Simulation of ionic transition-metal crystals: The cluster model and the cluster-lattice interaction in the light of the theory of electronic separability

The cluster model often applied to the study of transition-metal ions in ionic crystals is rigorously formulated in the framework of the theory of electronic separability (TES). This theory shows that the cluster-lattice coupling appearing in the effective cluster Hamiltonian should include two separate operators: (1) the lattice effective potential containing nuclear attraction, Coulomb, and exchange terms, and (2) a lattice projection operator enforcing the cluster-lattice orthogonality required by the Pauli principle. The analysis of the TES equations also suggests a hierarchy of lattice models for dealing with the cluster-lattice interaction in an approximate way. Using a Hartree-Fock-Roothaan description for the intracluster interactions, three of these models are investigated and illustrated by means of several examples. First, the familiar point-charge model is deduced from the TES equations. The main conceptual and practical deficiencies of this model are discussed. Then, a TES-consistent lattice model in which the cluster-lattice exchange interactions are approximated by Slater X{α} formula is presented. It is shown that this scheme, named the PX{α} model, does not suffer from most limitations of the point-charge approximation and gives a coherent and reasonable description of the equilibrium geometry of the (CrF₆{)}^{4{{-}}}$ unit in ${KCrF}₃$. Finally, the model potential (MP) lattice model is presented and discussed. In this model the lattice ions are described by accurate local model potentials without resorting to the X{α} approximation. The MP scheme gives satisfactory equilibrium geometries and relative stabilization energies for V, Cr, and Mn impurities in KMgF₃.