.. 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_04_lennard_jones_potential.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_04_lennard_jones_potential.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_md_pair_potentials_plot_04_lennard_jones_potential.py:


The Lennard-Jones 12-6 potential: repulsion, dispersion, and the minimum
===========================================================================

Lennard-Jones's 1924 pair potential combines a steep :math:`r^{-12}`
repulsion with the :math:`r^{-6}` London-dispersion attraction:

.. math::

   U(r) = 4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right].

:class:`~chemistrykit.md.systems.lj_fluid.LennardJones` evaluates it and
its force. The left panel splits the potential into its two terms; the
right panel shows the potential and force together, crossing zero at
:math:`r=\sigma` and :math:`r=2^{1/6}\sigma`
(:attr:`~chemistrykit.md.systems.lj_fluid.LennardJones.r_min`)
respectively. The curves are drawn for argon, with
:math:`\epsilon/k_B = 120\ \text{K}` and :math:`\sigma = 3.4\ \text{\AA}`.

.. GENERATED FROM PYTHON SOURCE LINES 22-58

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.md.systems.lj_fluid import LennardJones

    epsilon_K, sigma_A = 120.0, 3.4  # argon: epsilon/k_B in kelvin, sigma in angstrom
    lj = LennardJones(epsilon=epsilon_K, sigma=sigma_A)
    r = np.linspace(3.0, 9.0, 500)

    repulsion = 4.0 * epsilon_K * (sigma_A / r) ** 12
    attraction = -4.0 * epsilon_K * (sigma_A / r) ** 6

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    axes[0].plot(r, repulsion, color="firebrick", linestyle="--", label=r"repulsion $+4\epsilon(\sigma/r)^{12}$")
    axes[0].plot(r, attraction, color="steelblue", linestyle="--", label=r"dispersion $-4\epsilon(\sigma/r)^{6}$")
    axes[0].plot(r, lj.energy(r), color="black", label="sum U(r)")
    axes[0].axhline(0.0, color="gray", linewidth=0.8)
    axes[0].set_ylim(-250, 400)
    axes[0].set_xlabel(r"r ($\mathrm{\AA}$)")
    axes[0].set_ylabel(r"$U/k_B$ (K)")
    axes[0].set_title("Two terms of the 12-6 potential")
    axes[0].legend()

    axes[1].plot(r, lj.energy(r), color="black", label="U(r)")
    axes[1].plot(r, lj.force_scalar(r), color="darkorange", label="force -dU/dr")
    axes[1].axhline(0.0, color="gray", linewidth=0.8)
    axes[1].axvline(sigma_A, color="gray", linestyle=":", linewidth=0.8)
    axes[1].axvline(lj.r_min, color="gray", linestyle="--", linewidth=0.8)
    axes[1].annotate(r"$r=\sigma$", (sigma_A, 150), xytext=(3.45, 150))
    axes[1].annotate(r"$r_{min}=2^{1/6}\sigma$", (lj.r_min, -200), xytext=(lj.r_min + 0.1, -200))
    axes[1].set_ylim(-250, 250)
    axes[1].set_xlabel(r"r ($\mathrm{\AA}$)")
    axes[1].set_title("Potential and force")
    axes[1].legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/md/pair_potentials/images/sphx_glr_plot_04_lennard_jones_potential_001.png
   :alt: Two terms of the 12-6 potential, Potential and force
   :srcset: /api/gallery/md/pair_potentials/images/sphx_glr_plot_04_lennard_jones_potential_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 59-61

The well depth equals epsilon and sits at :math:`2^{1/6}\sigma`, where
the force vanishes:

.. GENERATED FROM PYTHON SOURCE LINES 61-66

.. code-block:: Python


    print(f"r_min = {lj.r_min:.3f} angstrom, U(r_min)/k_B = {float(lj.energy(lj.r_min)):.1f} K")
    print(f"U(sigma) = {float(lj.energy(sigma_A)):.1e} K, force at r_min = {float(lj.force_scalar(lj.r_min)):.1e}")

    plt.show()




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

 .. code-block:: none

    r_min = 3.816 angstrom, U(r_min)/k_B = -120.0 K
    U(sigma) = 0.0e+00 K, force at r_min = -8.4e-14





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

   **Total running time of the script:** (0 minutes 0.072 seconds)


.. _sphx_glr_download_api_gallery_md_pair_potentials_plot_04_lennard_jones_potential.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_lennard_jones_potential.ipynb <plot_04_lennard_jones_potential.ipynb>`

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

      :download:`Download Python source code: plot_04_lennard_jones_potential.py <plot_04_lennard_jones_potential.py>`

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

      :download:`Download zipped: plot_04_lennard_jones_potential.zip <plot_04_lennard_jones_potential.zip>`


.. only:: html

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

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