  The impulsive limit (the "sudden approximation") has been widely employed to
describe the interaction between molecules and short, far-off-resonant laser
pulses. This approximation assumes that the timescale of the laser--molecule
interaction is significantly shorter than the internal rotational period of the
molecule, resulting in the rotational motion being instantaneously "frozen"
during the interaction. This simplified description of laser-molecule
interaction is incorporated in various theoretical models predicting rotational
dynamics of molecules driven by short laser pulses. In this theoretical work,
we develop an effective theory for ultrashort laser pulses by examining the
full time-evolution operator and solving the time-dependent Schr\"odinger
equation at the operator level. Our findings reveal a critical angular
momentum, $l_\mathrm{crit}$, at which the impulsive limit breaks down. In other
words, the validity of the sudden approximation depends not only on the pulse
duration, but also on its intensity, since the latter determines how many
angular momentum states are populated. We explore both ultrashort multi-cycle
(Gaussian) pulses and the somewhat less studied half-cycle pulses, which
produce distinct effective potentials. We discuss the limitations of the
impulsive limit and propose a new method that rescales the effective matrix
elements, enabling an improved and more accurate description of laser-molecule
interactions.
