Hubbard and Heisenberg Models for Four-Site Four-Electron Systems. Group-Theoretical Interrelationships and Applications to Multinuclear Transition-Metal Clusters

Abstract

Abstract Hubbard models for four-site four-electron {4,4} systems are analytically solved to elucidate group-theoretical interrelationships between effective model Hamiltonians for radical clusters, metal clusters, Cu4O4, and so on. The group operations used for this purpose are permutation group (SN), spatial symmetry (PN), spin rotation (S), and time-reversal (T). The magnetic (color) group (T × S) is utilized for determination of the magnetic symmetry of spin structures obtained on the basis of the classical Heisenberg model; namely a vector representation of radical spins. The magnetic double group theory (T × S × PN) is used for characterization of general Hartree–Fock solutions (GHF) involving axial, helical, and torsional spin density waves for {4,4} systems. The spin-optimized SCF solutions are constructed from the spin projection of the GHF solutions (T × S × PN) by the use of the permutation symmetry (SN). The SN group is also used for the quantum Heisenberg model for {4,4} systems. In order to show the interrelationships between these model Hamiltonians, the electronic states of {4,4} systems with the D4h, Td, and D3h symmetries are constructed by the use of the magnetically ordered general spin orbitals which have been determined by the GHF calculations. The spin-optimized SCF solutions for {4,4} systems examined here are equivalent to the full CI wavefunctions satisfying both spatial and spin symmetries. The relative contributions of the spin polarization (SP) and doubly excited configurations in the GHF, projected GHF and SO-SCF wavefunctions are clarified to elucidate possible mechanisms of spin alignments and antiferromagnetic spin correlations. Implications of these computational results are discussed in relation to quantum and classical representations of spin alignments in molecular magnetic materials such as iron–sulfur clusters. Molecular magnets having helical and torsional spin structures are designed and discussed in relation to the most general spin alignment in the species.