Femtosecond nonlinear spectroscopy is the main tool for
the time-resolved
detection of photophysical and photochemical processes. Since most
systems of chemical interest are rather complex, theoretical support
is indispensable for the extraction of the intrinsic system dynamics
from the detected spectroscopic responses. There exist two alternative
theoretical formalisms for the calculation of spectroscopic signals,
the nonlinear response-function (NRF) approach and the spectroscopic
equation-of-motion (EOM) approach. In the NRF formalism, the system–field
interaction is assumed to be sufficiently weak and is treated in lowest-order
perturbation theory for each laser pulse interacting with the sample.
The conceptual alternative to the NRF method is the extraction of
the spectroscopic signals from the solutions of quantum mechanical,
semiclassical, or quasiclassical EOMs which govern the time evolution
of the material system interacting with the radiation field of the
laser pulses. The NRF formalism and its applications to a broad range
of material systems and spectroscopic signals have been comprehensively
reviewed in the literature. This article provides a detailed review
of the suite of EOM methods, including applications to 4-wave-mixing
and N-wave-mixing signals detected with weak or strong
fields. Under certain circumstances, the spectroscopic EOM methods
may be more efficient than the NRF method for the computation of various
nonlinear spectroscopic signals.