Theoretical study of the electronic states of hollandite vanadate K₂V₈O₁₆

Electronic states of hollandite vanadate K₂V₈O₁₆, a one-dimensional zigzag-chain system of t_2g orbitals in a mixed valent state, are considered. We calculate the Madelung energies to determine the most stable charge-ordering pattern that is consistent with the observed superlattice structure. We then develop the strong-coupling perturbation theory to derive the effective spin-orbit Hamiltonian, starting from the triply-degenerate t_2g orbitals in the VO₆ octahedral structure. An exact-diagonalization technique is used on small clusters of this Hamiltonian to determine the orbital-ordering pattern and spin structures in the ground state. We thereby discuss the electronic and magnetic properties of K₂ V₈O₁₆.