.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/md/pair_potentials/plot_03_buckingham_vs_lj.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_md_pair_potentials_plot_03_buckingham_vs_lj.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_md_pair_potentials_plot_03_buckingham_vs_lj.py:


Buckingham vs. Lennard-Jones: exponential repulsion and its inner turnover
============================================================================

The :class:`~chemistrykit.md.systems.pair_potentials.Buckingham` (exp-6)
potential replaces the Lennard-Jones :math:`r^{-12}` repulsive wall with
an exponential -- closer to the true quantum-mechanical electron-overlap
repulsion -- while keeping the same :math:`r^{-6}` dispersion attraction.
Matched to a Lennard-Jones potential's well depth, position, and
long-range tail, the two agree closely everywhere near and beyond the
minimum. But the exponential repulsion is eventually overwhelmed by the
unbounded :math:`-C/r^6` attractive term at short enough range, so the
Buckingham potential has a second, spurious turnover well inside the
true minimum: past that point it falls back toward :math:`-\infty` as
:math:`r\to0` instead of diverging to :math:`+\infty` the way
Lennard-Jones's pure :math:`r^{-12}` repulsion does.

.. GENERATED FROM PYTHON SOURCE LINES 20-40

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.md.systems.lj_fluid import LennardJones
    from chemistrykit.md.systems.pair_potentials import Buckingham

    lj = LennardJones(epsilon=1.0, sigma=1.0)

    # Match the Buckingham potential's dispersion coefficient exactly to
    # Lennard-Jones's (same -C/r**6 tail), and choose its exponential steepness
    # B so the two repulsive walls have the same local slope at LJ's own
    # minimum -- fixing A so the resulting potential shares that minimum too.
    C = 4.0 * lj.epsilon * lj.sigma**6
    B = 12.0 / lj.r_min
    A = 6.0 * C / (B * lj.r_min**7) * np.exp(B * lj.r_min)
    buckingham = Buckingham(A=A, B=B, C=C)

    print(f"Lennard-Jones well depth at r_min:  {lj.energy(lj.r_min):.4f}")
    print(f"Buckingham well depth at same r_min: {buckingham.energy(lj.r_min):.4f}")





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

 .. code-block:: none

    Lennard-Jones well depth at r_min:  -1.0000
    Buckingham well depth at same r_min: -1.0000




.. GENERATED FROM PYTHON SOURCE LINES 41-44

Away from the origin, the two potentials are nearly indistinguishable:
matching the dispersion tail and the minimum's location and slope pins
down the whole well shape almost completely.

.. GENERATED FROM PYTHON SOURCE LINES 44-57

.. code-block:: Python


    r_wide = np.linspace(0.85 * lj.r_min, 3.0 * lj.sigma, 400)

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    axes[0].plot(r_wide, lj.energy(r_wide), label="Lennard-Jones")
    axes[0].plot(r_wide, buckingham.energy(r_wide), label="Buckingham", linestyle="--")
    axes[0].axhline(-lj.epsilon, color="gray", linestyle=":", linewidth=0.8)
    axes[0].set_xlabel("r")
    axes[0].set_ylabel("U(r)")
    axes[0].set_ylim(-1.5, 2.0)
    axes[0].set_title("Near the well: the two potentials nearly coincide")
    axes[0].legend()




.. image-sg:: /api/gallery/md/pair_potentials/images/sphx_glr_plot_03_buckingham_vs_lj_001.png
   :alt: Near the well: the two potentials nearly coincide
   :srcset: /api/gallery/md/pair_potentials/images/sphx_glr_plot_03_buckingham_vs_lj_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11b8efb60>



.. GENERATED FROM PYTHON SOURCE LINES 58-63

At short enough range, though, the Buckingham potential's exponential
repulsion is overtaken by its own unbounded dispersion attraction: past
a spurious inner maximum, well inside the true minimum, it turns over
and plunges toward -infinity rather than diverging to +infinity the way
Lennard-Jones's pure r**-12 repulsion does.

.. GENERATED FROM PYTHON SOURCE LINES 63-77

.. code-block:: Python


    r_short = np.linspace(0.25, 0.9 * lj.r_min, 400)

    axes[1].plot(r_short, lj.energy(r_short), label="Lennard-Jones")
    axes[1].plot(r_short, buckingham.energy(r_short), label="Buckingham", linestyle="--")
    axes[1].axhline(0.0, color="gray", linestyle=":", linewidth=0.8)
    axes[1].set_xlabel("r")
    axes[1].set_ylabel("U(r)")
    axes[1].set_ylim(-20.0, 20.0)
    axes[1].set_title("Short range: Buckingham's unphysical inner turnover")
    axes[1].legend()

    fig.tight_layout()
    plt.show()








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

   **Total running time of the script:** (0 minutes 0.056 seconds)


.. _sphx_glr_download_api_gallery_md_pair_potentials_plot_03_buckingham_vs_lj.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_buckingham_vs_lj.ipynb <plot_03_buckingham_vs_lj.ipynb>`

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

      :download:`Download Python source code: plot_03_buckingham_vs_lj.py <plot_03_buckingham_vs_lj.py>`

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

      :download:`Download zipped: plot_03_buckingham_vs_lj.zip <plot_03_buckingham_vs_lj.zip>`


.. only:: html

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

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