.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/example_04_GS_analysis.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_example_04_GS_analysis.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_example_04_GS_analysis.py:


Ground state analysis for NiO
================================================================================
This example follows the :ref:`sphx_glr_auto_examples_example_03_AIM_XAS.py`
example and considers the same model. This time we show how to analyze
the eigenvectors in terms of a
:math:`\alpha |d^8L^{10}\rangle + \beta |d^9L^9\rangle
+ \gamma |d^{10}L^8\rangle`
representation.

.. GENERATED FROM PYTHON SOURCE LINES 14-17

Hamiltonian
------------------------------------------------------------------------------
We start by obtaining all eigenvectors of the previous Hamiltonian.

.. GENERATED FROM PYTHON SOURCE LINES 17-26

.. code-block:: Python


    import numpy as np
    import matplotlib.pyplot as plt
    import edrixs

    backend = 'scipy'
    eval_i, evec_i = edrixs.ed(hmat_i, shift=shift, num_evals=len(basis_i), backend=backend)
    eval_i = eval_i - eval_i.min()





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

 .. code-block:: none

    /home/runner/micromamba/envs/build-env/lib/python3.12/site-packages/edrixs/scipy_backend/scipy_backend.py:98: UserWarning: The problem size 190 minus the constraints size 0 is too small relative to the block size 190. Using a dense eigensolver instead of LOBPCG iterations.No output of the history of the iterations.
      eigenvalues, eigenvectors = lobpcg(




.. GENERATED FROM PYTHON SOURCE LINES 36-43

Number of d electrons
------------------------------------------------------------------------------
We can count the number of :math:`d` electrons by building a single-particle
operator :code:`single_particle_Nd` with ones on the diagonal at the nickel
:math:`d`-shell orbitals and zeros elsewhere. :func:`~edrixs.solvers.build_op`
transforms this into the many-body operator :code:`O`, whose expectation value
in each eigenvector is the :math:`d`-electron count of that state.

.. GENERATED FROM PYTHON SOURCE LINES 43-57

.. code-block:: Python


    single_particle_Nd = np.zeros_like(emat_i)
    single_particle_Nd[:norb_d, :norb_d] = np.eye(norb_d)

    O = edrixs.build_op(single_particle_Nd, None, basis_i, backend=backend)

    nd_expect = np.sum(evec_i.conj() * (O @ evec_i), axis=0).real

    fig, ax = plt.subplots()

    ax.set_xlabel('Energy (eV)')
    ax.set_ylabel('$d$ electron count')
    ax.plot(eval_i, nd_expect)




.. image-sg:: /auto_examples/images/sphx_glr_example_04_GS_analysis_001.png
   :alt: example 04 GS analysis
   :srcset: /auto_examples/images/sphx_glr_example_04_GS_analysis_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    [<matplotlib.lines.Line2D object at 0x7f65479c22d0>]



.. GENERATED FROM PYTHON SOURCE LINES 58-64

Configuration weights alpha, beta, gamma
------------------------------------------------------------------------------
Every Fock state has a definite :math:`d` occupation, so :math:`O` is diagonal
in the Fock basis. :code:`O.diagonal()` therefore gives the :math:`d`-electron
count of each Fock-basis component, in the same order as the amplitudes of the
eigenvectors returned by :func:`~edrixs.solvers.ed`.

.. GENERATED FROM PYTHON SOURCE LINES 64-81

.. code-block:: Python

    nd_per_component = np.rint(O.diagonal().real).astype(int)

    amp2 = np.abs(evec_i)**2
    alphas = amp2[nd_per_component == 8].sum(axis=0)
    betas = amp2[nd_per_component == 9].sum(axis=0)
    gammas = amp2[nd_per_component == 10].sum(axis=0)

    print("Ground state\nalpha={:.3f}\tbeta={:.3f}\tgamma={:.3f}".format(
        alphas[0], betas[0], gammas[0]))

    fig, ax = plt.subplots()
    ax.plot(eval_i, alphas, label=r'$\alpha$ $d^8L^{10}$')
    ax.plot(eval_i, betas, label=r'$\beta$ $d^9L^{9}$')
    ax.plot(eval_i, gammas, label=r'$\gamma$ $d^{10}L^{8}$')
    ax.set_xlabel('Energy (eV)')
    ax.set_ylabel('Population')
    ax.legend()



.. image-sg:: /auto_examples/images/sphx_glr_example_04_GS_analysis_002.png
   :alt: example 04 GS analysis
   :srcset: /auto_examples/images/sphx_glr_example_04_GS_analysis_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Ground state
    alpha=0.823     beta=0.172      gamma=0.004

    <matplotlib.legend.Legend object at 0x7f654739ffb0>




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

   **Total running time of the script:** (0 minutes 0.322 seconds)


.. _sphx_glr_download_auto_examples_example_04_GS_analysis.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: example_04_GS_analysis.ipynb <example_04_GS_analysis.ipynb>`

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

      :download:`Download Python source code: example_04_GS_analysis.py <example_04_GS_analysis.py>`

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

      :download:`Download zipped: example_04_GS_analysis.zip <example_04_GS_analysis.zip>`


.. only:: html

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

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