Reduction Symmetry Method in Crystal Field Theory (IV)

Abstract

The rotation groups of the type R[,lv and branch rules for the representations of the groups (unitary and point) are introduced. Relations between the SC and the 6U, 6r c0efficient.s are developed by using the graph technique. The main formulas for the matrix elements of oneparticle operators between states of complex configurations and the eigenvalues of the Casimir operators are given.Es werden Drehgruppen vom Typ R[,lv und Verzweigungsregeln fur die Darstellungen der Qruppen (unitLre und Punktgruppen) eingefiihrt. Beziehungen zwischen den Untergruppen-Koeffizienten und den 6 U , 6r-Koeffizienten werden mittels Graphentechnik abgeleitet. Die Hauptformeln fur die Matrixelemente der Einteilchenoperatoren zwischen ZustLnden komplexer Konfigurationen und die Eigenwerte der Casimir-Operatoren werden angegeben.

Energy levels of paramagnetic ions: Algebra. III. The case of dN ions in cubical symmetry

“…4), these theories have known considerable progress during the last three decades . Recent developments in crystal-and ligand-field theories have been chiefly achieved in five (not independent) directions: (i) the development, in algebraic and diagrammatic forms, of irreducible tensor methods for the strongfield model 11 -3,7,14,16], the weak-field model [5,6,8,15], and the medium-field model [9, 141; (ii) the adaptation of irreducible tensor methods to crystal-field approaches of photoelectron spectroscopy of partly filled shell ions systems [18,191; (iii) the consideration of chains of groups involving both finite and Lie groups [5-171; (iv) the use of Lie-like and second quantization techniques for the classification of state vectors and operators [6,7,10,15,17]; and (v) the introduction of more and more refined Hamiltonians [9,20,21] and more and more refined bases (as for example MO-LCAO bases) [22-241. To be complete, we should also mention the development of additive submodels (angular overlap model [25], superposition model 1261, and point-charge and point-dipole electrostatic model [U]) to interpret the global crystal-field parameters and thus to approach the metal-ligand bond in a semiempirical way. Our interest in this paper is directed towards global (or symmetry-adapted) rather than local (or additive) contributions to the crystal-field Hamiltonian so that we shall not deal with additive submodels.…”