.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/example_7_Coulomb.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_example_7_Coulomb.py: Coulomb interactions ===================================== In this example we provide more details on how Coulomb interactions are implemented in multiplet calculations and EDRIXS in particular. We aim to clarify the form of the matrices, how they are parametrized, and how the breaking of spherical symmetry can switch on additional elements that one might not anticipate. Our example is based on a :math:`d` atomic shell. .. GENERATED FROM PYTHON SOURCE LINES 13-50 Create matrix ------------------------------------------------------------------------------ The Coulomb interaction between two particles can be written as .. math:: \begin{equation} \hat{H} = \frac{1}{2} \int d\mathbf{r} \int d\mathbf{r}^\prime \Sigma_{\sigma, \sigma^\prime} |\hat{\psi}^\sigma(\mathbf{r})|^2 \frac{e^2}{R} |\hat{\psi}^{\sigma^\prime}(\mathbf{r})|^2, \end{equation} where :math:`\hat{\psi}^\sigma(\mathbf{r})` is the electron wavefunction, with spin :math:`\sigma`, and :math:`R=|r-r^\prime|` is the electron separation. Solving our problem in this form is difficult due to the need to symmeterize the wavefunction to follow fermionic statistics. Using second quantization, we can use operators to impose the required particle exchange statistics and write the equation in terms of a tensor :math:`U` .. math:: \begin{equation} \hat{H} = \sum_{\alpha,\beta,\gamma,\delta,\sigma,\sigma^\prime} U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} \hat{f}^{\dagger}_{\alpha\sigma} \hat{f}^{\dagger}_{\beta\sigma^\prime} \hat{f}_{t\sigma^\prime}\hat{f}_{\delta\sigma}, \end{equation} where :math:`\alpha`, :math:`\beta`, :math:`\gamma`, :math:`\delta` are orbital indices and :math:`\hat{f}^{\dagger}` (:math:`\hat{f}`) are the creation (anihilation) operators. For a :math:`d`-electron system, we have :math:`10` distinct spin-orbitals (:math:`5` orbitals each with :math:`2` spins), which makes matrix the :math:`10\times10\times10\times10` in total size. In EDRIXS the matrix can be created as follows: .. GENERATED FROM PYTHON SOURCE LINES 50-58 .. code-block:: default import edrixs import numpy as np import scipy import matplotlib.pyplot as plt import itertools F0, F2, F4 = 6.94, 14.7, 4.41 umat_chb = edrixs.get_umat_slater('d', F0, F2, F4) .. GENERATED FROM PYTHON SOURCE LINES 59-61 We stored this under variable :code:`umat_chb` where "cbh" stands for complex harmonic basis, which is the default basis in EDRIXS. .. GENERATED FROM PYTHON SOURCE LINES 63-92 Parameterizing interactions ------------------------------------------------------------------------------ EDRIXS parameterizes the interactions in :math:`U` via Slater integral parameters :math:`F^{k}`. These relate to integrals of various spherical Harmonics as well as Clebsch-Gordon coefficients, Gaunt coefficients, and Wigner 3J symbols. Textbooks such as [1]_ can be used for further reference. If you are interested in the details of how EDRIXS does this (and you probably aren't) function :func:`.umat_slater`, constructs the required matrix via Gaunt coeficents from :func:`.get_gaunt`. Two alternative parameterizations are common. The first are the Racah parameters, which are .. math:: \begin{eqnarray} A &=& F^0 - \frac{49}{441} F^4 \\ B &=& \frac{1}{49}F^2 - \frac{5}{441}F^4 \\ C &=& \frac{35}{441}F^4. \end{eqnarray} or an alternative form for the Slater integrals .. math:: \begin{eqnarray} F_0 &=& F^0 \\ F_2 &=& \frac{1}{49}F^2 \\ F_4 &=& \frac{1}{441}F^4, \end{eqnarray} which involves different normalization parameters. .. GENERATED FROM PYTHON SOURCE LINES 94-101 Basis transform ------------------------------------------------------------------------------ If we want to use the real harmonic basis, we can use a tensor transformation, which imposes the following orbital order :math:`3z^2-r^2, xz, yz, x^2-y^2, xy`, each of which involves :math:`\uparrow, \downarrow` spin pairs. Let's perform this transformation and store a list of these orbitals. .. GENERATED FROM PYTHON SOURCE LINES 101-104 .. code-block:: default umat = edrixs.transform_utensor(umat_chb, edrixs.tmat_c2r('d', True)) orbitals = ['3z^2-r^2', 'xz', 'yz', 'x^2-y^2', 'xy'] .. GENERATED FROM PYTHON SOURCE LINES 105-158 Interactions ------------------------------------------------------------------------------ Tensor :math:`U` is a series of matrix elements .. math:: \begin{equation} \langle\psi_{\gamma,\delta}^{\bar{\sigma},\bar{\sigma}^\prime} |\hat{H}| \psi_{\alpha,\beta}^{\sigma,\sigma^\prime}\rangle \end{equation} the combination of which defines the energetic cost of pairwise electron-electron interactions between states :math:`\alpha,\sigma` and :math:`\beta,\sigma^\prime`. In EDRIXS we follow the convention of summing over all orbital pairs. Some other texts count each pair of indices only once. The matrix elements here will consequently be half the magnitude of those in other references. We can express the interactions in terms of the orbitals involved. It is common to distinguish "direct Coulomb" and "exchange" interactions. The former come from electrons in the same orbital and the later involve swapping orbital labels for electrons. We will use :math:`U_0` and :math:`J` as a shorthand for distinguishing these. Before we describe the different types of interactions, we note that since the Coulomb interaction is real, and due to the spin symmmetry properties of the process :math:`U` always obeys .. math:: \begin{equation} U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma} = U_{\beta\sigma,\alpha\sigma^\prime,\delta\sigma^\prime,\gamma\sigma} = U_{\delta\sigma,\gamma\sigma^\prime,\beta\sigma^\prime,\alpha\sigma} = U_{\gamma\sigma,\delta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}. \end{equation} 1. Intra orbital ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The direct Coulomb energy cost to double-occupy an orbital comes from terms like :math:`U_{\alpha\sigma,\alpha\bar\sigma,\alpha\bar\sigma,\alpha\sigma}`. In this notation, we use :math:`\sigma^\prime` to denote that the matrix element is summed over all pairs and :math:`\bar{\sigma}` to denote sums over all opposite spin pairs. Due to rotational symmetry, all these elements are the same and equal to .. math:: \begin{eqnarray} U_0 &=& \frac{A}{2} + 2B + \frac{3C}{2}\\ &=& \frac{F_0}{2} + 2F_2 + 18F_4 \end{eqnarray} Let's print these to demonstrate where these live in the array .. GENERATED FROM PYTHON SOURCE LINES 158-162 .. code-block:: default for i in range(0, 5): val = umat[i*2, i*2 + 1, i*2 + 1, i*2].real print(f"{orbitals[i]:<8} \t {val:.3f}") .. rst-class:: sphx-glr-script-out .. code-block:: none 3z^2-r^2 4.250 xz 4.250 yz 4.250 x^2-y^2 4.250 xy 4.250 .. GENERATED FROM PYTHON SOURCE LINES 163-170 2. Inter orbital Coulomb interactions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Direct Coulomb repulsion between different orbitals depends on terms like :math:`U_{\alpha\sigma,\beta\sigma^\prime,\beta\sigma^\prime,\alpha\sigma}`. Expresions for these parameters are provided in column :math:`U` in :ref:`table_2_orbital`. We can print the values from :code:`umat` like this: .. GENERATED FROM PYTHON SOURCE LINES 170-174 .. code-block:: default for i, j in itertools.combinations(range(5), 2): val = umat[i*2, j*2 + 1, j*2 + 1, i*2].real print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") .. rst-class:: sphx-glr-script-out .. code-block:: none 3z^2-r^2 xz 3.650 3z^2-r^2 yz 3.650 3z^2-r^2 x^2-y^2 2.900 3z^2-r^2 