  We consider the possible charge and orbital ordering in a Hollandite compound
Bi_xV_8O_{16}, which is a new one-dimensional triangular-lattice t_{2g}-orbital
system. Using the strong-coupling perturbation theory, we derive the effective
spin-orbit Hamiltonian in the approximation neglecting the small off-diagonal
hopping parameters or orbital fluctuation, whereby we obtain the spin
Hamiltonians in the partial space of each orbital-ordering pattern. We then
apply an exact-diagonalization technique on small clusters to these spin
Hamiltonians and calculate the ground-state phase diagram. We find that a
variety of orbital-ordering patterns appear in the parameter space, which
include the state characterized by the partial singlet formation consistent
with recent NMR experiment.
