.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/quantum/huckel/plot_04_fukui_frontier_orbitals.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_04_fukui_frontier_orbitals.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_quantum_huckel_plot_04_fukui_frontier_orbitals.py:


Fukui's frontier orbitals: why naphthalene reacts at the alpha position
=========================================================================

Kenichi Fukui, Teijiro Yonezawa and Haruo Shingu (1952) proposed that an
aromatic hydrocarbon's site of electrophilic substitution is set by one
orbital alone, the highest occupied molecular orbital: the attacking
electrophile goes where the HOMO density :math:`2c_r^2` is largest.
For naphthalene the Huckel HOMO has coefficient 0.425 on the alpha
carbons (1, 4, 5, 8), 0.263 on the beta carbons (2, 3, 6, 7), and zero on
the bridgeheads, so :meth:`~chemistrykit.quantum.systems.huckel.HuckelSystem.frontier_electron_density`
predicts alpha substitution, which is what nitration and halogenation
mostly give.

.. GENERATED FROM PYTHON SOURCE LINES 17-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.quantum.systems.huckel import HuckelSystem

    # Atom order: C1, C2, C3, C4, C4a, C5, C6, C7, C8, C8a.
    labels = ["1", "2", "3", "4", "4a", "5", "6", "7", "8", "8a"]
    bonds = [(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9), (9, 0), (4, 9)]
    naphthalene = HuckelSystem(n_atoms=10, bonds=bonds, labels=labels)

    homo_density = naphthalene.frontier_electron_density(10, "homo")
    for label, f in zip(labels, homo_density):
        print(f"C{label:3s} HOMO density = {f:.3f}")





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

 .. code-block:: none

    C1   HOMO density = 0.362
    C2   HOMO density = 0.138
    C3   HOMO density = 0.138
    C4   HOMO density = 0.362
    C4a  HOMO density = 0.000
    C5   HOMO density = 0.362
    C6   HOMO density = 0.138
    C7   HOMO density = 0.138
    C8   HOMO density = 0.362
    C8a  HOMO density = 0.000




.. GENERATED FROM PYTHON SOURCE LINES 32-33

Draw the molecule with circles proportional to the HOMO density.

.. GENERATED FROM PYTHON SOURCE LINES 33-62

.. code-block:: Python


    s3 = np.sqrt(3.0) / 2.0
    xy = np.array(
        [
            [-s3, 1.5],  # 1
            [-2 * s3, 1.0],  # 2
            [-2 * s3, 0.0],  # 3
            [-s3, -0.5],  # 4
            [0.0, 0.0],  # 4a
            [s3, -0.5],  # 5
            [2 * s3, 0.0],  # 6
            [2 * s3, 1.0],  # 7
            [s3, 1.5],  # 8
            [0.0, 1.0],  # 8a
        ]
    )
    xy = xy[:, ::-1] * np.array([1.0, -1.0])  # lay the molecule on its side

    fig, ax = plt.subplots(figsize=(6, 5))
    for i, j in bonds:
        ax.plot(xy[[i, j], 0], xy[[i, j], 1], color="black", linewidth=1.5)
    ax.scatter(xy[:, 0], xy[:, 1], s=4000 * homo_density + 5, color="crimson", alpha=0.6, zorder=3)
    for (x, y), label in zip(xy, labels):
        ax.text(x, y, label, ha="center", va="center", fontsize=9, zorder=4)
    ax.set_aspect("equal")
    ax.axis("off")
    ax.set_title("Naphthalene HOMO density: largest at the alpha carbons")
    fig.tight_layout()




.. image-sg:: /api/gallery/quantum/huckel/images/sphx_glr_plot_04_fukui_frontier_orbitals_001.png
   :alt: Naphthalene HOMO density: largest at the alpha carbons
   :srcset: /api/gallery/quantum/huckel/images/sphx_glr_plot_04_fukui_frontier_orbitals_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 63-66

The LUMO has the same magnitudes (naphthalene is an alternant
hydrocarbon), so nucleophilic and radical attack are predicted at the
alpha positions too.

.. GENERATED FROM PYTHON SOURCE LINES 66-80

.. code-block:: Python


    lumo_density = naphthalene.frontier_electron_density(10, "lumo")
    fig2, ax2 = plt.subplots(figsize=(7, 4))
    x = np.arange(10)
    ax2.bar(x - 0.2, homo_density, width=0.4, label="HOMO (electrophilic attack)")
    ax2.bar(x + 0.2, lumo_density, width=0.4, label="LUMO (nucleophilic attack)")
    ax2.set_xticks(x)
    ax2.set_xticklabels([f"C{lab}" for lab in labels])
    ax2.set_ylabel(r"frontier density $2c_r^2$")
    ax2.set_title("Fukui frontier electron densities of naphthalene")
    ax2.legend()
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/quantum/huckel/images/sphx_glr_plot_04_fukui_frontier_orbitals_002.png
   :alt: Fukui frontier electron densities of naphthalene
   :srcset: /api/gallery/quantum/huckel/images/sphx_glr_plot_04_fukui_frontier_orbitals_002.png
   :class: sphx-glr-single-img






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

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


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.. only:: html

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