In publications [287], [216], and [211], among others, in collaboration with
Dr. Hélène Bolvin, we used the crystal field (CF) splitting of atomic 4f and 5f
orbitals in order to set up simple one-electron models to explain the observed
magnetism of f^{1} complexes. During the time when I got involved in this research, I
programmed the calculation of the simultaneous eigenfunctions of the CF and
the spin-orbit (SO) Hamiltonian for a set of atomic orbitals with a given
angular quantum number ℓ in Mathematica, essentially to learn how this stuff
works.

You can download an example Mathematica notebook (792 kBytes) here. (See
also our Downloads page). In there is a calculation for a D_{4h} symmetric crystal field,
for example to represent the ligand environment of the metal in a complex such as
[NpO_{2}Cl_{4}]^{2−}.

The calculation is done by setting up a basis of |ℓ,m_{ℓ},m_{s}⟩ spin functions for an
atomic shell with a given value of ℓ. The CF is then represented by actual point
charges, or by symbolic values, transformed to a representation in terms of spherical
harmonics, and the operator matrix is calculated in the |ℓ,m_{ℓ},m_{s}⟩ basis. We also
calculate the SO operator

in the same basis and determine the Ĥ^{SO} eigenfunctions |j,m_{j}⟩ in the basis
|ℓ,m_{ℓ},m_{s}⟩. Here, ζ is an empirical SO coupling constant. We then determine the
eigenfunctions of the model Hamiltonian

in the |ℓ,m_{ℓ},m_{s}⟩ basis as well as in the basis |j,m_{j}⟩ of the eigenfunctions of H^{SO}
(i.e. the atomic spinors). The eigenvalues of the model Hamiltonian can be plotted
for different values of the CF parameters and ζ.

From the eigenvectors of Ĥ^{m} and the matrix elements of the spin and angular
momentum operators one can also generate information such as the electron
magnetic moment (or rather the g factor) for a magnetic field oriented in different
directions relative to the molecule. The graphics at the top of this page are based
on Mathematica calculations with notebooks similar to the one provided
here, showing the electron g-factor of 5f^{1} actinide complexes for different
ratios of selected CF parameters and the SO coupling constant. Details, and
explanations of the notation, are provided in publications [286], [215], and
[210].

© 2018 – 2022 J. Autschbach. The material shown on this web page is based on the results of research funded by a grant from the US Department of Energy (Basic Energy Sciences, Heavy Element Chemistry program, grant DE-SC0001136). Any opinions, findings, and conclusions or recommendations expressed here are those of the author and do not necessarily reflect the views of this funding agency.