xy 2.900 xz yz 3.150 xz x^2-y^2 3.150 xz xy 3.150 yz x^2-y^2 3.150 yz xy 3.150 x^2-y^2 xy 3.900 .. GENERATED FROM PYTHON SOURCE LINES 175-181 3. Inter-orbital exchange interactions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Exchange terms exist with the form :math:`U_{\alpha\sigma,\beta\sigma^\prime,\alpha\sigma^\prime,\beta\sigma}`. Expresions for these parameters are provided in column :math:`J` of :ref:`table_2_orbital`. These come from terms like this in the matrix: .. GENERATED FROM PYTHON SOURCE LINES 181-185 .. code-block:: default for i, j in itertools.combinations(range(5), 2): val = umat[i*2, j*2 + 1, i*2 + 1, j*2].real print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") .. rst-class:: sphx-glr-script-out .. code-block:: none 3z^2-r^2 xz 0.300 3z^2-r^2 yz 0.300 3z^2-r^2 x^2-y^2 0.675 3z^2-r^2 xy 0.675 xz yz 0.550 xz x^2-y^2 0.550 xz xy 0.550 yz x^2-y^2 0.550 yz xy 0.550 x^2-y^2 xy 0.175 .. GENERATED FROM PYTHON SOURCE LINES 186-193 4. Pair hopping term ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Terms that swap pairs of electrons exist as :math:`(1-\delta_{\sigma\sigma'})U_{\alpha\sigma,\alpha\bar\sigma,\beta\bar\sigma,\beta\sigma}` and depend on exchange interactions column :math:`J` from :ref:`table_2_orbital` and here in the matrix. .. GENERATED FROM PYTHON SOURCE LINES 193-197 .. code-block:: default for i, j in itertools.combinations(range(5), 2): val = umat[i*2, i*2 + 1, j*2 + 1, j*2].real print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {val:.3f}") .. rst-class:: sphx-glr-script-out .. code-block:: none 3z^2-r^2 xz 0.300 3z^2-r^2 yz 0.300 3z^2-r^2 x^2-y^2 0.675 3z^2-r^2 xy 0.675 xz yz 0.550 xz x^2-y^2 0.550 xz xy 0.550 yz x^2-y^2 0.550 yz xy 0.550 x^2-y^2 xy 0.175 .. GENERATED FROM PYTHON SOURCE LINES 198-215 5. Three orbital ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Another set of terms that one might not immediately anticipate involve three orbitals like .. math:: \begin{equation} U_{\alpha\sigma, \gamma\sigma', \beta\sigma', \gamma\sigma} \\ U_{\alpha\sigma, \gamma\sigma', \gamma\sigma', \beta\sigma} \\ (1-\delta_{\sigma\sigma'}) U_{\alpha\sigma, \beta\sigma', \gamma\sigma', \gamma\sigma} \end{equation} for :math:`\alpha=3z^2-r^2, \beta=x^2-y^2, \gamma=xz/yz`. These are needed to maintain the rotational symmetry of the interations. See :ref:`table_3_orbital` for the expressions. We can print some of these via: .. GENERATED FROM PYTHON SOURCE LINES 215-227 .. code-block:: default ijkl = [[0, 1, 3, 1], [0, 2, 3, 2], [1, 0, 3, 1], [1, 1, 3, 0], [2, 0, 3, 2], [2, 2, 3, 0]] for i, j, k, l in ijkl: val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t" f"{orbitals[k]:<8} \t {orbitals[l]:<8} \t {val:.3f}") .. rst-class:: sphx-glr-script-out .. code-block:: none 3z^2-r^2 xz x^2-y^2 xz 0.217 3z^2-r^2 yz x^2-y^2 yz -0.217 xz 3z^2-r^2 x^2-y^2 xz -0.433 xz xz x^2-y^2 3z^2-r^2 0.217 yz 3z^2-r^2 x^2-y^2 yz 0.433 yz yz x^2-y^2 3z^2-r^2 -0.217 .. GENERATED FROM PYTHON SOURCE LINES 228-233 6. Four orbital ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Futher multi-orbital terms include :math:`U_{\alpha\sigma,\beta\sigma^\prime,\gamma\sigma^\prime,\delta\sigma}`. We can find these here in the matrix: .. GENERATED FROM PYTHON SOURCE LINES 233-249 .. code-block:: default ijkl = [[0, 1, 2, 4], [0, 1, 4, 2], [0, 2, 1, 4], [0, 2, 4, 1], [0, 4, 1, 2], [0, 4, 2, 1], [3, 1, 4, 2], [3, 2, 4, 1], [3, 4, 1, 2], [3, 4, 2, 1]] for i, j, k, l in ijkl: val = umat[i*2, j*2 + 1, k*2 + 1, l*2].real print(f"{orbitals[i]:<8} \t {orbitals[j]:<8} \t {orbitals[k]:<8}" f"\t {orbitals[l]:<8} \t {val:.3f}") .. rst-class:: sphx-glr-script-out .. code-block:: none 3z^2-r^2 xz yz xy -0.433 3z^2-r^2 xz xy yz 0.217 3z^2-r^2 yz xz xy -0.433 3z^2-r^2 yz xy xz 0.217 3z^2-r^2 xy xz yz 0.217 