  The explicit criterion is derived in detail for the convergence of the Magnus
expansion and the existence of the Magnus solution in the interaction picture,
i.e., the exponential propagator in the weakly coupled spin (I=1/2) system
SnAMX... (n=1,2,3,...) in which only spin group S is subjected to a
time-dependent shaped selective radiofrequency pulse. The derivation is built
on the same scheme as the Maricq's (J. Chem. Phys. 86 (1987) 5647). It is shown
that the criterion depends only upon amplitude of the time-dependent field
applied to the system, and the Magnus expansion converges and the Magnus
solution exists when the flip angle of a non-negative-amplitude shaped RF pulse
or a weak-amplitude shaped pulse is smaller than 2pi. The exponential
propagator then can be decomposed into a product of a series of elementary
propagators and can be used to determine time evolution of the spin system
under the shaped pulse. The linear differential equations to determine the
unknown parameters in the Magnus solution are obtained explicitly. An
alternative propagator in an expansion form for the Magnus solution is also
proposed to describe the time evolution when the criterion is not met.
