.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/spectro/vibrational/plot_04_linear_bend_degeneracy.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_vibrational_plot_04_linear_bend_degeneracy.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_spectro_vibrational_plot_04_linear_bend_degeneracy.py:


Herzberg's linear-molecule rules: CO2's doubly degenerate bend and 3N-5 vibrations
===================================================================================

Herzberg's *Infrared and Raman Spectra of Polyatomic Molecules* (1945)
set out the rules for counting and labelling vibrations of polyatomic
molecules. A nonlinear molecule of N atoms has :math:`3N-6` vibrations.
A linear molecule has :math:`3N-5`, because rotation about its axis
does not count as a degree of freedom. For CO2 this means four vibrations: the
symmetric stretch, the antisymmetric stretch, and a *doubly degenerate*
bend (in-plane and out-of-plane bending have the same frequency by
cylindrical symmetry). This example compares linear CO2 with bent H2O
using
:class:`~chemistrykit.spectro.systems.vibrational.TriatomicNormalModes`,
whose ``is_linear`` flag marks the bend as a degenerate pair.

.. GENERATED FROM PYTHON SOURCE LINES 19-39

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    import scipy.constants as sc

    from chemistrykit.spectro.systems.vibrational import TriatomicNormalModes

    u = sc.atomic_mass
    co2 = TriatomicNormalModes.linear(mass_terminal=15.999 * u, mass_central=12.011 * u, bond_length=116.3e-12, k_r=1600.0, k_theta=0.85e-18).solve()
    h2o = TriatomicNormalModes.bent(
        mass_terminal=1.008 * u, mass_central=15.999 * u, bond_length=95.8e-12, bond_angle_degrees=104.5, k_r=770.0, k_theta=0.7e-18
    ).solve()

    n_atoms = 3
    for name, res in (("CO2", co2), ("H2O", h2o)):
        degeneracies = np.array([2 if (res.is_linear and k == 0) else 1 for k in range(3)])
        rule = "3N-5" if res.is_linear else "3N-6"
        expected = 3 * n_atoms - (5 if res.is_linear else 6)
        print(f"{name}: is_linear={res.is_linear}; vibrations counted with degeneracy = {degeneracies.sum()} ({rule} = {expected})")
        print(f"   wavenumbers (cm^-1): {np.round(res.wavenumbers, 1)}, degeneracies {degeneracies.tolist()}")





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

 .. code-block:: none

    CO2: is_linear=True; vibrations counted with degeneracy = 4 (3N-5 = 4)
       wavenumbers (cm^-1): [ 699.  1302.8 2493.8], degeneracies [2, 1, 1]
    H2O: is_linear=False; vibrations counted with degeneracy = 3 (3N-6 = 3)
       wavenumbers (cm^-1): [1661.2 3686.3 3739.9], degeneracies [1, 1, 1]




.. GENERATED FROM PYTHON SOURCE LINES 40-43

In the linear molecule the bend coordinate is the transverse
displacement :math:`(x_0-2x_1+x_2)/r`. The out-of-plane (y) bend gives
the same frequency, so a stick for the bend carries twice the weight.

.. GENERATED FROM PYTHON SOURCE LINES 43-57

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    for ax, name, res in ((axes[0], "CO2 (linear, 3N-5 = 4)", co2), (axes[1], "H2O (bent, 3N-6 = 3)", h2o)):
        degeneracies = [2 if (res.is_linear and k == 0) else 1 for k in range(3)]
        ax.vlines(res.wavenumbers, 0.0, degeneracies, color="tab:purple", lw=3)
        for nu, g in zip(res.wavenumbers, degeneracies, strict=True):
            ax.annotate("doubly degenerate bend" if g == 2 else "", (nu, g), xytext=(5, -15), textcoords="offset points", fontsize=8)
        ax.set_ylim(0.0, 2.5)
        ax.set_ylabel("number of modes")
        ax.set_xlabel(r"wavenumber (cm$^{-1}$)")
        ax.set_title(name)
    fig.suptitle("Counting vibrations: a linear molecule's degenerate bend")
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/spectro/vibrational/images/sphx_glr_plot_04_linear_bend_degeneracy_001.png
   :alt: Counting vibrations: a linear molecule's degenerate bend, CO2 (linear, 3N-5 = 4), H2O (bent, 3N-6 = 3)
   :srcset: /api/gallery/spectro/vibrational/images/sphx_glr_plot_04_linear_bend_degeneracy_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.059 seconds)


.. _sphx_glr_download_api_gallery_spectro_vibrational_plot_04_linear_bend_degeneracy.py:

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