Noether’s theorem and conserved quantities for the crystal- and ligand-field Hamiltonians invariant under continuous rotational symmetry

Applications of Noether's theorem to crystal-field (CF) and ligand-field Hamiltonians invariant under continuous rotational symmetry are discussed. Deeper meaning of the seemingly unrelated concepts of (i) Noether's theorems, (ii) the algebraic symmetry of Hamiltonians, and (iii) the rotational invariants and moments of CF Hamiltonians is considered and their interrelationships unraveled. Our investigations enable formulation of an important theorem and a conjecture on the conserved quantities stipulated by Noether's theorem for the CF Hamiltonians in question. Geometrical meaning of the second-order conserved quantities suggests feasibility of derivation of a conservation law encompassing all the conserved quantities identified. The existence of the conserved quantities has profound implications for interpretation of experimental CF parameter data sets, which are encapsulated in five corollaries. Our considerations reveal that various aspects require reinterpretation. This includes, e.g., (i) the feasibility of determination of CF parameters from fitting experimental spectra, and (ii) the reduction of the existing higher-order rotational invariants for hexagonal type-II and cubic symmetries to combinations of primary lower-order invariants. The approach presented in this paper enables adoption of better-fitting strategies utilizing the well-defined conserved quantities, which are invariant under continuous rotational symmetry.

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