  We discuss the physical properties and accuracy of three distinct dynamical
(ie, frequency-dependent) kernels for the computation of optical excitations
within linear response theory: i) an a priori built kernel inspired by the
dressed time-dependent density-functional theory (TDDFT) kernel proposed by
Maitra and coworkers, ii) the dynamical kernel stemming from the Bethe-Salpeter
equation (BSE) formalism derived originally by Strinati , and iii) the
second-order BSE kernel derived by Yang and coworkers . The principal take-home
message of the present paper is that dynamical kernels can provide, thanks to
their frequency-dependent nature, additional excitations that can be associated
to higher-order excitations (such as the infamous double excitations), an
unappreciated feature of dynamical quantities. We also analyze, for each
kernel, the appearance of spurious excitations originating from the approximate
nature of the kernels, as first evidenced by Romaniello. Using a simple
two-level model, prototypical examples of valence, charge-transfer, and Rydberg
excited states are considered.
