Effective Charge and Spin Hamiltonian for the Quarter-Filled Ladder Compound $\alpha'$-NaV$_2$O$_5$

An effective intra- and inter-ladder charge-spin hamiltonian for the quarter-filled ladder compound $\alpha'$-NaV$_2$O$_5$ has been derived by using the standard canonical transformation method. In the derivation, it is clear that a finite inter-site Coulomb repulsion is needed to get a meaningful result otherwise the perturbation becomes ill-defined. Various limiting cases depending on the values of the model parameters have been analyzed in detail and the effective exchange couplings are estimated. We find that the effective intra-ladder exchange may become ferromagnetic for the case of zig-zag charge ordering in a purely electronic model. We estimate the magnitude of the effective inter-rung Coulomb repulsion in a ladder and find it to be about one-order of magnitude too small in order to stabilize charge-ordering.