\emph{Ab initio} derivation of the crystal field parameters for lanthanide ions: The f$^1$ case

The crystal field theory as explained by Abragam and Bleaney in their landmark 1970 book on transition-ion electron paramagnetic resonance remains a cornerstone in the development of luminescence applications and molecular magnets based on the $f$-elements. The modern numerical derivation of the 27 $B_k^q$ Stevens crystal field parameters (CFPs), which describe the splitting of the energy levels of a central ion, is traditionally achieved through the effective Hamiltonian theory and multiconfiguration wavefunction theory calculations, insofar as the lowest $J$ level fully captures the targeted low-energy physics. In this work, we present a novel theoretical approach for determining the CFPs. The procedure resembles the traditional extraction path but crucially accounts for the full $\ket{J,M_J}$ space of an ion configuration with $L=3$ and $S=\nicefrac{1}{2}$. By demonstrating the extraction procedure using the simplest case of a Ce$^\text{III}$ 4f$^1$ ion with a crystal-field split $J \in \{\nicefrac{5}{2}, \nicefrac{7}{2}\}$ manifold, it is shown for the first time that a unique set of CFPs describes the splitting and mixing both the $J$ manifolds. In fact, this $J/J^\prime$ mixing is analogous to the ``spin mixing'' in binuclear transition metal complexes. At the employed level of calculation, we demonstrate that there is no spin-orbit coupling influence on the CFP values, contrary to previous beliefs. This work represents the first step of a larger effort in reviewing the theory and extraction procedures of CFPs in f-element complexes.