  We study the one-band Hubbard model at half filling with hopping asymmetry
and its effective model at finite but large U up to the second order of tmix/U.
Two variational wave functions, the resonating valence bond (RVB) wave function
and antiferromagnetic (AF) RVB coexisted wave function, are studied by
variational Monte Carlo method on L*L square lattices up to L=12. Based on
these two wave functions, the phase diagrams for both models are presented. For
the Hubbard model, we find that there is a metal-insulator transition when the
hopping parameter tmix which changes the local double occupant vanishes while
only a metal-insulator crossover is explored for any finite tmix. For the
effective model in which the perturbation expansion is up to the second order
of tmix/U, a clear metal-insulator transition can be identified for both
variational wave functions and the phase diagram can be drawn accordingly. In
both models, we find that the systems are dominated by AF-RVB wave function
when U is large while the RVB wave function is favored when U is small.
