.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structure/bonding/plot_01_coulson_bond_order.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_structure_bonding_plot_01_coulson_bond_order.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_structure_bonding_plot_01_coulson_bond_order.py:


Coulson's molecular-orbital bond order in butadiene, hexatriene, and benzene
============================================================================

Charles Coulson (1939) defined the pi bond order between neighbouring
atoms from the Huckel molecular orbitals themselves,

.. math::

    p_{ij} = \sum_k n_k\, c_{ik} c_{jk},

summing over occupied orbitals :math:`k` with :math:`n_k` electrons. It
gives fractional orders that no single Lewis structure shows. In
butadiene the terminal bonds come out at 0.894 and the central bond at
0.447, so the "single" bond has real double-bond character; in benzene
every bond has exactly 2/3. This example computes these values with
:func:`~chemistrykit.structure.systems.bonding.coulson_pi_bond_order`
from :class:`~chemistrykit.quantum.systems.huckel.HuckelSystem` orbitals.

.. GENERATED FROM PYTHON SOURCE LINES 22-52

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.quantum.systems.huckel import HuckelSystem
    from chemistrykit.structure.systems.bonding import coulson_pi_bond_order


    def pi_bond_orders(system, n_atoms, bonds):
        """Coulson bond order of each bond, with the n_atoms pi electrons filling the lowest orbitals in pairs."""
        result = system.solve()
        order = np.argsort(result.energies)
        occupations = np.zeros(n_atoms)
        occupations[order[: n_atoms // 2]] = 2.0
        return [coulson_pi_bond_order(result.coefficients, occupations, i, j) for i, j in bonds]


    molecules = {
        "ethene": (HuckelSystem.linear_polyene(2), 2, [(0, 1)]),
        "butadiene": (HuckelSystem.linear_polyene(4), 4, [(i, i + 1) for i in range(3)]),
        "hexatriene": (HuckelSystem.linear_polyene(6), 6, [(i, i + 1) for i in range(5)]),
        "benzene": (HuckelSystem.cyclic_polyene(6), 6, [(i, (i + 1) % 6) for i in range(6)]),
    }
    orders = {}
    for name, (system, n, bonds) in molecules.items():
        orders[name] = pi_bond_orders(system, n, bonds)
        print(f"{name:10s}: pi bond orders " + ", ".join(f"{p:.3f}" for p in orders[name]))

    assert np.allclose(orders["butadiene"], [2 / np.sqrt(5), 1 / np.sqrt(5), 2 / np.sqrt(5)], atol=1e-3)
    assert np.allclose(orders["benzene"], 2 / 3)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    ethene    : pi bond orders 1.000
    butadiene : pi bond orders 0.894, 0.447, 0.894
    hexatriene: pi bond orders 0.871, 0.483, 0.785, 0.483, 0.871
    benzene   : pi bond orders 0.667, 0.667, 0.667, 0.667, 0.667, 0.667




.. GENERATED FROM PYTHON SOURCE LINES 53-56

The pi bond order adds to the one sigma bond, so benzene's total C-C
bond order is :math:`1 + 2/3 = 5/3` for every ring bond. Bars show the
pi order bond by bond:

.. GENERATED FROM PYTHON SOURCE LINES 56-67

.. code-block:: Python


    fig, axes = plt.subplots(1, 4, figsize=(13, 3.5), sharey=True)
    for ax, (name, p) in zip(axes, orders.items()):
        ax.bar([f"{i + 1}-{(i + 1) % len(p) + 1 if name == 'benzene' else i + 2}" for i in range(len(p))], p, color="C0")
        ax.axhline(1.0, color="gray", linestyle="--", linewidth=0.8)
        ax.set_title(name)
        ax.tick_params(axis="x", labelsize=8)
    axes[0].set_ylabel(r"Coulson $\pi$ bond order $p_{ij}$")
    fig.suptitle("Coulson bond orders from Huckel orbitals (dashed: a full double bond)")
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/structure/bonding/images/sphx_glr_plot_01_coulson_bond_order_001.png
   :alt: Coulson bond orders from Huckel orbitals (dashed: a full double bond), ethene, butadiene, hexatriene, benzene
   :srcset: /api/gallery/structure/bonding/images/sphx_glr_plot_01_coulson_bond_order_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.068 seconds)


.. _sphx_glr_download_api_gallery_structure_bonding_plot_01_coulson_bond_order.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_coulson_bond_order.ipynb <plot_01_coulson_bond_order.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_coulson_bond_order.py <plot_01_coulson_bond_order.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_coulson_bond_order.zip <plot_01_coulson_bond_order.zip>`


.. only:: html

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

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