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 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_