  A new approximating technique is developed so as to study the quantum
ferromagnetic spin-1 Blume-Capel model in the presence of a transverse crystal
field in the square lattice. Our proposal consists of approaching the spin
system by considering islands of finite clusters whose frontiers are surrounded
by non-interacting spins that are treated by the effective-field theory. The
resulting phase diagram is qualitatively correct, in contrast to most
effective-field treatments, in which the first-order line exhibits spurious
behavior by not being perpendicular to the anisotropy axis at low temperatures.
The effect of the transverse anisotropy is also verified by the presence of
quantum phase transitions. The possibility of using larger sizes constitutes an
advantage to other approaches where the implementation of larger sizes is
costly computationally.
