Evolution of Tesseral and Spherical Tensor Operators in EPR and Optical Spectroscopy: Legacy of K.W.H. Stevens

This review examines the theoretical frameworks underlying electron magnetic resonance (EMR) and optical spectroscopy, with particular focus on the historical development and applications of tensor operators in interpreting spectroscopic data. Our motivation is to establish unequivocally the legacy of K.W.H. Stevens as a brilliant scholar of his epoch. The evolution from physical Hamiltonians to effective spin Hamiltonians is delineated, focusing on systems exhibiting orthorhombic, monoclinic, and triclinic site symmetry, where precise theoretical formulations are essential for accurate data interpretation. It highlights J.R. Pilbrow's seminal contributions to EPR spectroscopy of 3dN transition ions and his classification of low symmetry effects. A comprehensive assessment of tesseral tensor operators (TTO) and spherical tensor operators (STO), clarifying their relationship and appropriate applications in describing the zero-field splitting (ZFS) is provided. The key features of triclinic ZFS Hamiltonian are explained. The theoretical foundations of various Stevens-type operators, referred to as usual (USO), extended Stevens-type operators (ESO), and generalized extended Stevens-type operators (GESO) operators are examined, demonstrating that the selection of TTO versus STO should be guided by practical considerations rather than preconceived notion. The significance and wide usage of the USO and/or ESO is pointed out by referencing relevant books and reviews. The terminological inconsistencies identified in the recent literature, concerning, e.g. the key notions, distinctions between ZFS and crystal/ligand field (CF/LF) frameworks, and interchangeability of TTO versus STO formalisms, are discussed in the Appendix. The historical developments and key notions are elucidated to provide valuable insights in EMR, optical spectroscopies, magnetism and related areas.