EMR-related problems at the interface between the crystal field Hamiltonians and the zero-field splitting Hamiltonians

The interface between optical spectroscopy, electron magnetic resonance (EMR), and magnetism of transition ions forms the intricate web of interrelated notions. Major notions are the physical Hamiltonians, which include the crystal field (CF) (or equivalently ligand field (LF)) Hamiltonians, and the effective spin Hamiltonians (SH), which include the zero-field splitting (ZFS) Hamiltonians as well as to a certain extent also the notion of magnetic anisotropy (MA). Survey of recent literature has revealed that this interface, denoted CF (LF) <-> SH (ZFS), has become dangerously entangled over the years. The same notion is referred to by three names that are not synonymous: CF (LF), SH (ZFS), and MA. In view of the strong need for systematization of nomenclature aimed at bringing order to the multitude of different Hamiltonians and the associated quantities, we have embarked on this systematization. In this article, we do an overview of our efforts aimed at providing a deeper understanding of the major intricacies occurring at the CF (LF) <-> SH (ZFS) interface with the focus on the EMR-related problems for transition ions.