.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/quantum/huckel/plot_01_huckel_aromaticity_rule.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_quantum_huckel_plot_01_huckel_aromaticity_rule.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_quantum_huckel_plot_01_huckel_aromaticity_rule.py:


Huckel's 4n+2 aromaticity rule from the computed pi spectrum
==============================================================

Erich Huckel's 1931 pi-electron theory reduces a planar conjugated ring to
a small matrix eigenvalue problem: one 2p_z orbital per carbon, Coulomb
integral :math:`\alpha` on the diagonal, and resonance integral
:math:`\beta` only between bonded neighbours
(:class:`~chemistrykit.quantum.systems.huckel.HuckelSystem`). Filling the
resulting levels two electrons at a time shows why rings with
:math:`4n+2` pi electrons (benzene, 6) form a closed shell while rings with
:math:`4n` (cyclobutadiene, 4) leave two electrons unpaired in a
degenerate pair. :func:`~chemistrykit.quantum.systems.huckel.is_aromatic_by_huckel_rule`
checks the rule against the *computed* degeneracies, not only by counting
electrons.

.. GENERATED FROM PYTHON SOURCE LINES 19-37

.. code-block:: Python

    import matplotlib.pyplot as plt

    from chemistrykit.quantum.systems.huckel import HuckelSystem, is_aromatic_by_huckel_rule
    from chemistrykit.quantum.visualizers.quantum_plots import plot_huckel_levels

    # alpha = 0, beta = -1: energies are in units of |beta|, bonding levels negative.
    benzene = HuckelSystem.cyclic_polyene(6)
    cyclobutadiene = HuckelSystem.cyclic_polyene(4)

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9, 5))
    plot_huckel_levels(benzene.solve(), n_pi_electrons=6, ax=ax1)
    ax1.set_title("Benzene, 6 pi electrons: closed shell")
    plot_huckel_levels(cyclobutadiene.solve(), n_pi_electrons=4, ax=ax2)
    ax2.set_title("Cyclobutadiene, 4 pi electrons: open shell")
    for ax in (ax1, ax2):
        ax.set_ylabel(r"orbital energy $(E-\alpha)/|\beta|$")
    fig.tight_layout()




.. image-sg:: /api/gallery/quantum/huckel/images/sphx_glr_plot_01_huckel_aromaticity_rule_001.png
   :alt: Benzene, 6 pi electrons: closed shell, Cyclobutadiene, 4 pi electrons: open shell
   :srcset: /api/gallery/quantum/huckel/images/sphx_glr_plot_01_huckel_aromaticity_rule_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 38-42

The same test applied to a family of rings and ring ions. The cyclopropenyl
cation (2 electrons), cyclopentadienyl anion (6) and tropylium cation (6)
are all 4n+2 and closed shell; cyclobutadiene and cyclooctatetraene (4n)
are not.

.. GENERATED FROM PYTHON SOURCE LINES 42-69

.. code-block:: Python


    cases = [
        ("cyclopropenyl cation", 3, 2),
        ("cyclobutadiene", 4, 4),
        ("cyclopentadienyl anion", 5, 6),
        ("benzene", 6, 6),
        ("tropylium cation", 7, 6),
        ("cyclooctatetraene", 8, 8),
        ("cyclooctatetraene dianion", 8, 10),
    ]
    names, verdicts = [], []
    for name, n_atoms, n_pi in cases:
        energies = HuckelSystem.cyclic_polyene(n_atoms).solve().energies
        aromatic = is_aromatic_by_huckel_rule(n_pi, energies)
        names.append(f"{name}\n({n_pi} pi e-)")
        verdicts.append(aromatic)
        print(f"{name:27s} ring={n_atoms} pi electrons={n_pi:2d} aromatic={aromatic}")

    fig2, ax3 = plt.subplots(figsize=(9, 4))
    ax3.bar(range(len(cases)), [1] * len(cases), color=["seagreen" if v else "lightgray" for v in verdicts])
    ax3.set_xticks(range(len(cases)))
    ax3.set_xticklabels(names, fontsize=8)
    ax3.set_yticks([])
    ax3.set_title("Huckel 4n+2 rule: green = aromatic (closed-shell 4n+2 ring)")
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/quantum/huckel/images/sphx_glr_plot_01_huckel_aromaticity_rule_002.png
   :alt: Huckel 4n+2 rule: green = aromatic (closed-shell 4n+2 ring)
   :srcset: /api/gallery/quantum/huckel/images/sphx_glr_plot_01_huckel_aromaticity_rule_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    cyclopropenyl cation        ring=3 pi electrons= 2 aromatic=True
    cyclobutadiene              ring=4 pi electrons= 4 aromatic=False
    cyclopentadienyl anion      ring=5 pi electrons= 6 aromatic=True
    benzene                     ring=6 pi electrons= 6 aromatic=True
    tropylium cation            ring=7 pi electrons= 6 aromatic=True
    cyclooctatetraene           ring=8 pi electrons= 8 aromatic=False
    cyclooctatetraene dianion   ring=8 pi electrons=10 aromatic=True





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

   **Total running time of the script:** (0 minutes 0.088 seconds)


.. _sphx_glr_download_api_gallery_quantum_huckel_plot_01_huckel_aromaticity_rule.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_huckel_aromaticity_rule.ipynb <plot_01_huckel_aromaticity_rule.ipynb>`

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