.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/quantum/hartree_fock/plot_01_lcao_molecular_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_hartree_fock_plot_01_lcao_molecular_orbitals.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_quantum_hartree_fock_plot_01_lcao_molecular_orbitals.py:


Hund-Mulliken LCAO molecular orbitals of H2+: bonding and antibonding
=======================================================================

Hund and Mulliken's molecular-orbital theory builds one-electron orbitals
spread over the whole molecule as a linear combination of atomic
orbitals, :math:`\psi_\pm=c_A\chi_A\pm c_B\chi_B`.
:class:`~chemistrykit.quantum.systems.hartree_fock.H2PlusVariational`
does this for H2+ with one Gaussian orbital per proton: diagonalizing the
2x2 problem gives the in-phase bonding orbital (equal coefficients, energy
below the isolated-atom level) and the out-of-phase antibonding orbital
(opposite coefficients, energy above it). Following both orbital energies
from short bond lengths out to separated atoms gives Mulliken's
correlation diagram for the simplest molecule.

.. GENERATED FROM PYTHON SOURCE LINES 18-32

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import ELECTRONVOLT
    from chemistrykit.quantum.systems.hartree_fock import H2PlusVariational

    BOHR_RADIUS = 5.29177e-11
    h2plus = H2PlusVariational(bond_length=106.0e-12)
    alpha = h2plus.optimize_exponent(1.0 / BOHR_RADIUS**2).optimized_alpha
    result = h2plus.solve(alpha)
    for i, name in enumerate(("bonding", "antibonding")):
        c_a, c_b = result.orbital_coefficients(i)
        print(f"{name:12s} E = {result.energies[i] / ELECTRONVOLT:8.3f} eV   (c_A, c_B) = ({c_a:+.4f}, {c_b:+.4f})")





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

 .. code-block:: none

    bonding      E =  -28.018 eV   (c_A, c_B) = (-0.5939, -0.5939)
    antibonding  E =  -15.437 eV   (c_A, c_B) = (+0.9264, -0.9264)




.. GENERATED FROM PYTHON SOURCE LINES 33-37

Orbital energies as a function of bond length, at a fixed exponent so the
separated-atom limit is a single atomic level. The bonding MO drops below
that level as the atoms approach; the antibonding MO rises above it, and
rises further than the bonding MO falls because of the overlap S_AB.

.. GENERATED FROM PYTHON SOURCE LINES 37-57

.. code-block:: Python


    R_pm = np.linspace(40.0, 500.0, 80)
    bonding, antibonding, overlap = [], [], []
    for R in R_pm:
        res = H2PlusVariational(bond_length=R * 1.0e-12).solve(alpha)
        bonding.append(res.energies[0] / ELECTRONVOLT)
        antibonding.append(res.energies[1] / ELECTRONVOLT)
        overlap.append(res.overlap[0, 1])
    atom_level = H2PlusVariational(bond_length=1.0e-8).solve(alpha).energies[0] / ELECTRONVOLT

    fig, ax = plt.subplots(figsize=(7, 5))
    ax.plot(R_pm, bonding, color="steelblue", label=r"bonding $\sigma_g$: $c_A=c_B$")
    ax.plot(R_pm, antibonding, color="crimson", label=r"antibonding $\sigma_u^*$: $c_A=-c_B$")
    ax.axhline(atom_level, color="gray", linestyle=":", label="separated-atom orbital level")
    ax.set_xlabel("bond length R (pm)")
    ax.set_ylabel("electronic orbital energy (eV)")
    ax.set_title("H2+ LCAO molecular orbitals vs. bond length")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/quantum/hartree_fock/images/sphx_glr_plot_01_lcao_molecular_orbitals_001.png
   :alt: H2+ LCAO molecular orbitals vs. bond length
   :srcset: /api/gallery/quantum/hartree_fock/images/sphx_glr_plot_01_lcao_molecular_orbitals_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 58-60

The splitting is driven by the atomic-orbital overlap: both vanish
together as the atoms separate.

.. GENERATED FROM PYTHON SOURCE LINES 60-72

.. code-block:: Python


    fig2, ax2 = plt.subplots(figsize=(6, 4))
    ax2.plot(R_pm, np.array(antibonding) - np.array(bonding), color="purple", label="MO splitting (eV)")
    ax2.set_xlabel("bond length R (pm)")
    ax2.set_ylabel("antibonding - bonding (eV)")
    ax3 = ax2.twinx()
    ax3.plot(R_pm, overlap, color="gray", linestyle="--", label=r"overlap $S_{AB}$")
    ax3.set_ylabel(r"$S_{AB}$")
    ax2.set_title("Bonding/antibonding splitting follows the overlap")
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/quantum/hartree_fock/images/sphx_glr_plot_01_lcao_molecular_orbitals_002.png
   :alt: Bonding/antibonding splitting follows the overlap
   :srcset: /api/gallery/quantum/hartree_fock/images/sphx_glr_plot_01_lcao_molecular_orbitals_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.095 seconds)


.. _sphx_glr_download_api_gallery_quantum_hartree_fock_plot_01_lcao_molecular_orbitals.py:

.. only:: html

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

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

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