3z^2-r^2 xy yz xz 0.217 x^2-y^2 xz xy yz -0.375 x^2-y^2 yz xy xz 0.375 x^2-y^2 xy xz yz -0.375 x^2-y^2 xy yz xz 0.375 .. GENERATED FROM PYTHON SOURCE LINES 250-256 Effects of multi-orbital terms ------------------------------------------------------------------------------ To test the effects of the multi-orbital terms, let's plot the eigenenergy spectra with and without multi-orbital terms switched on for system with and without a cubic crystal field. We will use a :math:`d`-shell with two electrons. .. GENERATED FROM PYTHON SOURCE LINES 256-292 .. code-block:: default ten_dqs = [0, 2, 4, 12] def diagonalize(ten_dq, umat): emat = edrixs.cb_op(edrixs.cf_cubic_d(ten_dq), edrixs.tmat_c2r('d', ispin=True)) H = (edrixs.build_opers(4, umat, basis) + edrixs.build_opers(2, emat, basis)) e, v = scipy.linalg.eigh(H) return e - e.min() basis = edrixs.get_fock_bin_by_N(10, 2) umat_no_multiorbital = np.copy(umat) B = F2/49 - 5*F4/441 for val in [np.sqrt(3)*B/2, np.sqrt(3)*B, 3*B/2]: umat_no_multiorbital[(np.abs(umat)- val) < 1e-6] = 0 fig, axs = plt.subplots(1, len(ten_dqs), figsize=(8, 3)) for cind, (ax, ten_dq) in enumerate(zip(axs, ten_dqs)): ax.hlines(diagonalize(ten_dq, umat), xmin=0, xmax=1, label='on', color=f'C{cind}') ax.hlines(diagonalize(ten_dq, umat_no_multiorbital), xmin=1.5, xmax=2.5, label='off', linestyle=':', color=f'C{cind}') ax.set_title(f"$10D_q={ten_dq}$") ax.set_ylim([-.5, 20]) ax.set_xticks([]) ax.legend() fig.suptitle("Eigenvalues with 3&4-orbital effects on/off") fig.subplots_adjust(wspace=.3) axs[0].set_ylabel('Eigenvalues (eV)') fig.subplots_adjust(top=.8) plt.show() .. image-sg:: /auto_examples/images/sphx_glr_example_7_Coulomb_001.png :alt: Eigenvalues with 3&4-orbital effects on/off, $10D_q=0$, $10D_q=2$, $10D_q=4$, $10D_q=12$ :srcset: /auto_examples/images/sphx_glr_example_7_Coulomb_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 293-382 On the left of the plot Coulomb interactions in spherical symmetry cause substantial mxing between :math:`t_{2g}` and :math:`e_{g}` orbitals in the eigenstates and 3 & 4 orbital orbital terms are crucial for obtaining the the right eigenenergies. As :math:`10D_q` get large, this mixing is switched off and the spectra start to become independent of whether the 3 & 4 orbital orbital terms are included or not. .. _table_2_orbital: .. table:: Table of 2 orbital interactions +-----------------------------+------------------+-----------------------+----------------+--------------------+ |Orbitals :math:`\alpha,\beta`|:math:`U_0` Racah | :math:`U_0` Slater |:math:`J` Racah |:math:`J` Slater | +=============================+==================+=======================+================+====================+ |:math:`3z^2-r^2, xz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | +-----------------------------+------------------+-----------------------+----------------+--------------------+ |:math:`3z^2-r^2, yz` |:math:`A/2+B+C/2` |:math:`F_0/2+F_2-12F_4`| :math:`B/2+C/2`|:math:`F_2/2+15F_4` | +-----------------------------+------------------+-----------------------+----------------+--------------------+ |:math:`3z^2-r^2, x^2-y^2` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| +-----------------------------+------------------+-----------------------+----------------+--------------------+ |:math:`3z^2-r^2, xy` |:math:`A/2-2B+C/2`|:math:`F_0/2-2F_2+3F_4`|:math:`2B+C/2` |:math:`2F_2+15F_4/2`| +-----------------------------+------------------+-----------------------+----------------+--------------------+ |:math:`xz, yz` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| +-----------------------------+------------------+-----------------------+----------------+--------------------+ |:math:`xz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| +-----------------------------+------------------+-----------------------+----------------+--------------------+ |:math:`xz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| +-----------------------------+------------------+-----------------------+----------------+--------------------+ |:math:`yz, x^2-y^2` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| +-----------------------------+------------------+-----------------------+----------------+--------------------+ |:math:`yz, xy` |:math:`A/2-B+C/2` |:math:`F_0/2-F_2-12F_4`|:math:`3B/2+C/2`|:math:`3F_2/2+10F_4`| +-----------------------------+------------------+-----------------------+----------------+--------------------+ |:math:`x^2-y^2, xy` |:math:`A/2+2B+C/2`|:math:`F_0+4F_2-34F_4` | :math:`C/2` |:math:`35F_4/2` | +-----------------------------+------------------+-----------------------+----------------+--------------------+ .. _table_3_orbital: .. table:: Table of 3 orbital interactions +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| +=============================+=============+====================================================+=====================================================+ |:math:`3z^2-r^2, xz, x^2-y^2, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`3z^2-r^2, yz, x^2-y^2, yz` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`xz, 3z^2-r^2, x^2-y^2, xz` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`xz, xz, x^2-y^2, 3z^2-r^2` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`yz, 3z^2-r^2, x^2-y^2, yz` | :math:`\sqrt{3}B` | :math:`\sqrt{3}F_2-5\sqrt{3}F_4` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`yz, yz, x^2-y^2, 3z^2-r^2` | :math:`-\sqrt{3}B/2` | :math:`-\sqrt{3}F_2/2+5\sqrt{3}F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ .. _table_4_orbital: .. table:: Table of 4 orbital interactions +-----------------------------+-------------+----------------------------------------------------+-----------------------------------------------------+ |Orbitals :math:`\alpha,\beta,\gamma,\delta`|:math:`\langle\alpha\beta|\gamma\delta\rangle` Racah|:math:`\langle\alpha\beta|\gamma\delta\rangle` Slater| +=============================+=============+====================================================+=====================================================+ |:math:`3z^2-r^2, xz, yz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`3z^2-r^2, xz, xy, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`3z^2-r^2, yz, xz, xy` | :math:`-\sqrt{3}B` | :math:`-\sqrt{3}F_2+5\sqrt{3}F_4` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`3z^2-r^2, yz, xy, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`3z^2-r^2, xy, xz, yz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`3z^2-r^2, xy, yz, xz` | :math:`\sqrt{3}B/2` | :math:`\sqrt{3}F_2/2-5\sqrt{3}F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`x^2-y^2 , xz, xy, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`x^2-y^2 , yz, xy, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`x^2-y^2 , xy, xz, yz` | :math:`-3B/2` | :math:`-3F_2/2+15F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ |:math:`x^2-y^2 , xy, yz, xz` | :math:`3B/2` | :math:`3F_2/2-15F_4/2` | +-------------------------------------------+----------------------------------------------------+-----------------------------------------------------+ .. rubric:: Footnotes .. [1] MSugano S, Tanabe Y and Kamimura H. 1970. Multiplets of Transition-Metal Ions in Crystals. Academic Press, New York and London. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.579 seconds) .. _sphx_glr_download_auto_examples_example_7_Coulomb.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: example_7_Coulomb.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: example_7_Coulomb.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_