.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/quantum/huckel/plot_02_pauling_resonance_energy.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_02_pauling_resonance_energy.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_quantum_huckel_plot_02_pauling_resonance_energy.py:


Pauling's resonance energy as Huckel delocalization energy
============================================================

Linus Pauling's *The Nature of the Chemical Bond* made "resonance energy"
the chemist's measure of the extra stability a molecule gains because its
true structure is a superposition of several Lewis structures (benzene's
two Kekule structures foremost). In Huckel theory that stabilization is
the delocalization energy
(:meth:`~chemistrykit.quantum.systems.huckel.HuckelSystem.delocalization_energy`):
the computed pi energy minus that of the same electrons held in isolated,
ethylene-like double bonds, :math:`2(\alpha+\beta)` per bond.

.. GENERATED FROM PYTHON SOURCE LINES 16-33

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.quantum.systems.huckel import HuckelSystem

    molecules = {
        "ethylene": (HuckelSystem.linear_polyene(2), 2),
        "butadiene": (HuckelSystem.linear_polyene(4), 4),
        "hexatriene": (HuckelSystem.linear_polyene(6), 6),
        "cyclobutadiene": (HuckelSystem.cyclic_polyene(4), 4),
        "benzene": (HuckelSystem.cyclic_polyene(6), 6),
    }
    resonance = {}
    for name, (system, n_pi) in molecules.items():
        resonance[name] = system.delocalization_energy(n_pi)
        print(f"{name:15s} pi energy = {system.pi_electron_energy(n_pi):7.4f} |beta|, resonance energy = {resonance[name]:7.4f}")





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

 .. code-block:: none

    ethylene        pi energy = -2.0000 |beta|, resonance energy =  0.0000
    butadiene       pi energy = -4.4721 |beta|, resonance energy = -0.4721
    hexatriene      pi energy = -6.9879 |beta|, resonance energy = -0.9879
    cyclobutadiene  pi energy = -4.0000 |beta|, resonance energy = -0.0000
    benzene         pi energy = -8.0000 |beta|, resonance energy = -2.0000




.. GENERATED FROM PYTHON SOURCE LINES 34-40

With :math:`\beta = -1`, benzene's resonance energy is exactly
:math:`2|\beta|`, far more
than open-chain hexatriene with the same six pi electrons, and
cyclobutadiene gets none at all. The thermochemical benzene
resonance energy of about 36 kcal/mol (from heats of
hydrogenation) corresponds to :math:`|\beta|` near 18 kcal/mol on this scale.

.. GENERATED FROM PYTHON SOURCE LINES 40-49

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(7, 4))
    names = list(resonance)
    values = -np.array([resonance[n] for n in names])
    ax.bar(names, values, color=["gray", "gray", "gray", "lightcoral", "seagreen"])
    ax.set_ylabel(r"resonance stabilization $(-E_{deloc})/|\beta|$")
    ax.set_title("Resonance (delocalization) energy: benzene stands out")
    fig.tight_layout()




.. image-sg:: /api/gallery/quantum/huckel/images/sphx_glr_plot_02_pauling_resonance_energy_001.png
   :alt: Resonance (delocalization) energy: benzene stands out
   :srcset: /api/gallery/quantum/huckel/images/sphx_glr_plot_02_pauling_resonance_energy_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 50-52

Resonance energy per pi electron in linear polyenes creeps up with chain
length but stays well below benzene's :math:`|\beta|/3` per electron.

.. GENERATED FROM PYTHON SOURCE LINES 52-65

.. code-block:: Python


    n_values = np.arange(2, 21, 2)
    per_electron = [-HuckelSystem.linear_polyene(n).delocalization_energy(n) / n for n in n_values]
    fig2, ax2 = plt.subplots(figsize=(6, 4))
    ax2.plot(n_values, per_electron, "o-", label="linear polyenes")
    ax2.axhline(-resonance["benzene"] / 6, color="seagreen", linestyle="--", label="benzene")
    ax2.set_xlabel("number of pi electrons")
    ax2.set_ylabel(r"resonance energy per electron $/|\beta|$")
    ax2.set_title("Resonance energy per pi electron")
    ax2.legend()
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/quantum/huckel/images/sphx_glr_plot_02_pauling_resonance_energy_002.png
   :alt: Resonance energy per pi electron
   :srcset: /api/gallery/quantum/huckel/images/sphx_glr_plot_02_pauling_resonance_energy_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.077 seconds)


.. _sphx_glr_download_api_gallery_quantum_huckel_plot_02_pauling_resonance_energy.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_pauling_resonance_energy.ipynb <plot_02_pauling_resonance_energy.ipynb>`

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

      :download:`Download Python source code: plot_02_pauling_resonance_energy.py <plot_02_pauling_resonance_energy.py>`

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

      :download:`Download zipped: plot_02_pauling_resonance_energy.zip <plot_02_pauling_resonance_energy.zip>`


.. only:: html

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

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