Calculating the |L2,3| XAS of |Ti4+|
====================================

In this tutorial, we will use the |L2,3| absorption spectrum of |Ti4+| to
illustrates some of the fundamental concepts in using multiplet calculations as
a tool to interpret experimental data. The absorption spectrum results from the
2p to 3d electronic dipole transitions. As can be seen from the figure below
the spectrum calculated using multiplet theory can be in very good agreement
with the experimental spectrum (dotted).

.. figure:: assets/degroot_fig1.png
    :width: 60 %
    :align: center

    de Groot et al., Phys. Chem. Minerals, 1992, 19, 140--147.

Atomic Multiplet
----------------
1. Calculate the |L2,3| spectrum of |Ti4+| in spherical symmetry, i.e. using an
atomic representation of the system. For this, you need to exclude the crystal
field contribution from the Hamiltonian. In addition, set the Lorentzian
broadening to 0.1, and the Gaussian broadening to 0.0. Leave all the other
parameters to their default values. How many transitions are visible in the
spectrum? Identify the |L2| and |L3| edges.

    **Note**: What is the initial electronic configuration of |Ti4+|? How many
    multi-electronic states does this configuration have? Write down their
    spectroscopic terms. Repeat the same exercise for the final electronic
    configuration. Using the selection rules for dipole transitions
    (Δ\ *J* = 0, ±1 except for *J* = *J*\ ' = 0) how many transitions
    do you expect?

2. In semi-empirical multiplet calculations, the Hamiltonian parameters are
varied to improve the agreement with the experiment. In the case of the atomic
parameters, i.e. Slater integrals and the spin-orbit coupling constants,
instead of modifying the parameters themselves, it is customary to use scaling
factors to change them. Run a calculation with the scaling of the 2p spin-orbit
coupling constant, ζ(2p), set to 0.5. After the calculation finishes, set this
value back to 1.0, and run a second calculation with the 3d spin-orbit
coupling, ζ(3d), in the *Final Hamiltonian* set to 0.5. Which interaction
affects the most the energy separation between the |L2| and |L3| edges? Check
that the energy separation between the two edges is close to 3/2·ζ(2p).

3. Change back the scaling parameter to 1.0 for both 2p and 3d spin-orbit
coupling. Perform three calculations using 0.8, 0.4, and 0.0 for the scaling of
the Slater integrals, |Fk| and |Gk|. Instead of changing each scaling value
individually as before, use the input boxes above the Hamiltonian terms. Plot
the three spectra. What is the influence of the electronic repulsions on the
spectrum? Check that in the last case, with the scaling factors set to zero,
the intensity ratio of the |L3|/|L2| also called the branching ratio, is close
to 2:1.

Crystal Field Multiplet
-----------------------
1. Set the scaling factors for the Slater integrals back to 0.8. Enable the
*Crystal Field* and change the 10Dq value to 2.0 eV for both the initial and
final Hamiltonian. In octahedral symmetry, the crystal field splitting 10Dq is
also written |DeltaO|. Run the calculation and compare it with the case of
spherical symmetry. How many transitions do you observe at the |L3| edge? How
many transitions at the |L2| edge?

2. In the previous calculation the Lorentzian broadening was set to 0.1 eV to
better identify the number of transitions. Change it to 0.2 eV and run the
calculation. Observe its effect on the final spectrum.

3. Run a set of calculations with 10Dq ranging from 0 to 2.0 eV, in steps of
0.5 eV. Plot the resulting spectra. What is the influence of the crystal field
splitting?

4. Set the 10Dq value to 2.0 eV and switch off the Slater integrals and the 3d
spin-orbit coupling. How many transitions does the calculated spectrum have?
Check if their intensity ratio is close to 6:4:3:2, i.e. the theoretical ratio
is given by the degeneracy of the 3d orbitals (3:2) and the branching ratio
discussed before (2:1). What is the energy separation between the first two
transitions? How does this compare to the energy separation between the last
two transitions?

Adding Tetragonal Distortion
----------------------------
1. Next we are going to study the influence of a tetragonal distortion, i.e. an
elongation or compression along one of the four-fold axes. Lowering the
symmetry from |Oh| to |D4h|, results in a different energy splitting of the 3d
orbitals as can be seen in the figure below. The relative energy position of
the orbitals depends on the distortion applied to the octahedron and is
determined by two parameters Ds and Dt, in addition to the Dq parameter.

.. image:: assets/orbitals_diagram.png
    :width: 60 %
    :align: center

2. Change the symmetry of the system to |D4h|. Note that by doing this all
parameters will be reset to their default values. Set the Dq value to 0.25 eV.
This is equivalent to setting the 10Dq value to 2.5 eV in the case of the |Oh|
symmetry. While keeping Dt zero, vary the value of Ds between -0.6 and 0.6 eV
in steps of 0.2 eV. Try to rationalize the changes you observe in the spectrum.
Do a similar test for Dt while keeping Ds zero.

.. |L2,3| replace:: L\ :sub:`2,3`\
.. |Ti4+| replace:: Ti\ :sup:`4+`\
.. |L2| replace:: L\ :sub:`2`\
.. |L3| replace:: L\ :sub:`3`\
.. |Fk| replace:: F\ :sub:`k`\
.. |Gk| replace:: G\ :sub:`k`\
.. |DeltaO| replace:: Δ\ :sub:`O`\
.. |2p3/2| replace:: 2p\ :sub:`3/2`\
.. |2p1/2| replace:: 2p\ :sub:`1/2`\
.. |3d(eg)| replace:: 3d(e\ :sub:`g`)\
.. |3d(t2g)| replace:: 3d(t\ :sub:`2g`)\
.. |Oh| replace:: O\ :sub:`h`\
.. |D4h| replace:: D\ :sub:`4h`\