.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/spectro/nmr/plot_05_abx_second_order_spectra.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_spectro_nmr_plot_05_abx_second_order_spectra.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_spectro_nmr_plot_05_abx_second_order_spectra.py:


Second-order NMR: from AX to AB, and the ABX system
===================================================

First-order rules (doublets of equal lines, spaced by :math:`J`, centered
on each shift) hold only while :math:`\Delta\nu \gg J`. As the two shifts
approach each other the doublets lean toward one another: the inner lines
grow and the outer ones shrink, the "roof effect". At :math:`\Delta\nu=0`
the two nuclei become equivalent and a single line remains.
:func:`~chemistrykit.spectro.second_order_spectrum` diagonalizes the full
spin Hamiltonian, and :func:`~chemistrykit.spectro.ab_quartet` gives the
closed-form AB result it must agree with. The right panel is an ABX
system (:func:`~chemistrykit.spectro.abx_spectrum`), the case Bernstein,
Pople and Schneider analyzed: a strongly coupled CH2 pair (AB) next to a
distant CH (X). Its AB part is no longer two simple doublets of doublets.

.. GENERATED FROM PYTHON SOURCE LINES 19-52

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.spectro import ab_quartet, abx_spectrum, second_order_spectrum

    nu0 = 400.0  # MHz
    J = 10.0  # Hz
    ratios = [5.0, 2.0, 1.0, 0.3, 0.0]  # delta_nu / J

    fig, axes = plt.subplots(1, 2, figsize=(12, 5.2))
    x = np.linspace(-40, 40, 4000)
    for k, ratio in enumerate(ratios):
        delta_ppm = ratio * J / nu0
        s = second_order_spectrum([2.0 - delta_ppm / 2, 2.0 + delta_ppm / 2], [[0, J], [J, 0]], nu0)
        hz = (s.positions - 2.0) * nu0
        line = sum(a * 0.3**2 / ((x - p) ** 2 + 0.3**2) for p, a in zip(hz, s.intensities, strict=True))
        axes[0].plot(x, line / 4 + 1.2 * k, color="black")
        axes[0].text(33, 1.2 * k + 0.25, rf"$\Delta\nu/J={ratio:g}$")
    axes[0].set_xlabel("offset from center (Hz)")
    axes[0].set_yticks([])
    axes[0].set_title("AX -> AB -> A2 at J = 10 Hz")

    abx = abx_spectrum([2.50, 2.55, 4.20], j_ab_hz=16.0, j_ax_hz=4.0, j_bx_hz=9.0, spectrometer_frequency_mhz=nu0)
    ppm = np.linspace(2.40, 2.65, 3000)
    ab_part = abx.positions < 3.0
    line = sum(a * 0.001**2 / ((ppm - p) ** 2 + 0.001**2) for p, a in zip(abx.positions[ab_part], abx.intensities[ab_part], strict=True))
    axes[1].plot(ppm, line, color="darkorange")
    axes[1].invert_xaxis()
    axes[1].set_xlabel("chemical shift (ppm)")
    axes[1].set_yticks([])
    axes[1].set_title(r"AB part of an ABX system (400 MHz, $J_{AB}$ = 16 Hz)")
    fig.tight_layout()




.. image-sg:: /api/gallery/spectro/nmr/images/sphx_glr_plot_05_abx_second_order_spectra_001.png
   :alt: AX -> AB -> A2 at J = 10 Hz, AB part of an ABX system (400 MHz, $J_{AB}$ = 16 Hz)
   :srcset: /api/gallery/spectro/nmr/images/sphx_glr_plot_05_abx_second_order_spectra_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 53-54

The exact diagonalization reproduces the closed-form AB quartet:

.. GENERATED FROM PYTHON SOURCE LINES 54-62

.. code-block:: Python


    s = second_order_spectrum([2.00, 2.02], [[0, J], [J, 0]], nu0)
    q = ab_quartet(2.00, 2.02, J, nu0)
    print("exact  :", np.round(s.intensities, 4))
    print("formula:", np.round(q.intensities, 4))
    print(f"ABX: {int(ab_part.sum())} AB lines, {int((~ab_part).sum())} X lines, total intensity {abx.intensities.sum():.1f}")

    plt.show()




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

 .. code-block:: none

    exact  : [0.2191 1.7809 1.7809 0.2191]
    formula: [0.2191 1.7809 1.7809 0.2191]
    ABX: 8 AB lines, 6 X lines, total intensity 12.0





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

   **Total running time of the script:** (0 minutes 0.945 seconds)


.. _sphx_glr_download_api_gallery_spectro_nmr_plot_05_abx_second_order_spectra.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_05_abx_second_order_spectra.ipynb <plot_05_abx_second_order_spectra.ipynb>`

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

      :download:`Download Python source code: plot_05_abx_second_order_spectra.py <plot_05_abx_second_order_spectra.py>`

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

      :download:`Download zipped: plot_05_abx_second_order_spectra.zip <plot_05_abx_second_order_spectra.zip>`


.. only:: html

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

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