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

TL;DR

This work develops a fully ab initio route to derive crystal-field parameters for the f$^1$ configuration, addressing whether spin–orbit coupling alters CFP values and whether truncating to a single $J$ manifold biases results. By constructing complete model CF matrices in both spin-free $|l,m_l\rangle$ and spin-orbit coupled $|j,m_j\rangle$ spaces and using state-of-the-art CASSCF/SOCI calculations, des Cloizeaux mappings, and irreducible tensor operator decompositions, the authors show that CFPs are identical when the full space is included; truncation to the ground $J$ manifold introduces bias and misses $J$-mixing. The cerocene anion Ce(C8H8)2$^-$ exemplifies realistic $J$-mixing and reproduces ground-state admixture and g-tensor anisotropy, validating the approach. The work provides a robust, open framework (NewMag) for first-principles CFP extraction with broad implications for lanthanide/actinide chemistry in luminescence, magnetism, and quantum technologies.

Abstract

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.