.. _sphx_glr_api_gallery_quantum_huckel:

Huckel molecular-orbital theory
---------------------------------

Huckel pi-electron theory for conjugated molecules: the 4n+2 aromaticity
rule checked against computed spectra, resonance (delocalization)
energies, the Coulson and Frost-circle closed forms, and Fukui's
frontier-orbital reactivity index.


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    <div class="sphx-glr-thumbcontainer" tooltip="Erich Huckel&#x27;s 1931 pi-electron theory reduces a planar conjugated ring to a small matrix eigenvalue problem: one 2p_z orbital per carbon, Coulomb integral \alpha on the diagonal, and resonance integral \beta only between bonded neighbours (~chemistrykit.quantum.systems.huckel.HuckelSystem). Filling the resulting levels two electrons at a time shows why rings with 4n+2 pi electrons (benzene, 6) form a closed shell while rings with 4n (cyclobutadiene, 4) leave two electrons unpaired in a degenerate pair. is_aromatic_by_huckel_rule checks the rule against the computed degeneracies, not only by counting electrons.">

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  .. image:: /api/gallery/quantum/huckel/images/thumb/sphx_glr_plot_01_huckel_aromaticity_rule_thumb.png
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  :doc:`/api/gallery/quantum/huckel/plot_01_huckel_aromaticity_rule`

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      <div class="sphx-glr-thumbnail-title">Huckel's 4n+2 aromaticity rule from the computed pi spectrum</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Linus Pauling&#x27;s The Nature of the Chemical Bond made &quot;resonance energy&quot; the chemist&#x27;s measure of the extra stability a molecule gains because its true structure is a superposition of several Lewis structures (benzene&#x27;s two Kekule structures foremost). In Huckel theory that stabilization is the delocalization energy (~chemistrykit.quantum.systems.huckel.HuckelSystem.delocalization_energy): the computed pi energy minus that of the same electrons held in isolated, ethylene-like double bonds, 2(\alpha+\beta) per bond.">

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  .. image:: /api/gallery/quantum/huckel/images/thumb/sphx_glr_plot_02_pauling_resonance_energy_thumb.png
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  :doc:`/api/gallery/quantum/huckel/plot_02_pauling_resonance_energy`

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      <div class="sphx-glr-thumbnail-title">Pauling's resonance energy as Huckel delocalization energy</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Coulson (1939) wrote the Huckel eigenvalues of a linear chain of n atoms in closed form, E_k=\alpha+2\beta\cos(k\pi/(n+1)), and Frost and Musulin (1953) turned the ring result E_k=\alpha+2\beta\cos(2\pi k/n) into a drawing: put a regular n-gon, one vertex pointing down, inside a circle of radius 2|\beta|; the heights of the vertices are the orbital energies. Both closed forms (~chemistrykit.quantum.systems.huckel.linear_polyene_eigenvalues, cyclic_polyene_eigenvalues) are checked here against numerical diagonalization.">

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  .. image:: /api/gallery/quantum/huckel/images/thumb/sphx_glr_plot_03_coulson_frost_closed_forms_thumb.png
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  :doc:`/api/gallery/quantum/huckel/plot_03_coulson_frost_closed_forms`

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      <div class="sphx-glr-thumbnail-title">Coulson's polyene formula and the Frost circle</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Kenichi Fukui, Teijiro Yonezawa and Haruo Shingu (1952) proposed that an aromatic hydrocarbon&#x27;s site of electrophilic substitution is set by one orbital alone, the highest occupied molecular orbital: the attacking electrophile goes where the HOMO density 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 frontier_electron_density predicts alpha substitution, which is what nitration and halogenation mostly give.">

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  .. image:: /api/gallery/quantum/huckel/images/thumb/sphx_glr_plot_04_fukui_frontier_orbitals_thumb.png
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  :doc:`/api/gallery/quantum/huckel/plot_04_fukui_frontier_orbitals`

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      <div class="sphx-glr-thumbnail-title">Fukui's frontier orbitals: why naphthalene reacts at the alpha position</div>
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.. toctree::
   :hidden:

   /api/gallery/quantum/huckel/plot_01_huckel_aromaticity_rule
   /api/gallery/quantum/huckel/plot_02_pauling_resonance_energy
   /api/gallery/quantum/huckel/plot_03_coulson_frost_closed_forms
   /api/gallery/quantum/huckel/plot_04_fukui_frontier_orbitals