  We propose an approximation scheme to describe the dynamics of the spin-boson
model when the spectral density of the environment shows a peak at a
characteristic frequency $\Omega$ which can be very close (or even equal) to
the spin Zeeman frequency $\Delta$. Mapping the problem onto a two-state system
(TSS) coupled to a harmonic oscillator (HO) with frequency $\omega_0$ we show
that the representation of displaced HO states provides an appropriate basis to
truncate the Hilbert space of the TSS-HO system and therefore a better picture
of the system dynamics. We derive an effective Hamiltonian for the TSS-HO
system, and show it furnishes a very good approximation for the system dynamics
even when its two subsystems are moderately coupled. Finally, assuming the
regime of weak HO-bath coupling and low temperatures, we are able to
analytically evaluate the dissipative TSS dynamics.
